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\"C:\\\\Users\\\\contr\\\\anaconda3\\\\Lib\\\\site-packages\\\\pandas\\\\io\\\\formats\\\\html.py\": 1718309385.5598285}\n" + ] + } + ], + "source": [ + "\n", + "# imports\n", + "import os\n", + "import sys\n", + "import types\n", + "import json\n", + "import base64\n", + "\n", + "# figure size/format\n", + "fig_width = 5\n", + "fig_height = 4\n", + "fig_format = 'png'\n", + "fig_dpi = 96\n", + "interactivity = ''\n", + "is_shiny = False\n", + "is_dashboard = False\n", + "plotly_connected = True\n", + "\n", + "# matplotlib defaults / format\n", + "try:\n", + " import matplotlib.pyplot as plt\n", + " plt.rcParams['figure.figsize'] = (fig_width, fig_height)\n", + " plt.rcParams['figure.dpi'] = fig_dpi\n", + " plt.rcParams['savefig.dpi'] = \"figure\"\n", + " from IPython.display import set_matplotlib_formats\n", + " set_matplotlib_formats(fig_format)\n", + "except Exception:\n", + " pass\n", + "\n", + "# plotly use connected mode\n", + "try:\n", + " import plotly.io as pio\n", + " if plotly_connected:\n", + " pio.renderers.default = \"notebook_connected\"\n", + " else:\n", + " pio.renderers.default = \"notebook\"\n", + " for template in pio.templates.keys():\n", + " pio.templates[template].layout.margin = dict(t=30,r=0,b=0,l=0)\n", + "except Exception:\n", + " pass\n", + "\n", + "# disable itables paging for dashboards\n", + "if is_dashboard:\n", + " try:\n", + " from itables import options\n", + " options.dom = 'fiBrtlp'\n", + " options.maxBytes = 1024 * 1024\n", + " options.language = dict(info = \"Showing _TOTAL_ entries\")\n", + " options.classes = \"display nowrap compact\"\n", + " options.paging = False\n", + " options.searching = True\n", + " options.ordering = True\n", + " options.info = True\n", + " options.lengthChange = False\n", + " options.autoWidth = False\n", + " options.responsive = True\n", + " options.keys = True\n", + " options.buttons = []\n", + " except Exception:\n", + " pass\n", + " \n", + " try:\n", + " import altair as alt\n", + " # By default, dashboards will have container sized\n", + " # vega visualizations which allows them to flow reasonably\n", + " theme_sentinel = '_quarto-dashboard-internal'\n", + " def make_theme(name):\n", + " nonTheme = alt.themes._plugins[name] \n", + " def patch_theme(*args, **kwargs):\n", + " existingTheme = nonTheme()\n", + " if 'height' not in existingTheme:\n", + " existingTheme['height'] = 'container'\n", + " if 'width' not in existingTheme:\n", + " existingTheme['width'] = 'container'\n", + "\n", + " if 'config' not in existingTheme:\n", + " existingTheme['config'] = dict()\n", + " \n", + " # Configure the default font sizes\n", + " title_font_size = 15\n", + " header_font_size = 13\n", + " axis_font_size = 12\n", + " legend_font_size = 12\n", + " mark_font_size = 12\n", + " tooltip = False\n", + "\n", + " config = existingTheme['config']\n", + "\n", + " # The Axis\n", + " if 'axis' not in config:\n", + " config['axis'] = dict()\n", + " axis = config['axis']\n", + " if 'labelFontSize' not in axis:\n", + " axis['labelFontSize'] = axis_font_size\n", + " if 'titleFontSize' not in axis:\n", + " axis['titleFontSize'] = axis_font_size \n", + "\n", + " # The legend\n", + " if 'legend' not in config:\n", + " config['legend'] = dict()\n", + " legend = config['legend']\n", + " if 'labelFontSize' not in legend:\n", + " legend['labelFontSize'] = legend_font_size\n", + " if 'titleFontSize' not in legend:\n", + " legend['titleFontSize'] = legend_font_size \n", + "\n", + " # The header\n", + " if 'header' not in config:\n", + " config['header'] = dict()\n", + " header = config['header']\n", + " if 'labelFontSize' not in header:\n", + " header['labelFontSize'] = header_font_size\n", + " if 'titleFontSize' not in header:\n", + " header['titleFontSize'] = header_font_size \n", + "\n", + " # Title\n", + " if 'title' not in config:\n", + " config['title'] = dict()\n", + " title = config['title']\n", + " if 'fontSize' not in title:\n", + " title['fontSize'] = title_font_size\n", + "\n", + " # Marks\n", + " if 'mark' not in config:\n", + " config['mark'] = dict()\n", + " mark = config['mark']\n", + " if 'fontSize' not in mark:\n", + " mark['fontSize'] = mark_font_size\n", + "\n", + " # Mark tooltips\n", + " if tooltip and 'tooltip' not in mark:\n", + " mark['tooltip'] = dict(content=\"encoding\")\n", + "\n", + " return existingTheme\n", + " \n", + " return patch_theme\n", + "\n", + " # We can only do this once per session\n", + " if theme_sentinel not in alt.themes.names():\n", + " for name in alt.themes.names():\n", + " alt.themes.register(name, make_theme(name))\n", + " \n", + " # register a sentinel theme so we only do this once\n", + " alt.themes.register(theme_sentinel, make_theme('default'))\n", + " alt.themes.enable('default')\n", + "\n", + " except Exception:\n", + " pass\n", + "\n", + "# enable pandas latex repr when targeting pdfs\n", + "try:\n", + " import pandas as pd\n", + " if fig_format == 'pdf':\n", + " pd.set_option('display.latex.repr', True)\n", + "except Exception:\n", + " pass\n", + "\n", + "# interactivity\n", + "if interactivity:\n", + " from IPython.core.interactiveshell import InteractiveShell\n", + " InteractiveShell.ast_node_interactivity = interactivity\n", + "\n", + "# NOTE: the kernel_deps code is repeated in the cleanup.py file\n", + "# (we can't easily share this code b/c of the way it is run).\n", + "# If you edit this code also edit the same code in cleanup.py!\n", + "\n", + "# output kernel dependencies\n", + "kernel_deps = dict()\n", + "for module in list(sys.modules.values()):\n", + " # Some modules play games with sys.modules (e.g. email/__init__.py\n", + " # in the standard library), and occasionally this can cause strange\n", + " # failures in getattr. Just ignore anything that's not an ordinary\n", + " # module.\n", + " if not isinstance(module, types.ModuleType):\n", + " continue\n", + " path = getattr(module, \"__file__\", None)\n", + " if not path:\n", + " continue\n", + " if path.endswith(\".pyc\") or path.endswith(\".pyo\"):\n", + " path = path[:-1]\n", + " if not os.path.exists(path):\n", + " continue\n", + " kernel_deps[path] = os.stat(path).st_mtime\n", + "print(json.dumps(kernel_deps))\n", + "\n", + "# set run_path if requested\n", + "run_path = 'QzpcVXNlcnNcY29udHJcT25lRHJpdmUgLSBJTlRFQ1xUUkkgOVxFQ09OT01cZml4ZWRfaW5jb21lX2dhcmNo'\n", + "if run_path:\n", + " # hex-decode the path\n", + " run_path = base64.b64decode(run_path.encode(\"utf-8\")).decode(\"utf-8\")\n", + " os.chdir(run_path)\n", + "\n", + "# reset state\n", + "%reset\n", + "\n", + "# shiny\n", + "# Checking for shiny by using False directly because we're after the %reset. We don't want\n", + "# to set a variable that stays in global scope.\n", + "if False:\n", + " try:\n", + " import htmltools as _htmltools\n", + " import ast as _ast\n", + "\n", + " _htmltools.html_dependency_render_mode = \"json\"\n", + "\n", + " # This decorator will be added to all function definitions\n", + " def _display_if_has_repr_html(x):\n", + " try:\n", + " # IPython 7.14 preferred import\n", + " from IPython.display import display, HTML\n", + " except:\n", + " from IPython.core.display import display, HTML\n", + "\n", + " if hasattr(x, '_repr_html_'):\n", + " display(HTML(x._repr_html_()))\n", + " return x\n", + "\n", + " # ideally we would undo the call to ast_transformers.append\n", + " # at the end of this block whenver an error occurs, we do \n", + " # this for now as it will only be a problem if the user \n", + " # switches from shiny to not-shiny mode (and even then likely\n", + " # won't matter)\n", + " import builtins\n", + " builtins._display_if_has_repr_html = _display_if_has_repr_html\n", + "\n", + " class _FunctionDefReprHtml(_ast.NodeTransformer):\n", + " def visit_FunctionDef(self, node):\n", + " node.decorator_list.insert(\n", + " 0,\n", + " _ast.Name(id=\"_display_if_has_repr_html\", ctx=_ast.Load())\n", + " )\n", + " return node\n", + "\n", + " def visit_AsyncFunctionDef(self, node):\n", + " node.decorator_list.insert(\n", + " 0,\n", + " _ast.Name(id=\"_display_if_has_repr_html\", ctx=_ast.Load())\n", + " )\n", + " return node\n", + "\n", + " ip = get_ipython()\n", + " ip.ast_transformers.append(_FunctionDefReprHtml())\n", + "\n", + " except:\n", + " pass\n", + "\n", + "def ojs_define(**kwargs):\n", + " import json\n", + " try:\n", + " # IPython 7.14 preferred import\n", + " from IPython.display import display, HTML\n", + " except:\n", + " from IPython.core.display import display, HTML\n", + "\n", + " # do some minor magic for convenience when handling pandas\n", + " # dataframes\n", + " def convert(v):\n", + " try:\n", + " import pandas as pd\n", + " except ModuleNotFoundError: # don't do the magic when pandas is not available\n", + " return v\n", + " if type(v) == pd.Series:\n", + " v = pd.DataFrame(v)\n", + " if type(v) == pd.DataFrame:\n", + " j = json.loads(v.T.to_json(orient='split'))\n", + " return dict((k,v) for (k,v) in zip(j[\"index\"], j[\"data\"]))\n", + " else:\n", + " return v\n", + "\n", + " v = dict(contents=list(dict(name=key, value=convert(value)) for (key, value) in kwargs.items()))\n", + " display(HTML(''), metadata=dict(ojs_define = True))\n", + "globals()[\"ojs_define\"] = ojs_define\n", + "# globals()[\"__spec__\"] = None" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "importing-libraries", + "metadata": {}, + "outputs": [], + "source": [ + "#| label: importing-libraries\n", + "# Manejo de datos y análisis\n", + "import numpy as np\n", + "import pandas as pd\n", + "import scipy.stats as stats\n", + "\n", + "# Modelos estadísticos y econométricos\n", + "import statsmodels.api as sm\n", + "from pmdarima.arima import auto_arima\n", + "from arch import arch_model" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "importing-data", + "metadata": {}, + "outputs": [], + "source": [ + "#| label: importing-data\n", + "df = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\n", + "returns = df['price_return']\n", + "split_date = '2020-12-31'\n", + "R_test = df[df.index >= split_date]['price_return'].rolling(\n", + " window=5).std().dropna()" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "return-descriptive-stats", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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ObservationsMeanMedianStd. DevSkewnessKurtosisJarque-BeraProb.
019411.353251-0.31804423.5029110.705277.5879954789.5323310.0
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" + ], + "text/plain": [ + " Observations Mean Median Std. Dev Skewness Kurtosis \\\n", + "0 1941 1.353251 -0.318044 23.502911 0.70527 7.587995 \n", + "\n", + " Jarque-Bera Prob. \n", + "0 4789.532331 0.0 " + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: return-descriptive-stats\n", + "#| fig-cap: Tabla de las estadísticas descriptiva de la serie de retornos\n", + "desc_stats = returns.describe()\n", + "\n", + "skewness = returns.skew()\n", + "kurtosis = returns.kurtosis()\n", + "jb_test = sm.stats.jarque_bera(returns)\n", + "\n", + "descriptive_table = pd.DataFrame({\n", + " 'Observations': [int(desc_stats['count'])],\n", + " 'Mean': [desc_stats['mean']],\n", + " 'Median': [desc_stats['50%']],\n", + " 'Std. Dev': [desc_stats['std']],\n", + " 'Skewness': [skewness],\n", + " 'Kurtosis': [kurtosis],\n", + " 'Jarque-Bera': [jb_test[0]],\n", + " 'Prob.': [jb_test[1]]\n", + "})\n", + "descriptive_table" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "arima-model", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:48:07 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
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coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
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Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step)." + ], + "text/latex": [ + "\\begin{center}\n", + "\\begin{tabular}{lclc}\n", + "\\toprule\n", + "\\textbf{Dep. Variable:} & y & \\textbf{ No. Observations: } & 1941 \\\\\n", + "\\textbf{Model:} & SARIMAX(3, 0, 4) & \\textbf{ Log Likelihood } & -8853.517 \\\\\n", + "\\textbf{Date:} & Sat, 19 Oct 2024 & \\textbf{ AIC } & 17725.034 \\\\\n", + "\\textbf{Time:} & 15:48:07 & \\textbf{ BIC } & 17775.173 \\\\\n", + "\\textbf{Sample:} & 0 & \\textbf{ HQIC } & 17743.472 \\\\\n", + "\\textbf{} & - 1941 & \\textbf{ } & \\\\\n", + "\\textbf{Covariance Type:} & opg & \\textbf{ } & \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lcccccc}\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{z} & \\textbf{P$> |$z$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", + "\\midrule\n", + "\\textbf{intercept} & 0.3602 & 0.273 & 1.317 & 0.188 & -0.176 & 0.896 \\\\\n", + "\\textbf{ar.L1} & 0.1366 & 0.065 & 2.106 & 0.035 & 0.009 & 0.264 \\\\\n", + "\\textbf{ar.L2} & -0.1931 & 0.066 & -2.934 & 0.003 & -0.322 & -0.064 \\\\\n", + "\\textbf{ar.L3} & 0.7691 & 0.061 & 12.644 & 0.000 & 0.650 & 0.888 \\\\\n", + "\\textbf{ma.L1} & -0.1139 & 0.065 & -1.752 & 0.080 & -0.241 & 0.014 \\\\\n", + "\\textbf{ma.L2} & 0.2163 & 0.067 & 3.220 & 0.001 & 0.085 & 0.348 \\\\\n", + "\\textbf{ma.L3} & -0.7032 & 0.062 & -11.315 & 0.000 & -0.825 & -0.581 \\\\\n", + "\\textbf{ma.L4} & 0.0750 & 0.020 & 3.715 & 0.000 & 0.035 & 0.115 \\\\\n", + "\\textbf{sigma2} & 536.2183 & 8.358 & 64.154 & 0.000 & 519.836 & 552.600 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lclc}\n", + "\\textbf{Ljung-Box (L1) (Q):} & 0.00 & \\textbf{ Jarque-Bera (JB): } & 4395.18 \\\\\n", + "\\textbf{Prob(Q):} & 1.00 & \\textbf{ Prob(JB): } & 0.00 \\\\\n", + "\\textbf{Heteroskedasticity (H):} & 1.11 & \\textbf{ Skew: } & 0.59 \\\\\n", + "\\textbf{Prob(H) (two-sided):} & 0.17 & \\textbf{ Kurtosis: } & 10.28 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "%\\caption{SARIMAX Results}\n", + "\\end{center}\n", + "\n", + "Warnings: \\newline\n", + " [1] Covariance matrix calculated using the outer product of gradients (complex-step)." + ], + "text/plain": [ + "\n", + "\"\"\"\n", + " SARIMAX Results \n", + "==============================================================================\n", + "Dep. Variable: y No. Observations: 1941\n", + "Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517\n", + "Date: Sat, 19 Oct 2024 AIC 17725.034\n", + "Time: 15:48:07 BIC 17775.173\n", + "Sample: 0 HQIC 17743.472\n", + " - 1941 \n", + "Covariance Type: opg \n", + "==============================================================================\n", + " coef std err z P>|z| [0.025 0.975]\n", + "------------------------------------------------------------------------------\n", + "intercept 0.3602 0.273 1.317 0.188 -0.176 0.896\n", + "ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264\n", + "ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064\n", + "ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888\n", + "ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014\n", + "ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348\n", + "ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581\n", + "ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115\n", + "sigma2 536.2183 8.358 64.154 0.000 519.836 552.600\n", + "===================================================================================\n", + "Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18\n", + "Prob(Q): 1.00 Prob(JB): 0.00\n", + "Heteroskedasticity (H): 1.11 Skew: 0.59\n", + "Prob(H) (two-sided): 0.17 Kurtosis: 10.28\n", + "===================================================================================\n", + "\n", + "Warnings:\n", + "[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n", + "\"\"\"" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: arima-model\n", + "#| fig-cap: Modelo ARIMA de maxima verosimilitud para la serie de retornos.\n", + "#| warning: false\n", + "model_auto = auto_arima(returns)\n", + "model_auto.summary()" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "garch-model-fitting", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Iteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\n", + "Iteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\n", + "Iteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\n", + "Iteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\n", + "Iteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\n", + "Iteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\n", + "Iteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\n", + "Iteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\n", + "Iteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\n", + "Iteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\n", + "Iteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\n", + "Iteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\n", + "Iteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\n", + "Iteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\n", + "Iteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\n", + "Iteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\n", + "Iteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\n", + "Iteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\n", + "Iteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\n", + "Iteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\n", + "Iteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\n", + "Iteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\n", + "Iteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\n", + "Iteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\n", + "Iteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\n", + "Iteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\n", + "Optimization terminated successfully (Exit mode 0)\n", + " Current function value: 7622.868827051428\n", + " Iterations: 26\n", + " Function evaluations: 139\n", + " Gradient evaluations: 26\n" + ] + } + ], + "source": [ + "#| label: garch-model-fitting\n", + "#| fig-cap: Ajuste del modelo Zero-Garch ala serie de retornos\n", + "#| warning: false\n", + "#| output: false\n", + "\n", + "ar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\n", + "res = ar.fit(last_obs=split_date)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "garch-model", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:48:07 Df Model: 0
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Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
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Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust" + ], + "text/latex": [ + "\\begin{center}\n", + "\\begin{tabular}{lclc}\n", + "\\toprule\n", + "\\textbf{Dep. Variable:} & price\\_return & \\textbf{ R-squared: } & 0.000 \\\\\n", + "\\textbf{Mean Model:} & Zero Mean & \\textbf{ Adj. R-squared: } & 0.001 \\\\\n", + "\\textbf{Vol Model:} & GARCH & \\textbf{ Log-Likelihood: } & -7622.87 \\\\\n", + "\\textbf{Distribution:} & Standardized Student's t & \\textbf{ AIC: } & 15253.7 \\\\\n", + "\\textbf{Method:} & Maximum Likelihood & \\textbf{ BIC: } & 15275.6 \\\\\n", + "\\textbf{} & & \\textbf{ No. Observations: } & 1753 \\\\\n", + "\\textbf{Date:} & Sat, Oct 19 2024 & \\textbf{ Df Residuals: } & 1753 \\\\\n", + "\\textbf{Time:} & 15:48:07 & \\textbf{ Df Model: } & 0 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lccccc}\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n", + "\\midrule\n", + "\\textbf{omega} & 58.8679 & 25.182 & 2.338 & 1.940e-02 & [ 9.512,1.082e+02] \\\\\n", + "\\textbf{alpha[1]} & 0.1892 & 6.657e-02 & 2.842 & 4.482e-03 & [5.873e-02, 0.320] \\\\\n", + "\\textbf{beta[1]} & 0.7737 & 6.832e-02 & 11.324 & 9.943e-30 & [ 0.640, 0.908] \\\\\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n", + "\\midrule\n", + "\\textbf{nu} & 2.7894 & 0.230 & 12.114 & 8.853e-34 & [ 2.338, 3.241] \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "%\\caption{Zero Mean - GARCH Model Results}\n", + "\\end{center}\n", + "\n", + "Covariance estimator: robust" + ], + "text/plain": [ + "\n", + "\"\"\"\n", + " Zero Mean - GARCH Model Results \n", + "====================================================================================\n", + "Dep. Variable: price_return R-squared: 0.000\n", + "Mean Model: Zero Mean Adj. R-squared: 0.001\n", + "Vol Model: GARCH Log-Likelihood: -7622.87\n", + "Distribution: Standardized Student's t AIC: 15253.7\n", + "Method: Maximum Likelihood BIC: 15275.6\n", + " No. Observations: 1753\n", + "Date: Sat, Oct 19 2024 Df Residuals: 1753\n", + "Time: 15:48:07 Df Model: 0\n", + " Volatility Model \n", + "==========================================================================\n", + " coef std err t P>|t| 95.0% Conf. Int.\n", + "--------------------------------------------------------------------------\n", + "omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]\n", + "alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]\n", + "beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]\n", + " Distribution \n", + "========================================================================\n", + " coef std err t P>|t| 95.0% Conf. Int.\n", + "------------------------------------------------------------------------\n", + "nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]\n", + "========================================================================\n", + "\n", + "Covariance estimator: robust\n", + "\"\"\"" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: garch-model\n", + "#| fig-cap: Modelo Zero-Garch de la serie de retornos\n", + "res.summary()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3", + "path": "C:\\Users\\contr\\anaconda3\\share\\jupyter\\kernels\\python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/.jupyter_cache/executed/eec3d60fed8af4482d94dfe3e0a71116/base.ipynb b/.jupyter_cache/executed/eec3d60fed8af4482d94dfe3e0a71116/base.ipynb new file mode 100644 index 0000000..d7941d4 --- /dev/null +++ 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+ "fig_format = 'pdf'\n", + "fig_dpi = 300\n", + "interactivity = ''\n", + "is_shiny = False\n", + "is_dashboard = False\n", + "plotly_connected = True\n", + "\n", + "# matplotlib defaults / format\n", + "try:\n", + " import matplotlib.pyplot as plt\n", + " plt.rcParams['figure.figsize'] = (fig_width, fig_height)\n", + " plt.rcParams['figure.dpi'] = fig_dpi\n", + " plt.rcParams['savefig.dpi'] = \"figure\"\n", + " from IPython.display import set_matplotlib_formats\n", + " set_matplotlib_formats(fig_format)\n", + "except Exception:\n", + " pass\n", + "\n", + "# plotly use connected mode\n", + "try:\n", + " import plotly.io as pio\n", + " if plotly_connected:\n", + " pio.renderers.default = \"notebook_connected\"\n", + " else:\n", + " pio.renderers.default = \"notebook\"\n", + " for template in pio.templates.keys():\n", + " pio.templates[template].layout.margin = dict(t=30,r=0,b=0,l=0)\n", + "except Exception:\n", + " pass\n", + "\n", + "# disable itables paging for dashboards\n", + "if is_dashboard:\n", + " try:\n", + " from itables import options\n", + " options.dom = 'fiBrtlp'\n", + " options.maxBytes = 1024 * 1024\n", + " options.language = dict(info = \"Showing _TOTAL_ entries\")\n", + " options.classes = \"display nowrap compact\"\n", + " options.paging = False\n", + " options.searching = True\n", + " options.ordering = True\n", + " options.info = True\n", + " options.lengthChange = False\n", + " options.autoWidth = False\n", + " options.responsive = True\n", + " options.keys = True\n", + " options.buttons = []\n", + " except Exception:\n", + " pass\n", + " \n", + " try:\n", + " import altair as alt\n", + " # By default, dashboards will have container sized\n", + " # vega visualizations which allows them to flow reasonably\n", + " theme_sentinel = '_quarto-dashboard-internal'\n", + " def make_theme(name):\n", + " nonTheme = alt.themes._plugins[name] \n", + " def patch_theme(*args, **kwargs):\n", + " existingTheme = nonTheme()\n", + " if 'height' not in existingTheme:\n", + " existingTheme['height'] = 'container'\n", + " if 'width' not in existingTheme:\n", + " existingTheme['width'] = 'container'\n", + "\n", + " if 'config' not in existingTheme:\n", + " existingTheme['config'] = dict()\n", + " \n", + " # Configure the default font sizes\n", + " title_font_size = 15\n", + " header_font_size = 13\n", + " axis_font_size = 12\n", + " legend_font_size = 12\n", + " mark_font_size = 12\n", + " tooltip = False\n", + "\n", + " config = existingTheme['config']\n", + "\n", + " # The Axis\n", + " if 'axis' not in config:\n", + " config['axis'] = dict()\n", + " axis = config['axis']\n", + " if 'labelFontSize' not in axis:\n", + " axis['labelFontSize'] = axis_font_size\n", + " if 'titleFontSize' not in axis:\n", + " axis['titleFontSize'] = axis_font_size \n", + "\n", + " # The legend\n", + " if 'legend' not in config:\n", + " config['legend'] = dict()\n", + " legend = config['legend']\n", + " if 'labelFontSize' not in legend:\n", + " legend['labelFontSize'] = legend_font_size\n", + " if 'titleFontSize' not in legend:\n", + " legend['titleFontSize'] = legend_font_size \n", + "\n", + " # The header\n", + " if 'header' not in config:\n", + " config['header'] = dict()\n", + " header = config['header']\n", + " if 'labelFontSize' not in header:\n", + " header['labelFontSize'] = header_font_size\n", + " if 'titleFontSize' not in header:\n", + " header['titleFontSize'] = header_font_size \n", + "\n", + " # Title\n", + " if 'title' not in config:\n", + " config['title'] = dict()\n", + " title = config['title']\n", + " if 'fontSize' not in title:\n", + " title['fontSize'] = title_font_size\n", + "\n", + " # Marks\n", + " if 'mark' not in config:\n", + " config['mark'] = dict()\n", + " mark = config['mark']\n", + " if 'fontSize' not in mark:\n", + " mark['fontSize'] = mark_font_size\n", + "\n", + " # Mark tooltips\n", + " if tooltip and 'tooltip' not in mark:\n", + " mark['tooltip'] = dict(content=\"encoding\")\n", + "\n", + " return existingTheme\n", + " \n", + " return patch_theme\n", + "\n", + " # We can only do this once per session\n", + " if theme_sentinel not in alt.themes.names():\n", + " for name in alt.themes.names():\n", + " alt.themes.register(name, make_theme(name))\n", + " \n", + " # register a sentinel theme so we only do this once\n", + " alt.themes.register(theme_sentinel, make_theme('default'))\n", + " alt.themes.enable('default')\n", + "\n", + " except Exception:\n", + " pass\n", + "\n", + "# enable pandas latex repr when targeting pdfs\n", + "try:\n", + " import pandas as pd\n", + " if fig_format == 'pdf':\n", + " pd.set_option('display.latex.repr', True)\n", + "except Exception:\n", + " pass\n", + "\n", + "# interactivity\n", + "if interactivity:\n", + " from IPython.core.interactiveshell import InteractiveShell\n", + " InteractiveShell.ast_node_interactivity = interactivity\n", + "\n", + "# NOTE: the kernel_deps code is repeated in the cleanup.py file\n", + "# (we can't easily share this code b/c of the way it is run).\n", + "# If you edit this code also edit the same code in cleanup.py!\n", + "\n", + "# output kernel dependencies\n", + "kernel_deps = dict()\n", + "for module in list(sys.modules.values()):\n", + " # Some modules play games with sys.modules (e.g. email/__init__.py\n", + " # in the standard library), and occasionally this can cause strange\n", + " # failures in getattr. Just ignore anything that's not an ordinary\n", + " # module.\n", + " if not isinstance(module, types.ModuleType):\n", + " continue\n", + " path = getattr(module, \"__file__\", None)\n", + " if not path:\n", + " continue\n", + " if path.endswith(\".pyc\") or path.endswith(\".pyo\"):\n", + " path = path[:-1]\n", + " if not os.path.exists(path):\n", + " continue\n", + " kernel_deps[path] = os.stat(path).st_mtime\n", + "print(json.dumps(kernel_deps))\n", + "\n", + "# set run_path if requested\n", + "run_path = 'QzpcVXNlcnNcY29udHJcT25lRHJpdmUgLSBJTlRFQ1xUUkkgOVxFQ09OT01cZml4ZWRfaW5jb21lX2dhcmNo'\n", + "if run_path:\n", + " # hex-decode the path\n", + " run_path = base64.b64decode(run_path.encode(\"utf-8\")).decode(\"utf-8\")\n", + " os.chdir(run_path)\n", + "\n", + "# reset state\n", + "%reset\n", + "\n", + "# shiny\n", + "# Checking for shiny by using False directly because we're after the %reset. We don't want\n", + "# to set a variable that stays in global scope.\n", + "if False:\n", + " try:\n", + " import htmltools as _htmltools\n", + " import ast as _ast\n", + "\n", + " _htmltools.html_dependency_render_mode = \"json\"\n", + "\n", + " # This decorator will be added to all function definitions\n", + " def _display_if_has_repr_html(x):\n", + " try:\n", + " # IPython 7.14 preferred import\n", + " from IPython.display import display, HTML\n", + " except:\n", + " from IPython.core.display import display, HTML\n", + "\n", + " if hasattr(x, '_repr_html_'):\n", + " display(HTML(x._repr_html_()))\n", + " return x\n", + "\n", + " # ideally we would undo the call to ast_transformers.append\n", + " # at the end of this block whenver an error occurs, we do \n", + " # this for now as it will only be a problem if the user \n", + " # switches from shiny to not-shiny mode (and even then likely\n", + " # won't matter)\n", + " import builtins\n", + " builtins._display_if_has_repr_html = _display_if_has_repr_html\n", + "\n", + " class _FunctionDefReprHtml(_ast.NodeTransformer):\n", + " def visit_FunctionDef(self, node):\n", + " node.decorator_list.insert(\n", + " 0,\n", + " _ast.Name(id=\"_display_if_has_repr_html\", ctx=_ast.Load())\n", + " )\n", + " return node\n", + "\n", + " def visit_AsyncFunctionDef(self, node):\n", + " node.decorator_list.insert(\n", + " 0,\n", + " _ast.Name(id=\"_display_if_has_repr_html\", ctx=_ast.Load())\n", + " )\n", + " return node\n", + "\n", + " ip = get_ipython()\n", + " ip.ast_transformers.append(_FunctionDefReprHtml())\n", + "\n", + " except:\n", + " pass\n", + "\n", + "def ojs_define(**kwargs):\n", + " import json\n", + " try:\n", + " # IPython 7.14 preferred import\n", + " from IPython.display import display, HTML\n", + " except:\n", + " from IPython.core.display import display, HTML\n", + "\n", + " # do some minor magic for convenience when handling pandas\n", + " # dataframes\n", + " def convert(v):\n", + " try:\n", + " import pandas as pd\n", + " except ModuleNotFoundError: # don't do the magic when pandas is not available\n", + " return v\n", + " if type(v) == pd.Series:\n", + " v = pd.DataFrame(v)\n", + " if type(v) == pd.DataFrame:\n", + " j = json.loads(v.T.to_json(orient='split'))\n", + " return dict((k,v) for (k,v) in zip(j[\"index\"], j[\"data\"]))\n", + " else:\n", + " return v\n", + "\n", + " v = dict(contents=list(dict(name=key, value=convert(value)) for (key, value) in kwargs.items()))\n", + " display(HTML(''), metadata=dict(ojs_define = True))\n", + "globals()[\"ojs_define\"] = ojs_define\n", + "# globals()[\"__spec__\"] = None" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "importing-libraries", + "metadata": {}, + "outputs": [], + "source": [ + "#| label: importing-libraries\n", + "# Manejo de datos y análisis\n", + "import numpy as np\n", + "import pandas as pd\n", + "import scipy.stats as stats\n", + "\n", + "# Modelos estadísticos y econométricos\n", + "import statsmodels.api as sm\n", + "from pmdarima.arima import auto_arima\n", + "from arch import arch_model" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "importing-data", + "metadata": {}, + "outputs": [], + "source": [ + "#| label: importing-data\n", + "df = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\n", + "returns = df['price_return']\n", + "split_date = '2020-12-31'\n", + "R_test = df[df.index >= split_date]['price_return'].rolling(\n", + " window=5).std().dropna()" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "return-descriptive-stats", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " Observations Mean Median Std. Dev Skewness Kurtosis \\\n", + "0 1941 1.353251 -0.318044 23.502911 0.70527 7.587995 \n", + "\n", + " Jarque-Bera Prob. \n", + "0 4789.532331 0.0 " + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: return-descriptive-stats\n", + "#| fig-cap: Tabla de las estadísticas descriptiva de la serie de retornos\n", + "desc_stats = returns.describe()\n", + "\n", + "skewness = returns.skew()\n", + "kurtosis = returns.kurtosis()\n", + "jb_test = sm.stats.jarque_bera(returns)\n", + "\n", + "descriptive_table = pd.DataFrame({\n", + " 'Observations': [int(desc_stats['count'])],\n", + " 'Mean': [desc_stats['mean']],\n", + " 'Median': [desc_stats['50%']],\n", + " 'Std. Dev': [desc_stats['std']],\n", + " 'Skewness': [skewness],\n", + " 'Kurtosis': [kurtosis],\n", + " 'Jarque-Bera': [jb_test[0]],\n", + " 'Prob.': [jb_test[1]]\n", + "})\n", + "descriptive_table" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "arima-model", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:47:38 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
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coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
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Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step)." + ], + "text/latex": [ + "\\begin{center}\n", + "\\begin{tabular}{lclc}\n", + "\\toprule\n", + "\\textbf{Dep. Variable:} & y & \\textbf{ No. Observations: } & 1941 \\\\\n", + "\\textbf{Model:} & SARIMAX(3, 0, 4) & \\textbf{ Log Likelihood } & -8853.517 \\\\\n", + "\\textbf{Date:} & Sat, 19 Oct 2024 & \\textbf{ AIC } & 17725.034 \\\\\n", + "\\textbf{Time:} & 15:47:38 & \\textbf{ BIC } & 17775.173 \\\\\n", + "\\textbf{Sample:} & 0 & \\textbf{ HQIC } & 17743.472 \\\\\n", + "\\textbf{} & - 1941 & \\textbf{ } & \\\\\n", + "\\textbf{Covariance Type:} & opg & \\textbf{ } & \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lcccccc}\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{z} & \\textbf{P$> |$z$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", + "\\midrule\n", + "\\textbf{intercept} & 0.3602 & 0.273 & 1.317 & 0.188 & -0.176 & 0.896 \\\\\n", + "\\textbf{ar.L1} & 0.1366 & 0.065 & 2.106 & 0.035 & 0.009 & 0.264 \\\\\n", + "\\textbf{ar.L2} & -0.1931 & 0.066 & -2.934 & 0.003 & -0.322 & -0.064 \\\\\n", + "\\textbf{ar.L3} & 0.7691 & 0.061 & 12.644 & 0.000 & 0.650 & 0.888 \\\\\n", + "\\textbf{ma.L1} & -0.1139 & 0.065 & -1.752 & 0.080 & -0.241 & 0.014 \\\\\n", + "\\textbf{ma.L2} & 0.2163 & 0.067 & 3.220 & 0.001 & 0.085 & 0.348 \\\\\n", + "\\textbf{ma.L3} & -0.7032 & 0.062 & -11.315 & 0.000 & -0.825 & -0.581 \\\\\n", + "\\textbf{ma.L4} & 0.0750 & 0.020 & 3.715 & 0.000 & 0.035 & 0.115 \\\\\n", + "\\textbf{sigma2} & 536.2183 & 8.358 & 64.154 & 0.000 & 519.836 & 552.600 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lclc}\n", + "\\textbf{Ljung-Box (L1) (Q):} & 0.00 & \\textbf{ Jarque-Bera (JB): } & 4395.18 \\\\\n", + "\\textbf{Prob(Q):} & 1.00 & \\textbf{ Prob(JB): } & 0.00 \\\\\n", + "\\textbf{Heteroskedasticity (H):} & 1.11 & \\textbf{ Skew: } & 0.59 \\\\\n", + "\\textbf{Prob(H) (two-sided):} & 0.17 & \\textbf{ Kurtosis: } & 10.28 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "%\\caption{SARIMAX Results}\n", + "\\end{center}\n", + "\n", + "Warnings: \\newline\n", + " [1] Covariance matrix calculated using the outer product of gradients (complex-step)." + ], + "text/plain": [ + "\n", + "\"\"\"\n", + " SARIMAX Results \n", + "==============================================================================\n", + "Dep. Variable: y No. Observations: 1941\n", + "Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517\n", + "Date: Sat, 19 Oct 2024 AIC 17725.034\n", + "Time: 15:47:38 BIC 17775.173\n", + "Sample: 0 HQIC 17743.472\n", + " - 1941 \n", + "Covariance Type: opg \n", + "==============================================================================\n", + " coef std err z P>|z| [0.025 0.975]\n", + "------------------------------------------------------------------------------\n", + "intercept 0.3602 0.273 1.317 0.188 -0.176 0.896\n", + "ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264\n", + "ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064\n", + "ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888\n", + "ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014\n", + "ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348\n", + "ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581\n", + "ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115\n", + "sigma2 536.2183 8.358 64.154 0.000 519.836 552.600\n", + "===================================================================================\n", + "Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18\n", + "Prob(Q): 1.00 Prob(JB): 0.00\n", + "Heteroskedasticity (H): 1.11 Skew: 0.59\n", + "Prob(H) (two-sided): 0.17 Kurtosis: 10.28\n", + "===================================================================================\n", + "\n", + "Warnings:\n", + "[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n", + "\"\"\"" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: arima-model\n", + "#| fig-cap: Modelo ARIMA de maxima verosimilitud para la serie de retornos.\n", + "#| warning: false\n", + "model_auto = auto_arima(returns)\n", + "model_auto.summary()" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "garch-model-fitting", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Iteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\n", + "Iteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\n", + "Iteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\n", + "Iteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\n", + "Iteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\n", + "Iteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\n", + "Iteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\n", + "Iteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\n", + "Iteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\n", + "Iteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\n", + "Iteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\n", + "Iteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\n", + "Iteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\n", + "Iteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\n", + "Iteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\n", + "Iteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\n", + "Iteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\n", + "Iteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\n", + "Iteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\n", + "Iteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\n", + "Iteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\n", + "Iteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\n", + "Iteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\n", + "Iteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\n", + "Iteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\n", + "Iteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\n", + "Optimization terminated successfully (Exit mode 0)\n", + " Current function value: 7622.868827051428\n", + " Iterations: 26\n", + " Function evaluations: 139\n", + " Gradient evaluations: 26\n" + ] + } + ], + "source": [ + "#| label: garch-model-fitting\n", + "#| fig-cap: Ajuste del modelo Zero-Garch ala serie de retornos\n", + "#| warning: false\n", + "#| output: false\n", + "\n", + "ar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\n", + "res = ar.fit(last_obs=split_date)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "garch-model", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:47:38 Df Model: 0
