Chapter 3 Linear Regression Exercise.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.api as sm\n",
    "from statsmodels.formula.api import ols\n",
    "from statsmodels.tools.eval_measures import rmse, mse\n",
    "\n",
    "from sklearn.model_selection import train_test_split\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_1 = np.random.normal(loc=5, scale=7, size=1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 16.,  41.,  96., 201., 246., 211., 132.,  42.,  11.,   4.]),\n",
       " array([-14.69619077, -10.3840988 ,  -6.07200683,  -1.75991486,\n",
       "          2.55217711,   6.86426908,  11.17636105,  15.48845302,\n",
       "         19.80054499,  24.11263696,  28.42472893]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# normally distributed with mean 5 and standard deviation 7\n",
    "plt.hist(x_1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "true_error = np.random.