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Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
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Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust" + ], + "text/latex": [ + "\\begin{center}\n", + "\\begin{tabular}{lclc}\n", + "\\toprule\n", + "\\textbf{Dep. Variable:} & price\\_return & \\textbf{ R-squared: } & 0.000 \\\\\n", + "\\textbf{Mean Model:} & Zero Mean & \\textbf{ Adj. R-squared: } & 0.001 \\\\\n", + "\\textbf{Vol Model:} & GARCH & \\textbf{ Log-Likelihood: } & -7622.87 \\\\\n", + "\\textbf{Distribution:} & Standardized Student's t & \\textbf{ AIC: } & 15253.7 \\\\\n", + "\\textbf{Method:} & Maximum Likelihood & \\textbf{ BIC: } & 15275.6 \\\\\n", + "\\textbf{} & & \\textbf{ No. Observations: } & 1753 \\\\\n", + "\\textbf{Date:} & Sat, Oct 19 2024 & \\textbf{ Df Residuals: } & 1753 \\\\\n", + "\\textbf{Time:} & 15:47:38 & \\textbf{ Df Model: } & 0 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lccccc}\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n", + "\\midrule\n", + "\\textbf{omega} & 58.8679 & 25.182 & 2.338 & 1.940e-02 & [ 9.512,1.082e+02] \\\\\n", + "\\textbf{alpha[1]} & 0.1892 & 6.657e-02 & 2.842 & 4.482e-03 & [5.873e-02, 0.320] \\\\\n", + "\\textbf{beta[1]} & 0.7737 & 6.832e-02 & 11.324 & 9.943e-30 & [ 0.640, 0.908] \\\\\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n", + "\\midrule\n", + "\\textbf{nu} & 2.7894 & 0.230 & 12.114 & 8.853e-34 & [ 2.338, 3.241] \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "%\\caption{Zero Mean - GARCH Model Results}\n", + "\\end{center}\n", + "\n", + "Covariance estimator: robust" + ], + "text/plain": [ + "\n", + "\"\"\"\n", + " Zero Mean - GARCH Model Results \n", + "====================================================================================\n", + "Dep. Variable: price_return R-squared: 0.000\n", + "Mean Model: Zero Mean Adj. R-squared: 0.001\n", + "Vol Model: GARCH Log-Likelihood: -7622.87\n", + "Distribution: Standardized Student's t AIC: 15253.7\n", + "Method: Maximum Likelihood BIC: 15275.6\n", + " No. Observations: 1753\n", + "Date: Sat, Oct 19 2024 Df Residuals: 1753\n", + "Time: 15:47:38 Df Model: 0\n", + " Volatility Model \n", + "==========================================================================\n", + " coef std err t P>|t| 95.0% Conf. Int.\n", + "--------------------------------------------------------------------------\n", + "omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]\n", + "alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]\n", + "beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]\n", + " Distribution \n", + "========================================================================\n", + " coef std err t P>|t| 95.0% Conf. Int.\n", + "------------------------------------------------------------------------\n", + "nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]\n", + "========================================================================\n", + "\n", + "Covariance estimator: robust\n", + "\"\"\"" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: garch-model\n", + "#| fig-cap: Modelo Zero-Garch de la serie de retornos\n", + "res.summary()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3", + "path": "C:\\Users\\contr\\anaconda3\\share\\jupyter\\kernels\\python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + 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\"C:\\\\Users\\\\contr\\\\anaconda3\\\\Lib\\\\site-packages\\\\pandas\\\\io\\\\formats\\\\html.py\": 1718309385.5598285}\n" + ] + } + ], + "source": [ + "\n", + "# imports\n", + "import os\n", + "import sys\n", + "import types\n", + "import json\n", + "import base64\n", + "\n", + "# figure size/format\n", + "fig_width = 7\n", + "fig_height = 5\n", + "fig_format = 'png'\n", + "fig_dpi = 96\n", + "interactivity = ''\n", + "is_shiny = False\n", + "is_dashboard = False\n", + "plotly_connected = True\n", + "\n", + "# matplotlib defaults / format\n", + "try:\n", + " import matplotlib.pyplot as plt\n", + " plt.rcParams['figure.figsize'] = (fig_width, fig_height)\n", + " plt.rcParams['figure.dpi'] = fig_dpi\n", + " plt.rcParams['savefig.dpi'] = \"figure\"\n", + " from IPython.display import set_matplotlib_formats\n", + " set_matplotlib_formats(fig_format)\n", + "except Exception:\n", + " pass\n", + "\n", + "# plotly use connected mode\n", + "try:\n", + " import plotly.io as pio\n", + " if plotly_connected:\n", + " pio.renderers.default = \"notebook_connected\"\n", + " else:\n", + " pio.renderers.default = \"notebook\"\n", + " for template in pio.templates.keys():\n", + " pio.templates[template].layout.margin = dict(t=30,r=0,b=0,l=0)\n", + "except Exception:\n", + " pass\n", + "\n", + "# disable itables paging for dashboards\n", + "if is_dashboard:\n", + " try:\n", + " from itables import options\n", + " options.dom = 'fiBrtlp'\n", + " options.maxBytes = 1024 * 1024\n", + " options.language = dict(info = \"Showing _TOTAL_ entries\")\n", + " options.classes = \"display nowrap compact\"\n", + " options.paging = False\n", + " options.searching = True\n", + " options.ordering = True\n", + " options.info = True\n", + " options.lengthChange = False\n", + " options.autoWidth = False\n", + " options.responsive = True\n", + " options.keys = True\n", + " options.buttons = []\n", + " except Exception:\n", + " pass\n", + " \n", + " try:\n", + " import altair as alt\n", + " # By default, dashboards will have container sized\n", + " # vega visualizations which allows them to flow reasonably\n", + " theme_sentinel = '_quarto-dashboard-internal'\n", + " def make_theme(name):\n", + " nonTheme = alt.themes._plugins[name] \n", + " def patch_theme(*args, **kwargs):\n", + " existingTheme = nonTheme()\n", + " if 'height' not in existingTheme:\n", + " existingTheme['height'] = 'container'\n", + " if 'width' not in existingTheme:\n", + " existingTheme['width'] = 'container'\n", + "\n", + " if 'config' not in existingTheme:\n", + " existingTheme['config'] = dict()\n", + " \n", + " # Configure the default font sizes\n", + " title_font_size = 15\n", + " header_font_size = 13\n", + " axis_font_size = 12\n", + " legend_font_size = 12\n", + " mark_font_size = 12\n", + " tooltip = False\n", + "\n", + " config = existingTheme['config']\n", + "\n", + " # The Axis\n", + " if 'axis' not in config:\n", + " config['axis'] = dict()\n", + " axis = config['axis']\n", + " if 'labelFontSize' not in axis:\n", + " axis['labelFontSize'] = axis_font_size\n", + " if 'titleFontSize' not in axis:\n", + " axis['titleFontSize'] = axis_font_size \n", + "\n", + " # The legend\n", + " if 'legend' not in config:\n", + " config['legend'] = dict()\n", + " legend = config['legend']\n", + " if 'labelFontSize' not in legend:\n", + " legend['labelFontSize'] = legend_font_size\n", + " if 'titleFontSize' not in legend:\n", + " legend['titleFontSize'] = legend_font_size \n", + "\n", + " # The header\n", + " if 'header' not in config:\n", + " config['header'] = dict()\n", + " header = config['header']\n", + " if 'labelFontSize' not in header:\n", + " header['labelFontSize'] = header_font_size\n", + " if 'titleFontSize' not in header:\n", + " header['titleFontSize'] = header_font_size \n", + "\n", + " # Title\n", + " if 'title' not in config:\n", + " config['title'] = dict()\n", + " title = config['title']\n", + " if 'fontSize' not in title:\n", + " title['fontSize'] = title_font_size\n", + "\n", + " # Marks\n", + " if 'mark' not in config:\n", + " config['mark'] = dict()\n", + " mark = config['mark']\n", + " if 'fontSize' not in mark:\n", + " mark['fontSize'] = mark_font_size\n", + "\n", + " # Mark tooltips\n", + " if tooltip and 'tooltip' not in mark:\n", + " mark['tooltip'] = dict(content=\"encoding\")\n", + "\n", + " return existingTheme\n", + " \n", + " return patch_theme\n", + "\n", + " # We can only do this once per session\n", + " if theme_sentinel not in alt.themes.names():\n", + " for name in alt.themes.names():\n", + " alt.themes.register(name, make_theme(name))\n", + " \n", + " # register a sentinel theme so we only do this once\n", + " alt.themes.register(theme_sentinel, make_theme('default'))\n", + " alt.themes.enable('default')\n", + "\n", + " except Exception:\n", + " pass\n", + "\n", + "# enable pandas latex repr when targeting pdfs\n", + "try:\n", + " import pandas as pd\n", + " if fig_format == 'pdf':\n", + " pd.set_option('display.latex.repr', True)\n", + "except Exception:\n", + " pass\n", + "\n", + "# interactivity\n", + "if interactivity:\n", + " from IPython.core.interactiveshell import InteractiveShell\n", + " InteractiveShell.ast_node_interactivity = interactivity\n", + "\n", + "# NOTE: the kernel_deps code is repeated in the cleanup.py file\n", + "# (we can't easily share this code b/c of the way it is run).\n", + "# If you edit this code also edit the same code in cleanup.py!\n", + "\n", + "# output kernel dependencies\n", + "kernel_deps = dict()\n", + "for module in list(sys.modules.values()):\n", + " # Some modules play games with sys.modules (e.g. email/__init__.py\n", + " # in the standard library), and occasionally this can cause strange\n", + " # failures in getattr. Just ignore anything that's not an ordinary\n", + " # module.\n", + " if not isinstance(module, types.ModuleType):\n", + " continue\n", + " path = getattr(module, \"__file__\", None)\n", + " if not path:\n", + " continue\n", + " if path.endswith(\".pyc\") or path.endswith(\".pyo\"):\n", + " path = path[:-1]\n", + " if not os.path.exists(path):\n", + " continue\n", + " kernel_deps[path] = os.stat(path).st_mtime\n", + "print(json.dumps(kernel_deps))\n", + "\n", + "# set run_path if requested\n", + "run_path = 'QzpcVXNlcnNcY29udHJcT25lRHJpdmUgLSBJTlRFQ1xUUkkgOVxFQ09OT01cZml4ZWRfaW5jb21lX2dhcmNo'\n", + "if run_path:\n", + " # hex-decode the path\n", + " run_path = base64.b64decode(run_path.encode(\"utf-8\")).decode(\"utf-8\")\n", + " os.chdir(run_path)\n", + "\n", + "# reset state\n", + "%reset\n", + "\n", + "# shiny\n", + "# Checking for shiny by using False directly because we're after the %reset. We don't want\n", + "# to set a variable that stays in global scope.\n", + "if False:\n", + " try:\n", + " import htmltools as _htmltools\n", + " import ast as _ast\n", + "\n", + " _htmltools.html_dependency_render_mode = \"json\"\n", + "\n", + " # This decorator will be added to all function definitions\n", + " def _display_if_has_repr_html(x):\n", + " try:\n", + " # IPython 7.14 preferred import\n", + " from IPython.display import display, HTML\n", + " except:\n", + " from IPython.core.display import display, HTML\n", + "\n", + " if hasattr(x, '_repr_html_'):\n", + " display(HTML(x._repr_html_()))\n", + " return x\n", + "\n", + " # ideally we would undo the call to ast_transformers.append\n", + " # at the end of this block whenver an error occurs, we do \n", + " # this for now as it will only be a problem if the user \n", + " # switches from shiny to not-shiny mode (and even then likely\n", + " # won't matter)\n", + " import builtins\n", + " builtins._display_if_has_repr_html = _display_if_has_repr_html\n", + "\n", + " class _FunctionDefReprHtml(_ast.NodeTransformer):\n", + " def visit_FunctionDef(self, node):\n", + " node.decorator_list.insert(\n", + " 0,\n", + " _ast.Name(id=\"_display_if_has_repr_html\", ctx=_ast.Load())\n", + " )\n", + " return node\n", + "\n", + " def visit_AsyncFunctionDef(self, node):\n", + " node.decorator_list.insert(\n", + " 0,\n", + " _ast.Name(id=\"_display_if_has_repr_html\", ctx=_ast.Load())\n", + " )\n", + " return node\n", + "\n", + " ip = get_ipython()\n", + " ip.ast_transformers.append(_FunctionDefReprHtml())\n", + "\n", + " except:\n", + " pass\n", + "\n", + "def ojs_define(**kwargs):\n", + " import json\n", + " try:\n", + " # IPython 7.14 preferred import\n", + " from IPython.display import display, HTML\n", + " except:\n", + " from IPython.core.display import display, HTML\n", + "\n", + " # do some minor magic for convenience when handling pandas\n", + " # dataframes\n", + " def convert(v):\n", + " try:\n", + " import pandas as pd\n", + " except ModuleNotFoundError: # don't do the magic when pandas is not available\n", + " return v\n", + " if type(v) == pd.Series:\n", + " v = pd.DataFrame(v)\n", + " if type(v) == pd.DataFrame:\n", + " j = json.loads(v.T.to_json(orient='split'))\n", + " return dict((k,v) for (k,v) in zip(j[\"index\"], j[\"data\"]))\n", + " else:\n", + " return v\n", + "\n", + " v = dict(contents=list(dict(name=key, value=convert(value)) for (key, value) in kwargs.items()))\n", + " display(HTML(''), metadata=dict(ojs_define = True))\n", + "globals()[\"ojs_define\"] = ojs_define\n", + "# globals()[\"__spec__\"] = None" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "importing-libraries", + "metadata": {}, + "outputs": [], + "source": [ + "#| label: importing-libraries\n", + "# Manejo de datos y análisis\n", + "import numpy as np\n", + "import pandas as pd\n", + "import scipy.stats as stats\n", + "\n", + "# Modelos estadísticos y econométricos\n", + "import statsmodels.api as sm\n", + "from pmdarima.arima import auto_arima\n", + "from arch import arch_model" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "importing-data", + "metadata": {}, + "outputs": [], + "source": [ + "#| label: importing-data\n", + "df = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\n", + "returns = df['price_return']\n", + "split_date = '2020-12-31'\n", + "R_test = df[df.index >= split_date]['price_return'].rolling(\n", + " window=5).std().dropna()" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "return-descriptive-stats", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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ObservationsMeanMedianStd. DevSkewnessKurtosisJarque-BeraProb.
019411.353251-0.31804423.5029110.705277.5879954789.5323310.0
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" + ], + "text/plain": [ + " Observations Mean Median Std. Dev Skewness Kurtosis \\\n", + "0 1941 1.353251 -0.318044 23.502911 0.70527 7.587995 \n", + "\n", + " Jarque-Bera Prob. \n", + "0 4789.532331 0.0 " + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: return-descriptive-stats\n", + "#| fig-cap: Tabla de las estadísticas descriptiva de la serie de retornos\n", + "desc_stats = returns.describe()\n", + "\n", + "skewness = returns.skew()\n", + "kurtosis = returns.kurtosis()\n", + "jb_test = sm.stats.jarque_bera(returns)\n", + "\n", + "descriptive_table = pd.DataFrame({\n", + " 'Observations': [int(desc_stats['count'])],\n", + " 'Mean': [desc_stats['mean']],\n", + " 'Median': [desc_stats['50%']],\n", + " 'Std. Dev': [desc_stats['std']],\n", + " 'Skewness': [skewness],\n", + " 'Kurtosis': [kurtosis],\n", + " 'Jarque-Bera': [jb_test[0]],\n", + " 'Prob.': [jb_test[1]]\n", + "})\n", + "descriptive_table" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "arima-model", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:48:37 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
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coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
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Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step)." + ], + "text/latex": [ + "\\begin{center}\n", + "\\begin{tabular}{lclc}\n", + "\\toprule\n", + "\\textbf{Dep. Variable:} & y & \\textbf{ No. Observations: } & 1941 \\\\\n", + "\\textbf{Model:} & SARIMAX(3, 0, 4) & \\textbf{ Log Likelihood } & -8853.517 \\\\\n", + "\\textbf{Date:} & Sat, 19 Oct 2024 & \\textbf{ AIC } & 17725.034 \\\\\n", + "\\textbf{Time:} & 15:48:37 & \\textbf{ BIC } & 17775.173 \\\\\n", + "\\textbf{Sample:} & 0 & \\textbf{ HQIC } & 17743.472 \\\\\n", + "\\textbf{} & - 1941 & \\textbf{ } & \\\\\n", + "\\textbf{Covariance Type:} & opg & \\textbf{ } & \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lcccccc}\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{z} & \\textbf{P$> |$z$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", + "\\midrule\n", + "\\textbf{intercept} & 0.3602 & 0.273 & 1.317 & 0.188 & -0.176 & 0.896 \\\\\n", + "\\textbf{ar.L1} & 0.1366 & 0.065 & 2.106 & 0.035 & 0.009 & 0.264 \\\\\n", + "\\textbf{ar.L2} & -0.1931 & 0.066 & -2.934 & 0.003 & -0.322 & -0.064 \\\\\n", + "\\textbf{ar.L3} & 0.7691 & 0.061 & 12.644 & 0.000 & 0.650 & 0.888 \\\\\n", + "\\textbf{ma.L1} & -0.1139 & 0.065 & -1.752 & 0.080 & -0.241 & 0.014 \\\\\n", + "\\textbf{ma.L2} & 0.2163 & 0.067 & 3.220 & 0.001 & 0.085 & 0.348 \\\\\n", + "\\textbf{ma.L3} & -0.7032 & 0.062 & -11.315 & 0.000 & -0.825 & -0.581 \\\\\n", + "\\textbf{ma.L4} & 0.0750 & 0.020 & 3.715 & 0.000 & 0.035 & 0.115 \\\\\n", + "\\textbf{sigma2} & 536.2183 & 8.358 & 64.154 & 0.000 & 519.836 & 552.600 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lclc}\n", + "\\textbf{Ljung-Box (L1) (Q):} & 0.00 & \\textbf{ Jarque-Bera (JB): } & 4395.18 \\\\\n", + "\\textbf{Prob(Q):} & 1.00 & \\textbf{ Prob(JB): } & 0.00 \\\\\n", + "\\textbf{Heteroskedasticity (H):} & 1.11 & \\textbf{ Skew: } & 0.59 \\\\\n", + "\\textbf{Prob(H) (two-sided):} & 0.17 & \\textbf{ Kurtosis: } & 10.28 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "%\\caption{SARIMAX Results}\n", + "\\end{center}\n", + "\n", + "Warnings: \\newline\n", + " [1] Covariance matrix calculated using the outer product of gradients (complex-step)." + ], + "text/plain": [ + "\n", + "\"\"\"\n", + " SARIMAX Results \n", + "==============================================================================\n", + "Dep. Variable: y No. Observations: 1941\n", + "Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517\n", + "Date: Sat, 19 Oct 2024 AIC 17725.034\n", + "Time: 15:48:37 BIC 17775.173\n", + "Sample: 0 HQIC 17743.472\n", + " - 1941 \n", + "Covariance Type: opg \n", + "==============================================================================\n", + " coef std err z P>|z| [0.025 0.975]\n", + "------------------------------------------------------------------------------\n", + "intercept 0.3602 0.273 1.317 0.188 -0.176 0.896\n", + "ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264\n", + "ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064\n", + "ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888\n", + "ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014\n", + "ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348\n", + "ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581\n", + "ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115\n", + "sigma2 536.2183 8.358 64.154 0.000 519.836 552.600\n", + "===================================================================================\n", + "Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18\n", + "Prob(Q): 1.00 Prob(JB): 0.00\n", + "Heteroskedasticity (H): 1.11 Skew: 0.59\n", + "Prob(H) (two-sided): 0.17 Kurtosis: 10.28\n", + "===================================================================================\n", + "\n", + "Warnings:\n", + "[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n", + "\"\"\"" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: arima-model\n", + "#| fig-cap: Modelo ARIMA de maxima verosimilitud para la serie de retornos.\n", + "#| warning: false\n", + "model_auto = auto_arima(returns)\n", + "model_auto.summary()" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "garch-model-fitting", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Iteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\n", + "Iteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\n", + "Iteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\n", + "Iteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\n", + "Iteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\n", + "Iteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\n", + "Iteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\n", + "Iteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\n", + "Iteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\n", + "Iteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\n", + "Iteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\n", + "Iteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\n", + "Iteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\n", + "Iteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\n", + "Iteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\n", + "Iteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\n", + "Iteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\n", + "Iteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\n", + "Iteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\n", + "Iteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\n", + "Iteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\n", + "Iteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\n", + "Iteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\n", + "Iteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\n", + "Iteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\n", + "Iteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\n", + "Optimization terminated successfully (Exit mode 0)\n", + " Current function value: 7622.868827051428\n", + " Iterations: 26\n", + " Function evaluations: 139\n", + " Gradient evaluations: 26\n" + ] + } + ], + "source": [ + "#| label: garch-model-fitting\n", + "#| fig-cap: Ajuste del modelo Zero-Garch ala serie de retornos\n", + "#| warning: false\n", + "#| output: false\n", + "\n", + "ar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\n", + "res = ar.fit(last_obs=split_date)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "garch-model", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "\n", + " \n", + "\n", + "
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:48:37 Df Model: 0
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Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
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Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust" + ], + "text/latex": [ + "\\begin{center}\n", + "\\begin{tabular}{lclc}\n", + "\\toprule\n", + "\\textbf{Dep. Variable:} & price\\_return & \\textbf{ R-squared: } & 0.000 \\\\\n", + "\\textbf{Mean Model:} & Zero Mean & \\textbf{ Adj. R-squared: } & 0.001 \\\\\n", + "\\textbf{Vol Model:} & GARCH & \\textbf{ Log-Likelihood: } & -7622.87 \\\\\n", + "\\textbf{Distribution:} & Standardized Student's t & \\textbf{ AIC: } & 15253.7 \\\\\n", + "\\textbf{Method:} & Maximum Likelihood & \\textbf{ BIC: } & 15275.6 \\\\\n", + "\\textbf{} & & \\textbf{ No. Observations: } & 1753 \\\\\n", + "\\textbf{Date:} & Sat, Oct 19 2024 & \\textbf{ Df Residuals: } & 1753 \\\\\n", + "\\textbf{Time:} & 15:48:37 & \\textbf{ Df Model: } & 0 \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "\\begin{tabular}{lccccc}\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n", + "\\midrule\n", + "\\textbf{omega} & 58.8679 & 25.182 & 2.338 & 1.940e-02 & [ 9.512,1.082e+02] \\\\\n", + "\\textbf{alpha[1]} & 0.1892 & 6.657e-02 & 2.842 & 4.482e-03 & [5.873e-02, 0.320] \\\\\n", + "\\textbf{beta[1]} & 0.7737 & 6.832e-02 & 11.324 & 9.943e-30 & [ 0.640, 0.908] \\\\\n", + " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n", + "\\midrule\n", + "\\textbf{nu} & 2.7894 & 0.230 & 12.114 & 8.853e-34 & [ 2.338, 3.241] \\\\\n", + "\\bottomrule\n", + "\\end{tabular}\n", + "%\\caption{Zero Mean - GARCH Model Results}\n", + "\\end{center}\n", + "\n", + "Covariance estimator: robust" + ], + "text/plain": [ + "\n", + "\"\"\"\n", + " Zero Mean - GARCH Model Results \n", + "====================================================================================\n", + "Dep. Variable: price_return R-squared: 0.000\n", + "Mean Model: Zero Mean Adj. R-squared: 0.001\n", + "Vol Model: GARCH Log-Likelihood: -7622.87\n", + "Distribution: Standardized Student's t AIC: 15253.7\n", + "Method: Maximum Likelihood BIC: 15275.6\n", + " No. Observations: 1753\n", + "Date: Sat, Oct 19 2024 Df Residuals: 1753\n", + "Time: 15:48:37 Df Model: 0\n", + " Volatility Model \n", + "==========================================================================\n", + " coef std err t P>|t| 95.0% Conf. Int.\n", + "--------------------------------------------------------------------------\n", + "omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]\n", + "alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]\n", + "beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]\n", + " Distribution \n", + "========================================================================\n", + " coef std err t P>|t| 95.0% Conf. Int.\n", + "------------------------------------------------------------------------\n", + "nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]\n", + "========================================================================\n", + "\n", + "Covariance estimator: robust\n", + "\"\"\"" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#| label: garch-model\n", + "#| fig-cap: Modelo Zero-Garch de la serie de retornos\n", + "res.summary()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3", + "path": "C:\\Users\\contr\\anaconda3\\share\\jupyter\\kernels\\python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/.jupyter_cache/global.db b/.jupyter_cache/global.db index 446f20ceba3e2c88c6c0b26debc380e97b8aba86..4234c3ce963578b82257e417d3fe34c75565d880 100644 GIT binary patch delta 477 zcmaiwJ5Iwu6h&i^SV+j3q)I`GNTEPTv1aBy&x~CW(a=OlD0RYkK2!+rfUE!k-T_LL z4pG1gSOrB&mO)&EZ{hkL9o>5u>tL}C?)HP;eg7orJzXr%I$<A-r z&T({9gu9Iii@M{n_qj+62Q<vlR9HcEyIU1hx1w1=K&@rDg2ZZCK3>m00gcReng9R* delta 75 zcmZp8z}WDBae_3X@I)DBM&XSKOZ>T*`BpLT_wmc{t=cRoFo$pQst diff --git a/_freeze/index/execute-results/docx.json b/_freeze/index/execute-results/docx.json index 61fb894..6b6638b 100644 --- a/_freeze/index/execute-results/docx.json +++ b/_freeze/index/execute-results/docx.json @@ -1,9 +1,10 @@ { - "hash": "8ccdb86df4a0f84495987258f19b87bf", + "hash": "22289e552d482a5d6266287352c6d0f1", "result": { - "markdown": "---\ntitle: La Palma Earthquakes\nauthor:\n - name: Steve Purves\n orcid: 0000-0002-0760-5497\n corresponding: true\n email: steve@curvenote.com\n roles:\n - Investigation\n - Project administration\n - Software\n - Visualization\n affiliations:\n - Curvenote\n - name: Rowan Cockett\n orcid: 0000-0002-7859-8394\n corresponding: false\n roles: []\n affiliations:\n - Curvenote\nkeywords:\n - La Palma\n - Earthquakes\nabstract: |\n In September 2021, a significant jump in seismic activity on the island of La Palma (Canary Islands, Spain) signaled the start of a volcanic crisis that still continues at the time of writing. Earthquake data is continually collected and published by the Instituto Geográphico Nacional (IGN). ...\nplain-language-summary: |\n Earthquake data for the island of La Palma from the September 2021 eruption is found ...\nkey-points:\n - A web scraping script was developed to pull data from the Instituto Geogràphico Nacional into a machine-readable form for analysis\n - Earthquake events on La Palma are consistent with the presence of both mantle and crustal reservoirs.\ndate: last-modified\nbibliography: references.bib\ncitation:\n container-title: Earth and Space Science\nnumber-sections: true\n---\n\n:::{#9f477374 .cell .markdown}\n## Introduction\n:::\n\n::: {.cell execution_count=1}\n``` {.python .cell-code .hidden}\nimport matplotlib.pyplot as plt\nimport numpy as np\neruptions = [1492, 1585, 1646, 1677, 1712, 1949, 1971, 2021]\n```\n:::\n\n\n::: {.cell execution_count=2}\n``` {.python .cell-code .hidden}\nplt.figure(figsize=(6, 1))\nplt.eventplot(eruptions, lineoffsets=0, linelengths=0.1, color='black')\nplt.gca().axes.get_yaxis().set_visible(False)\nplt.ylabel('')\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Timeline of recent earthquakes on La Palma](index_files/figure-docx/fig-timeline-output-1.png){#fig-timeline fig-alt='An event plot of the years of the last 8 eruptions on La Palma.'}\n:::\n:::\n\n\n::: {.cell execution_count=3}\n``` {.python .cell-code .hidden}\navg_years_between_eruptions = np.mean(np.diff(eruptions[:-1]))\navg_years_between_eruptions\n```\n\n::: {.cell-output .cell-output-display .hidden execution_count=6}\n```\n79.83333333333333\n```\n:::\n:::\n\n\n:::{#f319e497 .cell .markdown}\nBased on data up to and including 1971, eruptions on La Palma happen every 79.8 years on average.\n\nStudies of the magma systems feeding the volcano, such as @marrero2019, have proposed that there are two main magma reservoirs feeding the Cumbre Vieja volcano; one in the mantle (30-40km depth) which charges and in turn feeds a shallower crustal reservoir (10-20km depth).\n\nEight eruptions have been recorded since the late 1400s (@fig-timeline).\n\nData and methods are discussed in @sec-data-methods.\n\nLet $x$ denote the number of eruptions in a year. Then, $x$ can be modeled by a Poisson distribution\n\n$$\np(x) = \\frac{e^{-\\lambda} \\lambda^{x}}{x !}\n$$ {#eq-poisson}\n\nwhere $\\lambda$ is the rate of eruptions per year. Using @eq-poisson, the probability of an eruption in the next $t$ years can be calculated.\n\n| Name | Year |\n|---------------------|------|\n| Current | 2021 |\n| Teneguía | 1971 |\n| Nambroque | 1949 |\n| El Charco | 1712 |\n| Volcán San Antonio | 1677 |\n| Volcán San Martin | 1646 |\n| Tajuya near El Paso | 1585 |\n| Montaña Quemada | 1492 |\n\n: Recent historic eruptions on La Palma {#tbl-history}\n\n@tbl-history summarises the eruptions recorded since the colonization of the islands by Europeans in the late 1400s.\n\n![Map of La Palma](images/la-palma-map.png){#fig-map}\n\nLa Palma is one of the west most islands in the Volcanic Archipelago of the Canary Islands (@fig-map).\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-spatial-plot >}}\n\n\n\n\n\n@fig-spatial-plot shows the location of recent Earthquakes on La Palma.\n\n## Data & Methods {#sec-data-methods}\n\n## Conclusion\n\n## References {.unnumbered}\n\n::: {#refs}\n:::\n:::\n\n", + "engine": "jupyter", + "markdown": "---\ntitle: Un estudio sobre los retornos del mercado de renta fija dominicano utilizando basados en modelos AR-GARCH\nsubtitle: Trabajo final de tópicos de econometría\nauthor:\n - name: Ian Contreras\n email: 1116048@intec.edu.do\n affiliations:\n - INSTITUTO TECNOLÓGICO DE SANTO DOMINGO (INTEC)\nkeywords:\n - Mercado de renta fija\n - Deuda pública\n - Modelos AR-GARCH\nabstract: |\n Este estudio tiene como objetivo analizar los retornos y la volatilidad del mercado de deuda pública dominicana mediante la aplicación de modelos ARMA-GARCH. A través de estos modelos, se busca identificar patrones en el comportamiento de los retornos y predecir su evolución futura. \ndate: last-modified\nbibliography: references.bib\nexecute:\n echo: false\nnocite: |\n @*\ncitation:\n container-title: Referencias\nnumber-sections: true\n---\n\n::: {#importing-libraries .cell execution_count=1}\n``` {.python .cell-code .hidden}\n# Manejo de datos y análisis\nimport numpy as np\nimport pandas as pd\nimport scipy.stats as stats\n\n# Modelos estadísticos y econométricos\nimport statsmodels.api as sm\nfrom pmdarima.arima import auto_arima\nfrom arch import arch_model\n```\n:::\n\n\n::: {#importing-data .cell execution_count=2}\n``` {.python .cell-code .hidden}\ndf = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\nreturns = df['price_return']\nsplit_date = '2020-12-31'\nR_test = df[df.index >= split_date]['price_return'].rolling(\n window=5).std().dropna()\n```\n:::\n\n\n:::{#e2a8e1fe .cell .markdown}\n# Introducción\n\nEl mercado de renta fija en la República Dominicana ha emergido como uno de los sectores más importantes dentro de la economía, especialmente en lo que respecta a la deuda pública, la cual juega un papel crucial en la financiación de proyectos gubernamentales y el sostenimiento de la política fiscal. En los últimos cincuenta años, el país ha experimentado un crecimiento económico significativo, consolidándose como una de las economías más dinámicas de América Latina. Este crecimiento ha estado acompañado por una mayor complejidad económica, lo que ha incrementado la relevancia del mercado financiero dominicano, en particular el mercado de deuda pública, que ha sido un activo clave para fondos de pensiones y fondos de inversión.\n\nSin embargo, este mercado también presenta importantes desafíos, especialmente en términos de baja liquidez en el mercado secundario y episodios de alta volatilidad. Estos problemas dificultan la previsión precisa de los movimientos del mercado y complican la toma de decisiones estratégicas para los inversores. En este contexto, surge la necesidad de utilizar herramientas econométricas que puedan capturar con mayor precisión las dinámicas subyacentes de los retornos y la volatilidad del mercado de deuda pública dominicano.\n\nLa **pregunta de investigación** que guía este estudio es: ¿Es posible modelar adecuadamente los retornos y la volatilidad del mercado de deuda pública dominicano mediante modelos ARIMA-GARCH?, y ¿Qué tan efectivas son estas técnicas para predecir el comportamiento futuro del mercado, especialmente durante periodos de alta volatilidad?\n\nEl **objetivo principal** de este estudio es aplicar modelos ARMA-GARCH para analizar y predecir el comportamiento de los retornos del índice de deuda pública IRP-GOBIX, considerando tanto la tendencia como los patrones de volatilidad del mercado. A través de la estimación de estos modelos, se busca identificar los principales factores que afectan la dinámica de los retornos y proporcionar información valiosa para los actores del mercado, facilitando una gestión más eficiente del riesgo.\n\nAdemás, este estudio pretende evaluar la **consistencia de los modelos** y determinar si los modelos de volatilidad más sofisticados, como el Zero-GARCH, ofrecen una mejor capacidad de predicción que los modelos tradicionales basados únicamente en niveles y diferencias, como el ARIMA. Un aspecto clave será analizar si el modelo GARCH captura adecuadamente los choques bruscos y los episodios de volatilidad extrema, o si suaviza excesivamente estos movimientos.\n\nLa **relevancia** de este estudio radica en la creciente importancia del mercado de deuda pública dominicano, tanto para la estabilidad financiera del país como para los inversores extranjeros que buscan oportunidades de inversión en mercados emergentes. Comprender los patrones de retorno y volatilidad es esencial para diseñar estrategias de inversión que minimicen el riesgo y maximicen el rendimiento en un entorno económico globalizado y altamente incierto.\n\n# Metodología\n\nEl modelo ARIMA(p,d,q), donde AR(p) es el componente autorregresivo de orden p, d denota el orden de la diferencia aplicada para transformar una serie temporal no estacionaria en estacionaria, y MA(q) representa el promedio móvil de orden q. La esencia del modelo ARIMA radica en la combinación de la operación de diferenciación y el modelo ARMA. Cualquier serie no estacionaria puede volverse estacionaria mediante una diferenciación de orden adecuado, permitiendo así ajustar un modelo ARMA a la serie transformada.\n\nEl modelo de promedio móvil autorregresivo ARMA(p,q) incluye tanto un componente autorregresivo de orden p como uno de promedio móvil de orden q, y se estructura de la siguiente forma:\n\n$$\nX_t = \\phi_1 X_{t-1} + \\phi_2 X_{t-2} + \\dots + \\phi_p X_{t-p} + \\epsilon_t + \\theta_1 \\epsilon_{t-1} + \\theta_2 \\epsilon_{t-2} + \\dots + \\theta_q \\epsilon_{t-q}\n$$ {eq: arma}\n\nEstos modelos son adecuados solo para describir series temporales suaves, mientras que en la práctica, a menudo se trabaja con series temporales no suaves. Para estos casos, aplicar una o dos diferenciaciones puede transformar los datos en series suaves.\n\nSi no se cumple el supuesto de homogeneidad de la varianza, puede ocurrir heterocedasticidad. Dado que los datos de la muestra son suaves, en este documento se establece un modelo ARMA(p,q) para la serie de rendimientos con el fin de describir las características de volatilidad del índice GOBIX. Se realizan pruebas de efecto ARCH y se resuelve el problema de la heterocedasticidad mediante un modelo GARCH.\n\nEl modelo GARCH extiende el modelo ARCH al considerar la autocorrelación de orden p en la función de heterocedasticidad, lo que permite ajustar de manera efectiva una función de heterocedasticidad con memoria a largo plazo. El modelo GARCH se define de la siguiente manera:\n\n$$\na_t = \\sigma_t \\epsilon_t, \\quad \\sigma_t^2 = \\alpha_0 + \\sum_{i=1}^{p} \\alpha_i a_{t-i}^2 + \\sum_{j=1}^{q} \\beta_j \\sigma_{t-j}^2\n$$ {#eq: garch}\n\nEl modelo GARCH, propuesto por Bollerslev, atribuye la volatilidad actual tanto a la volatilidad de momentos pasados como a los errores de momentos pasados, lo que le permite explicar el fenómeno de agrupación en la volatilidad de los rendimientos financieros.\n\n# Datos\n\n## Fuente de datos\nUn Índice Financiero es una medida estadística que refleja el valor de un conjunto de activos financieros agrupados de acuerdo con ciertos criterios.Estos se utilizan como objetivo de rendimiento de portafolios, referencias sobre las características retorno/riesgo de una clase de activo y como referencia para productos anclados a índices.\n\nEn el mercado de renta fija de la República Dominicana, el único índice público es el GOBIX, que representa la deuda gubernamental consolidada en peso. Este índice está compuesto por títulos emitidos localmente por el Banco Central y el Ministerio de Hacienda, y se publica en dos modalidades: índice de precio limpio e índice de retorno-precio. Para este estudio, se utilizará el índice de retorno precio @sec-retorno-precio, ya que aisla el componente de retorno de mercado el cual es el objetivo de estudio. \n\nEl GOBIX se caracteriza por los siguientes criterios:\n\n- **Elegibilidad**: Solo se incluyen títulos \"bullet\" emitidos en pesos dominicanos con al menos 60 días desde su emisión.\n- **Liquidez**: Los títulos se seleccionan en función del Índice de Bursatilidad publicado por la Proveedora de Precios, garantizando la inclusión de los instrumentos más negociados.\n- **Ponderación**: La ponderación se realiza por capitalización de mercado, lo que asegura que los títulos más relevantes tengan mayor peso en el índice.\n- **Rebalanceo**: Se efectúa un rebalanceo mensual para ajustar el índice según la disponibilidad de inversión en los bonos.\n- **Fuentes de información**: Los datos provienen de la Bolsa de Valores de la República Dominicana y la Proveedora de Precios.\n\n## Índice de Retorno-Precio {#sec-retorno-precio}\n\nEl índice de retorno-precio es una metodología utilizada para calcular el rendimiento de un índice de renta fija, centrándose exclusivamente en las variaciones del precio limpio de los títulos, sin tener en cuenta los intereses acumulados ni el pago de cupones durante el período de cálculo. En el contexto del GOBIX, este índice refleja las fluctuaciones en los precios de los bonos dominicanos debido a las variaciones en las tasas de interés del mercado.\n\n**Características del Índice de Retorno-Precio del GOBIX**:\n\n- **Enfoque en el Precio Limpio**: El cálculo del índice se basa en el precio limpio de los bonos, excluyendo los intereses acumulados.\n- **Reflejo de la Volatilidad**: Este índice es un indicador del riesgo asociado a las fluctuaciones en las tasas de interés y su impacto en el capital invertido.\n- **Fórmula de Cálculo**:\n\n $$\n IRP_t = IRP_{m-1} \\times (1 + RPI_t)\n $$\n\n Donde:\n - $IRP_0 = 100$ es el valor base del índice en la fecha de inicio.\n - $IRP_t$ es el valor del índice en el día $t$.\n - $IRP_{m-1}$ es el valor del índice en el último día hábil del mes anterior.\n - $RPI_t$ es el retorno-precio del índice en $t$.\n\nEl retorno-precio del índice ($RPI_t$) se calcula como la suma ponderada del retorno-precio de cada bono ($RPI_{i,t}$), donde la ponderación ($\\omega_i$) se determina según la capitalización de mercado de cada bono.\n\nEn este análisis, expresaremos la serie de retornos en puntos básicos para evitar problemas de escalamiento de los datos y posibles desbordamientos decimales (overflow decimal), que pueden ocurrir cuando se manejan fluctuaciones muy pequeñas con alta precisión. Un punto básico equivale a 0.01% o 1/100 de un porcentaje. Este enfoque nos permitirá manejar los datos con mayor estabilidad y evitar posibles errores numéricos derivados del escalamiento inapropiado. No tiene efectos estadísticos en las distribución. \n\n## Selección de muestra de entrenamiento/prueba {#sec-test-train}\n\nDurante la pandemia, se experimentó uno de los ciclos de tasas de interés más pronunciados de las últimas décadas. Este periodo presentó características estructurales distintas en comparación con los ciclos históricos previos. Para evitar sesgos en las estimaciones debido a la alta volatilidad observada durante ese tiempo, evaluaremos únicamente la serie de retornos entre el periodo 2014-2021, que se considera un ciclo de tasas más regular.\n\nLa metodología empleada para dividir el conjunto de datos en muestras de entrenamiento y prueba será el **Time Series Cross-Validator**. A diferencia de la validación cruzada tradicional, que aleatoriza los datos, esta técnica respeta el orden temporal, lo cual es crucial en el análisis de series de tiempo.\n\nEn cada partición (k-ésimo split), la técnica funciona de la siguiente manera:\n\n- **Conjunto de Entrenamiento**: Incluye los primeros *k* pliegues (folds).\n- **Conjunto de Prueba**: Incluye el pliegue *(k+1)*.\n\n# Análisis Exploratorio {#sec-eda}\n\nEl objetivo de este análisis exploratorio es examinar la serie de retornos diarios del IRP-GOBIX para identificar patrones de volatilidad, evaluar la distribución de los retornos, y explorar la autocorrelación en los datos. Utilizando herramientas estadísticas como la distribución de los retornos, el test de estacionariedad Dickey-Fuller, y los gráficos de autocorrelación (ACF) y autocorrelación parcial (PACF), se busca comprender la dinámica temporal de los retornos y validar la aplicación de modelos ARMA-GARCH para capturar su comportamiento futuro. Adicionalmente, se ajustarán diversas distribuciones teóricas, como la distribución normal y la t de Student, para evaluar su capacidad de describir las características empíricas de los datos.\n\n## Histórico de retornos precio del mercado de deuda pública dominicano {#sec-historico-retornos}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-price-return-series >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis de la serie de retornos diarios del IRP-GOBIX revela un comportamiento interesante, con periodos prolongados de baja volatilidad seguidos por episodios de alta volatilidad. Entre 2015 y 2022, los retornos se mantuvieron generalmente dentro de un rango de ±50 puntos básicos, lo que indica una estabilidad relativa. Sin embargo, en momentos clave, como el ciclo de tasas de 2018, la volatilidad se incrementó drásticamente, alcanzando picos de hasta 200 puntos básicos. Este patrón es característico del fenómeno conocido como **\"volatility clustering\"**, donde periodos tranquilos son seguidos por fases de mayor volatilidad. La naturaleza de estos picos parece estar relacionada con factores externos, como cambios en las tasas de interés y la incertidumbre macroeconómica global, lo que subraya la importancia de emplear modelos ARMA-GARCH para capturar estas dinámicas y prever comportamientos futuros en el mercado de deuda pública dominicana.\n\n## Análisis descriptivo de la muestra {#sec-estadistica-descriptiva}\n:::\n\n::: {.cell execution_count=3}\n``` {.python .cell-code .hidden}\ndesc_stats = returns.describe()\n\nskewness = returns.skew()\nkurtosis = returns.kurtosis()\njb_test = sm.stats.jarque_bera(returns)\n\ndescriptive_table = pd.DataFrame({\n 'Observations': [int(desc_stats['count'])],\n 'Mean': [desc_stats['mean']],\n 'Median': [desc_stats['50%']],\n 'Std. Dev': [desc_stats['std']],\n 'Skewness': [skewness],\n 'Kurtosis': [kurtosis],\n 'Jarque-Bera': [jb_test[0]],\n 'Prob.': [jb_test[1]]\n})\ndescriptive_table\n```\n\n::: {#return-descriptive-stats .cell-output .cell-output-display execution_count=9}\n```{=html}\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ObservationsMeanMedianStd. DevSkewnessKurtosisJarque-BeraProb.
019411.353251-0.31804423.5029110.705277.5879954789.5323310.0
\n
\n```\n\nTabla de las estadísticas descriptiva de la serie de retornos\n:::\n:::\n\n\n:::{#e3661616 .cell .markdown}\nEl análisis descriptivo de los retornos ofrece una perspectiva más detallada sobre la distribución de la serie. Con un total de 1,941 observaciones, la **media** de los retornos es de $1.353251$, lo que indica un rendimiento promedio positivo. Sin embargo, la **mediana** de $-0.318044$ sugiere un leve sesgo hacia valores negativos, lo que refleja que la mayoría de los retornos tienden a ser ligeramente inferiores a la media. La **desviación estándar**, con un valor de $23.502911$, revela una amplia dispersión de los datos, lo que confirma la presencia de una volatilidad significativa en los rendimientos, muy superior al valor medio.\n\nEn cuanto a la **asimetría** ($0.70527$), se observa una ligera inclinación hacia valores extremos positivos, lo que implica la existencia de eventos fuera de lo común que afectan los rendimientos. La **kurtosis** de $7.587995$ indica una distribución **leptocúrtica**, caracterizada por una alta concentración de valores alrededor de la media y colas más gruesas que una distribución normal, lo que es típico en datos financieros que presentan eventos extremos. El **test de Jarque-Bera**, con un valor de $4,789.532331$, confirma que la serie no sigue una distribución normal, validando la presencia de asimetría y colas pesadas en los datos. Estos resultados resaltan la necesidad de emplear modelos que puedan capturar adecuadamente estos comportamientos no lineales, esenciales para el análisis del mercado.\n\n## Análisis de la distribución de los retornos {#sec-distribucion-retornos}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-distribution-fitting >}}\n\n\n\n\n\n\n\n\n\n\nAl ajustar diferentes distribuciones a los retornos, observamos que la distribución empírica (en azul) tiene una alta concentración en torno a cero, con colas más gruesas de lo que se esperaría bajo una distribución normal. Entre las distribuciones probadas, la t de Student (en rojo) es la que mejor captura los valores extremos, con colas más largas y una mayor concentración en el centro, lo que es típico en series financieras que experimentan episodios de alta volatilidad. En contraste, la distribución normal (en naranja) y la lognormal (en verde) subestiman las colas, demostrando su ineficacia para modelar adecuadamente los valores atípicos. Esto refuerza la idea de que la t de Student es una mejor candidata para modelar los retornos, dado que puede ajustarse mejor a la leptocurtosis observada en los datos.\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-tqq-plot >}}\n\n\n\n\n\n\n\n\n\n\nLa gráfica Q-Q (Quantile-Quantile) compara los cuantiles teóricos de la distribución t de Student con los cuantiles observados de los retornos del GOBIX. En general, la mayoría de los puntos se alinean bien con la línea roja, lo que sugiere que los retornos de precios siguen razonablemente esta distribución, particularmente en las partes centrales. Sin embargo, en las colas extremas, se observan algunas desviaciones, lo que indica que, aunque la t de Student es un buen ajuste, no es perfecta para todos los escenarios.\n\nEl **estadístico de Kolmogorov-Smirnov** es de $0.0286$, con un **p-valor** de $0.0827$. Aunque esto sugiere que la t de Student captura bien la mayor parte de los retornos, las discrepancias en las colas extremas muestran que la distribución teórica no es un ajuste exacto a los datos reales.\n\n## Evaluación de supuestos para análisis AR-GARCH\n\n### Test de estacionariedad {#sec-estacionariedad}\n\nPara garantizar la validez de los modelos ARMA y GARCH, se evaluó la estacionariedad de la serie de retornos utilizando el test de Dickey-Fuller aumentado (ADF). Los resultados obtenidos muestran un **ADF Statistic** de $-13.1741$ y un **p-valor** de $1.2333 \\times 10^{-24}$, lo que permite rechazar la hipótesis nula de raíz unitaria con un nivel de significancia del 5%. Esto confirma que la serie es estacionaria, lo que significa que sus fluctuaciones se distribuyen alrededor de una media constante en el tiempo, sin tendencia significativa. Este hallazgo es consistente con el comportamiento típico de las series de retornos financieros, lo que habilita la correcta estimación de modelos ARMA-GARCH para capturar las dinámicas del mercado.\n\n### Test de autocorrelación {#sec-autocorrelacion}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-acf-pacf >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis de autocorrelación nos proporciona más detalles sobre la estructura interna de la serie. El gráfico de la **Función de Autocorrelación (ACF)** muestra un pico significativo en el primer rezago, seguido de valores cercanos a cero para los rezagos posteriores. Este patrón es indicativo de un proceso de media móvil de primer orden (**MA(1)**), donde los choques aleatorios tienen un impacto significativo en el primer rezago pero no en los siguientes. La **Función de Autocorrelación Parcial (PACF)** refuerza esta conclusión, mostrando un comportamiento similar con un pico en el primer rezago y valores prácticamente nulos a partir del segundo rezago. Esto sugiere que un modelo MA(1) sería apropiado para capturar las dinámicas de corto plazo en la serie.\n\nEn conjunto, este análisis exploratorio sugiere que la serie de retornos del IRP-GOBIX presenta características complejas, como **volatility clustering**, alta **leptocurtosis**, y una estructura de autocorrelación que se ajusta bien a un modelo MA(1). Estos hallazgos proporcionan una base sólida para la estimación de modelos ARMA-GARCH, los cuales podrán capturar de manera efectiva las dinámicas de volatilidad y retornos en el mercado de deuda pública dominicano.\n\n# Resultados\n\nEl presente análisis tiene como objetivo modelar la serie temporal de retornos del índice IRP-GOBIX utilizando inicialmente un modelo ARIMA automático (auto-ARIMA) para determinar si es posible capturar la dinámica de los retornos con un modelo basado únicamente en niveles y diferencias de los datos. Adicionalmente, se evaluará la presencia de problemas estructurales, como la heterocedasticidad, que podrían afectar la consistencia del modelo ARIMA. En caso de que este enfoque no sea adecuado, recurriremos a un modelo Zero-GARCH, un tipo específico de GARCH que asume una media cero para los retornos. Este modelo es particularmente útil en mercados de tasas de interés, donde se ha demostrado que los retornos tienden a cero en el largo plazo debido a la fuerte regresión hacia la media de estas variables. \n\n## Análisis del modelo ARIMA\n:::\n\n::: {.cell execution_count=4}\n``` {.python .cell-code .hidden}\nmodel_auto = auto_arima(returns)\nmodel_auto.summary()\n```\n\n::: {#arima-model .cell-output .cell-output-display execution_count=10}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:48:07 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n```\n\nModelo ARIMA de maxima verosimilitud para la serie de retornos.\n:::\n:::\n\n\n:::{#66537cbf .cell .markdown}\nEl modelo SARIMAX(3, 0, 4) se estimó con el objetivo de capturar la dinámica subyacente de la serie de retornos. Aunque los resultados muestran que algunos de los coeficientes del componente autorregresivo (AR) y del promedio móvil (MA) son estadísticamente significativos (e.g., $AR(3)$ con un coeficiente de $0.7691$, $p < 0.001$ y $MA(2)$ con $p < 0.01$), el desempeño general del modelo presenta ciertas limitaciones. El test de heterocedasticidad muestra un valor de 1.11, lo que indica la presencia de heterocedasticidad en la serie, un problema común en series financieras, donde la varianza de los errores no es constante a lo largo del tiempo.\n\nAdemás, el estadístico de Jarque-Bera de 4395.18, con una probabilidad de $p = 0.00$, confirma que los residuos no siguen una distribución normal, lo que refuerza la idea de que un modelo basado únicamente en niveles, como el ARIMA, podría no ser suficiente para capturar la volatilidad inherente a los retornos del mercado de deuda pública dominicano. Dado que los modelos ARIMA no están diseñados para abordar adecuadamente la heterocedasticidad, pasamos a estimar un modelo GARCH, que es más apropiado para capturar los cambios en la volatilidad.\n\n## Modelo Zero-GARCH\n\nPara abordar las limitaciones del modelo ARIMA, se estimó un modelo GARCH(1,1) con media cero, siguiendo la estructura propuesta por [@miah_rahman_2016], quienes demostrarón que los modelos GARCH(1,1) son altamente efectivos para capturar la volatilidad en los retornos de mercados financieros. La elección de un modelo con media cero se justifica porque en mercados de deuda pública y tasas de interés, los retornos tienden a revertir a la media, y en el largo plazo se espera que los retornos por apreciación de capital converjan a cero. [@fabozzi_fixed_income]\n:::\n\n::: {#garch-model-fitting .cell execution_count=5}\n``` {.python .cell-code .hidden}\nar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\nres = ar.fit(last_obs=split_date)\n```\n\n::: {.cell-output .cell-output-stdout .hidden}\n```\nIteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\nIteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\nIteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\nIteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\nIteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\nIteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\nIteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\nIteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\nIteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\nIteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\nIteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\nIteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\nIteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\nIteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\nIteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\nIteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\nIteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\nIteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\nIteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\nIteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\nIteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\nIteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\nIteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\nIteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\nIteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\nIteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\nOptimization terminated successfully (Exit mode 0)\n Current function value: 7622.868827051428\n Iterations: 26\n Function evaluations: 139\n Gradient evaluations: 26\n```\n:::\n:::\n\n\n::: {.cell execution_count=6}\n``` {.python .cell-code .hidden}\nres.summary()\n```\n\n::: {#garch-model .cell-output .cell-output-display execution_count=12}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:48:07 Df Model: 0
\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
\n\n\n\n \n\n\n \n\n
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust\n```\n\nModelo Zero-Garch de la serie de retornos\n:::\n:::\n\n\n:::{#8cea5158 .cell .markdown}\nLos resultados del modelo Zero-GARCH confirman un buen ajuste a la volatilidad de la serie de retornos. El coeficiente $\\omega$ del proceso de volatilidad es de $58.8679$ ($p < 0.05$), lo que indica un nivel base significativo de volatilidad en la serie. El parámetro $\\alpha_1$, que mide la influencia de los shocks pasados en la volatilidad actual, es positivo y significativo ($0.1892$, $p < 0.01$), lo que sugiere que los shocks pasados tienen un impacto considerable en la volatilidad presente. Por otro lado, $\\beta_1$ ($0.7737$, $p < 0.001$) indica que existe una fuerte persistencia en la volatilidad, característica común en los mercados financieros, donde las fases de alta volatilidad tienden a durar varios periodos.\n\nEl uso de la distribución t de Student para los residuos estandarizados, con un parámetro $\\nu$ de $2.7894$ ($p < 0.001$), confirma la presencia de colas más gruesas en la distribución de los retornos, lo que es consistente con la leptocurtosis observada en la serie de retornos.\n\n### Residuos del Zero-GARCH\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-garch-residuals >}}\n\n\n\n\n\n\n\n\n\n\nAl analizar los residuos estandarizados y la volatilidad condicional en el modelo Zero-GARCH, observamos cómo la volatilidad responde de manera dinámica a los choques en los retornos. En la primera gráfica, se aprecia que la volatilidad condicional es mayor durante episodios donde los residuos estandarizados alcanzan picos extremos, especialmente en periodos de alta volatilidad como 2018. Esto refuerza la capacidad del modelo GARCH para capturar **volatility clustering**, un fenómeno donde la alta volatilidad tiende a agruparse en ciertos periodos. Además, la persistencia de la volatilidad condicional a lo largo del tiempo confirma que los choques pasados tienen un efecto prolongado, lo cual es coherente con los resultados obtenidos en los coeficientes del modelo GARCH.\n\n### Residuos contra volatilidad condicional\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-residuals-vs-volatility >}}\n\n\n\n\n\n\n\n\n\n\nEn cuanto a la comparación entre los residuos estandarizados y los residuos de varianza unitaria, la segunda gráfica revela que ambos conjuntos de residuos presentan distribuciones similares, con una alta concentración alrededor de cero. Sin embargo, las colas más gruesas en los residuos estandarizados indican la presencia de eventos extremos más pronunciados, los cuales son típicos en series financieras que presentan leptocurtosis. Esta comparación subraya la importancia de utilizar distribuciones robustas, como la t de Student, para capturar adecuadamente las colas de la distribución, tal como lo hace el modelo GARCH en este caso.\n\nLos resultados del modelo Zero-GARCH confirman su capacidad para modelar de manera efectiva la volatilidad en los retornos del IRP-GOBIX. Aunque el modelo auto-ARIMA capturó algunos patrones de la serie, su inconsistencia en la presencia de heterocedasticidad subraya la necesidad de un enfoque basado en modelos GARCH. El modelo Zero-GARCH no solo demuestra un buen ajuste a los datos, sino que también captura la dinámica de la volatilidad condicional y los residuos estandarizados, revelando la presencia de **volatility clustering** y la persistencia en la volatilidad.\n\nLa comparación entre los residuos estandarizados y los residuos de varianza unitaria destaca la capacidad del modelo para manejar eventos extremos, una característica esencial en el análisis de series financieras. En resumen, el modelo Zero-GARCH es más robusto y adecuado para describir las dinámicas de volatilidad en el mercado de deuda pública dominicano, brindando información crucial sobre la persistencia y el comportamiento de la volatilidad a lo largo del tiempo.\n\n## Evaluación Retrospectiva (Backtesting)\n\nPara evaluar la capacidad predictiva del modelo Zero-GARCH, se realizó un análisis de backtesting sobre el periodo comprendido durante el año 2021. Se emplearon dos métodos principales de evaluación: el MAPE (Mean Absolute Percentage Error) y el análisis gráfico de los valores predichos frente a los valores actuales de la desviación estándar móvil. \n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-predicted-vs-actual-backtest >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis gráfico (Predichos vs Actuales) revela que el modelo captura de manera excelente la **tendencia** general de la serie, lo que implica que el modelo es eficiente a la hora de predecir los movimientos de largo plazo en la volatilidad. Sin embargo, un problema importante que surge es que el modelo tiende a suavizar los movimientos bruscos, especialmente en momentos donde la volatilidad sufre caídas o picos repentinos. Este fenómeno de **sobre-suavización** implica que el modelo no es completamente reactivo a los choques regresivos de gran magnitud, lo que puede deberse a la estructura propia del GARCH, la cual tiende a modelar la volatilidad alrededor de la media del periodo. \n\nEl MAPE calculado para el conjunto de test arroja un valor elevado de **739.70%**, lo cual podría sugerir en un análisis superficial que el modelo no es adecuado. Sin embargo, es importante considerar el tipo de mercado con el que estamos trabajando, caracterizado por una volatilidad intrínsecamente elevada y movimientos extremos. En estos contextos, valores altos de MAPE son comunes debido a la alta variabilidad en los retornos, lo que no necesariamente implica una mala capacidad predictiva, sino que refleja la naturaleza errática de los datos financieros. En este sentido, el MAPE alto está más relacionado con la complejidad del mercado que con la falta de ajuste del modelo.\n\nEl backtesting del modelo Zero-GARCH sobre la serie de retornos del IRP-GOBIX para el periodo de 2021 confirma que, si bien el modelo captura correctamente la tendencia general de la volatilidad, su capacidad para reaccionar ante movimientos abruptos es limitada. A pesar de esto, dado el contexto del mercado de deuda pública dominicano, la sobre-suavización en los niveles es un comportamiento esperado en modelos de volatilidad como el GARCH. En resumen, el modelo es útil para predecir tendencias a largo plazo, pero su precisión en niveles podría mejorarse si se ajusta para reaccionar mejor a eventos extremos.\n\n# Discusión y Conclusión\n\n## Discusión\n\nEste estudio ha permitido modelar los retornos y la volatilidad del mercado de deuda pública dominicano utilizando modelos ARIMA-GARCH. A pesar de los hallazgos significativos, existen algunas limitaciones que deben considerarse al interpretar los resultados:\n\n- **Problemas con la sobre-suavización**: Como se observó en el análisis retrospectivo, el modelo Zero-GARCH tiende a suavizar los movimientos bruscos en la volatilidad, lo que puede limitar su capacidad de capturar adecuadamente choques repentinos en el mercado. Aunque el modelo refleja bien la tendencia general, su capacidad de reacción frente a eventos extremos debe ser mejorada.\n\n- **MAPE elevado**: El MAPE obtenido durante la evaluación retrospectiva fue considerablemente alto. Si bien esto puede justificarse por la naturaleza volátil del mercado de deuda pública, sugiere que el modelo puede tener problemas para ajustarse a los movimientos de corto plazo. La alta volatilidad y los choques repentinos no fueron capturados con precisión, afectando el rendimiento del modelo.\n\n- **Limitaciones en los datos**: El análisis se centró en un período específico (2014-2021), excluyendo la pandemia debido a sus características atípicas. Si bien esto ayudó a evitar sesgos, la exclusión de este período crítico puede haber omitido dinámicas importantes del mercado en tiempos de crisis, limitando la generalización de los resultados.\n\n## Conclusión\n\nEn conclusión, este estudio ha demostrado que el modelo ARIMA, aunque útil para capturar la estructura de los retornos del mercado de deuda pública dominicano, presenta limitaciones cuando se enfrenta a problemas de heterocedasticidad. El modelo Zero-GARCH, por su parte, ha demostrado ser más adecuado para describir la dinámica de la volatilidad, capturando fenómenos como la persistencia en la volatilidad y el clustering. Además, la capacidad del modelo para manejar eventos extremos a través de la distribución t de Student resulta valiosa en este tipo de mercados caracterizados por alta volatilidad y choques inesperados.\n\nEntre los principales hallazgos destacan:\n\n- **Volatility clustering**: El análisis confirmó la presencia de agrupación de volatilidad, donde periodos de baja volatilidad son seguidos por periodos de alta volatilidad, un fenómeno común en los mercados financieros.\n\n- **Persistencia de la volatilidad**: El coeficiente $\\beta_1$ significativo en el modelo GARCH sugiere que los shocks en la volatilidad tienen un efecto prolongado en el tiempo, lo que es consistente con los patrones observados en mercados financieros emergentes como el dominicano.\n\n- **Dificultad para capturar movimientos extremos**: Aunque el modelo Zero-GARCH es efectivo en capturar la tendencia general, su capacidad para reaccionar ante cambios bruscos en la volatilidad sigue siendo limitada, lo cual es un área de mejora para futuras investigaciones.\n\n- **Implicaciones para los inversores**: Los resultados ofrecen una herramienta útil para la predicción de tendencias de volatilidad a largo plazo en el mercado de deuda pública dominicano, proporcionando una base para la toma de decisiones estratégicas en la gestión del riesgo.\n\nEste estudio no solo contribuye al entendimiento del comportamiento de los retornos y la volatilidad en el mercado de deuda pública dominicano.\n\n# Referencias\n\n::: {#refs}\n:::\n:::\n\n", "supporting": [ - "index_files/figure-docx" + "index_files\\figure-docx" ], "filters": [] } diff --git a/_freeze/index/execute-results/html.json b/_freeze/index/execute-results/html.json index 52a1235..52cc760 100644 --- a/_freeze/index/execute-results/html.json +++ b/_freeze/index/execute-results/html.json @@ -1,8 +1,8 @@ { - "hash": "8802eb87c1cc642b07fcebf8057b0508", + "hash": "22289e552d482a5d6266287352c6d0f1", "result": { "engine": "jupyter", - "markdown": "---\ntitle: Un estudio sobre los retornos del mercado de renta fija dominicano utilizando basados en modelos AR-GARCH\nsubtitle: Trabajo final de tópicos de econometría\nauthor:\n - name: Ian Contreras\n email: 1116048@intec.edu.do\n affiliations:\n - INSTITUTO TECNOLÓGICO DE SANTO DOMINGO (INTEC)\nkeywords:\n - Mercado de renta fija\n - Deuda pública\n - Modelos AR-GARCH\nabstract: |\n Este estudio tiene como objetivo analizar los retornos y la volatilidad del mercado de deuda pública dominicana mediante la aplicación de modelos ARMA-GARCH. A través de estos modelos, se busca identificar patrones en el comportamiento de los retornos y predecir su evolución futura. \ndate: last-modified\nbibliography: references.bib\nexecute:\n echo: false\nnocite: |\n @*\ncitation:\n container-title: Referencias\nnumber-sections: true\n---\n\n::: {#importing-libraries .cell execution_count=2}\n``` {.python .cell-code .hidden}\n# Manejo de datos y análisis\nimport numpy as np\nimport pandas as pd\nimport scipy.stats as stats\n\n# Modelos estadísticos y econométricos\nimport statsmodels.api as sm\nfrom pmdarima.arima import auto_arima\nfrom arch import arch_model\n```\n:::\n\n\n::: {#importing-data .cell execution_count=3}\n``` {.python .cell-code .hidden}\ndf = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\nreturns = df['price_return']\nsplit_date = '2020-12-31'\nR_test = df[df.index >= split_date]['price_return'].rolling(\n window=5).std().dropna()\n```\n:::\n\n\n:::{#06406d32 .cell .markdown}\n# Introducción\n\nEl mercado de renta fija en la República Dominicana ha emergido como uno de los sectores más importantes dentro de la economía, especialmente en lo que respecta a la deuda pública, la cual juega un papel crucial en la financiación de proyectos gubernamentales y el sostenimiento de la política fiscal. En los últimos cincuenta años, el país ha experimentado un crecimiento económico significativo, consolidándose como una de las economías más dinámicas de América Latina. Este crecimiento ha estado acompañado por una mayor complejidad económica, lo que ha incrementado la relevancia del mercado financiero dominicano, en particular el mercado de deuda pública, que ha sido un activo clave para fondos de pensiones y fondos de inversión.\n\nSin embargo, este mercado también presenta importantes desafíos, especialmente en términos de baja liquidez en el mercado secundario y episodios de alta volatilidad. Estos problemas dificultan la previsión precisa de los movimientos del mercado y complican la toma de decisiones estratégicas para los inversores. En este contexto, surge la necesidad de utilizar herramientas econométricas que puedan capturar con mayor precisión las dinámicas subyacentes de los retornos y la volatilidad del mercado de deuda pública dominicano.\n\nLa **pregunta de investigación** que guía este estudio es: ¿Es posible modelar adecuadamente los retornos y la volatilidad del mercado de deuda pública dominicano mediante modelos ARIMA-GARCH?, y ¿Qué tan efectivas son estas técnicas para predecir el comportamiento futuro del mercado, especialmente durante periodos de alta volatilidad?\n\nEl **objetivo principal** de este estudio es aplicar modelos ARMA-GARCH para analizar y predecir el comportamiento de los retornos del índice de deuda pública IRP-GOBIX, considerando tanto la tendencia como los patrones de volatilidad del mercado. A través de la estimación de estos modelos, se busca identificar los principales factores que afectan la dinámica de los retornos y proporcionar información valiosa para los actores del mercado, facilitando una gestión más eficiente del riesgo.\n\nAdemás, este estudio pretende evaluar la **consistencia de los modelos** y determinar si los modelos de volatilidad más sofisticados, como el Zero-GARCH, ofrecen una mejor capacidad de predicción que los modelos tradicionales basados únicamente en niveles y diferencias, como el ARIMA. Un aspecto clave será analizar si el modelo GARCH captura adecuadamente los choques bruscos y los episodios de volatilidad extrema, o si suaviza excesivamente estos movimientos.\n\nLa **relevancia** de este estudio radica en la creciente importancia del mercado de deuda pública dominicano, tanto para la estabilidad financiera del país como para los inversores extranjeros que buscan oportunidades de inversión en mercados emergentes. Comprender los patrones de retorno y volatilidad es esencial para diseñar estrategias de inversión que minimicen el riesgo y maximicen el rendimiento en un entorno económico globalizado y altamente incierto.\n\n# Metodología\n\nEl modelo ARIMA(p,d,q), donde AR(p) es el componente autorregresivo de orden p, d denota el orden de la diferencia aplicada para transformar una serie temporal no estacionaria en estacionaria, y MA(q) representa el promedio móvil de orden q. La esencia del modelo ARIMA radica en la combinación de la operación de diferenciación y el modelo ARMA. Cualquier serie no estacionaria puede volverse estacionaria mediante una diferenciación de orden adecuado, permitiendo así ajustar un modelo ARMA a la serie transformada.\n\nEl modelo de promedio móvil autorregresivo ARMA(p,q) incluye tanto un componente autorregresivo de orden p como uno de promedio móvil de orden q, y se estructura de la siguiente forma:\n\n$$\nX_t = \\phi_1 X_{t-1} + \\phi_2 X_{t-2} + \\dots + \\phi_p X_{t-p} + \\epsilon_t + \\theta_1 \\epsilon_{t-1} + \\theta_2 \\epsilon_{t-2} + \\dots + \\theta_q \\epsilon_{t-q}\n$$ {eq: arma}\n\nEstos modelos son adecuados solo para describir series temporales suaves, mientras que en la práctica, a menudo se trabaja con series temporales no suaves. Para estos casos, aplicar una o dos diferenciaciones puede transformar los datos en series suaves.\n\nSi no se cumple el supuesto de homogeneidad de la varianza, puede ocurrir heterocedasticidad. Dado que los datos de la muestra son suaves, en este documento se establece un modelo ARMA(p,q) para la serie de rendimientos con el fin de describir las características de volatilidad del índice GOBIX. Se realizan pruebas de efecto ARCH y se resuelve el problema de la heterocedasticidad mediante un modelo GARCH.\n\nEl modelo GARCH extiende el modelo ARCH al considerar la autocorrelación de orden p en la función de heterocedasticidad, lo que permite ajustar de manera efectiva una función de heterocedasticidad con memoria a largo plazo. El modelo GARCH se define de la siguiente manera:\n\n$$\na_t = \\sigma_t \\epsilon_t, \\quad \\sigma_t^2 = \\alpha_0 + \\sum_{i=1}^{p} \\alpha_i a_{t-i}^2 + \\sum_{j=1}^{q} \\beta_j \\sigma_{t-j}^2\n$$ {#eq: garch}\n\nEl modelo GARCH, propuesto por Bollerslev, atribuye la volatilidad actual tanto a la volatilidad de momentos pasados como a los errores de momentos pasados, lo que le permite explicar el fenómeno de agrupación en la volatilidad de los rendimientos financieros.\n\n# Datos\n\n## Fuente de datos\nUn Índice Financiero es una medida estadística que refleja el valor de un conjunto de activos financieros agrupados de acuerdo con ciertos criterios.Estos se utilizan como objetivo de rendimiento de portafolios, referencias sobre las características retorno/riesgo de una clase de activo y como referencia para productos anclados a índices.\n\nEn el mercado de renta fija de la República Dominicana, el único índice público es el GOBIX, que representa la deuda gubernamental consolidada en peso. Este índice está compuesto por títulos emitidos localmente por el Banco Central y el Ministerio de Hacienda, y se publica en dos modalidades: índice de precio limpio e índice de retorno-precio. Para este estudio, se utilizará el índice de retorno precio @sec-retorno-precio, ya que aisla el componente de retorno de mercado el cual es el objetivo de estudio. \n\nEl GOBIX se caracteriza por los siguientes criterios:\n\n- **Elegibilidad**: Solo se incluyen títulos \"bullet\" emitidos en pesos dominicanos con al menos 60 días desde su emisión.\n- **Liquidez**: Los títulos se seleccionan en función del Índice de Bursatilidad publicado por la Proveedora de Precios, garantizando la inclusión de los instrumentos más negociados.\n- **Ponderación**: La ponderación se realiza por capitalización de mercado, lo que asegura que los títulos más relevantes tengan mayor peso en el índice.\n- **Rebalanceo**: Se efectúa un rebalanceo mensual para ajustar el índice según la disponibilidad de inversión en los bonos.\n- **Fuentes de información**: Los datos provienen de la Bolsa de Valores de la República Dominicana y la Proveedora de Precios.\n\n## Índice de Retorno-Precio {#sec-retorno-precio}\n\nEl índice de retorno-precio es una metodología utilizada para calcular el rendimiento de un índice de renta fija, centrándose exclusivamente en las variaciones del precio limpio de los títulos, sin tener en cuenta los intereses acumulados ni el pago de cupones durante el período de cálculo. En el contexto del GOBIX, este índice refleja las fluctuaciones en los precios de los bonos dominicanos debido a las variaciones en las tasas de interés del mercado.\n\n**Características del Índice de Retorno-Precio del GOBIX**:\n\n- **Enfoque en el Precio Limpio**: El cálculo del índice se basa en el precio limpio de los bonos, excluyendo los intereses acumulados.\n- **Reflejo de la Volatilidad**: Este índice es un indicador del riesgo asociado a las fluctuaciones en las tasas de interés y su impacto en el capital invertido.\n- **Fórmula de Cálculo**:\n\n $$\n IRP_t = IRP_{m-1} \\times (1 + RPI_t)\n $$\n\n Donde:\n - $IRP_0 = 100$ es el valor base del índice en la fecha de inicio.\n - $IRP_t$ es el valor del índice en el día $t$.\n - $IRP_{m-1}$ es el valor del índice en el último día hábil del mes anterior.\n - $RPI_t$ es el retorno-precio del índice en $t$.\n\nEl retorno-precio del índice ($RPI_t$) se calcula como la suma ponderada del retorno-precio de cada bono ($RPI_{i,t}$), donde la ponderación ($\\omega_i$) se determina según la capitalización de mercado de cada bono.\n\nEn este análisis, expresaremos la serie de retornos en puntos básicos para evitar problemas de escalamiento de los datos y posibles desbordamientos decimales (overflow decimal), que pueden ocurrir cuando se manejan fluctuaciones muy pequeñas con alta precisión. Un punto básico equivale a 0.01% o 1/100 de un porcentaje. Este enfoque nos permitirá manejar los datos con mayor estabilidad y evitar posibles errores numéricos derivados del escalamiento inapropiado. No tiene efectos estadísticos en las distribución. \n\n## Selección de muestra de entrenamiento/prueba {#sec-test-train}\n\nDurante la pandemia, se experimentó uno de los ciclos de tasas de interés más pronunciados de las últimas décadas. Este periodo presentó características estructurales distintas en comparación con los ciclos históricos previos. Para evitar sesgos en las estimaciones debido a la alta volatilidad observada durante ese tiempo, evaluaremos únicamente la serie de retornos entre el periodo 2014-2021, que se considera un ciclo de tasas más regular.\n\nLa metodología empleada para dividir el conjunto de datos en muestras de entrenamiento y prueba será el **Time Series Cross-Validator**. A diferencia de la validación cruzada tradicional, que aleatoriza los datos, esta técnica respeta el orden temporal, lo cual es crucial en el análisis de series de tiempo.\n\nEn cada partición (k-ésimo split), la técnica funciona de la siguiente manera:\n\n- **Conjunto de Entrenamiento**: Incluye los primeros *k* pliegues (folds).\n- **Conjunto de Prueba**: Incluye el pliegue *(k+1)*.\n\n# Análisis Exploratorio {#sec-eda}\n\nEl objetivo de este análisis exploratorio es examinar la serie de retornos diarios del IRP-GOBIX para identificar patrones de volatilidad, evaluar la distribución de los retornos, y explorar la autocorrelación en los datos. Utilizando herramientas estadísticas como la distribución de los retornos, el test de estacionariedad Dickey-Fuller, y los gráficos de autocorrelación (ACF) y autocorrelación parcial (PACF), se busca comprender la dinámica temporal de los retornos y validar la aplicación de modelos ARMA-GARCH para capturar su comportamiento futuro. Adicionalmente, se ajustarán diversas distribuciones teóricas, como la distribución normal y la t de Student, para evaluar su capacidad de describir las características empíricas de los datos.\n\n## Histórico de retornos precio del mercado de deuda pública dominicano {#sec-historico-retornos}\n\n\n{{< embed notebooks/data-screening.qmd#fig-price-return-series >}}\n\n\nEl análisis de la serie de retornos diarios del IRP-GOBIX revela un comportamiento interesante, con periodos prolongados de baja volatilidad seguidos por episodios de alta volatilidad. Entre 2015 y 2022, los retornos se mantuvieron generalmente dentro de un rango de ±50 puntos básicos, lo que indica una estabilidad relativa. Sin embargo, en momentos clave, como el ciclo de tasas de 2018, la volatilidad se incrementó drásticamente, alcanzando picos de hasta 200 puntos básicos. Este patrón es característico del fenómeno conocido como **\"volatility clustering\"**, donde periodos tranquilos son seguidos por fases de mayor volatilidad. La naturaleza de estos picos parece estar relacionada con factores externos, como cambios en las tasas de interés y la incertidumbre macroeconómica global, lo que subraya la importancia de emplear modelos ARMA-GARCH para capturar estas dinámicas y prever comportamientos futuros en el mercado de deuda pública dominicana.\n\n## Análisis descriptivo de la muestra {#sec-estadistica-descriptiva}\n:::\n\n::: {#cell-return-descriptive-stats .cell execution_count=4}\n``` {.python .cell-code .hidden}\ndesc_stats = returns.describe()\n\nskewness = returns.skew()\nkurtosis = returns.kurtosis()\njb_test = sm.stats.jarque_bera(returns)\n\ndescriptive_table = pd.DataFrame({\n 'Observations': [int(desc_stats['count'])],\n 'Mean': [desc_stats['mean']],\n 'Median': [desc_stats['50%']],\n 'Std. Dev': [desc_stats['std']],\n 'Skewness': [skewness],\n 'Kurtosis': [kurtosis],\n 'Jarque-Bera': [jb_test[0]],\n 'Prob.': [jb_test[1]]\n})\ndescriptive_table\n```\n\n::: {#return-descriptive-stats .cell-output .cell-output-display execution_count=3}\n```{=html}\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ObservationsMeanMedianStd. DevSkewnessKurtosisJarque-BeraProb.
019411.353251-0.31804423.5029110.705277.5879954789.5323310.0
\n
\n```\n\nTabla de las estadísticas descriptiva de la serie de retornos\n:::\n:::\n\n\n:::{#d640e5c2 .cell .markdown}\nEl análisis descriptivo de los retornos ofrece una perspectiva más detallada sobre la distribución de la serie. Con un total de 1,941 observaciones, la **media** de los retornos es de $1.353251$, lo que indica un rendimiento promedio positivo. Sin embargo, la **mediana** de $-0.318044$ sugiere un leve sesgo hacia valores negativos, lo que refleja que la mayoría de los retornos tienden a ser ligeramente inferiores a la media. La **desviación estándar**, con un valor de $23.502911$, revela una amplia dispersión de los datos, lo que confirma la presencia de una volatilidad significativa en los rendimientos, muy superior al valor medio.\n\nEn cuanto a la **asimetría** ($0.70527$), se observa una ligera inclinación hacia valores extremos positivos, lo que implica la existencia de eventos fuera de lo común que afectan los rendimientos. La **kurtosis** de $7.587995$ indica una distribución **leptocúrtica**, caracterizada por una alta concentración de valores alrededor de la media y colas más gruesas que una distribución normal, lo que es típico en datos financieros que presentan eventos extremos. El **test de Jarque-Bera**, con un valor de $4,789.532331$, confirma que la serie no sigue una distribución normal, validando la presencia de asimetría y colas pesadas en los datos. Estos resultados resaltan la necesidad de emplear modelos que puedan capturar adecuadamente estos comportamientos no lineales, esenciales para el análisis del mercado.\n\n## Análisis de la distribución de los retornos {#sec-distribucion-retornos}\n\n\n{{< embed notebooks/data-screening.qmd#fig-distribution-fitting >}}\n\n\nAl ajustar diferentes distribuciones a los retornos, observamos que la distribución empírica (en azul) tiene una alta concentración en torno a cero, con colas más gruesas de lo que se esperaría bajo una distribución normal. Entre las distribuciones probadas, la t de Student (en rojo) es la que mejor captura los valores extremos, con colas más largas y una mayor concentración en el centro, lo que es típico en series financieras que experimentan episodios de alta volatilidad. En contraste, la distribución normal (en naranja) y la lognormal (en verde) subestiman las colas, demostrando su ineficacia para modelar adecuadamente los valores atípicos. Esto refuerza la idea de que la t de Student es una mejor candidata para modelar los retornos, dado que puede ajustarse mejor a la leptocurtosis observada en los datos.\n\n\n{{< embed notebooks/data-screening.qmd#fig-tqq-plot >}}\n\n\nLa gráfica Q-Q (Quantile-Quantile) compara los cuantiles teóricos de la distribución t de Student con los cuantiles observados de los retornos del GOBIX. En general, la mayoría de los puntos se alinean bien con la línea roja, lo que sugiere que los retornos de precios siguen razonablemente esta distribución, particularmente en las partes centrales. Sin embargo, en las colas extremas, se observan algunas desviaciones, lo que indica que, aunque la t de Student es un buen ajuste, no es perfecta para todos los escenarios.\n\nEl **estadístico de Kolmogorov-Smirnov** es de $0.0286$, con un **p-valor** de $0.0827$. Aunque esto sugiere que la t de Student captura bien la mayor parte de los retornos, las discrepancias en las colas extremas muestran que la distribución teórica no es un ajuste exacto a los datos reales.\n\n## Evaluación de supuestos para análisis AR-GARCH\n\n### Test de estacionariedad {#sec-estacionariedad}\n\nPara garantizar la validez de los modelos ARMA y GARCH, se evaluó la estacionariedad de la serie de retornos utilizando el test de Dickey-Fuller aumentado (ADF). Los resultados obtenidos muestran un **ADF Statistic** de $-13.1741$ y un **p-valor** de $1.2333 \\times 10^{-24}$, lo que permite rechazar la hipótesis nula de raíz unitaria con un nivel de significancia del 5%. Esto confirma que la serie es estacionaria, lo que significa que sus fluctuaciones se distribuyen alrededor de una media constante en el tiempo, sin tendencia significativa. Este hallazgo es consistente con el comportamiento típico de las series de retornos financieros, lo que habilita la correcta estimación de modelos ARMA-GARCH para capturar las dinámicas del mercado.\n\n### Test de autocorrelación {#sec-autocorrelacion}\n\n\n{{< embed notebooks/data-screening.qmd#fig-acf-pacf >}}\n\n\nEl análisis de autocorrelación nos proporciona más detalles sobre la estructura interna de la serie. El gráfico de la **Función de Autocorrelación (ACF)** muestra un pico significativo en el primer rezago, seguido de valores cercanos a cero para los rezagos posteriores. Este patrón es indicativo de un proceso de media móvil de primer orden (**MA(1)**), donde los choques aleatorios tienen un impacto significativo en el primer rezago pero no en los siguientes. La **Función de Autocorrelación Parcial (PACF)** refuerza esta conclusión, mostrando un comportamiento similar con un pico en el primer rezago y valores prácticamente nulos a partir del segundo rezago. Esto sugiere que un modelo MA(1) sería apropiado para capturar las dinámicas de corto plazo en la serie.\n\nEn conjunto, este análisis exploratorio sugiere que la serie de retornos del IRP-GOBIX presenta características complejas, como **volatility clustering**, alta **leptocurtosis**, y una estructura de autocorrelación que se ajusta bien a un modelo MA(1). Estos hallazgos proporcionan una base sólida para la estimación de modelos ARMA-GARCH, los cuales podrán capturar de manera efectiva las dinámicas de volatilidad y retornos en el mercado de deuda pública dominicano.\n\n# Resultados\n\nEl presente análisis tiene como objetivo modelar la serie temporal de retornos del índice IRP-GOBIX utilizando inicialmente un modelo ARIMA automático (auto-ARIMA) para determinar si es posible capturar la dinámica de los retornos con un modelo basado únicamente en niveles y diferencias de los datos. Adicionalmente, se evaluará la presencia de problemas estructurales, como la heterocedasticidad, que podrían afectar la consistencia del modelo ARIMA. En caso de que este enfoque no sea adecuado, recurriremos a un modelo Zero-GARCH, un tipo específico de GARCH que asume una media cero para los retornos. Este modelo es particularmente útil en mercados de tasas de interés, donde se ha demostrado que los retornos tienden a cero en el largo plazo debido a la fuerte regresión hacia la media de estas variables. \n\n## Análisis del modelo ARIMA\n:::\n\n::: {#cell-arima-model .cell execution_count=5}\n``` {.python .cell-code .hidden}\nmodel_auto = auto_arima(returns)\nmodel_auto.summary()\n```\n\n::: {#arima-model .cell-output .cell-output-display execution_count=4}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:09:21 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n```\n\nModelo ARIMA de maxima verosimilitud para la serie de retornos.\n:::\n:::\n\n\n:::{#460e23ba .cell .markdown}\nEl modelo SARIMAX(3, 0, 4) se estimó con el objetivo de capturar la dinámica subyacente de la serie de retornos. Aunque los resultados muestran que algunos de los coeficientes del componente autorregresivo (AR) y del promedio móvil (MA) son estadísticamente significativos (e.g., $AR(3)$ con un coeficiente de $0.7691$, $p < 0.001$ y $MA(2)$ con $p < 0.01$), el desempeño general del modelo presenta ciertas limitaciones. El test de heterocedasticidad muestra un valor de 1.11, lo que indica la presencia de heterocedasticidad en la serie, un problema común en series financieras, donde la varianza de los errores no es constante a lo largo del tiempo.\n\nAdemás, el estadístico de Jarque-Bera de 4395.18, con una probabilidad de $p = 0.00$, confirma que los residuos no siguen una distribución normal, lo que refuerza la idea de que un modelo basado únicamente en niveles, como el ARIMA, podría no ser suficiente para capturar la volatilidad inherente a los retornos del mercado de deuda pública dominicano. Dado que los modelos ARIMA no están diseñados para abordar adecuadamente la heterocedasticidad, pasamos a estimar un modelo GARCH, que es más apropiado para capturar los cambios en la volatilidad.\n\n## Modelo Zero-GARCH\n\nPara abordar las limitaciones del modelo ARIMA, se estimó un modelo GARCH(1,1) con media cero, siguiendo la estructura propuesta por [@miah_rahman_2016], quienes demostrarón que los modelos GARCH(1,1) son altamente efectivos para capturar la volatilidad en los retornos de mercados financieros. La elección de un modelo con media cero se justifica porque en mercados de deuda pública y tasas de interés, los retornos tienden a revertir a la media, y en el largo plazo se espera que los retornos por apreciación de capital converjan a cero. [@fabozzi_fixed_income]\n:::\n\n::: {#garch-model-fitting .cell execution_count=6}\n``` {.python .cell-code .hidden}\nar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\nres = ar.fit(last_obs=split_date)\n```\n\n::: {.cell-output .cell-output-stdout .hidden}\n```\nIteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\nIteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\nIteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\nIteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\nIteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\nIteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\nIteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\nIteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\nIteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\nIteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\nIteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\nIteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\nIteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\nIteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\nIteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\nIteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\nIteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\nIteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\nIteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\nIteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\nIteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\nIteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\nIteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\nIteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\nIteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\nIteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\nOptimization terminated successfully (Exit mode 0)\n Current function value: 7622.868827051428\n Iterations: 26\n Function evaluations: 139\n Gradient evaluations: 26\n```\n:::\n:::\n\n\n::: {#cell-garch-model .cell execution_count=7}\n``` {.python .cell-code .hidden}\nres.summary()\n```\n\n::: {#garch-model .cell-output .cell-output-display execution_count=6}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:09:21 Df Model: 0
\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
\n\n\n\n \n\n\n \n\n
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust\n```\n\nModelo Zero-Garch de la serie de retornos\n:::\n:::\n\n\n:::{#67ad3e26 .cell .markdown}\nLos resultados del modelo Zero-GARCH confirman un buen ajuste a la volatilidad de la serie de retornos. El coeficiente $\\omega$ del proceso de volatilidad es de $58.8679$ ($p < 0.05$), lo que indica un nivel base significativo de volatilidad en la serie. El parámetro $\\alpha_1$, que mide la influencia de los shocks pasados en la volatilidad actual, es positivo y significativo ($0.1892$, $p < 0.01$), lo que sugiere que los shocks pasados tienen un impacto considerable en la volatilidad presente. Por otro lado, $\\beta_1$ ($0.7737$, $p < 0.001$) indica que existe una fuerte persistencia en la volatilidad, característica común en los mercados financieros, donde las fases de alta volatilidad tienden a durar varios periodos.\n\nEl uso de la distribución t de Student para los residuos estandarizados, con un parámetro $\\nu$ de $2.7894$ ($p < 0.001$), confirma la presencia de colas más gruesas en la distribución de los retornos, lo que es consistente con la leptocurtosis observada en la serie de retornos.\n\n### Residuos del Zero-GARCH\n\n\n{{< embed notebooks/data-screening.qmd#fig-garch-residuals >}}\n\n\nAl analizar los residuos estandarizados y la volatilidad condicional en el modelo Zero-GARCH, observamos cómo la volatilidad responde de manera dinámica a los choques en los retornos. En la primera gráfica, se aprecia que la volatilidad condicional es mayor durante episodios donde los residuos estandarizados alcanzan picos extremos, especialmente en periodos de alta volatilidad como 2018. Esto refuerza la capacidad del modelo GARCH para capturar **volatility clustering**, un fenómeno donde la alta volatilidad tiende a agruparse en ciertos periodos. Además, la persistencia de la volatilidad condicional a lo largo del tiempo confirma que los choques pasados tienen un efecto prolongado, lo cual es coherente con los resultados obtenidos en los coeficientes del modelo GARCH.\n\n### Residuos contra volatilidad condicional\n\n\n{{< embed notebooks/data-screening.qmd#fig-residuals-vs-volatility >}}\n\n\nEn cuanto a la comparación entre los residuos estandarizados y los residuos de varianza unitaria, la segunda gráfica revela que ambos conjuntos de residuos presentan distribuciones similares, con una alta concentración alrededor de cero. Sin embargo, las colas más gruesas en los residuos estandarizados indican la presencia de eventos extremos más pronunciados, los cuales son típicos en series financieras que presentan leptocurtosis. Esta comparación subraya la importancia de utilizar distribuciones robustas, como la t de Student, para capturar adecuadamente las colas de la distribución, tal como lo hace el modelo GARCH en este caso.\n\nLos resultados del modelo Zero-GARCH confirman su capacidad para modelar de manera efectiva la volatilidad en los retornos del IRP-GOBIX. Aunque el modelo auto-ARIMA capturó algunos patrones de la serie, su inconsistencia en la presencia de heterocedasticidad subraya la necesidad de un enfoque basado en modelos GARCH. El modelo Zero-GARCH no solo demuestra un buen ajuste a los datos, sino que también captura la dinámica de la volatilidad condicional y los residuos estandarizados, revelando la presencia de **volatility clustering** y la persistencia en la volatilidad.\n\nLa comparación entre los residuos estandarizados y los residuos de varianza unitaria destaca la capacidad del modelo para manejar eventos extremos, una característica esencial en el análisis de series financieras. En resumen, el modelo Zero-GARCH es más robusto y adecuado para describir las dinámicas de volatilidad en el mercado de deuda pública dominicano, brindando información crucial sobre la persistencia y el comportamiento de la volatilidad a lo largo del tiempo.\n\n## Evaluación Retrospectiva (Backtesting)\n\nPara evaluar la capacidad predictiva del modelo Zero-GARCH, se realizó un análisis de backtesting sobre el periodo comprendido durante el año 2021. Se emplearon dos métodos principales de evaluación: el MAPE (Mean Absolute Percentage Error) y el análisis gráfico de los valores predichos frente a los valores actuales de la desviación estándar móvil. \n\n\n{{< embed notebooks/data-screening.qmd#fig-predicted-vs-actual-backtest >}}\n\n\nEl análisis gráfico (Predichos vs Actuales) revela que el modelo captura de manera excelente la **tendencia** general de la serie, lo que implica que el modelo es eficiente a la hora de predecir los movimientos de largo plazo en la volatilidad. Sin embargo, un problema importante que surge es que el modelo tiende a suavizar los movimientos bruscos, especialmente en momentos donde la volatilidad sufre caídas o picos repentinos. Este fenómeno de **sobre-suavización** implica que el modelo no es completamente reactivo a los choques regresivos de gran magnitud, lo que puede deberse a la estructura propia del GARCH, la cual tiende a modelar la volatilidad alrededor de la media del periodo. \n\nEl MAPE calculado para el conjunto de test arroja un valor elevado de **739.70%**, lo cual podría sugerir en un análisis superficial que el modelo no es adecuado. Sin embargo, es importante considerar el tipo de mercado con el que estamos trabajando, caracterizado por una volatilidad intrínsecamente elevada y movimientos extremos. En estos contextos, valores altos de MAPE son comunes debido a la alta variabilidad en los retornos, lo que no necesariamente implica una mala capacidad predictiva, sino que refleja la naturaleza errática de los datos financieros. En este sentido, el MAPE alto está más relacionado con la complejidad del mercado que con la falta de ajuste del modelo.\n\nEl backtesting del modelo Zero-GARCH sobre la serie de retornos del IRP-GOBIX para el periodo de 2021 confirma que, si bien el modelo captura correctamente la tendencia general de la volatilidad, su capacidad para reaccionar ante movimientos abruptos es limitada. A pesar de esto, dado el contexto del mercado de deuda pública dominicano, la sobre-suavización en los niveles es un comportamiento esperado en modelos de volatilidad como el GARCH. En resumen, el modelo es útil para predecir tendencias a largo plazo, pero su precisión en niveles podría mejorarse si se ajusta para reaccionar mejor a eventos extremos.\n\n# Discusión y Conclusión\n\n## Discusión\n\nEste estudio ha permitido modelar los retornos y la volatilidad del mercado de deuda pública dominicano utilizando modelos ARIMA-GARCH. A pesar de los hallazgos significativos, existen algunas limitaciones que deben considerarse al interpretar los resultados:\n\n- **Problemas con la sobre-suavización**: Como se observó en el análisis retrospectivo, el modelo Zero-GARCH tiende a suavizar los movimientos bruscos en la volatilidad, lo que puede limitar su capacidad de capturar adecuadamente choques repentinos en el mercado. Aunque el modelo refleja bien la tendencia general, su capacidad de reacción frente a eventos extremos debe ser mejorada.\n\n- **MAPE elevado**: El MAPE obtenido durante la evaluación retrospectiva fue considerablemente alto. Si bien esto puede justificarse por la naturaleza volátil del mercado de deuda pública, sugiere que el modelo puede tener problemas para ajustarse a los movimientos de corto plazo. La alta volatilidad y los choques repentinos no fueron capturados con precisión, afectando el rendimiento del modelo.\n\n- **Limitaciones en los datos**: El análisis se centró en un período específico (2014-2021), excluyendo la pandemia debido a sus características atípicas. Si bien esto ayudó a evitar sesgos, la exclusión de este período crítico puede haber omitido dinámicas importantes del mercado en tiempos de crisis, limitando la generalización de los resultados.\n\n## Conclusión\n\nEn conclusión, este estudio ha demostrado que el modelo ARIMA, aunque útil para capturar la estructura de los retornos del mercado de deuda pública dominicano, presenta limitaciones cuando se enfrenta a problemas de heterocedasticidad. El modelo Zero-GARCH, por su parte, ha demostrado ser más adecuado para describir la dinámica de la volatilidad, capturando fenómenos como la persistencia en la volatilidad y el clustering. Además, la capacidad del modelo para manejar eventos extremos a través de la distribución t de Student resulta valiosa en este tipo de mercados caracterizados por alta volatilidad y choques inesperados.\n\nEntre los principales hallazgos destacan:\n\n- **Volatility clustering**: El análisis confirmó la presencia de agrupación de volatilidad, donde periodos de baja volatilidad son seguidos por periodos de alta volatilidad, un fenómeno común en los mercados financieros.\n\n- **Persistencia de la volatilidad**: El coeficiente $\\beta_1$ significativo en el modelo GARCH sugiere que los shocks en la volatilidad tienen un efecto prolongado en el tiempo, lo que es consistente con los patrones observados en mercados financieros emergentes como el dominicano.\n\n- **Dificultad para capturar movimientos extremos**: Aunque el modelo Zero-GARCH es efectivo en capturar la tendencia general, su capacidad para reaccionar ante cambios bruscos en la volatilidad sigue siendo limitada, lo cual es un área de mejora para futuras investigaciones.\n\n- **Implicaciones para los inversores**: Los resultados ofrecen una herramienta útil para la predicción de tendencias de volatilidad a largo plazo en el mercado de deuda pública dominicano, proporcionando una base para la toma de decisiones estratégicas en la gestión del riesgo.\n\nEste estudio no solo contribuye al entendimiento del comportamiento de los retornos y la volatilidad en el mercado de deuda pública dominicano Futuros trabajos podrían explorar la integración de modelos más sofisticados que respondan mejor a choques repentinos y extremos, o bien considerar variables exógenas que puedan influir en la volatilidad.\n\n# Referencias\n\n::: {#refs}\n:::\n:::\n\n", + "markdown": "---\ntitle: Un estudio sobre los retornos del mercado de renta fija dominicano utilizando basados en modelos AR-GARCH\nsubtitle: Trabajo final de tópicos de econometría\nauthor:\n - name: Ian Contreras\n email: 1116048@intec.edu.do\n affiliations:\n - INSTITUTO TECNOLÓGICO DE SANTO DOMINGO (INTEC)\nkeywords:\n - Mercado de renta fija\n - Deuda pública\n - Modelos AR-GARCH\nabstract: |\n Este estudio tiene como objetivo analizar los retornos y la volatilidad del mercado de deuda pública dominicana mediante la aplicación de modelos ARMA-GARCH. A través de estos modelos, se busca identificar patrones en el comportamiento de los retornos y predecir su evolución futura. \ndate: last-modified\nbibliography: references.bib\nexecute:\n echo: false\nnocite: |\n @*\ncitation:\n container-title: Referencias\nnumber-sections: true\n---\n\n::: {#importing-libraries .cell execution_count=2}\n``` {.python .cell-code .hidden}\n# Manejo de datos y análisis\nimport numpy as np\nimport pandas as pd\nimport scipy.stats as stats\n\n# Modelos estadísticos y econométricos\nimport statsmodels.api as sm\nfrom pmdarima.arima import auto_arima\nfrom arch import arch_model\n```\n:::\n\n\n::: {#importing-data .cell execution_count=3}\n``` {.python .cell-code .hidden}\ndf = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\nreturns = df['price_return']\nsplit_date = '2020-12-31'\nR_test = df[df.index >= split_date]['price_return'].rolling(\n window=5).std().dropna()\n```\n:::\n\n\n:::{#cb3e10a3 .cell .markdown}\n# Introducción\n\nEl mercado de renta fija en la República Dominicana ha emergido como uno de los sectores más importantes dentro de la economía, especialmente en lo que respecta a la deuda pública, la cual juega un papel crucial en la financiación de proyectos gubernamentales y el sostenimiento de la política fiscal. En los últimos cincuenta años, el país ha experimentado un crecimiento económico significativo, consolidándose como una de las economías más dinámicas de América Latina. Este crecimiento ha estado acompañado por una mayor complejidad económica, lo que ha incrementado la relevancia del mercado financiero dominicano, en particular el mercado de deuda pública, que ha sido un activo clave para fondos de pensiones y fondos de inversión.\n\nSin embargo, este mercado también presenta importantes desafíos, especialmente en términos de baja liquidez en el mercado secundario y episodios de alta volatilidad. Estos problemas dificultan la previsión precisa de los movimientos del mercado y complican la toma de decisiones estratégicas para los inversores. En este contexto, surge la necesidad de utilizar herramientas econométricas que puedan capturar con mayor precisión las dinámicas subyacentes de los retornos y la volatilidad del mercado de deuda pública dominicano.\n\nLa **pregunta de investigación** que guía este estudio es: ¿Es posible modelar adecuadamente los retornos y la volatilidad del mercado de deuda pública dominicano mediante modelos ARIMA-GARCH?, y ¿Qué tan efectivas son estas técnicas para predecir el comportamiento futuro del mercado, especialmente durante periodos de alta volatilidad?\n\nEl **objetivo principal** de este estudio es aplicar modelos ARMA-GARCH para analizar y predecir el comportamiento de los retornos del índice de deuda pública IRP-GOBIX, considerando tanto la tendencia como los patrones de volatilidad del mercado. A través de la estimación de estos modelos, se busca identificar los principales factores que afectan la dinámica de los retornos y proporcionar información valiosa para los actores del mercado, facilitando una gestión más eficiente del riesgo.\n\nAdemás, este estudio pretende evaluar la **consistencia de los modelos** y determinar si los modelos de volatilidad más sofisticados, como el Zero-GARCH, ofrecen una mejor capacidad de predicción que los modelos tradicionales basados únicamente en niveles y diferencias, como el ARIMA. Un aspecto clave será analizar si el modelo GARCH captura adecuadamente los choques bruscos y los episodios de volatilidad extrema, o si suaviza excesivamente estos movimientos.\n\nLa **relevancia** de este estudio radica en la creciente importancia del mercado de deuda pública dominicano, tanto para la estabilidad financiera del país como para los inversores extranjeros que buscan oportunidades de inversión en mercados emergentes. Comprender los patrones de retorno y volatilidad es esencial para diseñar estrategias de inversión que minimicen el riesgo y maximicen el rendimiento en un entorno económico globalizado y altamente incierto.\n\n# Metodología\n\nEl modelo ARIMA(p,d,q), donde AR(p) es el componente autorregresivo de orden p, d denota el orden de la diferencia aplicada para transformar una serie temporal no estacionaria en estacionaria, y MA(q) representa el promedio móvil de orden q. La esencia del modelo ARIMA radica en la combinación de la operación de diferenciación y el modelo ARMA. Cualquier serie no estacionaria puede volverse estacionaria mediante una diferenciación de orden adecuado, permitiendo así ajustar un modelo ARMA a la serie transformada.\n\nEl modelo de promedio móvil autorregresivo ARMA(p,q) incluye tanto un componente autorregresivo de orden p como uno de promedio móvil de orden q, y se estructura de la siguiente forma:\n\n$$\nX_t = \\phi_1 X_{t-1} + \\phi_2 X_{t-2} + \\dots + \\phi_p X_{t-p} + \\epsilon_t + \\theta_1 \\epsilon_{t-1} + \\theta_2 \\epsilon_{t-2} + \\dots + \\theta_q \\epsilon_{t-q}\n$$ {eq: arma}\n\nEstos modelos son adecuados solo para describir series temporales suaves, mientras que en la práctica, a menudo se trabaja con series temporales no suaves. Para estos casos, aplicar una o dos diferenciaciones puede transformar los datos en series suaves.\n\nSi no se cumple el supuesto de homogeneidad de la varianza, puede ocurrir heterocedasticidad. Dado que los datos de la muestra son suaves, en este documento se establece un modelo ARMA(p,q) para la serie de rendimientos con el fin de describir las características de volatilidad del índice GOBIX. Se realizan pruebas de efecto ARCH y se resuelve el problema de la heterocedasticidad mediante un modelo GARCH.\n\nEl modelo GARCH extiende el modelo ARCH al considerar la autocorrelación de orden p en la función de heterocedasticidad, lo que permite ajustar de manera efectiva una función de heterocedasticidad con memoria a largo plazo. El modelo GARCH se define de la siguiente manera:\n\n$$\na_t = \\sigma_t \\epsilon_t, \\quad \\sigma_t^2 = \\alpha_0 + \\sum_{i=1}^{p} \\alpha_i a_{t-i}^2 + \\sum_{j=1}^{q} \\beta_j \\sigma_{t-j}^2\n$$ {#eq: garch}\n\nEl modelo GARCH, propuesto por Bollerslev, atribuye la volatilidad actual tanto a la volatilidad de momentos pasados como a los errores de momentos pasados, lo que le permite explicar el fenómeno de agrupación en la volatilidad de los rendimientos financieros.\n\n# Datos\n\n## Fuente de datos\nUn Índice Financiero es una medida estadística que refleja el valor de un conjunto de activos financieros agrupados de acuerdo con ciertos criterios.Estos se utilizan como objetivo de rendimiento de portafolios, referencias sobre las características retorno/riesgo de una clase de activo y como referencia para productos anclados a índices.\n\nEn el mercado de renta fija de la República Dominicana, el único índice público es el GOBIX, que representa la deuda gubernamental consolidada en peso. Este índice está compuesto por títulos emitidos localmente por el Banco Central y el Ministerio de Hacienda, y se publica en dos modalidades: índice de precio limpio e índice de retorno-precio. Para este estudio, se utilizará el índice de retorno precio @sec-retorno-precio, ya que aisla el componente de retorno de mercado el cual es el objetivo de estudio. \n\nEl GOBIX se caracteriza por los siguientes criterios:\n\n- **Elegibilidad**: Solo se incluyen títulos \"bullet\" emitidos en pesos dominicanos con al menos 60 días desde su emisión.\n- **Liquidez**: Los títulos se seleccionan en función del Índice de Bursatilidad publicado por la Proveedora de Precios, garantizando la inclusión de los instrumentos más negociados.\n- **Ponderación**: La ponderación se realiza por capitalización de mercado, lo que asegura que los títulos más relevantes tengan mayor peso en el índice.\n- **Rebalanceo**: Se efectúa un rebalanceo mensual para ajustar el índice según la disponibilidad de inversión en los bonos.\n- **Fuentes de información**: Los datos provienen de la Bolsa de Valores de la República Dominicana y la Proveedora de Precios.\n\n## Índice de Retorno-Precio {#sec-retorno-precio}\n\nEl índice de retorno-precio es una metodología utilizada para calcular el rendimiento de un índice de renta fija, centrándose exclusivamente en las variaciones del precio limpio de los títulos, sin tener en cuenta los intereses acumulados ni el pago de cupones durante el período de cálculo. En el contexto del GOBIX, este índice refleja las fluctuaciones en los precios de los bonos dominicanos debido a las variaciones en las tasas de interés del mercado.\n\n**Características del Índice de Retorno-Precio del GOBIX**:\n\n- **Enfoque en el Precio Limpio**: El cálculo del índice se basa en el precio limpio de los bonos, excluyendo los intereses acumulados.\n- **Reflejo de la Volatilidad**: Este índice es un indicador del riesgo asociado a las fluctuaciones en las tasas de interés y su impacto en el capital invertido.\n- **Fórmula de Cálculo**:\n\n $$\n IRP_t = IRP_{m-1} \\times (1 + RPI_t)\n $$\n\n Donde:\n - $IRP_0 = 100$ es el valor base del índice en la fecha de inicio.\n - $IRP_t$ es el valor del índice en el día $t$.\n - $IRP_{m-1}$ es el valor del índice en el último día hábil del mes anterior.\n - $RPI_t$ es el retorno-precio del índice en $t$.\n\nEl retorno-precio del índice ($RPI_t$) se calcula como la suma ponderada del retorno-precio de cada bono ($RPI_{i,t}$), donde la ponderación ($\\omega_i$) se determina según la capitalización de mercado de cada bono.\n\nEn este análisis, expresaremos la serie de retornos en puntos básicos para evitar problemas de escalamiento de los datos y posibles desbordamientos decimales (overflow decimal), que pueden ocurrir cuando se manejan fluctuaciones muy pequeñas con alta precisión. Un punto básico equivale a 0.01% o 1/100 de un porcentaje. Este enfoque nos permitirá manejar los datos con mayor estabilidad y evitar posibles errores numéricos derivados del escalamiento inapropiado. No tiene efectos estadísticos en las distribución. \n\n## Selección de muestra de entrenamiento/prueba {#sec-test-train}\n\nDurante la pandemia, se experimentó uno de los ciclos de tasas de interés más pronunciados de las últimas décadas. Este periodo presentó características estructurales distintas en comparación con los ciclos históricos previos. Para evitar sesgos en las estimaciones debido a la alta volatilidad observada durante ese tiempo, evaluaremos únicamente la serie de retornos entre el periodo 2014-2021, que se considera un ciclo de tasas más regular.\n\nLa metodología empleada para dividir el conjunto de datos en muestras de entrenamiento y prueba será el **Time Series Cross-Validator**. A diferencia de la validación cruzada tradicional, que aleatoriza los datos, esta técnica respeta el orden temporal, lo cual es crucial en el análisis de series de tiempo.\n\nEn cada partición (k-ésimo split), la técnica funciona de la siguiente manera:\n\n- **Conjunto de Entrenamiento**: Incluye los primeros *k* pliegues (folds).\n- **Conjunto de Prueba**: Incluye el pliegue *(k+1)*.\n\n# Análisis Exploratorio {#sec-eda}\n\nEl objetivo de este análisis exploratorio es examinar la serie de retornos diarios del IRP-GOBIX para identificar patrones de volatilidad, evaluar la distribución de los retornos, y explorar la autocorrelación en los datos. Utilizando herramientas estadísticas como la distribución de los retornos, el test de estacionariedad Dickey-Fuller, y los gráficos de autocorrelación (ACF) y autocorrelación parcial (PACF), se busca comprender la dinámica temporal de los retornos y validar la aplicación de modelos ARMA-GARCH para capturar su comportamiento futuro. Adicionalmente, se ajustarán diversas distribuciones teóricas, como la distribución normal y la t de Student, para evaluar su capacidad de describir las características empíricas de los datos.\n\n## Histórico de retornos precio del mercado de deuda pública dominicano {#sec-historico-retornos}\n\n\n{{< embed notebooks/data-screening.qmd#fig-price-return-series >}}\n\n\nEl análisis de la serie de retornos diarios del IRP-GOBIX revela un comportamiento interesante, con periodos prolongados de baja volatilidad seguidos por episodios de alta volatilidad. Entre 2015 y 2022, los retornos se mantuvieron generalmente dentro de un rango de ±50 puntos básicos, lo que indica una estabilidad relativa. Sin embargo, en momentos clave, como el ciclo de tasas de 2018, la volatilidad se incrementó drásticamente, alcanzando picos de hasta 200 puntos básicos. Este patrón es característico del fenómeno conocido como **\"volatility clustering\"**, donde periodos tranquilos son seguidos por fases de mayor volatilidad. La naturaleza de estos picos parece estar relacionada con factores externos, como cambios en las tasas de interés y la incertidumbre macroeconómica global, lo que subraya la importancia de emplear modelos ARMA-GARCH para capturar estas dinámicas y prever comportamientos futuros en el mercado de deuda pública dominicana.\n\n## Análisis descriptivo de la muestra {#sec-estadistica-descriptiva}\n:::\n\n::: {#cell-return-descriptive-stats .cell execution_count=4}\n``` {.python .cell-code .hidden}\ndesc_stats = returns.describe()\n\nskewness = returns.skew()\nkurtosis = returns.kurtosis()\njb_test = sm.stats.jarque_bera(returns)\n\ndescriptive_table = pd.DataFrame({\n 'Observations': [int(desc_stats['count'])],\n 'Mean': [desc_stats['mean']],\n 'Median': [desc_stats['50%']],\n 'Std. Dev': [desc_stats['std']],\n 'Skewness': [skewness],\n 'Kurtosis': [kurtosis],\n 'Jarque-Bera': [jb_test[0]],\n 'Prob.': [jb_test[1]]\n})\ndescriptive_table\n```\n\n::: {#return-descriptive-stats .cell-output .cell-output-display execution_count=3}\n```{=html}\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ObservationsMeanMedianStd. DevSkewnessKurtosisJarque-BeraProb.
019411.353251-0.31804423.5029110.705277.5879954789.5323310.0
\n
\n```\n\nTabla de las estadísticas descriptiva de la serie de retornos\n:::\n:::\n\n\n:::{#7cf7bee8 .cell .markdown}\nEl análisis descriptivo de los retornos ofrece una perspectiva más detallada sobre la distribución de la serie. Con un total de 1,941 observaciones, la **media** de los retornos es de $1.353251$, lo que indica un rendimiento promedio positivo. Sin embargo, la **mediana** de $-0.318044$ sugiere un leve sesgo hacia valores negativos, lo que refleja que la mayoría de los retornos tienden a ser ligeramente inferiores a la media. La **desviación estándar**, con un valor de $23.502911$, revela una amplia dispersión de los datos, lo que confirma la presencia de una volatilidad significativa en los rendimientos, muy superior al valor medio.\n\nEn cuanto a la **asimetría** ($0.70527$), se observa una ligera inclinación hacia valores extremos positivos, lo que implica la existencia de eventos fuera de lo común que afectan los rendimientos. La **kurtosis** de $7.587995$ indica una distribución **leptocúrtica**, caracterizada por una alta concentración de valores alrededor de la media y colas más gruesas que una distribución normal, lo que es típico en datos financieros que presentan eventos extremos. El **test de Jarque-Bera**, con un valor de $4,789.532331$, confirma que la serie no sigue una distribución normal, validando la presencia de asimetría y colas pesadas en los datos. Estos resultados resaltan la necesidad de emplear modelos que puedan capturar adecuadamente estos comportamientos no lineales, esenciales para el análisis del mercado.\n\n## Análisis de la distribución de los retornos {#sec-distribucion-retornos}\n\n\n{{< embed notebooks/data-screening.qmd#fig-distribution-fitting >}}\n\n\nAl ajustar diferentes distribuciones a los retornos, observamos que la distribución empírica (en azul) tiene una alta concentración en torno a cero, con colas más gruesas de lo que se esperaría bajo una distribución normal. Entre las distribuciones probadas, la t de Student (en rojo) es la que mejor captura los valores extremos, con colas más largas y una mayor concentración en el centro, lo que es típico en series financieras que experimentan episodios de alta volatilidad. En contraste, la distribución normal (en naranja) y la lognormal (en verde) subestiman las colas, demostrando su ineficacia para modelar adecuadamente los valores atípicos. Esto refuerza la idea de que la t de Student es una mejor candidata para modelar los retornos, dado que puede ajustarse mejor a la leptocurtosis observada en los datos.\n\n\n{{< embed notebooks/data-screening.qmd#fig-tqq-plot >}}\n\n\nLa gráfica Q-Q (Quantile-Quantile) compara los cuantiles teóricos de la distribución t de Student con los cuantiles observados de los retornos del GOBIX. En general, la mayoría de los puntos se alinean bien con la línea roja, lo que sugiere que los retornos de precios siguen razonablemente esta distribución, particularmente en las partes centrales. Sin embargo, en las colas extremas, se observan algunas desviaciones, lo que indica que, aunque la t de Student es un buen ajuste, no es perfecta para todos los escenarios.\n\nEl **estadístico de Kolmogorov-Smirnov** es de $0.0286$, con un **p-valor** de $0.0827$. Aunque esto sugiere que la t de Student captura bien la mayor parte de los retornos, las discrepancias en las colas extremas muestran que la distribución teórica no es un ajuste exacto a los datos reales.\n\n## Evaluación de supuestos para análisis AR-GARCH\n\n### Test de estacionariedad {#sec-estacionariedad}\n\nPara garantizar la validez de los modelos ARMA y GARCH, se evaluó la estacionariedad de la serie de retornos utilizando el test de Dickey-Fuller aumentado (ADF). Los resultados obtenidos muestran un **ADF Statistic** de $-13.1741$ y un **p-valor** de $1.2333 \\times 10^{-24}$, lo que permite rechazar la hipótesis nula de raíz unitaria con un nivel de significancia del 5%. Esto confirma que la serie es estacionaria, lo que significa que sus fluctuaciones se distribuyen alrededor de una media constante en el tiempo, sin tendencia significativa. Este hallazgo es consistente con el comportamiento típico de las series de retornos financieros, lo que habilita la correcta estimación de modelos ARMA-GARCH para capturar las dinámicas del mercado.\n\n### Test de autocorrelación {#sec-autocorrelacion}\n\n\n{{< embed notebooks/data-screening.qmd#fig-acf-pacf >}}\n\n\nEl análisis de autocorrelación nos proporciona más detalles sobre la estructura interna de la serie. El gráfico de la **Función de Autocorrelación (ACF)** muestra un pico significativo en el primer rezago, seguido de valores cercanos a cero para los rezagos posteriores. Este patrón es indicativo de un proceso de media móvil de primer orden (**MA(1)**), donde los choques aleatorios tienen un impacto significativo en el primer rezago pero no en los siguientes. La **Función de Autocorrelación Parcial (PACF)** refuerza esta conclusión, mostrando un comportamiento similar con un pico en el primer rezago y valores prácticamente nulos a partir del segundo rezago. Esto sugiere que un modelo MA(1) sería apropiado para capturar las dinámicas de corto plazo en la serie.\n\nEn conjunto, este análisis exploratorio sugiere que la serie de retornos del IRP-GOBIX presenta características complejas, como **volatility clustering**, alta **leptocurtosis**, y una estructura de autocorrelación que se ajusta bien a un modelo MA(1). Estos hallazgos proporcionan una base sólida para la estimación de modelos ARMA-GARCH, los cuales podrán capturar de manera efectiva las dinámicas de volatilidad y retornos en el mercado de deuda pública dominicano.\n\n# Resultados\n\nEl presente análisis tiene como objetivo modelar la serie temporal de retornos del índice IRP-GOBIX utilizando inicialmente un modelo ARIMA automático (auto-ARIMA) para determinar si es posible capturar la dinámica de los retornos con un modelo basado únicamente en niveles y diferencias de los datos. Adicionalmente, se evaluará la presencia de problemas estructurales, como la heterocedasticidad, que podrían afectar la consistencia del modelo ARIMA. En caso de que este enfoque no sea adecuado, recurriremos a un modelo Zero-GARCH, un tipo específico de GARCH que asume una media cero para los retornos. Este modelo es particularmente útil en mercados de tasas de interés, donde se ha demostrado que los retornos tienden a cero en el largo plazo debido a la fuerte regresión hacia la media de estas variables. \n\n## Análisis del modelo ARIMA\n:::\n\n::: {#cell-arima-model .cell execution_count=5}\n``` {.python .cell-code .hidden}\nmodel_auto = auto_arima(returns)\nmodel_auto.summary()\n```\n\n::: {#arima-model .cell-output .cell-output-display execution_count=4}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:09:21 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n```\n\nModelo ARIMA de maxima verosimilitud para la serie de retornos.\n:::\n:::\n\n\n:::{#f0d39e9b .cell .markdown}\nEl modelo SARIMAX(3, 0, 4) se estimó con el objetivo de capturar la dinámica subyacente de la serie de retornos. Aunque los resultados muestran que algunos de los coeficientes del componente autorregresivo (AR) y del promedio móvil (MA) son estadísticamente significativos (e.g., $AR(3)$ con un coeficiente de $0.7691$, $p < 0.001$ y $MA(2)$ con $p < 0.01$), el desempeño general del modelo presenta ciertas limitaciones. El test de heterocedasticidad muestra un valor de 1.11, lo que indica la presencia de heterocedasticidad en la serie, un problema común en series financieras, donde la varianza de los errores no es constante a lo largo del tiempo.\n\nAdemás, el estadístico de Jarque-Bera de 4395.18, con una probabilidad de $p = 0.00$, confirma que los residuos no siguen una distribución normal, lo que refuerza la idea de que un modelo basado únicamente en niveles, como el ARIMA, podría no ser suficiente para capturar la volatilidad inherente a los retornos del mercado de deuda pública dominicano. Dado que los modelos ARIMA no están diseñados para abordar adecuadamente la heterocedasticidad, pasamos a estimar un modelo GARCH, que es más apropiado para capturar los cambios en la volatilidad.\n\n## Modelo Zero-GARCH\n\nPara abordar las limitaciones del modelo ARIMA, se estimó un modelo GARCH(1,1) con media cero, siguiendo la estructura propuesta por [@miah_rahman_2016], quienes demostrarón que los modelos GARCH(1,1) son altamente efectivos para capturar la volatilidad en los retornos de mercados financieros. La elección de un modelo con media cero se justifica porque en mercados de deuda pública y tasas de interés, los retornos tienden a revertir a la media, y en el largo plazo se espera que los retornos por apreciación de capital converjan a cero. [@fabozzi_fixed_income]\n:::\n\n::: {#garch-model-fitting .cell execution_count=6}\n``` {.python .cell-code .hidden}\nar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\nres = ar.fit(last_obs=split_date)\n```\n\n::: {.cell-output .cell-output-stdout .hidden}\n```\nIteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\nIteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\nIteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\nIteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\nIteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\nIteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\nIteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\nIteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\nIteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\nIteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\nIteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\nIteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\nIteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\nIteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\nIteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\nIteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\nIteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\nIteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\nIteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\nIteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\nIteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\nIteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\nIteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\nIteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\nIteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\nIteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\nOptimization terminated successfully (Exit mode 0)\n Current function value: 7622.868827051428\n Iterations: 26\n Function evaluations: 139\n Gradient evaluations: 26\n```\n:::\n:::\n\n\n::: {#cell-garch-model .cell execution_count=7}\n``` {.python .cell-code .hidden}\nres.summary()\n```\n\n::: {#garch-model .cell-output .cell-output-display execution_count=6}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:09:21 Df Model: 0
\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
\n\n\n\n \n\n\n \n\n
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust\n```\n\nModelo Zero-Garch de la serie de retornos\n:::\n:::\n\n\n:::{#d5cfedd9 .cell .markdown}\nLos resultados del modelo Zero-GARCH confirman un buen ajuste a la volatilidad de la serie de retornos. El coeficiente $\\omega$ del proceso de volatilidad es de $58.8679$ ($p < 0.05$), lo que indica un nivel base significativo de volatilidad en la serie. El parámetro $\\alpha_1$, que mide la influencia de los shocks pasados en la volatilidad actual, es positivo y significativo ($0.1892$, $p < 0.01$), lo que sugiere que los shocks pasados tienen un impacto considerable en la volatilidad presente. Por otro lado, $\\beta_1$ ($0.7737$, $p < 0.001$) indica que existe una fuerte persistencia en la volatilidad, característica común en los mercados financieros, donde las fases de alta volatilidad tienden a durar varios periodos.\n\nEl uso de la distribución t de Student para los residuos estandarizados, con un parámetro $\\nu$ de $2.7894$ ($p < 0.001$), confirma la presencia de colas más gruesas en la distribución de los retornos, lo que es consistente con la leptocurtosis observada en la serie de retornos.\n\n### Residuos del Zero-GARCH\n\n\n{{< embed notebooks/data-screening.qmd#fig-garch-residuals >}}\n\n\nAl analizar los residuos estandarizados y la volatilidad condicional en el modelo Zero-GARCH, observamos cómo la volatilidad responde de manera dinámica a los choques en los retornos. En la primera gráfica, se aprecia que la volatilidad condicional es mayor durante episodios donde los residuos estandarizados alcanzan picos extremos, especialmente en periodos de alta volatilidad como 2018. Esto refuerza la capacidad del modelo GARCH para capturar **volatility clustering**, un fenómeno donde la alta volatilidad tiende a agruparse en ciertos periodos. Además, la persistencia de la volatilidad condicional a lo largo del tiempo confirma que los choques pasados tienen un efecto prolongado, lo cual es coherente con los resultados obtenidos en los coeficientes del modelo GARCH.\n\n### Residuos contra volatilidad condicional\n\n\n{{< embed notebooks/data-screening.qmd#fig-residuals-vs-volatility >}}\n\n\nEn cuanto a la comparación entre los residuos estandarizados y los residuos de varianza unitaria, la segunda gráfica revela que ambos conjuntos de residuos presentan distribuciones similares, con una alta concentración alrededor de cero. Sin embargo, las colas más gruesas en los residuos estandarizados indican la presencia de eventos extremos más pronunciados, los cuales son típicos en series financieras que presentan leptocurtosis. Esta comparación subraya la importancia de utilizar distribuciones robustas, como la t de Student, para capturar adecuadamente las colas de la distribución, tal como lo hace el modelo GARCH en este caso.\n\nLos resultados del modelo Zero-GARCH confirman su capacidad para modelar de manera efectiva la volatilidad en los retornos del IRP-GOBIX. Aunque el modelo auto-ARIMA capturó algunos patrones de la serie, su inconsistencia en la presencia de heterocedasticidad subraya la necesidad de un enfoque basado en modelos GARCH. El modelo Zero-GARCH no solo demuestra un buen ajuste a los datos, sino que también captura la dinámica de la volatilidad condicional y los residuos estandarizados, revelando la presencia de **volatility clustering** y la persistencia en la volatilidad.\n\nLa comparación entre los residuos estandarizados y los residuos de varianza unitaria destaca la capacidad del modelo para manejar eventos extremos, una característica esencial en el análisis de series financieras. En resumen, el modelo Zero-GARCH es más robusto y adecuado para describir las dinámicas de volatilidad en el mercado de deuda pública dominicano, brindando información crucial sobre la persistencia y el comportamiento de la volatilidad a lo largo del tiempo.\n\n## Evaluación Retrospectiva (Backtesting)\n\nPara evaluar la capacidad predictiva del modelo Zero-GARCH, se realizó un análisis de backtesting sobre el periodo comprendido durante el año 2021. Se emplearon dos métodos principales de evaluación: el MAPE (Mean Absolute Percentage Error) y el análisis gráfico de los valores predichos frente a los valores actuales de la desviación estándar móvil. \n\n\n{{< embed notebooks/data-screening.qmd#fig-predicted-vs-actual-backtest >}}\n\n\nEl análisis gráfico (Predichos vs Actuales) revela que el modelo captura de manera excelente la **tendencia** general de la serie, lo que implica que el modelo es eficiente a la hora de predecir los movimientos de largo plazo en la volatilidad. Sin embargo, un problema importante que surge es que el modelo tiende a suavizar los movimientos bruscos, especialmente en momentos donde la volatilidad sufre caídas o picos repentinos. Este fenómeno de **sobre-suavización** implica que el modelo no es completamente reactivo a los choques regresivos de gran magnitud, lo que puede deberse a la estructura propia del GARCH, la cual tiende a modelar la volatilidad alrededor de la media del periodo. \n\nEl MAPE calculado para el conjunto de test arroja un valor elevado de **739.70%**, lo cual podría sugerir en un análisis superficial que el modelo no es adecuado. Sin embargo, es importante considerar el tipo de mercado con el que estamos trabajando, caracterizado por una volatilidad intrínsecamente elevada y movimientos extremos. En estos contextos, valores altos de MAPE son comunes debido a la alta variabilidad en los retornos, lo que no necesariamente implica una mala capacidad predictiva, sino que refleja la naturaleza errática de los datos financieros. En este sentido, el MAPE alto está más relacionado con la complejidad del mercado que con la falta de ajuste del modelo.\n\nEl backtesting del modelo Zero-GARCH sobre la serie de retornos del IRP-GOBIX para el periodo de 2021 confirma que, si bien el modelo captura correctamente la tendencia general de la volatilidad, su capacidad para reaccionar ante movimientos abruptos es limitada. A pesar de esto, dado el contexto del mercado de deuda pública dominicano, la sobre-suavización en los niveles es un comportamiento esperado en modelos de volatilidad como el GARCH. En resumen, el modelo es útil para predecir tendencias a largo plazo, pero su precisión en niveles podría mejorarse si se ajusta para reaccionar mejor a eventos extremos.\n\n# Discusión y Conclusión\n\n## Discusión\n\nEste estudio ha permitido modelar los retornos y la volatilidad del mercado de deuda pública dominicano utilizando modelos ARIMA-GARCH. A pesar de los hallazgos significativos, existen algunas limitaciones que deben considerarse al interpretar los resultados:\n\n- **Problemas con la sobre-suavización**: Como se observó en el análisis retrospectivo, el modelo Zero-GARCH tiende a suavizar los movimientos bruscos en la volatilidad, lo que puede limitar su capacidad de capturar adecuadamente choques repentinos en el mercado. Aunque el modelo refleja bien la tendencia general, su capacidad de reacción frente a eventos extremos debe ser mejorada.\n\n- **MAPE elevado**: El MAPE obtenido durante la evaluación retrospectiva fue considerablemente alto. Si bien esto puede justificarse por la naturaleza volátil del mercado de deuda pública, sugiere que el modelo puede tener problemas para ajustarse a los movimientos de corto plazo. La alta volatilidad y los choques repentinos no fueron capturados con precisión, afectando el rendimiento del modelo.\n\n- **Limitaciones en los datos**: El análisis se centró en un período específico (2014-2021), excluyendo la pandemia debido a sus características atípicas. Si bien esto ayudó a evitar sesgos, la exclusión de este período crítico puede haber omitido dinámicas importantes del mercado en tiempos de crisis, limitando la generalización de los resultados.\n\n## Conclusión\n\nEn conclusión, este estudio ha demostrado que el modelo ARIMA, aunque útil para capturar la estructura de los retornos del mercado de deuda pública dominicano, presenta limitaciones cuando se enfrenta a problemas de heterocedasticidad. El modelo Zero-GARCH, por su parte, ha demostrado ser más adecuado para describir la dinámica de la volatilidad, capturando fenómenos como la persistencia en la volatilidad y el clustering. Además, la capacidad del modelo para manejar eventos extremos a través de la distribución t de Student resulta valiosa en este tipo de mercados caracterizados por alta volatilidad y choques inesperados.\n\nEntre los principales hallazgos destacan:\n\n- **Volatility clustering**: El análisis confirmó la presencia de agrupación de volatilidad, donde periodos de baja volatilidad son seguidos por periodos de alta volatilidad, un fenómeno común en los mercados financieros.\n\n- **Persistencia de la volatilidad**: El coeficiente $\\beta_1$ significativo en el modelo GARCH sugiere que los shocks en la volatilidad tienen un efecto prolongado en el tiempo, lo que es consistente con los patrones observados en mercados financieros emergentes como el dominicano.\n\n- **Dificultad para capturar movimientos extremos**: Aunque el modelo Zero-GARCH es efectivo en capturar la tendencia general, su capacidad para reaccionar ante cambios bruscos en la volatilidad sigue siendo limitada, lo cual es un área de mejora para futuras investigaciones.\n\n- **Implicaciones para los inversores**: Los resultados ofrecen una herramienta útil para la predicción de tendencias de volatilidad a largo plazo en el mercado de deuda pública dominicano, proporcionando una base para la toma de decisiones estratégicas en la gestión del riesgo.\n\nEste estudio no solo contribuye al entendimiento del comportamiento de los retornos y la volatilidad en el mercado de deuda pública dominicano.\n\n# Referencias\n\n::: {#refs}\n:::\n:::\n\n", "supporting": [ "index_files\\figure-html" ], diff --git a/_freeze/index/execute-results/tex.json b/_freeze/index/execute-results/tex.json index 0c51bcb..3c5b3c3 100644 --- a/_freeze/index/execute-results/tex.json +++ b/_freeze/index/execute-results/tex.json @@ -1,9 +1,10 @@ { - "hash": "8ccdb86df4a0f84495987258f19b87bf", + "hash": "22289e552d482a5d6266287352c6d0f1", "result": { - "markdown": "---\ntitle: La Palma Earthquakes\nauthor:\n - name: Steve Purves\n orcid: 0000-0002-0760-5497\n corresponding: true\n email: steve@curvenote.com\n roles:\n - Investigation\n - Project administration\n - Software\n - Visualization\n affiliations:\n - Curvenote\n - name: Rowan Cockett\n orcid: 0000-0002-7859-8394\n corresponding: false\n roles: []\n affiliations:\n - Curvenote\nkeywords:\n - La Palma\n - Earthquakes\nabstract: |\n In September 2021, a significant jump in seismic activity on the island of La Palma (Canary Islands, Spain) signaled the start of a volcanic crisis that still continues at the time of writing. Earthquake data is continually collected and published by the Instituto Geográphico Nacional (IGN). ...\nplain-language-summary: |\n Earthquake data for the island of La Palma from the September 2021 eruption is found ...\nkey-points:\n - A web scraping script was developed to pull data from the Instituto Geogràphico Nacional into a machine-readable form for analysis\n - Earthquake events on La Palma are consistent with the presence of both mantle and crustal reservoirs.\ndate: last-modified\nbibliography: references.bib\ncitation:\n container-title: Earth and Space Science\nnumber-sections: true\n---\n\n:::{#b61468ee .cell .markdown}\n## Introduction\n:::\n\n::: {.cell execution_count=1}\n``` {.python .cell-code .hidden}\nimport matplotlib.pyplot as plt\nimport numpy as np\neruptions = [1492, 1585, 1646, 1677, 1712, 1949, 1971, 2021]\n```\n:::\n\n\n::: {.cell execution_count=2}\n``` {.python .cell-code .hidden}\nplt.figure(figsize=(6, 1))\nplt.eventplot(eruptions, lineoffsets=0, linelengths=0.1, color='black')\nplt.gca().axes.get_yaxis().set_visible(False)\nplt.ylabel('')\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Timeline of recent earthquakes on La Palma](index_files/figure-pdf/fig-timeline-output-1.pdf){#fig-timeline fig-alt='An event plot of the years of the last 8 eruptions on La Palma.' fig-pos='H'}\n:::\n:::\n\n\n::: {.cell execution_count=3}\n``` {.python .cell-code .hidden}\navg_years_between_eruptions = np.mean(np.diff(eruptions[:-1]))\navg_years_between_eruptions\n```\n\n::: {.cell-output .cell-output-display .hidden execution_count=12}\n```\n79.83333333333333\n```\n:::\n:::\n\n\n:::{#7d4d67ef .cell .markdown}\nBased on data up to and including 1971, eruptions on La Palma happen every 79.8 years on average.\n\nStudies of the magma systems feeding the volcano, such as @marrero2019, have proposed that there are two main magma reservoirs feeding the Cumbre Vieja volcano; one in the mantle (30-40km depth) which charges and in turn feeds a shallower crustal reservoir (10-20km depth).\n\nEight eruptions have been recorded since the late 1400s (@fig-timeline).\n\nData and methods are discussed in @sec-data-methods.\n\nLet $x$ denote the number of eruptions in a year. Then, $x$ can be modeled by a Poisson distribution\n\n$$\np(x) = \\frac{e^{-\\lambda} \\lambda^{x}}{x !}\n$$ {#eq-poisson}\n\nwhere $\\lambda$ is the rate of eruptions per year. Using @eq-poisson, the probability of an eruption in the next $t$ years can be calculated.\n\n| Name | Year |\n|---------------------|------|\n| Current | 2021 |\n| Teneguía | 1971 |\n| Nambroque | 1949 |\n| El Charco | 1712 |\n| Volcán San Antonio | 1677 |\n| Volcán San Martin | 1646 |\n| Tajuya near El Paso | 1585 |\n| Montaña Quemada | 1492 |\n\n: Recent historic eruptions on La Palma {#tbl-history}\n\n@tbl-history summarises the eruptions recorded since the colonization of the islands by Europeans in the late 1400s.\n\n![Map of La Palma](images/la-palma-map.png){#fig-map}\n\nLa Palma is one of the west most islands in the Volcanic Archipelago of the Canary Islands (@fig-map).\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-spatial-plot >}}\n\n\n\n\n\n\n\n\n\n@fig-spatial-plot shows the location of recent Earthquakes on La Palma.\n\n## Data & Methods {#sec-data-methods}\n\n## Conclusion\n\n## References {.unnumbered}\n\n::: {#refs}\n:::\n:::\n\n", + "engine": "jupyter", + "markdown": "---\ntitle: Un estudio sobre los retornos del mercado de renta fija dominicano utilizando basados en modelos AR-GARCH\nsubtitle: Trabajo final de tópicos de econometría\nauthor:\n - name: Ian Contreras\n email: 1116048@intec.edu.do\n affiliations:\n - INSTITUTO TECNOLÓGICO DE SANTO DOMINGO (INTEC)\nkeywords:\n - Mercado de renta fija\n - Deuda pública\n - Modelos AR-GARCH\nabstract: |\n Este estudio tiene como objetivo analizar los retornos y la volatilidad del mercado de deuda pública dominicana mediante la aplicación de modelos ARMA-GARCH. A través de estos modelos, se busca identificar patrones en el comportamiento de los retornos y predecir su evolución futura. \ndate: last-modified\nbibliography: references.bib\nexecute:\n echo: false\nnocite: |\n @*\ncitation:\n container-title: Referencias\nnumber-sections: true\n---\n\n::: {#importing-libraries .cell execution_count=1}\n``` {.python .cell-code .hidden}\n# Manejo de datos y análisis\nimport numpy as np\nimport pandas as pd\nimport scipy.stats as stats\n\n# Modelos estadísticos y econométricos\nimport statsmodels.api as sm\nfrom pmdarima.arima import auto_arima\nfrom arch import arch_model\n```\n:::\n\n\n::: {#importing-data .cell execution_count=2}\n``` {.python .cell-code .hidden}\ndf = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\nreturns = df['price_return']\nsplit_date = '2020-12-31'\nR_test = df[df.index >= split_date]['price_return'].rolling(\n window=5).std().dropna()\n```\n:::\n\n\n:::{#e4f79b8b .cell .markdown}\n# Introducción\n\nEl mercado de renta fija en la República Dominicana ha emergido como uno de los sectores más importantes dentro de la economía, especialmente en lo que respecta a la deuda pública, la cual juega un papel crucial en la financiación de proyectos gubernamentales y el sostenimiento de la política fiscal. En los últimos cincuenta años, el país ha experimentado un crecimiento económico significativo, consolidándose como una de las economías más dinámicas de América Latina. Este crecimiento ha estado acompañado por una mayor complejidad económica, lo que ha incrementado la relevancia del mercado financiero dominicano, en particular el mercado de deuda pública, que ha sido un activo clave para fondos de pensiones y fondos de inversión.\n\nSin embargo, este mercado también presenta importantes desafíos, especialmente en términos de baja liquidez en el mercado secundario y episodios de alta volatilidad. Estos problemas dificultan la previsión precisa de los movimientos del mercado y complican la toma de decisiones estratégicas para los inversores. En este contexto, surge la necesidad de utilizar herramientas econométricas que puedan capturar con mayor precisión las dinámicas subyacentes de los retornos y la volatilidad del mercado de deuda pública dominicano.\n\nLa **pregunta de investigación** que guía este estudio es: ¿Es posible modelar adecuadamente los retornos y la volatilidad del mercado de deuda pública dominicano mediante modelos ARIMA-GARCH?, y ¿Qué tan efectivas son estas técnicas para predecir el comportamiento futuro del mercado, especialmente durante periodos de alta volatilidad?\n\nEl **objetivo principal** de este estudio es aplicar modelos ARMA-GARCH para analizar y predecir el comportamiento de los retornos del índice de deuda pública IRP-GOBIX, considerando tanto la tendencia como los patrones de volatilidad del mercado. A través de la estimación de estos modelos, se busca identificar los principales factores que afectan la dinámica de los retornos y proporcionar información valiosa para los actores del mercado, facilitando una gestión más eficiente del riesgo.\n\nAdemás, este estudio pretende evaluar la **consistencia de los modelos** y determinar si los modelos de volatilidad más sofisticados, como el Zero-GARCH, ofrecen una mejor capacidad de predicción que los modelos tradicionales basados únicamente en niveles y diferencias, como el ARIMA. Un aspecto clave será analizar si el modelo GARCH captura adecuadamente los choques bruscos y los episodios de volatilidad extrema, o si suaviza excesivamente estos movimientos.\n\nLa **relevancia** de este estudio radica en la creciente importancia del mercado de deuda pública dominicano, tanto para la estabilidad financiera del país como para los inversores extranjeros que buscan oportunidades de inversión en mercados emergentes. Comprender los patrones de retorno y volatilidad es esencial para diseñar estrategias de inversión que minimicen el riesgo y maximicen el rendimiento en un entorno económico globalizado y altamente incierto.\n\n# Metodología\n\nEl modelo ARIMA(p,d,q), donde AR(p) es el componente autorregresivo de orden p, d denota el orden de la diferencia aplicada para transformar una serie temporal no estacionaria en estacionaria, y MA(q) representa el promedio móvil de orden q. La esencia del modelo ARIMA radica en la combinación de la operación de diferenciación y el modelo ARMA. Cualquier serie no estacionaria puede volverse estacionaria mediante una diferenciación de orden adecuado, permitiendo así ajustar un modelo ARMA a la serie transformada.\n\nEl modelo de promedio móvil autorregresivo ARMA(p,q) incluye tanto un componente autorregresivo de orden p como uno de promedio móvil de orden q, y se estructura de la siguiente forma:\n\n$$\nX_t = \\phi_1 X_{t-1} + \\phi_2 X_{t-2} + \\dots + \\phi_p X_{t-p} + \\epsilon_t + \\theta_1 \\epsilon_{t-1} + \\theta_2 \\epsilon_{t-2} + \\dots + \\theta_q \\epsilon_{t-q}\n$$ {eq: arma}\n\nEstos modelos son adecuados solo para describir series temporales suaves, mientras que en la práctica, a menudo se trabaja con series temporales no suaves. Para estos casos, aplicar una o dos diferenciaciones puede transformar los datos en series suaves.\n\nSi no se cumple el supuesto de homogeneidad de la varianza, puede ocurrir heterocedasticidad. Dado que los datos de la muestra son suaves, en este documento se establece un modelo ARMA(p,q) para la serie de rendimientos con el fin de describir las características de volatilidad del índice GOBIX. Se realizan pruebas de efecto ARCH y se resuelve el problema de la heterocedasticidad mediante un modelo GARCH.\n\nEl modelo GARCH extiende el modelo ARCH al considerar la autocorrelación de orden p en la función de heterocedasticidad, lo que permite ajustar de manera efectiva una función de heterocedasticidad con memoria a largo plazo. El modelo GARCH se define de la siguiente manera:\n\n$$\na_t = \\sigma_t \\epsilon_t, \\quad \\sigma_t^2 = \\alpha_0 + \\sum_{i=1}^{p} \\alpha_i a_{t-i}^2 + \\sum_{j=1}^{q} \\beta_j \\sigma_{t-j}^2\n$$ {#eq: garch}\n\nEl modelo GARCH, propuesto por Bollerslev, atribuye la volatilidad actual tanto a la volatilidad de momentos pasados como a los errores de momentos pasados, lo que le permite explicar el fenómeno de agrupación en la volatilidad de los rendimientos financieros.\n\n# Datos\n\n## Fuente de datos\nUn Índice Financiero es una medida estadística que refleja el valor de un conjunto de activos financieros agrupados de acuerdo con ciertos criterios.Estos se utilizan como objetivo de rendimiento de portafolios, referencias sobre las características retorno/riesgo de una clase de activo y como referencia para productos anclados a índices.\n\nEn el mercado de renta fija de la República Dominicana, el único índice público es el GOBIX, que representa la deuda gubernamental consolidada en peso. Este índice está compuesto por títulos emitidos localmente por el Banco Central y el Ministerio de Hacienda, y se publica en dos modalidades: índice de precio limpio e índice de retorno-precio. Para este estudio, se utilizará el índice de retorno precio @sec-retorno-precio, ya que aisla el componente de retorno de mercado el cual es el objetivo de estudio. \n\nEl GOBIX se caracteriza por los siguientes criterios:\n\n- **Elegibilidad**: Solo se incluyen títulos \"bullet\" emitidos en pesos dominicanos con al menos 60 días desde su emisión.\n- **Liquidez**: Los títulos se seleccionan en función del Índice de Bursatilidad publicado por la Proveedora de Precios, garantizando la inclusión de los instrumentos más negociados.\n- **Ponderación**: La ponderación se realiza por capitalización de mercado, lo que asegura que los títulos más relevantes tengan mayor peso en el índice.\n- **Rebalanceo**: Se efectúa un rebalanceo mensual para ajustar el índice según la disponibilidad de inversión en los bonos.\n- **Fuentes de información**: Los datos provienen de la Bolsa de Valores de la República Dominicana y la Proveedora de Precios.\n\n## Índice de Retorno-Precio {#sec-retorno-precio}\n\nEl índice de retorno-precio es una metodología utilizada para calcular el rendimiento de un índice de renta fija, centrándose exclusivamente en las variaciones del precio limpio de los títulos, sin tener en cuenta los intereses acumulados ni el pago de cupones durante el período de cálculo. En el contexto del GOBIX, este índice refleja las fluctuaciones en los precios de los bonos dominicanos debido a las variaciones en las tasas de interés del mercado.\n\n**Características del Índice de Retorno-Precio del GOBIX**:\n\n- **Enfoque en el Precio Limpio**: El cálculo del índice se basa en el precio limpio de los bonos, excluyendo los intereses acumulados.\n- **Reflejo de la Volatilidad**: Este índice es un indicador del riesgo asociado a las fluctuaciones en las tasas de interés y su impacto en el capital invertido.\n- **Fórmula de Cálculo**:\n\n $$\n IRP_t = IRP_{m-1} \\times (1 + RPI_t)\n $$\n\n Donde:\n - $IRP_0 = 100$ es el valor base del índice en la fecha de inicio.\n - $IRP_t$ es el valor del índice en el día $t$.\n - $IRP_{m-1}$ es el valor del índice en el último día hábil del mes anterior.\n - $RPI_t$ es el retorno-precio del índice en $t$.\n\nEl retorno-precio del índice ($RPI_t$) se calcula como la suma ponderada del retorno-precio de cada bono ($RPI_{i,t}$), donde la ponderación ($\\omega_i$) se determina según la capitalización de mercado de cada bono.\n\nEn este análisis, expresaremos la serie de retornos en puntos básicos para evitar problemas de escalamiento de los datos y posibles desbordamientos decimales (overflow decimal), que pueden ocurrir cuando se manejan fluctuaciones muy pequeñas con alta precisión. Un punto básico equivale a 0.01% o 1/100 de un porcentaje. Este enfoque nos permitirá manejar los datos con mayor estabilidad y evitar posibles errores numéricos derivados del escalamiento inapropiado. No tiene efectos estadísticos en las distribución. \n\n## Selección de muestra de entrenamiento/prueba {#sec-test-train}\n\nDurante la pandemia, se experimentó uno de los ciclos de tasas de interés más pronunciados de las últimas décadas. Este periodo presentó características estructurales distintas en comparación con los ciclos históricos previos. Para evitar sesgos en las estimaciones debido a la alta volatilidad observada durante ese tiempo, evaluaremos únicamente la serie de retornos entre el periodo 2014-2021, que se considera un ciclo de tasas más regular.\n\nLa metodología empleada para dividir el conjunto de datos en muestras de entrenamiento y prueba será el **Time Series Cross-Validator**. A diferencia de la validación cruzada tradicional, que aleatoriza los datos, esta técnica respeta el orden temporal, lo cual es crucial en el análisis de series de tiempo.\n\nEn cada partición (k-ésimo split), la técnica funciona de la siguiente manera:\n\n- **Conjunto de Entrenamiento**: Incluye los primeros *k* pliegues (folds).\n- **Conjunto de Prueba**: Incluye el pliegue *(k+1)*.\n\n# Análisis Exploratorio {#sec-eda}\n\nEl objetivo de este análisis exploratorio es examinar la serie de retornos diarios del IRP-GOBIX para identificar patrones de volatilidad, evaluar la distribución de los retornos, y explorar la autocorrelación en los datos. Utilizando herramientas estadísticas como la distribución de los retornos, el test de estacionariedad Dickey-Fuller, y los gráficos de autocorrelación (ACF) y autocorrelación parcial (PACF), se busca comprender la dinámica temporal de los retornos y validar la aplicación de modelos ARMA-GARCH para capturar su comportamiento futuro. Adicionalmente, se ajustarán diversas distribuciones teóricas, como la distribución normal y la t de Student, para evaluar su capacidad de describir las características empíricas de los datos.\n\n## Histórico de retornos precio del mercado de deuda pública dominicano {#sec-historico-retornos}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-price-return-series >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis de la serie de retornos diarios del IRP-GOBIX revela un comportamiento interesante, con periodos prolongados de baja volatilidad seguidos por episodios de alta volatilidad. Entre 2015 y 2022, los retornos se mantuvieron generalmente dentro de un rango de ±50 puntos básicos, lo que indica una estabilidad relativa. Sin embargo, en momentos clave, como el ciclo de tasas de 2018, la volatilidad se incrementó drásticamente, alcanzando picos de hasta 200 puntos básicos. Este patrón es característico del fenómeno conocido como **\"volatility clustering\"**, donde periodos tranquilos son seguidos por fases de mayor volatilidad. La naturaleza de estos picos parece estar relacionada con factores externos, como cambios en las tasas de interés y la incertidumbre macroeconómica global, lo que subraya la importancia de emplear modelos ARMA-GARCH para capturar estas dinámicas y prever comportamientos futuros en el mercado de deuda pública dominicana.\n\n## Análisis descriptivo de la muestra {#sec-estadistica-descriptiva}\n:::\n\n::: {.cell execution_count=3}\n``` {.python .cell-code .hidden}\ndesc_stats = returns.describe()\n\nskewness = returns.skew()\nkurtosis = returns.kurtosis()\njb_test = sm.stats.jarque_bera(returns)\n\ndescriptive_table = pd.DataFrame({\n 'Observations': [int(desc_stats['count'])],\n 'Mean': [desc_stats['mean']],\n 'Median': [desc_stats['50%']],\n 'Std. Dev': [desc_stats['std']],\n 'Skewness': [skewness],\n 'Kurtosis': [kurtosis],\n 'Jarque-Bera': [jb_test[0]],\n 'Prob.': [jb_test[1]]\n})\ndescriptive_table\n```\n\n::: {#return-descriptive-stats .cell-output .cell-output-display execution_count=3}\n```{=html}\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ObservationsMeanMedianStd. DevSkewnessKurtosisJarque-BeraProb.
019411.353251-0.31804423.5029110.705277.5879954789.5323310.0
\n
\n```\n\nTabla de las estadísticas descriptiva de la serie de retornos\n:::\n:::\n\n\n:::{#712a9981 .cell .markdown}\nEl análisis descriptivo de los retornos ofrece una perspectiva más detallada sobre la distribución de la serie. Con un total de 1,941 observaciones, la **media** de los retornos es de $1.353251$, lo que indica un rendimiento promedio positivo. Sin embargo, la **mediana** de $-0.318044$ sugiere un leve sesgo hacia valores negativos, lo que refleja que la mayoría de los retornos tienden a ser ligeramente inferiores a la media. La **desviación estándar**, con un valor de $23.502911$, revela una amplia dispersión de los datos, lo que confirma la presencia de una volatilidad significativa en los rendimientos, muy superior al valor medio.\n\nEn cuanto a la **asimetría** ($0.70527$), se observa una ligera inclinación hacia valores extremos positivos, lo que implica la existencia de eventos fuera de lo común que afectan los rendimientos. La **kurtosis** de $7.587995$ indica una distribución **leptocúrtica**, caracterizada por una alta concentración de valores alrededor de la media y colas más gruesas que una distribución normal, lo que es típico en datos financieros que presentan eventos extremos. El **test de Jarque-Bera**, con un valor de $4,789.532331$, confirma que la serie no sigue una distribución normal, validando la presencia de asimetría y colas pesadas en los datos. Estos resultados resaltan la necesidad de emplear modelos que puedan capturar adecuadamente estos comportamientos no lineales, esenciales para el análisis del mercado.\n\n## Análisis de la distribución de los retornos {#sec-distribucion-retornos}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-distribution-fitting >}}\n\n\n\n\n\n\n\n\n\n\nAl ajustar diferentes distribuciones a los retornos, observamos que la distribución empírica (en azul) tiene una alta concentración en torno a cero, con colas más gruesas de lo que se esperaría bajo una distribución normal. Entre las distribuciones probadas, la t de Student (en rojo) es la que mejor captura los valores extremos, con colas más largas y una mayor concentración en el centro, lo que es típico en series financieras que experimentan episodios de alta volatilidad. En contraste, la distribución normal (en naranja) y la lognormal (en verde) subestiman las colas, demostrando su ineficacia para modelar adecuadamente los valores atípicos. Esto refuerza la idea de que la t de Student es una mejor candidata para modelar los retornos, dado que puede ajustarse mejor a la leptocurtosis observada en los datos.\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-tqq-plot >}}\n\n\n\n\n\n\n\n\n\n\nLa gráfica Q-Q (Quantile-Quantile) compara los cuantiles teóricos de la distribución t de Student con los cuantiles observados de los retornos del GOBIX. En general, la mayoría de los puntos se alinean bien con la línea roja, lo que sugiere que los retornos de precios siguen razonablemente esta distribución, particularmente en las partes centrales. Sin embargo, en las colas extremas, se observan algunas desviaciones, lo que indica que, aunque la t de Student es un buen ajuste, no es perfecta para todos los escenarios.\n\nEl **estadístico de Kolmogorov-Smirnov** es de $0.0286$, con un **p-valor** de $0.0827$. Aunque esto sugiere que la t de Student captura bien la mayor parte de los retornos, las discrepancias en las colas extremas muestran que la distribución teórica no es un ajuste exacto a los datos reales.\n\n## Evaluación de supuestos para análisis AR-GARCH\n\n### Test de estacionariedad {#sec-estacionariedad}\n\nPara garantizar la validez de los modelos ARMA y GARCH, se evaluó la estacionariedad de la serie de retornos utilizando el test de Dickey-Fuller aumentado (ADF). Los resultados obtenidos muestran un **ADF Statistic** de $-13.1741$ y un **p-valor** de $1.2333 \\times 10^{-24}$, lo que permite rechazar la hipótesis nula de raíz unitaria con un nivel de significancia del 5%. Esto confirma que la serie es estacionaria, lo que significa que sus fluctuaciones se distribuyen alrededor de una media constante en el tiempo, sin tendencia significativa. Este hallazgo es consistente con el comportamiento típico de las series de retornos financieros, lo que habilita la correcta estimación de modelos ARMA-GARCH para capturar las dinámicas del mercado.\n\n### Test de autocorrelación {#sec-autocorrelacion}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-acf-pacf >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis de autocorrelación nos proporciona más detalles sobre la estructura interna de la serie. El gráfico de la **Función de Autocorrelación (ACF)** muestra un pico significativo en el primer rezago, seguido de valores cercanos a cero para los rezagos posteriores. Este patrón es indicativo de un proceso de media móvil de primer orden (**MA(1)**), donde los choques aleatorios tienen un impacto significativo en el primer rezago pero no en los siguientes. La **Función de Autocorrelación Parcial (PACF)** refuerza esta conclusión, mostrando un comportamiento similar con un pico en el primer rezago y valores prácticamente nulos a partir del segundo rezago. Esto sugiere que un modelo MA(1) sería apropiado para capturar las dinámicas de corto plazo en la serie.\n\nEn conjunto, este análisis exploratorio sugiere que la serie de retornos del IRP-GOBIX presenta características complejas, como **volatility clustering**, alta **leptocurtosis**, y una estructura de autocorrelación que se ajusta bien a un modelo MA(1). Estos hallazgos proporcionan una base sólida para la estimación de modelos ARMA-GARCH, los cuales podrán capturar de manera efectiva las dinámicas de volatilidad y retornos en el mercado de deuda pública dominicano.\n\n# Resultados\n\nEl presente análisis tiene como objetivo modelar la serie temporal de retornos del índice IRP-GOBIX utilizando inicialmente un modelo ARIMA automático (auto-ARIMA) para determinar si es posible capturar la dinámica de los retornos con un modelo basado únicamente en niveles y diferencias de los datos. Adicionalmente, se evaluará la presencia de problemas estructurales, como la heterocedasticidad, que podrían afectar la consistencia del modelo ARIMA. En caso de que este enfoque no sea adecuado, recurriremos a un modelo Zero-GARCH, un tipo específico de GARCH que asume una media cero para los retornos. Este modelo es particularmente útil en mercados de tasas de interés, donde se ha demostrado que los retornos tienden a cero en el largo plazo debido a la fuerte regresión hacia la media de estas variables. \n\n## Análisis del modelo ARIMA\n:::\n\n::: {.cell execution_count=4}\n``` {.python .cell-code .hidden}\nmodel_auto = auto_arima(returns)\nmodel_auto.summary()\n```\n\n::: {#arima-model .cell-output .cell-output-display execution_count=4}\n```{=tex}\n\\begin{center}\n\\begin{tabular}{lclc}\n\\toprule\n\\textbf{Dep. Variable:} & y & \\textbf{ No. Observations: } & 1941 \\\\\n\\textbf{Model:} & SARIMAX(3, 0, 4) & \\textbf{ Log Likelihood } & -8853.517 \\\\\n\\textbf{Date:} & Sat, 19 Oct 2024 & \\textbf{ AIC } & 17725.034 \\\\\n\\textbf{Time:} & 15:47:38 & \\textbf{ BIC } & 17775.173 \\\\\n\\textbf{Sample:} & 0 & \\textbf{ HQIC } & 17743.472 \\\\\n\\textbf{} & - 1941 & \\textbf{ } & \\\\\n\\textbf{Covariance Type:} & opg & \\textbf{ } & \\\\\n\\bottomrule\n\\end{tabular}\n\\begin{tabular}{lcccccc}\n & \\textbf{coef} & \\textbf{std err} & \\textbf{z} & \\textbf{P$> |$z$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n\\midrule\n\\textbf{intercept} & 0.3602 & 0.273 & 1.317 & 0.188 & -0.176 & 0.896 \\\\\n\\textbf{ar.L1} & 0.1366 & 0.065 & 2.106 & 0.035 & 0.009 & 0.264 \\\\\n\\textbf{ar.L2} & -0.1931 & 0.066 & -2.934 & 0.003 & -0.322 & -0.064 \\\\\n\\textbf{ar.L3} & 0.7691 & 0.061 & 12.644 & 0.000 & 0.650 & 0.888 \\\\\n\\textbf{ma.L1} & -0.1139 & 0.065 & -1.752 & 0.080 & -0.241 & 0.014 \\\\\n\\textbf{ma.L2} & 0.2163 & 0.067 & 3.220 & 0.001 & 0.085 & 0.348 \\\\\n\\textbf{ma.L3} & -0.7032 & 0.062 & -11.315 & 0.000 & -0.825 & -0.581 \\\\\n\\textbf{ma.L4} & 0.0750 & 0.020 & 3.715 & 0.000 & 0.035 & 0.115 \\\\\n\\textbf{sigma2} & 536.2183 & 8.358 & 64.154 & 0.000 & 519.836 & 552.600 \\\\\n\\bottomrule\n\\end{tabular}\n\\begin{tabular}{lclc}\n\\textbf{Ljung-Box (L1) (Q):} & 0.00 & \\textbf{ Jarque-Bera (JB): } & 4395.18 \\\\\n\\textbf{Prob(Q):} & 1.00 & \\textbf{ Prob(JB): } & 0.00 \\\\\n\\textbf{Heteroskedasticity (H):} & 1.11 & \\textbf{ Skew: } & 0.59 \\\\\n\\textbf{Prob(H) (two-sided):} & 0.17 & \\textbf{ Kurtosis: } & 10.28 \\\\\n\\bottomrule\n\\end{tabular}\n%\\caption{SARIMAX Results}\n\\end{center}\n\nWarnings: \\newline\n [1] Covariance matrix calculated using the outer product of gradients (complex-step).\n```\n\nModelo ARIMA de maxima verosimilitud para la serie de retornos.\n:::\n:::\n\n\n:::{#5cc2ce27 .cell .markdown}\nEl modelo SARIMAX(3, 0, 4) se estimó con el objetivo de capturar la dinámica subyacente de la serie de retornos. Aunque los resultados muestran que algunos de los coeficientes del componente autorregresivo (AR) y del promedio móvil (MA) son estadísticamente significativos (e.g., $AR(3)$ con un coeficiente de $0.7691$, $p < 0.001$ y $MA(2)$ con $p < 0.01$), el desempeño general del modelo presenta ciertas limitaciones. El test de heterocedasticidad muestra un valor de 1.11, lo que indica la presencia de heterocedasticidad en la serie, un problema común en series financieras, donde la varianza de los errores no es constante a lo largo del tiempo.\n\nAdemás, el estadístico de Jarque-Bera de 4395.18, con una probabilidad de $p = 0.00$, confirma que los residuos no siguen una distribución normal, lo que refuerza la idea de que un modelo basado únicamente en niveles, como el ARIMA, podría no ser suficiente para capturar la volatilidad inherente a los retornos del mercado de deuda pública dominicano. Dado que los modelos ARIMA no están diseñados para abordar adecuadamente la heterocedasticidad, pasamos a estimar un modelo GARCH, que es más apropiado para capturar los cambios en la volatilidad.\n\n## Modelo Zero-GARCH\n\nPara abordar las limitaciones del modelo ARIMA, se estimó un modelo GARCH(1,1) con media cero, siguiendo la estructura propuesta por [@miah_rahman_2016], quienes demostrarón que los modelos GARCH(1,1) son altamente efectivos para capturar la volatilidad en los retornos de mercados financieros. La elección de un modelo con media cero se justifica porque en mercados de deuda pública y tasas de interés, los retornos tienden a revertir a la media, y en el largo plazo se espera que los retornos por apreciación de capital converjan a cero. [@fabozzi_fixed_income]\n:::\n\n::: {#garch-model-fitting .cell execution_count=5}\n``` {.python .cell-code .hidden}\nar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\nres = ar.fit(last_obs=split_date)\n```\n\n::: {.cell-output .cell-output-stdout .hidden}\n```\nIteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\nIteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\nIteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\nIteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\nIteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\nIteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\nIteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\nIteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\nIteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\nIteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\nIteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\nIteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\nIteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\nIteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\nIteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\nIteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\nIteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\nIteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\nIteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\nIteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\nIteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\nIteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\nIteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\nIteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\nIteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\nIteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\nOptimization terminated successfully (Exit mode 0)\n Current function value: 7622.868827051428\n Iterations: 26\n Function evaluations: 139\n Gradient evaluations: 26\n```\n:::\n:::\n\n\n::: {.cell execution_count=6}\n``` {.python .cell-code .hidden}\nres.summary()\n```\n\n::: {#garch-model .cell-output .cell-output-display execution_count=6}\n```{=tex}\n\\begin{center}\n\\begin{tabular}{lclc}\n\\toprule\n\\textbf{Dep. Variable:} & price\\_return & \\textbf{ R-squared: } & 0.000 \\\\\n\\textbf{Mean Model:} & Zero Mean & \\textbf{ Adj. R-squared: } & 0.001 \\\\\n\\textbf{Vol Model:} & GARCH & \\textbf{ Log-Likelihood: } & -7622.87 \\\\\n\\textbf{Distribution:} & Standardized Student's t & \\textbf{ AIC: } & 15253.7 \\\\\n\\textbf{Method:} & Maximum Likelihood & \\textbf{ BIC: } & 15275.6 \\\\\n\\textbf{} & & \\textbf{ No. Observations: } & 1753 \\\\\n\\textbf{Date:} & Sat, Oct 19 2024 & \\textbf{ Df Residuals: } & 1753 \\\\\n\\textbf{Time:} & 15:47:38 & \\textbf{ Df Model: } & 0 \\\\\n\\bottomrule\n\\end{tabular}\n\\begin{tabular}{lccccc}\n & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n\\midrule\n\\textbf{omega} & 58.8679 & 25.182 & 2.338 & 1.940e-02 & [ 9.512,1.082e+02] \\\\\n\\textbf{alpha[1]} & 0.1892 & 6.657e-02 & 2.842 & 4.482e-03 & [5.873e-02, 0.320] \\\\\n\\textbf{beta[1]} & 0.7737 & 6.832e-02 & 11.324 & 9.943e-30 & [ 0.640, 0.908] \\\\\n & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{95.0\\% Conf. Int.} \\\\\n\\midrule\n\\textbf{nu} & 2.7894 & 0.230 & 12.114 & 8.853e-34 & [ 2.338, 3.241] \\\\\n\\bottomrule\n\\end{tabular}\n%\\caption{Zero Mean - GARCH Model Results}\n\\end{center}\n\nCovariance estimator: robust\n```\n\nModelo Zero-Garch de la serie de retornos\n:::\n:::\n\n\n:::{#59534dc4 .cell .markdown}\nLos resultados del modelo Zero-GARCH confirman un buen ajuste a la volatilidad de la serie de retornos. El coeficiente $\\omega$ del proceso de volatilidad es de $58.8679$ ($p < 0.05$), lo que indica un nivel base significativo de volatilidad en la serie. El parámetro $\\alpha_1$, que mide la influencia de los shocks pasados en la volatilidad actual, es positivo y significativo ($0.1892$, $p < 0.01$), lo que sugiere que los shocks pasados tienen un impacto considerable en la volatilidad presente. Por otro lado, $\\beta_1$ ($0.7737$, $p < 0.001$) indica que existe una fuerte persistencia en la volatilidad, característica común en los mercados financieros, donde las fases de alta volatilidad tienden a durar varios periodos.\n\nEl uso de la distribución t de Student para los residuos estandarizados, con un parámetro $\\nu$ de $2.7894$ ($p < 0.001$), confirma la presencia de colas más gruesas en la distribución de los retornos, lo que es consistente con la leptocurtosis observada en la serie de retornos.\n\n### Residuos del Zero-GARCH\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-garch-residuals >}}\n\n\n\n\n\n\n\n\n\n\nAl analizar los residuos estandarizados y la volatilidad condicional en el modelo Zero-GARCH, observamos cómo la volatilidad responde de manera dinámica a los choques en los retornos. En la primera gráfica, se aprecia que la volatilidad condicional es mayor durante episodios donde los residuos estandarizados alcanzan picos extremos, especialmente en periodos de alta volatilidad como 2018. Esto refuerza la capacidad del modelo GARCH para capturar **volatility clustering**, un fenómeno donde la alta volatilidad tiende a agruparse en ciertos periodos. Además, la persistencia de la volatilidad condicional a lo largo del tiempo confirma que los choques pasados tienen un efecto prolongado, lo cual es coherente con los resultados obtenidos en los coeficientes del modelo GARCH.\n\n### Residuos contra volatilidad condicional\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-residuals-vs-volatility >}}\n\n\n\n\n\n\n\n\n\n\nEn cuanto a la comparación entre los residuos estandarizados y los residuos de varianza unitaria, la segunda gráfica revela que ambos conjuntos de residuos presentan distribuciones similares, con una alta concentración alrededor de cero. Sin embargo, las colas más gruesas en los residuos estandarizados indican la presencia de eventos extremos más pronunciados, los cuales son típicos en series financieras que presentan leptocurtosis. Esta comparación subraya la importancia de utilizar distribuciones robustas, como la t de Student, para capturar adecuadamente las colas de la distribución, tal como lo hace el modelo GARCH en este caso.\n\nLos resultados del modelo Zero-GARCH confirman su capacidad para modelar de manera efectiva la volatilidad en los retornos del IRP-GOBIX. Aunque el modelo auto-ARIMA capturó algunos patrones de la serie, su inconsistencia en la presencia de heterocedasticidad subraya la necesidad de un enfoque basado en modelos GARCH. El modelo Zero-GARCH no solo demuestra un buen ajuste a los datos, sino que también captura la dinámica de la volatilidad condicional y los residuos estandarizados, revelando la presencia de **volatility clustering** y la persistencia en la volatilidad.\n\nLa comparación entre los residuos estandarizados y los residuos de varianza unitaria destaca la capacidad del modelo para manejar eventos extremos, una característica esencial en el análisis de series financieras. En resumen, el modelo Zero-GARCH es más robusto y adecuado para describir las dinámicas de volatilidad en el mercado de deuda pública dominicano, brindando información crucial sobre la persistencia y el comportamiento de la volatilidad a lo largo del tiempo.\n\n## Evaluación Retrospectiva (Backtesting)\n\nPara evaluar la capacidad predictiva del modelo Zero-GARCH, se realizó un análisis de backtesting sobre el periodo comprendido durante el año 2021. Se emplearon dos métodos principales de evaluación: el MAPE (Mean Absolute Percentage Error) y el análisis gráfico de los valores predichos frente a los valores actuales de la desviación estándar móvil. \n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-predicted-vs-actual-backtest >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis gráfico (Predichos vs Actuales) revela que el modelo captura de manera excelente la **tendencia** general de la serie, lo que implica que el modelo es eficiente a la hora de predecir los movimientos de largo plazo en la volatilidad. Sin embargo, un problema importante que surge es que el modelo tiende a suavizar los movimientos bruscos, especialmente en momentos donde la volatilidad sufre caídas o picos repentinos. Este fenómeno de **sobre-suavización** implica que el modelo no es completamente reactivo a los choques regresivos de gran magnitud, lo que puede deberse a la estructura propia del GARCH, la cual tiende a modelar la volatilidad alrededor de la media del periodo. \n\nEl MAPE calculado para el conjunto de test arroja un valor elevado de **739.70%**, lo cual podría sugerir en un análisis superficial que el modelo no es adecuado. Sin embargo, es importante considerar el tipo de mercado con el que estamos trabajando, caracterizado por una volatilidad intrínsecamente elevada y movimientos extremos. En estos contextos, valores altos de MAPE son comunes debido a la alta variabilidad en los retornos, lo que no necesariamente implica una mala capacidad predictiva, sino que refleja la naturaleza errática de los datos financieros. En este sentido, el MAPE alto está más relacionado con la complejidad del mercado que con la falta de ajuste del modelo.\n\nEl backtesting del modelo Zero-GARCH sobre la serie de retornos del IRP-GOBIX para el periodo de 2021 confirma que, si bien el modelo captura correctamente la tendencia general de la volatilidad, su capacidad para reaccionar ante movimientos abruptos es limitada. A pesar de esto, dado el contexto del mercado de deuda pública dominicano, la sobre-suavización en los niveles es un comportamiento esperado en modelos de volatilidad como el GARCH. En resumen, el modelo es útil para predecir tendencias a largo plazo, pero su precisión en niveles podría mejorarse si se ajusta para reaccionar mejor a eventos extremos.\n\n# Discusión y Conclusión\n\n## Discusión\n\nEste estudio ha permitido modelar los retornos y la volatilidad del mercado de deuda pública dominicano utilizando modelos ARIMA-GARCH. A pesar de los hallazgos significativos, existen algunas limitaciones que deben considerarse al interpretar los resultados:\n\n- **Problemas con la sobre-suavización**: Como se observó en el análisis retrospectivo, el modelo Zero-GARCH tiende a suavizar los movimientos bruscos en la volatilidad, lo que puede limitar su capacidad de capturar adecuadamente choques repentinos en el mercado. Aunque el modelo refleja bien la tendencia general, su capacidad de reacción frente a eventos extremos debe ser mejorada.\n\n- **MAPE elevado**: El MAPE obtenido durante la evaluación retrospectiva fue considerablemente alto. Si bien esto puede justificarse por la naturaleza volátil del mercado de deuda pública, sugiere que el modelo puede tener problemas para ajustarse a los movimientos de corto plazo. La alta volatilidad y los choques repentinos no fueron capturados con precisión, afectando el rendimiento del modelo.\n\n- **Limitaciones en los datos**: El análisis se centró en un período específico (2014-2021), excluyendo la pandemia debido a sus características atípicas. Si bien esto ayudó a evitar sesgos, la exclusión de este período crítico puede haber omitido dinámicas importantes del mercado en tiempos de crisis, limitando la generalización de los resultados.\n\n## Conclusión\n\nEn conclusión, este estudio ha demostrado que el modelo ARIMA, aunque útil para capturar la estructura de los retornos del mercado de deuda pública dominicano, presenta limitaciones cuando se enfrenta a problemas de heterocedasticidad. El modelo Zero-GARCH, por su parte, ha demostrado ser más adecuado para describir la dinámica de la volatilidad, capturando fenómenos como la persistencia en la volatilidad y el clustering. Además, la capacidad del modelo para manejar eventos extremos a través de la distribución t de Student resulta valiosa en este tipo de mercados caracterizados por alta volatilidad y choques inesperados.\n\nEntre los principales hallazgos destacan:\n\n- **Volatility clustering**: El análisis confirmó la presencia de agrupación de volatilidad, donde periodos de baja volatilidad son seguidos por periodos de alta volatilidad, un fenómeno común en los mercados financieros.\n\n- **Persistencia de la volatilidad**: El coeficiente $\\beta_1$ significativo en el modelo GARCH sugiere que los shocks en la volatilidad tienen un efecto prolongado en el tiempo, lo que es consistente con los patrones observados en mercados financieros emergentes como el dominicano.\n\n- **Dificultad para capturar movimientos extremos**: Aunque el modelo Zero-GARCH es efectivo en capturar la tendencia general, su capacidad para reaccionar ante cambios bruscos en la volatilidad sigue siendo limitada, lo cual es un área de mejora para futuras investigaciones.\n\n- **Implicaciones para los inversores**: Los resultados ofrecen una herramienta útil para la predicción de tendencias de volatilidad a largo plazo en el mercado de deuda pública dominicano, proporcionando una base para la toma de decisiones estratégicas en la gestión del riesgo.\n\nEste estudio no solo contribuye al entendimiento del comportamiento de los retornos y la volatilidad en el mercado de deuda pública dominicano.\n\n# Referencias\n\n::: {#refs}\n:::\n:::\n\n", "supporting": [ - "index_files/figure-pdf" + "index_files\\figure-pdf" ], "filters": [] } diff --git a/_freeze/index/execute-results/xml.json b/_freeze/index/execute-results/xml.json index ec89c5f..54a7326 100644 --- a/_freeze/index/execute-results/xml.json +++ b/_freeze/index/execute-results/xml.json @@ -1,9 +1,10 @@ { - "hash": "8ccdb86df4a0f84495987258f19b87bf", + "hash": "22289e552d482a5d6266287352c6d0f1", "result": { - "markdown": "---\ntitle: La Palma Earthquakes\nauthor:\n - name: Steve Purves\n orcid: 0000-0002-0760-5497\n corresponding: true\n email: steve@curvenote.com\n roles:\n - Investigation\n - Project administration\n - Software\n - Visualization\n affiliations:\n - Curvenote\n - name: Rowan Cockett\n orcid: 0000-0002-7859-8394\n corresponding: false\n roles: []\n affiliations:\n - Curvenote\nkeywords:\n - La Palma\n - Earthquakes\nabstract: |\n In September 2021, a significant jump in seismic activity on the island of La Palma (Canary Islands, Spain) signaled the start of a volcanic crisis that still continues at the time of writing. Earthquake data is continually collected and published by the Instituto Geográphico Nacional (IGN). ...\nplain-language-summary: |\n Earthquake data for the island of La Palma from the September 2021 eruption is found ...\nkey-points:\n - A web scraping script was developed to pull data from the Instituto Geogràphico Nacional into a machine-readable form for analysis\n - Earthquake events on La Palma are consistent with the presence of both mantle and crustal reservoirs.\ndate: last-modified\nbibliography: references.bib\ncitation:\n container-title: Earth and Space Science\nnumber-sections: true\n---\n\n:::{#67a3a4e8 .cell .markdown}\n## Introduction\n:::\n\n::: {#c69ee907 .cell execution_count=1}\n``` {.python .cell-code .hidden}\nimport matplotlib.pyplot as plt\nimport numpy as np\neruptions = [1492, 1585, 1646, 1677, 1712, 1949, 1971, 2021]\n```\n:::\n\n\n::: {#cell-fig-timeline .cell execution_count=2}\n``` {.python .cell-code .hidden}\nplt.figure(figsize=(6, 1))\nplt.eventplot(eruptions, lineoffsets=0, linelengths=0.1, color='black')\nplt.gca().axes.get_yaxis().set_visible(False)\nplt.ylabel('')\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Timeline of recent earthquakes on La Palma](index_files/figure-jats/fig-timeline-output-1.png){#fig-timeline fig-alt='An event plot of the years of the last 8 eruptions on La Palma.'}\n:::\n:::\n\n\n::: {#b397d1c8 .cell execution_count=3}\n``` {.python .cell-code .hidden}\navg_years_between_eruptions = np.mean(np.diff(eruptions[:-1]))\navg_years_between_eruptions\n```\n\n::: {.cell-output .cell-output-display .hidden execution_count=9}\n```\n79.83333333333333\n```\n:::\n:::\n\n\n:::{#1962cec2 .cell .markdown}\nBased on data up to and including 1971, eruptions on La Palma happen every 79.8 years on average.\n\nStudies of the magma systems feeding the volcano, such as @marrero2019, have proposed that there are two main magma reservoirs feeding the Cumbre Vieja volcano; one in the mantle (30-40km depth) which charges and in turn feeds a shallower crustal reservoir (10-20km depth).\n\nEight eruptions have been recorded since the late 1400s (@fig-timeline).\n\nData and methods are discussed in @sec-data-methods.\n\nLet $x$ denote the number of eruptions in a year. Then, $x$ can be modeled by a Poisson distribution\n\n$$\np(x) = \\frac{e^{-\\lambda} \\lambda^{x}}{x !}\n$$ {#eq-poisson}\n\nwhere $\\lambda$ is the rate of eruptions per year. Using @eq-poisson, the probability of an eruption in the next $t$ years can be calculated.\n\n| Name | Year |\n|---------------------|------|\n| Current | 2021 |\n| Teneguía | 1971 |\n| Nambroque | 1949 |\n| El Charco | 1712 |\n| Volcán San Antonio | 1677 |\n| Volcán San Martin | 1646 |\n| Tajuya near El Paso | 1585 |\n| Montaña Quemada | 1492 |\n\n: Recent historic eruptions on La Palma {#tbl-history}\n\n@tbl-history summarises the eruptions recorded since the colonization of the islands by Europeans in the late 1400s.\n\n![Map of La Palma](images/la-palma-map.png){#fig-map}\n\nLa Palma is one of the west most islands in the Volcanic Archipelago of the Canary Islands (@fig-map).\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-spatial-plot >}}\n\n\n\n\n\n\n\n@fig-spatial-plot shows the location of recent Earthquakes on La Palma.\n\n## Data & Methods {#sec-data-methods}\n\n## Conclusion\n\n## References {.unnumbered}\n\n::: {#refs}\n:::\n:::\n\n", + "engine": "jupyter", + "markdown": "---\ntitle: Un estudio sobre los retornos del mercado de renta fija dominicano utilizando basados en modelos AR-GARCH\nsubtitle: Trabajo final de tópicos de econometría\nauthor:\n - name: Ian Contreras\n email: 1116048@intec.edu.do\n affiliations:\n - INSTITUTO TECNOLÓGICO DE SANTO DOMINGO (INTEC)\nkeywords:\n - Mercado de renta fija\n - Deuda pública\n - Modelos AR-GARCH\nabstract: |\n Este estudio tiene como objetivo analizar los retornos y la volatilidad del mercado de deuda pública dominicana mediante la aplicación de modelos ARMA-GARCH. A través de estos modelos, se busca identificar patrones en el comportamiento de los retornos y predecir su evolución futura. \ndate: last-modified\nbibliography: references.bib\nexecute:\n echo: false\nnocite: |\n @*\ncitation:\n container-title: Referencias\nnumber-sections: true\n---\n\n::: {#importing-libraries .cell execution_count=1}\n``` {.python .cell-code .hidden}\n# Manejo de datos y análisis\nimport numpy as np\nimport pandas as pd\nimport scipy.stats as stats\n\n# Modelos estadísticos y econométricos\nimport statsmodels.api as sm\nfrom pmdarima.arima import auto_arima\nfrom arch import arch_model\n```\n:::\n\n\n::: {#importing-data .cell execution_count=2}\n``` {.python .cell-code .hidden}\ndf = pd.read_csv('data\\csv\\irp.csv', parse_dates=['date'], index_col='date')\nreturns = df['price_return']\nsplit_date = '2020-12-31'\nR_test = df[df.index >= split_date]['price_return'].rolling(\n window=5).std().dropna()\n```\n:::\n\n\n:::{#0f672cb6 .cell .markdown}\n# Introducción\n\nEl mercado de renta fija en la República Dominicana ha emergido como uno de los sectores más importantes dentro de la economía, especialmente en lo que respecta a la deuda pública, la cual juega un papel crucial en la financiación de proyectos gubernamentales y el sostenimiento de la política fiscal. En los últimos cincuenta años, el país ha experimentado un crecimiento económico significativo, consolidándose como una de las economías más dinámicas de América Latina. Este crecimiento ha estado acompañado por una mayor complejidad económica, lo que ha incrementado la relevancia del mercado financiero dominicano, en particular el mercado de deuda pública, que ha sido un activo clave para fondos de pensiones y fondos de inversión.\n\nSin embargo, este mercado también presenta importantes desafíos, especialmente en términos de baja liquidez en el mercado secundario y episodios de alta volatilidad. Estos problemas dificultan la previsión precisa de los movimientos del mercado y complican la toma de decisiones estratégicas para los inversores. En este contexto, surge la necesidad de utilizar herramientas econométricas que puedan capturar con mayor precisión las dinámicas subyacentes de los retornos y la volatilidad del mercado de deuda pública dominicano.\n\nLa **pregunta de investigación** que guía este estudio es: ¿Es posible modelar adecuadamente los retornos y la volatilidad del mercado de deuda pública dominicano mediante modelos ARIMA-GARCH?, y ¿Qué tan efectivas son estas técnicas para predecir el comportamiento futuro del mercado, especialmente durante periodos de alta volatilidad?\n\nEl **objetivo principal** de este estudio es aplicar modelos ARMA-GARCH para analizar y predecir el comportamiento de los retornos del índice de deuda pública IRP-GOBIX, considerando tanto la tendencia como los patrones de volatilidad del mercado. A través de la estimación de estos modelos, se busca identificar los principales factores que afectan la dinámica de los retornos y proporcionar información valiosa para los actores del mercado, facilitando una gestión más eficiente del riesgo.\n\nAdemás, este estudio pretende evaluar la **consistencia de los modelos** y determinar si los modelos de volatilidad más sofisticados, como el Zero-GARCH, ofrecen una mejor capacidad de predicción que los modelos tradicionales basados únicamente en niveles y diferencias, como el ARIMA. Un aspecto clave será analizar si el modelo GARCH captura adecuadamente los choques bruscos y los episodios de volatilidad extrema, o si suaviza excesivamente estos movimientos.\n\nLa **relevancia** de este estudio radica en la creciente importancia del mercado de deuda pública dominicano, tanto para la estabilidad financiera del país como para los inversores extranjeros que buscan oportunidades de inversión en mercados emergentes. Comprender los patrones de retorno y volatilidad es esencial para diseñar estrategias de inversión que minimicen el riesgo y maximicen el rendimiento en un entorno económico globalizado y altamente incierto.\n\n# Metodología\n\nEl modelo ARIMA(p,d,q), donde AR(p) es el componente autorregresivo de orden p, d denota el orden de la diferencia aplicada para transformar una serie temporal no estacionaria en estacionaria, y MA(q) representa el promedio móvil de orden q. La esencia del modelo ARIMA radica en la combinación de la operación de diferenciación y el modelo ARMA. Cualquier serie no estacionaria puede volverse estacionaria mediante una diferenciación de orden adecuado, permitiendo así ajustar un modelo ARMA a la serie transformada.\n\nEl modelo de promedio móvil autorregresivo ARMA(p,q) incluye tanto un componente autorregresivo de orden p como uno de promedio móvil de orden q, y se estructura de la siguiente forma:\n\n$$\nX_t = \\phi_1 X_{t-1} + \\phi_2 X_{t-2} + \\dots + \\phi_p X_{t-p} + \\epsilon_t + \\theta_1 \\epsilon_{t-1} + \\theta_2 \\epsilon_{t-2} + \\dots + \\theta_q \\epsilon_{t-q}\n$$ {eq: arma}\n\nEstos modelos son adecuados solo para describir series temporales suaves, mientras que en la práctica, a menudo se trabaja con series temporales no suaves. Para estos casos, aplicar una o dos diferenciaciones puede transformar los datos en series suaves.\n\nSi no se cumple el supuesto de homogeneidad de la varianza, puede ocurrir heterocedasticidad. Dado que los datos de la muestra son suaves, en este documento se establece un modelo ARMA(p,q) para la serie de rendimientos con el fin de describir las características de volatilidad del índice GOBIX. Se realizan pruebas de efecto ARCH y se resuelve el problema de la heterocedasticidad mediante un modelo GARCH.\n\nEl modelo GARCH extiende el modelo ARCH al considerar la autocorrelación de orden p en la función de heterocedasticidad, lo que permite ajustar de manera efectiva una función de heterocedasticidad con memoria a largo plazo. El modelo GARCH se define de la siguiente manera:\n\n$$\na_t = \\sigma_t \\epsilon_t, \\quad \\sigma_t^2 = \\alpha_0 + \\sum_{i=1}^{p} \\alpha_i a_{t-i}^2 + \\sum_{j=1}^{q} \\beta_j \\sigma_{t-j}^2\n$$ {#eq: garch}\n\nEl modelo GARCH, propuesto por Bollerslev, atribuye la volatilidad actual tanto a la volatilidad de momentos pasados como a los errores de momentos pasados, lo que le permite explicar el fenómeno de agrupación en la volatilidad de los rendimientos financieros.\n\n# Datos\n\n## Fuente de datos\nUn Índice Financiero es una medida estadística que refleja el valor de un conjunto de activos financieros agrupados de acuerdo con ciertos criterios.Estos se utilizan como objetivo de rendimiento de portafolios, referencias sobre las características retorno/riesgo de una clase de activo y como referencia para productos anclados a índices.\n\nEn el mercado de renta fija de la República Dominicana, el único índice público es el GOBIX, que representa la deuda gubernamental consolidada en peso. Este índice está compuesto por títulos emitidos localmente por el Banco Central y el Ministerio de Hacienda, y se publica en dos modalidades: índice de precio limpio e índice de retorno-precio. Para este estudio, se utilizará el índice de retorno precio @sec-retorno-precio, ya que aisla el componente de retorno de mercado el cual es el objetivo de estudio. \n\nEl GOBIX se caracteriza por los siguientes criterios:\n\n- **Elegibilidad**: Solo se incluyen títulos \"bullet\" emitidos en pesos dominicanos con al menos 60 días desde su emisión.\n- **Liquidez**: Los títulos se seleccionan en función del Índice de Bursatilidad publicado por la Proveedora de Precios, garantizando la inclusión de los instrumentos más negociados.\n- **Ponderación**: La ponderación se realiza por capitalización de mercado, lo que asegura que los títulos más relevantes tengan mayor peso en el índice.\n- **Rebalanceo**: Se efectúa un rebalanceo mensual para ajustar el índice según la disponibilidad de inversión en los bonos.\n- **Fuentes de información**: Los datos provienen de la Bolsa de Valores de la República Dominicana y la Proveedora de Precios.\n\n## Índice de Retorno-Precio {#sec-retorno-precio}\n\nEl índice de retorno-precio es una metodología utilizada para calcular el rendimiento de un índice de renta fija, centrándose exclusivamente en las variaciones del precio limpio de los títulos, sin tener en cuenta los intereses acumulados ni el pago de cupones durante el período de cálculo. En el contexto del GOBIX, este índice refleja las fluctuaciones en los precios de los bonos dominicanos debido a las variaciones en las tasas de interés del mercado.\n\n**Características del Índice de Retorno-Precio del GOBIX**:\n\n- **Enfoque en el Precio Limpio**: El cálculo del índice se basa en el precio limpio de los bonos, excluyendo los intereses acumulados.\n- **Reflejo de la Volatilidad**: Este índice es un indicador del riesgo asociado a las fluctuaciones en las tasas de interés y su impacto en el capital invertido.\n- **Fórmula de Cálculo**:\n\n $$\n IRP_t = IRP_{m-1} \\times (1 + RPI_t)\n $$\n\n Donde:\n - $IRP_0 = 100$ es el valor base del índice en la fecha de inicio.\n - $IRP_t$ es el valor del índice en el día $t$.\n - $IRP_{m-1}$ es el valor del índice en el último día hábil del mes anterior.\n - $RPI_t$ es el retorno-precio del índice en $t$.\n\nEl retorno-precio del índice ($RPI_t$) se calcula como la suma ponderada del retorno-precio de cada bono ($RPI_{i,t}$), donde la ponderación ($\\omega_i$) se determina según la capitalización de mercado de cada bono.\n\nEn este análisis, expresaremos la serie de retornos en puntos básicos para evitar problemas de escalamiento de los datos y posibles desbordamientos decimales (overflow decimal), que pueden ocurrir cuando se manejan fluctuaciones muy pequeñas con alta precisión. Un punto básico equivale a 0.01% o 1/100 de un porcentaje. Este enfoque nos permitirá manejar los datos con mayor estabilidad y evitar posibles errores numéricos derivados del escalamiento inapropiado. No tiene efectos estadísticos en las distribución. \n\n## Selección de muestra de entrenamiento/prueba {#sec-test-train}\n\nDurante la pandemia, se experimentó uno de los ciclos de tasas de interés más pronunciados de las últimas décadas. Este periodo presentó características estructurales distintas en comparación con los ciclos históricos previos. Para evitar sesgos en las estimaciones debido a la alta volatilidad observada durante ese tiempo, evaluaremos únicamente la serie de retornos entre el periodo 2014-2021, que se considera un ciclo de tasas más regular.\n\nLa metodología empleada para dividir el conjunto de datos en muestras de entrenamiento y prueba será el **Time Series Cross-Validator**. A diferencia de la validación cruzada tradicional, que aleatoriza los datos, esta técnica respeta el orden temporal, lo cual es crucial en el análisis de series de tiempo.\n\nEn cada partición (k-ésimo split), la técnica funciona de la siguiente manera:\n\n- **Conjunto de Entrenamiento**: Incluye los primeros *k* pliegues (folds).\n- **Conjunto de Prueba**: Incluye el pliegue *(k+1)*.\n\n# Análisis Exploratorio {#sec-eda}\n\nEl objetivo de este análisis exploratorio es examinar la serie de retornos diarios del IRP-GOBIX para identificar patrones de volatilidad, evaluar la distribución de los retornos, y explorar la autocorrelación en los datos. Utilizando herramientas estadísticas como la distribución de los retornos, el test de estacionariedad Dickey-Fuller, y los gráficos de autocorrelación (ACF) y autocorrelación parcial (PACF), se busca comprender la dinámica temporal de los retornos y validar la aplicación de modelos ARMA-GARCH para capturar su comportamiento futuro. Adicionalmente, se ajustarán diversas distribuciones teóricas, como la distribución normal y la t de Student, para evaluar su capacidad de describir las características empíricas de los datos.\n\n## Histórico de retornos precio del mercado de deuda pública dominicano {#sec-historico-retornos}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-price-return-series >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis de la serie de retornos diarios del IRP-GOBIX revela un comportamiento interesante, con periodos prolongados de baja volatilidad seguidos por episodios de alta volatilidad. Entre 2015 y 2022, los retornos se mantuvieron generalmente dentro de un rango de ±50 puntos básicos, lo que indica una estabilidad relativa. Sin embargo, en momentos clave, como el ciclo de tasas de 2018, la volatilidad se incrementó drásticamente, alcanzando picos de hasta 200 puntos básicos. Este patrón es característico del fenómeno conocido como **\"volatility clustering\"**, donde periodos tranquilos son seguidos por fases de mayor volatilidad. La naturaleza de estos picos parece estar relacionada con factores externos, como cambios en las tasas de interés y la incertidumbre macroeconómica global, lo que subraya la importancia de emplear modelos ARMA-GARCH para capturar estas dinámicas y prever comportamientos futuros en el mercado de deuda pública dominicana.\n\n## Análisis descriptivo de la muestra {#sec-estadistica-descriptiva}\n:::\n\n::: {#cell-return-descriptive-stats .cell execution_count=3}\n``` {.python .cell-code .hidden}\ndesc_stats = returns.describe()\n\nskewness = returns.skew()\nkurtosis = returns.kurtosis()\njb_test = sm.stats.jarque_bera(returns)\n\ndescriptive_table = pd.DataFrame({\n 'Observations': [int(desc_stats['count'])],\n 'Mean': [desc_stats['mean']],\n 'Median': [desc_stats['50%']],\n 'Std. Dev': [desc_stats['std']],\n 'Skewness': [skewness],\n 'Kurtosis': [kurtosis],\n 'Jarque-Bera': [jb_test[0]],\n 'Prob.': [jb_test[1]]\n})\ndescriptive_table\n```\n\n::: {#return-descriptive-stats .cell-output .cell-output-display execution_count=15}\n```{=html}\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
ObservationsMeanMedianStd. DevSkewnessKurtosisJarque-BeraProb.
019411.353251-0.31804423.5029110.705277.5879954789.5323310.0
\n
\n```\n\nTabla de las estadísticas descriptiva de la serie de retornos\n:::\n:::\n\n\n:::{#6e93a8dc .cell .markdown}\nEl análisis descriptivo de los retornos ofrece una perspectiva más detallada sobre la distribución de la serie. Con un total de 1,941 observaciones, la **media** de los retornos es de $1.353251$, lo que indica un rendimiento promedio positivo. Sin embargo, la **mediana** de $-0.318044$ sugiere un leve sesgo hacia valores negativos, lo que refleja que la mayoría de los retornos tienden a ser ligeramente inferiores a la media. La **desviación estándar**, con un valor de $23.502911$, revela una amplia dispersión de los datos, lo que confirma la presencia de una volatilidad significativa en los rendimientos, muy superior al valor medio.\n\nEn cuanto a la **asimetría** ($0.70527$), se observa una ligera inclinación hacia valores extremos positivos, lo que implica la existencia de eventos fuera de lo común que afectan los rendimientos. La **kurtosis** de $7.587995$ indica una distribución **leptocúrtica**, caracterizada por una alta concentración de valores alrededor de la media y colas más gruesas que una distribución normal, lo que es típico en datos financieros que presentan eventos extremos. El **test de Jarque-Bera**, con un valor de $4,789.532331$, confirma que la serie no sigue una distribución normal, validando la presencia de asimetría y colas pesadas en los datos. Estos resultados resaltan la necesidad de emplear modelos que puedan capturar adecuadamente estos comportamientos no lineales, esenciales para el análisis del mercado.\n\n## Análisis de la distribución de los retornos {#sec-distribucion-retornos}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-distribution-fitting >}}\n\n\n\n\n\n\n\n\n\n\nAl ajustar diferentes distribuciones a los retornos, observamos que la distribución empírica (en azul) tiene una alta concentración en torno a cero, con colas más gruesas de lo que se esperaría bajo una distribución normal. Entre las distribuciones probadas, la t de Student (en rojo) es la que mejor captura los valores extremos, con colas más largas y una mayor concentración en el centro, lo que es típico en series financieras que experimentan episodios de alta volatilidad. En contraste, la distribución normal (en naranja) y la lognormal (en verde) subestiman las colas, demostrando su ineficacia para modelar adecuadamente los valores atípicos. Esto refuerza la idea de que la t de Student es una mejor candidata para modelar los retornos, dado que puede ajustarse mejor a la leptocurtosis observada en los datos.\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-tqq-plot >}}\n\n\n\n\n\n\n\n\n\n\nLa gráfica Q-Q (Quantile-Quantile) compara los cuantiles teóricos de la distribución t de Student con los cuantiles observados de los retornos del GOBIX. En general, la mayoría de los puntos se alinean bien con la línea roja, lo que sugiere que los retornos de precios siguen razonablemente esta distribución, particularmente en las partes centrales. Sin embargo, en las colas extremas, se observan algunas desviaciones, lo que indica que, aunque la t de Student es un buen ajuste, no es perfecta para todos los escenarios.\n\nEl **estadístico de Kolmogorov-Smirnov** es de $0.0286$, con un **p-valor** de $0.0827$. Aunque esto sugiere que la t de Student captura bien la mayor parte de los retornos, las discrepancias en las colas extremas muestran que la distribución teórica no es un ajuste exacto a los datos reales.\n\n## Evaluación de supuestos para análisis AR-GARCH\n\n### Test de estacionariedad {#sec-estacionariedad}\n\nPara garantizar la validez de los modelos ARMA y GARCH, se evaluó la estacionariedad de la serie de retornos utilizando el test de Dickey-Fuller aumentado (ADF). Los resultados obtenidos muestran un **ADF Statistic** de $-13.1741$ y un **p-valor** de $1.2333 \\times 10^{-24}$, lo que permite rechazar la hipótesis nula de raíz unitaria con un nivel de significancia del 5%. Esto confirma que la serie es estacionaria, lo que significa que sus fluctuaciones se distribuyen alrededor de una media constante en el tiempo, sin tendencia significativa. Este hallazgo es consistente con el comportamiento típico de las series de retornos financieros, lo que habilita la correcta estimación de modelos ARMA-GARCH para capturar las dinámicas del mercado.\n\n### Test de autocorrelación {#sec-autocorrelacion}\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-acf-pacf >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis de autocorrelación nos proporciona más detalles sobre la estructura interna de la serie. El gráfico de la **Función de Autocorrelación (ACF)** muestra un pico significativo en el primer rezago, seguido de valores cercanos a cero para los rezagos posteriores. Este patrón es indicativo de un proceso de media móvil de primer orden (**MA(1)**), donde los choques aleatorios tienen un impacto significativo en el primer rezago pero no en los siguientes. La **Función de Autocorrelación Parcial (PACF)** refuerza esta conclusión, mostrando un comportamiento similar con un pico en el primer rezago y valores prácticamente nulos a partir del segundo rezago. Esto sugiere que un modelo MA(1) sería apropiado para capturar las dinámicas de corto plazo en la serie.\n\nEn conjunto, este análisis exploratorio sugiere que la serie de retornos del IRP-GOBIX presenta características complejas, como **volatility clustering**, alta **leptocurtosis**, y una estructura de autocorrelación que se ajusta bien a un modelo MA(1). Estos hallazgos proporcionan una base sólida para la estimación de modelos ARMA-GARCH, los cuales podrán capturar de manera efectiva las dinámicas de volatilidad y retornos en el mercado de deuda pública dominicano.\n\n# Resultados\n\nEl presente análisis tiene como objetivo modelar la serie temporal de retornos del índice IRP-GOBIX utilizando inicialmente un modelo ARIMA automático (auto-ARIMA) para determinar si es posible capturar la dinámica de los retornos con un modelo basado únicamente en niveles y diferencias de los datos. Adicionalmente, se evaluará la presencia de problemas estructurales, como la heterocedasticidad, que podrían afectar la consistencia del modelo ARIMA. En caso de que este enfoque no sea adecuado, recurriremos a un modelo Zero-GARCH, un tipo específico de GARCH que asume una media cero para los retornos. Este modelo es particularmente útil en mercados de tasas de interés, donde se ha demostrado que los retornos tienden a cero en el largo plazo debido a la fuerte regresión hacia la media de estas variables. \n\n## Análisis del modelo ARIMA\n:::\n\n::: {#cell-arima-model .cell execution_count=4}\n``` {.python .cell-code .hidden}\nmodel_auto = auto_arima(returns)\nmodel_auto.summary()\n```\n\n::: {#arima-model .cell-output .cell-output-display execution_count=16}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:48:37 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n```\n\nModelo ARIMA de maxima verosimilitud para la serie de retornos.\n:::\n:::\n\n\n:::{#2badc302 .cell .markdown}\nEl modelo SARIMAX(3, 0, 4) se estimó con el objetivo de capturar la dinámica subyacente de la serie de retornos. Aunque los resultados muestran que algunos de los coeficientes del componente autorregresivo (AR) y del promedio móvil (MA) son estadísticamente significativos (e.g., $AR(3)$ con un coeficiente de $0.7691$, $p < 0.001$ y $MA(2)$ con $p < 0.01$), el desempeño general del modelo presenta ciertas limitaciones. El test de heterocedasticidad muestra un valor de 1.11, lo que indica la presencia de heterocedasticidad en la serie, un problema común en series financieras, donde la varianza de los errores no es constante a lo largo del tiempo.\n\nAdemás, el estadístico de Jarque-Bera de 4395.18, con una probabilidad de $p = 0.00$, confirma que los residuos no siguen una distribución normal, lo que refuerza la idea de que un modelo basado únicamente en niveles, como el ARIMA, podría no ser suficiente para capturar la volatilidad inherente a los retornos del mercado de deuda pública dominicano. Dado que los modelos ARIMA no están diseñados para abordar adecuadamente la heterocedasticidad, pasamos a estimar un modelo GARCH, que es más apropiado para capturar los cambios en la volatilidad.\n\n## Modelo Zero-GARCH\n\nPara abordar las limitaciones del modelo ARIMA, se estimó un modelo GARCH(1,1) con media cero, siguiendo la estructura propuesta por [@miah_rahman_2016], quienes demostrarón que los modelos GARCH(1,1) son altamente efectivos para capturar la volatilidad en los retornos de mercados financieros. La elección de un modelo con media cero se justifica porque en mercados de deuda pública y tasas de interés, los retornos tienden a revertir a la media, y en el largo plazo se espera que los retornos por apreciación de capital converjan a cero. [@fabozzi_fixed_income]\n:::\n\n::: {#garch-model-fitting .cell execution_count=5}\n``` {.python .cell-code .hidden}\nar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\nres = ar.fit(last_obs=split_date)\n```\n\n::: {.cell-output .cell-output-stdout .hidden}\n```\nIteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\nIteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\nIteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\nIteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\nIteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\nIteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\nIteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\nIteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\nIteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\nIteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\nIteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\nIteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\nIteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\nIteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\nIteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\nIteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\nIteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\nIteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\nIteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\nIteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\nIteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\nIteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\nIteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\nIteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\nIteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\nIteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\nOptimization terminated successfully (Exit mode 0)\n Current function value: 7622.868827051428\n Iterations: 26\n Function evaluations: 139\n Gradient evaluations: 26\n```\n:::\n:::\n\n\n::: {#cell-garch-model .cell execution_count=6}\n``` {.python .cell-code .hidden}\nres.summary()\n```\n\n::: {#garch-model .cell-output .cell-output-display execution_count=18}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:48:37 Df Model: 0
\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
\n\n\n\n \n\n\n \n\n
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust\n```\n\nModelo Zero-Garch de la serie de retornos\n:::\n:::\n\n\n:::{#e8cc15d4 .cell .markdown}\nLos resultados del modelo Zero-GARCH confirman un buen ajuste a la volatilidad de la serie de retornos. El coeficiente $\\omega$ del proceso de volatilidad es de $58.8679$ ($p < 0.05$), lo que indica un nivel base significativo de volatilidad en la serie. El parámetro $\\alpha_1$, que mide la influencia de los shocks pasados en la volatilidad actual, es positivo y significativo ($0.1892$, $p < 0.01$), lo que sugiere que los shocks pasados tienen un impacto considerable en la volatilidad presente. Por otro lado, $\\beta_1$ ($0.7737$, $p < 0.001$) indica que existe una fuerte persistencia en la volatilidad, característica común en los mercados financieros, donde las fases de alta volatilidad tienden a durar varios periodos.\n\nEl uso de la distribución t de Student para los residuos estandarizados, con un parámetro $\\nu$ de $2.7894$ ($p < 0.001$), confirma la presencia de colas más gruesas en la distribución de los retornos, lo que es consistente con la leptocurtosis observada en la serie de retornos.\n\n### Residuos del Zero-GARCH\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-garch-residuals >}}\n\n\n\n\n\n\n\n\n\n\nAl analizar los residuos estandarizados y la volatilidad condicional en el modelo Zero-GARCH, observamos cómo la volatilidad responde de manera dinámica a los choques en los retornos. En la primera gráfica, se aprecia que la volatilidad condicional es mayor durante episodios donde los residuos estandarizados alcanzan picos extremos, especialmente en periodos de alta volatilidad como 2018. Esto refuerza la capacidad del modelo GARCH para capturar **volatility clustering**, un fenómeno donde la alta volatilidad tiende a agruparse en ciertos periodos. Además, la persistencia de la volatilidad condicional a lo largo del tiempo confirma que los choques pasados tienen un efecto prolongado, lo cual es coherente con los resultados obtenidos en los coeficientes del modelo GARCH.\n\n### Residuos contra volatilidad condicional\n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-residuals-vs-volatility >}}\n\n\n\n\n\n\n\n\n\n\nEn cuanto a la comparación entre los residuos estandarizados y los residuos de varianza unitaria, la segunda gráfica revela que ambos conjuntos de residuos presentan distribuciones similares, con una alta concentración alrededor de cero. Sin embargo, las colas más gruesas en los residuos estandarizados indican la presencia de eventos extremos más pronunciados, los cuales son típicos en series financieras que presentan leptocurtosis. Esta comparación subraya la importancia de utilizar distribuciones robustas, como la t de Student, para capturar adecuadamente las colas de la distribución, tal como lo hace el modelo GARCH en este caso.\n\nLos resultados del modelo Zero-GARCH confirman su capacidad para modelar de manera efectiva la volatilidad en los retornos del IRP-GOBIX. Aunque el modelo auto-ARIMA capturó algunos patrones de la serie, su inconsistencia en la presencia de heterocedasticidad subraya la necesidad de un enfoque basado en modelos GARCH. El modelo Zero-GARCH no solo demuestra un buen ajuste a los datos, sino que también captura la dinámica de la volatilidad condicional y los residuos estandarizados, revelando la presencia de **volatility clustering** y la persistencia en la volatilidad.\n\nLa comparación entre los residuos estandarizados y los residuos de varianza unitaria destaca la capacidad del modelo para manejar eventos extremos, una característica esencial en el análisis de series financieras. En resumen, el modelo Zero-GARCH es más robusto y adecuado para describir las dinámicas de volatilidad en el mercado de deuda pública dominicano, brindando información crucial sobre la persistencia y el comportamiento de la volatilidad a lo largo del tiempo.\n\n## Evaluación Retrospectiva (Backtesting)\n\nPara evaluar la capacidad predictiva del modelo Zero-GARCH, se realizó un análisis de backtesting sobre el periodo comprendido durante el año 2021. Se emplearon dos métodos principales de evaluación: el MAPE (Mean Absolute Percentage Error) y el análisis gráfico de los valores predichos frente a los valores actuales de la desviación estándar móvil. \n\n\n\n\n\n\n{{< embed notebooks/data-screening.qmd#fig-predicted-vs-actual-backtest >}}\n\n\n\n\n\n\n\n\n\n\nEl análisis gráfico (Predichos vs Actuales) revela que el modelo captura de manera excelente la **tendencia** general de la serie, lo que implica que el modelo es eficiente a la hora de predecir los movimientos de largo plazo en la volatilidad. Sin embargo, un problema importante que surge es que el modelo tiende a suavizar los movimientos bruscos, especialmente en momentos donde la volatilidad sufre caídas o picos repentinos. Este fenómeno de **sobre-suavización** implica que el modelo no es completamente reactivo a los choques regresivos de gran magnitud, lo que puede deberse a la estructura propia del GARCH, la cual tiende a modelar la volatilidad alrededor de la media del periodo. \n\nEl MAPE calculado para el conjunto de test arroja un valor elevado de **739.70%**, lo cual podría sugerir en un análisis superficial que el modelo no es adecuado. Sin embargo, es importante considerar el tipo de mercado con el que estamos trabajando, caracterizado por una volatilidad intrínsecamente elevada y movimientos extremos. En estos contextos, valores altos de MAPE son comunes debido a la alta variabilidad en los retornos, lo que no necesariamente implica una mala capacidad predictiva, sino que refleja la naturaleza errática de los datos financieros. En este sentido, el MAPE alto está más relacionado con la complejidad del mercado que con la falta de ajuste del modelo.\n\nEl backtesting del modelo Zero-GARCH sobre la serie de retornos del IRP-GOBIX para el periodo de 2021 confirma que, si bien el modelo captura correctamente la tendencia general de la volatilidad, su capacidad para reaccionar ante movimientos abruptos es limitada. A pesar de esto, dado el contexto del mercado de deuda pública dominicano, la sobre-suavización en los niveles es un comportamiento esperado en modelos de volatilidad como el GARCH. En resumen, el modelo es útil para predecir tendencias a largo plazo, pero su precisión en niveles podría mejorarse si se ajusta para reaccionar mejor a eventos extremos.\n\n# Discusión y Conclusión\n\n## Discusión\n\nEste estudio ha permitido modelar los retornos y la volatilidad del mercado de deuda pública dominicano utilizando modelos ARIMA-GARCH. A pesar de los hallazgos significativos, existen algunas limitaciones que deben considerarse al interpretar los resultados:\n\n- **Problemas con la sobre-suavización**: Como se observó en el análisis retrospectivo, el modelo Zero-GARCH tiende a suavizar los movimientos bruscos en la volatilidad, lo que puede limitar su capacidad de capturar adecuadamente choques repentinos en el mercado. Aunque el modelo refleja bien la tendencia general, su capacidad de reacción frente a eventos extremos debe ser mejorada.\n\n- **MAPE elevado**: El MAPE obtenido durante la evaluación retrospectiva fue considerablemente alto. Si bien esto puede justificarse por la naturaleza volátil del mercado de deuda pública, sugiere que el modelo puede tener problemas para ajustarse a los movimientos de corto plazo. La alta volatilidad y los choques repentinos no fueron capturados con precisión, afectando el rendimiento del modelo.\n\n- **Limitaciones en los datos**: El análisis se centró en un período específico (2014-2021), excluyendo la pandemia debido a sus características atípicas. Si bien esto ayudó a evitar sesgos, la exclusión de este período crítico puede haber omitido dinámicas importantes del mercado en tiempos de crisis, limitando la generalización de los resultados.\n\n## Conclusión\n\nEn conclusión, este estudio ha demostrado que el modelo ARIMA, aunque útil para capturar la estructura de los retornos del mercado de deuda pública dominicano, presenta limitaciones cuando se enfrenta a problemas de heterocedasticidad. El modelo Zero-GARCH, por su parte, ha demostrado ser más adecuado para describir la dinámica de la volatilidad, capturando fenómenos como la persistencia en la volatilidad y el clustering. Además, la capacidad del modelo para manejar eventos extremos a través de la distribución t de Student resulta valiosa en este tipo de mercados caracterizados por alta volatilidad y choques inesperados.\n\nEntre los principales hallazgos destacan:\n\n- **Volatility clustering**: El análisis confirmó la presencia de agrupación de volatilidad, donde periodos de baja volatilidad son seguidos por periodos de alta volatilidad, un fenómeno común en los mercados financieros.\n\n- **Persistencia de la volatilidad**: El coeficiente $\\beta_1$ significativo en el modelo GARCH sugiere que los shocks en la volatilidad tienen un efecto prolongado en el tiempo, lo que es consistente con los patrones observados en mercados financieros emergentes como el dominicano.\n\n- **Dificultad para capturar movimientos extremos**: Aunque el modelo Zero-GARCH es efectivo en capturar la tendencia general, su capacidad para reaccionar ante cambios bruscos en la volatilidad sigue siendo limitada, lo cual es un área de mejora para futuras investigaciones.\n\n- **Implicaciones para los inversores**: Los resultados ofrecen una herramienta útil para la predicción de tendencias de volatilidad a largo plazo en el mercado de deuda pública dominicano, proporcionando una base para la toma de decisiones estratégicas en la gestión del riesgo.\n\nEste estudio no solo contribuye al entendimiento del comportamiento de los retornos y la volatilidad en el mercado de deuda pública dominicano.\n\n# Referencias\n\n::: {#refs}\n:::\n:::\n\n", "supporting": [ - "index_files/figure-jats" + "index_files\\figure-jats" ], "filters": [] } diff --git a/_freeze/index/figure-docx/notebooks-data-screening-fig-acf-pacf-output-1.png b/_freeze/index/figure-docx/notebooks-data-screening-fig-acf-pacf-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..cdade3fc84c82149b4876fc28fe7da2798ddd12d GIT binary patch literal 35738 zcmdqK2UJzrmNk4WvlyNc1cWNZ1PBHYNm3>dQIMQLL_l)R$saYgF6h`;n3&s`7+(I_PS48P 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b/_freeze/notebooks/data-extract/execute-results/html.json @@ -1,10 +1,10 @@ { - "hash": "94c8977f205a7f9bfc4d718ceb2d9d25", + "hash": "40777f2cba5bca41531f7008934bd098", "result": { "engine": "jupyter", - "markdown": "---\ntitle: Data Extract\ndescription: Extracción, transformación y almacenamiento de las fuentes de datos para ser analizadas por Data-Screening\nauthors: Ian Contreras\n---\n\n::: {#importing-libraries .cell execution_count=1}\n``` {}\n#| label: importing-libraries\nimport pandas as pd\nimport os\n```\n:::\n\n\n::: {#irp .cell execution_count=2}\n``` {}\n#| label: irp\nprint(\"Current Working Directory:\", os.getcwd())\ndf = pd.read_excel(r\"..\\\\data\\\\inputs\\\\Historico del Indicador GOBIX DR.xlsx\",\n sheet_name=\"Indice Retorno Precio\")\ndf = df.rename(columns={'Fecha': 'date', 'Valor': 'irp'})\ndf['date'] = pd.to_datetime(df['date'])\ndf = df.set_index('date')\ndf[\"price_return\"] = df[\"irp\"].pct_change()\ndf = df.dropna()\ndf = df[df.index <= '2021-09-30']\ndf['price_return'] = df['price_return'] * 10_000\ndf.to_csv('..\\\\data\\\\csv\\\\irp.csv')\n```\n\n::: {.cell-output .cell-output-stdout}\n```\nCurrent Working Directory: C:\\Users\\contr\\OneDrive - INTEC\\TRI 9\\ECONOM\\fixed_income_garch\\notebooks\n```\n:::\n:::\n\n\n", + "markdown": "---\ntitle: Data Extract\ndescription: Extracción, transformación y almacenamiento de las fuentes de datos para ser analizadas por Data-Screening\nauthors: Ian Contreras\n---\n\n::: {#importing-libraries .cell execution_count=1}\n``` {}\n#| label: importing-libraries\nimport pandas as pd\nimport os\n```\n:::\n\n\n::: {#irp .cell execution_count=2}\n``` {}\n#| label: irp\nprint(\"Current Working Directory:\", os.getcwd())\ndf = pd.read_excel(r\"..\\\\data\\\\inputs\\\\historico_gobix_dr.xlsx\",\n sheet_name=\"Indice Retorno Precio\")\ndf = df.rename(columns={'Fecha': 'date', 'Valor': 'irp'})\ndf['date'] = pd.to_datetime(df['date'])\ndf = df.set_index('date')\ndf[\"price_return\"] = df[\"irp\"].pct_change()\ndf = df.dropna()\ndf = df[df.index <= '2021-09-30']\ndf['price_return'] = df['price_return'] * 10_000\ndf.to_csv('..\\\\data\\\\csv\\\\irp.csv')\n```\n\n::: {.cell-output .cell-output-stdout}\n```\nCurrent Working Directory: C:\\Users\\contr\\OneDrive - INTEC\\TRI 9\\ECONOM\\fixed_income_garch\\notebooks\n```\n:::\n:::\n\n\n", "supporting": [ - "data-extract_files" + "data-extract_files\\figure-html" ], "filters": [], "includes": {} diff --git a/_freeze/notebooks/data-extract/execute-results/xml.json b/_freeze/notebooks/data-extract/execute-results/xml.json new file mode 100644 index 0000000..73dd7e1 --- /dev/null +++ b/_freeze/notebooks/data-extract/execute-results/xml.json @@ -0,0 +1,11 @@ +{ + "hash": "40777f2cba5bca41531f7008934bd098", + "result": { + "engine": "jupyter", + "markdown": "---\ntitle: Data Extract\ndescription: Extracción, transformación y almacenamiento de las fuentes de datos para ser analizadas por Data-Screening\nauthors: Ian Contreras\n---\n\n::: {#importing-libraries .cell execution_count=1}\n``` {}\n#| label: importing-libraries\nimport pandas as pd\nimport os\n```\n:::\n\n\n::: {#irp .cell execution_count=2}\n``` {}\n#| label: irp\nprint(\"Current Working Directory:\", os.getcwd())\ndf = pd.read_excel(r\"..\\\\data\\\\inputs\\\\historico_gobix_dr.xlsx\",\n sheet_name=\"Indice Retorno Precio\")\ndf = df.rename(columns={'Fecha': 'date', 'Valor': 'irp'})\ndf['date'] = pd.to_datetime(df['date'])\ndf = df.set_index('date')\ndf[\"price_return\"] = df[\"irp\"].pct_change()\ndf = df.dropna()\ndf = df[df.index <= '2021-09-30']\ndf['price_return'] = df['price_return'] * 10_000\ndf.to_csv('..\\\\data\\\\csv\\\\irp.csv')\n```\n\n::: {.cell-output .cell-output-stdout}\n```\nCurrent Working Directory: C:\\Users\\contr\\OneDrive - INTEC\\TRI 9\\ECONOM\\fixed_income_garch\\notebooks\n```\n:::\n:::\n\n\n", + "supporting": [ + "data-extract_files\\figure-jats" + ], + "filters": [] + } +} \ No newline at end of file diff --git a/_freeze/notebooks/data-screening/execute-results/xml.json b/_freeze/notebooks/data-screening/execute-results/xml.json index 6d4c893..3d69eef 100644 --- a/_freeze/notebooks/data-screening/execute-results/xml.json +++ b/_freeze/notebooks/data-screening/execute-results/xml.json @@ -1,9 +1,10 @@ { - "hash": "d87b6d9ac2228b736962871bfd9167c0", + "hash": "0859b7da7aa9039b87fc8a12d84f2b3c", "result": { - "markdown": "---\ntitle: Data Screening\ndescription: Screening and filtering data from the IGN catalog down to the relevant events for La Palma\nauthors:\n - name: Steve Purves\n orcid: 0000-0002-0760-5497\n - name: Charlotte Wickham\n note: Edited original notebook by Steve Purves\n---\n\n::: {#43daa9a5 .cell execution_count=1}\n``` {}\nimport pandas as pd\nimport matplotlib.pyplot as plt\n```\n:::\n\n\n:::{#25be6d6d .cell .markdown}\nImport `la-palma.csv` - a clean subset of [`catalogoComunSV_1663233588717.csv`](https://github.com/Notebooks-Now/submission-quarto-full/blob/main/data/catalogoComunSV_1663233588717.csv):\n:::\n\n::: {#7e7b2b9e .cell execution_count=2}\n``` {}\ndf = pd.read_csv('la-palma.csv')\ndf['DateTime'] = pd.to_datetime(df['DateTime'])\ndf = df.sort_values(by=['DateTime'], ascending=True)\ndf.head()\n```\n\n::: {.cell-output .cell-output-display execution_count=23}\n```{=html}\n

\n```\n:::\n:::\n\n\n::: {#cca50ca5 .cell execution_count=3}\n``` {}\ndf.describe()\n```\n\n::: {.cell-output .cell-output-display execution_count=24}\n```{=html}\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
LongitudeLatitudeDepth(km)Magnitude
count11347.00000011347.00000011347.00000011347.000000
mean-17.84406228.56889314.0893452.432423
std0.0273450.0218998.6247870.705282
min-17.99580028.3102000.0000000.200000
25%-17.85410028.55800010.1000001.900000
50%-17.83870028.56400011.3000002.600000
75%-17.82970028.57430013.9000002.900000
max-17.70570028.69780046.9000005.100000
\n
\n```\n:::\n:::\n\n\n::: {#3f3b8176 .cell execution_count=4}\n``` {}\ndf.plot.scatter(x=\"Longitude\", y=\"Latitude\", figsize=(12,12), grid=\"on\");\n```\n\n::: {.cell-output .cell-output-display}\n![](data-screening_files/figure-jats/cell-5-output-1.png){}\n:::\n:::\n\n\n:::{#319030b4 .cell .markdown}\n### Filter \n\nAlso some rows seem to be missing depth measurements, (see differences in the **count** for columns above) - take only rows with valid depths\n:::\n\n::: {#f02e8517 .cell execution_count=5}\n``` {}\ndf = df[df['Depth(km)'].notna()]\ndf.describe()\n```\n\n::: {.cell-output .cell-output-display execution_count=26}\n```{=html}\n
\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
LongitudeLatitudeDepth(km)Magnitude
count11347.00000011347.00000011347.00000011347.000000
mean-17.84406228.56889314.0893452.432423
std0.0273450.0218998.6247870.705282
min-17.99580028.3102000.0000000.200000
25%-17.85410028.55800010.1000001.900000
50%-17.83870028.56400011.3000002.600000
75%-17.82970028.57430013.9000002.900000
max-17.70570028.69780046.9000005.100000
\n
\n```\n:::\n:::\n\n\n::: {#f31daa87 .cell execution_count=6}\n``` {}\nplt.figure(figsize=(20,6))\nplt.subplot(1,3,1)\ndf[\"Latitude\"].plot.hist(bins=20, density=True)\nplt.subplot(1,3,2)\ndf[\"Longitude\"].plot.hist(bins=20, density=True)\nplt.subplot(1,3,3)\ndf[\"Depth(km)\"].plot.hist(bins=50, density=True)\n```\n\n::: {.cell-output .cell-output-display execution_count=27}\n```\n\n```\n:::\n\n::: {.cell-output .cell-output-display}\n![](data-screening_files/figure-jats/cell-7-output-2.png){}\n:::\n:::\n\n\n:::{#2e9530fb .cell .markdown}\n## Spatial Plot\n\nScatter plot the spatial locations of events\n:::\n\n::: {#cell-fig-spatial-plot .cell execution_count=7}\n``` {}\n#| label: fig-spatial-plot\n#| fig-cap: Locations of earthquakes on La Palma since 2017.\n#| fig-alt: A scatterplot of earthquake locations plotting latitude against longitude.\nfrom matplotlib import colormaps\ncmap = colormaps['viridis_r']\nax = df.plot.scatter(x=\"Longitude\", y=\"Latitude\", \n s=47-df[\"Depth(km)\"], c=df[\"Magnitude\"], \n figsize=(12,10), grid=\"on\", cmap=cmap)\ncolorbar = ax.collections[0].colorbar\ncolorbar.set_label(\"Magnitude\")\n\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Locations of earthquakes on La Palma since 2017.](data-screening_files/figure-jats/fig-spatial-plot-output-1.png){#fig-spatial-plot fig-alt='A scatterplot of earthquake locations plotting latitude against longitude.'}\n:::\n:::\n\n\n:::{#f0aaa739 .cell .markdown}\n### Timeline Plot\n\nScatter plot the event time series and look for any quantization issues. Have times & dates been loaded correctly?\n\n\nax = df.plot.scatter(x='DateTime', y='Depth(km)', figsize=(20,8))\nax.set_ylim(50,0);\n:::\n\n", + "engine": "jupyter", + "markdown": "---\ntitle: Data Screening\ndescription: Visualización y análisis estadístico del feature dataset\nauthors: Ian Contreras\n---\n\n\n\n\n\n\n\n\n\n\n## Importaciones\n\n### Librerías\n\n::: {#importing-libraries .cell execution_count=2}\n``` {}\n#| label: importing-libraries\n# Manejo de datos y análisis\nimport numpy as np\nimport pandas as pd\nimport scipy.stats as stats\n\n# Modelos estadísticos y econométricos\nimport statsmodels.api as sm\nfrom statsmodels.graphics.tsaplots import plot_acf, plot_pacf\nfrom statsmodels.tsa.stattools import adfuller\nfrom pmdarima.arima import auto_arima\nfrom arch import arch_model\n\n# Visualización\nimport matplotlib.pyplot as plt\n\n# Presentación en IPython\nfrom IPython.display import Markdown, HTML\n```\n:::\n\n\n:::{#e84ff242 .cell .markdown}\n### Data\n:::\n\n::: {#importing-data .cell execution_count=3}\n``` {}\n#| label: importing-data\ndf = pd.read_csv(r'..\\data\\csv\\irp.csv', parse_dates=[\n 'date'], index_col='date')\n# df = pd.read_csv(r'.\\data\\csv\\irp.csv', parse_dates=[\n# 'date'], index_col='date')\nreturns = df['price_return']\nsplit_date = '2020-12-31'\nR_test = df[df.index >= split_date]['price_return'].rolling(\n window=5).std().dropna()\n```\n:::\n\n\n:::{#47dbdaea .cell .markdown}\n## Análisis Exploratorio. \n\n### Serie Histórica\n:::\n\n::: {#cell-fig-price-return-series .cell execution_count=4}\n``` {}\n#| label: fig-price-return-series\n#| fig-cap: Serie de retornos diarios IRP-GOBIX.\nfig, ax = plt.subplots(figsize=(12, 10))\nax.plot(df.index, df['price_return'])\nax.set_ylabel('Retorno Precio (BPS)')\nax.set_xlim([df.index.min(), df.index.max()])\nax.legend()\nplt.show()\n```\n\n::: {.cell-output .cell-output-stderr .hidden}\n```\nNo artists with labels found to put in legend. Note that artists whose label start with an underscore are ignored when legend() is called with no argument.\n```\n:::\n\n::: {.cell-output .cell-output-display}\n![Serie de retornos diarios IRP-GOBIX.](data-screening_files/figure-jats/fig-price-return-series-output-2.png){#fig-price-return-series}\n:::\n:::\n\n\n:::{#29f7e77f .cell .markdown}\n### Análisis descriptivo de la muestra\n:::\n\n::: {#cell-return-descriptive-stats .cell execution_count=5}\n``` {}\n#| label: return-descriptive-stats\n#| fig-cap: Tabla de las estadísticas descriptiva de la serie de retornos\ndesc_stats = returns.describe()\n\nskewness = returns.skew()\nkurtosis = returns.kurtosis()\njb_test = sm.stats.jarque_bera(returns)\n\ndescriptive_table = pd.DataFrame({\n 'Observations': [int(desc_stats['count'])],\n 'Mean': [desc_stats['mean']],\n 'Median': [desc_stats['50%']],\n 'Std. Dev': [desc_stats['std']],\n 'Skewness': [skewness],\n 'Kurtosis': [kurtosis],\n 'Jarque-Bera': [jb_test[0]],\n 'Prob.': [jb_test[1]]\n})\nMarkdown(descriptive_table.to_markdown(index=False))\n```\n\n::: {#return-descriptive-stats .cell-output .cell-output-display .cell-output-markdown execution_count=10}\n| Observations | Mean | Median | Std. Dev | Skewness | Kurtosis | Jarque-Bera | Prob. |\n|---------------:|--------:|----------:|-----------:|-----------:|-----------:|--------------:|--------:|\n| 1941 | 1.35325 | -0.318044 | 23.5029 | 0.70527 | 7.588 | 4789.53 | 0 |\n\nTabla de las estadísticas descriptiva de la serie de retornos\n:::\n:::\n\n\n:::{#98395acd .cell .markdown}\n### Ajuste de distribuciones\n:::\n\n::: {#cell-fig-distribution-fitting .cell execution_count=6}\n``` {}\n#| label: fig-distribution-fitting\n#| fig-cap: Ajuste de distribuciones a los retornos de precio.\ndistributions = [stats.norm, stats.t]\n\nplt.figure(figsize=(10, 6))\nplt.hist(returns, bins=50, density=True, alpha=0.6, label='Resultados Empíricos')\n\nfor dist in distributions:\n params = dist.fit(returns)\n x = np.linspace(returns.min(), returns.max(), 100)\n plt.plot(x, dist.pdf(x, *params), label=f'{dist.name} fit')\n\nplt.xlabel('Retorno Precio (BPS)')\nplt.ylabel('Verosimilitud')\nplt.legend()\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Ajuste de distribuciones a los retornos de precio.](data-screening_files/figure-jats/fig-distribution-fitting-output-1.png){#fig-distribution-fitting}\n:::\n:::\n\n\n:::{#b52e104a .cell .markdown}\n### Q-Q plot con respecto a distribución t\n:::\n\n::: {#cell-fig-tqq-plot .cell execution_count=7}\n``` {}\n#| label: fig-tqq-plot\n#| fig-cap: Q-Q Plot de retornos de precio contra la distribución t.\nt_params = stats.t.fit(returns)\n\nfig, ax = plt.subplots(figsize=(12, 10))\nstats.probplot(returns, dist=\"t\", sparams=t_params, plot=plt)\nplt.grid(True)\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Q-Q Plot de retornos de precio contra la distribución t.](data-screening_files/figure-jats/fig-tqq-plot-output-1.png){#fig-tqq-plot}\n:::\n:::\n\n\n:::{#d730a61a .cell .markdown}\n### Kolmorov Smirnov test para distribución t\n:::\n\n::: {#kolmorov-smirnov .cell execution_count=8}\n``` {}\n#| label: kolmorov-smirnov\nks_stat, p_value = stats.kstest(returns, 't', args=t_params)\nprint(f\"Kolmogorov-Smirnov statistic: {ks_stat}\")\nprint(f\"P-value: {p_value}\")\n```\n\n::: {.cell-output .cell-output-stdout}\n```\nKolmogorov-Smirnov statistic: 0.028557323574938343\nP-value: 0.0827419967953622\n```\n:::\n:::\n\n\n:::{#91c6f81d .cell .markdown}\n### Test de estacionariedad/ robustez\n:::\n\n::: {#adf .cell execution_count=9}\n``` {}\n#| label: adf\nresult = adfuller(returns)\nprint('ADF Statistic:', result[0])\nprint('p-value:', result[1])\n```\n\n::: {.cell-output .cell-output-stdout}\n```\nADF Statistic: -13.174114850736395\np-value: 1.2332976276203498e-24\n```\n:::\n:::\n\n\n:::{#7d50eeb6 .cell .markdown}\n### Test de autocorrelación\n:::\n\n::: {#cell-fig-acf-pacf .cell execution_count=10}\n``` {}\n#| label: fig-acf-pacf\n#| fig-cap: Función de autocorrelación (ACF) y función de autocorrelación parcial (PACF) de los retornos.\nfig, axes = plt.subplots(2, 1, figsize=(10, 8))\nplot_acf(returns, lags=24, ax=axes[0])\naxes[0].set_title('Función de Autocorrelación (ACF)')\n\nplot_pacf(returns, lags=12, ax=axes[1])\naxes[1].set_title('Función de Autocorrelación Parcial (PACF)')\n\nplt.tight_layout()\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Función de autocorrelación (ACF) y función de autocorrelación parcial (PACF) de los retornos.](data-screening_files/figure-jats/fig-acf-pacf-output-1.png){#fig-acf-pacf}\n:::\n:::\n\n\n:::{#e3ed5779 .cell .markdown}\n## Resultados \n### Estimación del modelo ARIMA \n:::\n\n::: {#cell-arima-model .cell execution_count=11}\n``` {}\n#| label: arima-model\n#| fig-cap: Modelo ARIMA de maxima verosimilitud para la serie de retornos.\nmodel_auto = auto_arima(returns)\nmodel_auto.summary()\n```\n\n::: {#arima-model .cell-output .cell-output-display execution_count=16}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
SARIMAX Results
Dep. Variable: y No. Observations: 1941
Model: SARIMAX(3, 0, 4) Log Likelihood -8853.517
Date: Sat, 19 Oct 2024 AIC 17725.034
Time: 15:10:18 BIC 17775.173
Sample: 0 HQIC 17743.472
- 1941
Covariance Type: opg
\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
coef std err z P>|z| [0.025 0.975]
intercept 0.3602 0.273 1.317 0.188 -0.176 0.896
ar.L1 0.1366 0.065 2.106 0.035 0.009 0.264
ar.L2 -0.1931 0.066 -2.934 0.003 -0.322 -0.064
ar.L3 0.7691 0.061 12.644 0.000 0.650 0.888
ma.L1 -0.1139 0.065 -1.752 0.080 -0.241 0.014
ma.L2 0.2163 0.067 3.220 0.001 0.085 0.348
ma.L3 -0.7032 0.062 -11.315 0.000 -0.825 -0.581
ma.L4 0.0750 0.020 3.715 0.000 0.035 0.115
sigma2 536.2183 8.358 64.154 0.000 519.836 552.600
\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 4395.18
Prob(Q): 1.00 Prob(JB): 0.00
Heteroskedasticity (H): 1.11 Skew: 0.59
Prob(H) (two-sided): 0.17 Kurtosis: 10.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).\n```\n\nModelo ARIMA de maxima verosimilitud para la serie de retornos.\n:::\n:::\n\n\n:::{#a6702049 .cell .markdown}\n### Estimación del AR-GARCH\n:::\n\n::: {#garch-model-fitting .cell execution_count=12}\n``` {}\n#| label: garch-model-fitting\n#| fig-cap: Ajuste del modelo Zero-Garch ala serie de retornos\nar = arch_model(returns, mean='Zero', vol='GARCH', dist='t')\nres = ar.fit(last_obs=split_date)\n```\n\n::: {.cell-output .cell-output-stdout}\n```\nIteration: 1, Func. Count: 6, Neg. LLF: 13422.240578344366\nIteration: 2, Func. Count: 13, Neg. LLF: 9134.230609110135\nIteration: 3, Func. Count: 19, Neg. LLF: 8666.557319151609\nIteration: 4, Func. Count: 25, Neg. LLF: 8791.275490419279\nIteration: 5, Func. Count: 31, Neg. LLF: 8082.081172585756\nIteration: 6, Func. Count: 37, Neg. LLF: 8739.332058159429\nIteration: 7, Func. Count: 43, Neg. LLF: 8658.860183258712\nIteration: 8, Func. Count: 49, Neg. LLF: 7638.6451038734085\nIteration: 9, Func. Count: 55, Neg. LLF: 7626.627815474433\nIteration: 10, Func. Count: 60, Neg. LLF: 7624.619361750198\nIteration: 11, Func. Count: 65, Neg. LLF: 7623.76815711753\nIteration: 12, Func. Count: 70, Neg. LLF: 7623.223103893486\nIteration: 13, Func. Count: 75, Neg. LLF: 7623.048366501091\nIteration: 14, Func. Count: 80, Neg. LLF: 7622.998678403758\nIteration: 15, Func. Count: 85, Neg. LLF: 7622.986214899395\nIteration: 16, Func. Count: 90, Neg. LLF: 7622.975788918075\nIteration: 17, Func. Count: 95, Neg. LLF: 7622.9611635250085\nIteration: 18, Func. Count: 100, Neg. LLF: 7622.937610163095\nIteration: 19, Func. Count: 105, Neg. LLF: 7622.908361612077\nIteration: 20, Func. Count: 110, Neg. LLF: 7622.881904359898\nIteration: 21, Func. Count: 115, Neg. LLF: 7622.870928301752\nIteration: 22, Func. Count: 120, Neg. LLF: 7622.868967833485\nIteration: 23, Func. Count: 125, Neg. LLF: 7622.868843481545\nIteration: 24, Func. Count: 130, Neg. LLF: 7622.868828856013\nIteration: 25, Func. Count: 135, Neg. LLF: 7622.868827051428\nIteration: 26, Func. Count: 139, Neg. LLF: 7622.868827051394\nOptimization terminated successfully (Exit mode 0)\n Current function value: 7622.868827051428\n Iterations: 26\n Function evaluations: 139\n Gradient evaluations: 26\n```\n:::\n:::\n\n\n:::{#4147ef24 .cell .markdown}\n### Resultados del AR-GARCH\n:::\n\n::: {#cell-garch-model .cell execution_count=13}\n``` {}\n#| label: garch-model\n#| fig-cap: Modelo Zero-Garch de la serie de retornos\nHTML(res.summary().as_html())\n```\n\n::: {#garch-model .cell-output .cell-output-display execution_count=18}\n```{=html}\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n\n \n\n
Zero Mean - GARCH Model Results
Dep. Variable: price_return R-squared: 0.000
Mean Model: Zero Mean Adj. R-squared: 0.001
Vol Model: GARCH Log-Likelihood: -7622.87
Distribution: Standardized Student's t AIC: 15253.7
Method: Maximum Likelihood BIC: 15275.6
No. Observations: 1753
Date: Sat, Oct 19 2024 Df Residuals: 1753
Time: 15:10:18 Df Model: 0
\n\n\n\n \n\n\n \n\n\n \n\n\n \n\n
Volatility Model
coef std err t P>|t| 95.0% Conf. Int.
omega 58.8679 25.182 2.338 1.940e-02 [ 9.512,1.082e+02]
alpha[1] 0.1892 6.657e-02 2.842 4.482e-03 [5.873e-02, 0.320]
beta[1] 0.7737 6.832e-02 11.324 9.943e-30 [ 0.640, 0.908]
\n\n\n\n \n\n\n \n\n
Distribution
coef std err t P>|t| 95.0% Conf. Int.
nu 2.7894 0.230 12.114 8.853e-34 [ 2.338, 3.241]


Covariance estimator: robust\n```\n\nModelo Zero-Garch de la serie de retornos\n:::\n:::\n\n\n:::{#1e6cbbd7 .cell .markdown}\n### Residuos del AR-GARCH\n:::\n\n::: {#cell-fig-garch-residuals .cell execution_count=14}\n``` {}\n#| label: fig-garch-residuals\n#| fig-cap: Residuos del modelo AR-GARCH.\nfig = res.plot()\n```\n\n::: {.cell-output .cell-output-display}\n![Residuos del modelo AR-GARCH.](data-screening_files/figure-jats/fig-garch-residuals-output-1.png){#fig-garch-residuals}\n:::\n:::\n\n\n:::{#5b5ebbe1 .cell .markdown}\n### Residuos contra volatilidad condicional\n:::\n\n::: {#cell-fig-residuals-vs-volatility .cell execution_count=15}\n``` {}\n#| label: fig-residuals-vs-volatility\n#| fig-cap: Residuos contra la volatilidad condicional del AR-GARCH.\nstd_resid = res.resid / res.conditional_volatility\nunit_var_resid = res.resid / res.resid.std()\ndf = pd.concat([std_resid, unit_var_resid], axis=1)\ndf.columns = [\"Residuos Estandarizados\", \"Residuos de Varianza Unitaria\"]\nsubplot = df.plot(kind=\"kde\", xlim=(-4, 4))\n```\n\n::: {.cell-output .cell-output-display}\n![Residuos contra la volatilidad condicional del AR-GARCH.](data-screening_files/figure-jats/fig-residuals-vs-volatility-output-1.png){#fig-residuals-vs-volatility}\n:::\n:::\n\n\n:::{#b8442420 .cell .markdown}\n## Evaluación Retrospectiva (Backtesting)\n\n### Conversion de varianza predicha. \n:::\n\n::: {#forecast .cell execution_count=16}\n``` {}\n#| label: forecast\nforecasts = res.forecast(horizon=1)\nforecasted_std = np.sqrt(forecasts.variance)\nforecasted_std = forecasted_std[forecasted_std.index >\n split_date].loc[R_test.index, 'h.1']\n```\n:::\n\n\n:::{#eb1363fd .cell .markdown}\n### MAPE del modelo\n:::\n\n::: {#mape .cell execution_count=17}\n``` {}\n#| label: mape\ndef MAPE(actual, predicted):\n actual_std = actual.rolling(window=5).std().dropna()\n return np.mean(np.abs((actual_std - predicted) / actual_std)) * 100\n\nmape_value = MAPE(R_test, forecasted_std)\nprint(f\"MAPE on Test Set (based on moving std): {mape_value:.2f}%\")\n```\n\n::: {.cell-output .cell-output-stdout}\n```\nMAPE on Test Set (based on moving std): 739.70%\n```\n:::\n:::\n\n\n:::{#b313f2e3 .cell .markdown}\n### Predichos vs Actuales Plot-Backtested\n:::\n\n::: {#cell-fig-predicted-vs-actual-backtest .cell execution_count=18}\n``` {}\n#| label: fig-predicted-vs-actual-backtest\n#| fig-cap: Desviación Estándar Móvil Real vs Pronosticada - Modelo Zero-GARCH\nfig, ax = plt.subplots(figsize=(10, 6))\nax.plot(R_test.index, R_test,\n label='Desviación Estándar Móvil Real', color='blue')\nax.plot(forecasted_std.index,\n forecasted_std, label='Desviación Estándar Móvil Pronosticada', color='red', linestyle='--')\nax.set_xlim([forecasted_std.index.min(), forecasted_std.index.max()])\nplt.legend()\nplt.show()\n```\n\n::: {.cell-output .cell-output-display}\n![Desviación Estándar Móvil Real vs Pronosticada - Modelo Zero-GARCH](data-screening_files/figure-jats/fig-predicted-vs-actual-backtest-output-1.png){#fig-predicted-vs-actual-backtest}\n:::\n:::\n\n\n", "supporting": [ - "data-screening_files/figure-jats" + "data-screening_files\\figure-jats" ], "filters": [] } diff --git a/_freeze/notebooks/data-screening/figure-jats/fig-acf-pacf-output-1.png b/_freeze/notebooks/data-screening/figure-jats/fig-acf-pacf-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..cdade3fc84c82149b4876fc28fe7da2798ddd12d GIT binary patch literal 35738 zcmdqK2UJzrmNk4WvlyNc1cWNZ1PBHYNm3>dQIMQLL_l)R$saYgF6h`;n3&s`7+(I_PS48P z(A?|@&oLf8?w_yP*jQK#@$#Df^#LApD+6Bk-MegYk<}JwRjnzM^&iN8OMJQ3vrs7a zR!W{eq2S=x*Wj$7VB5MdG(pL&WcYk0ST5snaEuSLiqXESAy15JV?#niW77;HlJsKj zMiOe3Xtf{bd^>I?SRJStR(oHM==Dx;`>$+tA#;e|ngp!tM0^p+uxf?!mPOR1;1|L`2-Z zefvSBQKh1?W{7itn#A_)+r6B82VT+is$(^Mf=5SfK7RNR9vX_>4|G8!W2sC z!NMm8%JCiT3RRJcT)bl)71Um*v7WTYEFQ0ZIpuNn+Z!F`<1VR;PIg1VAt4d3U!UB% 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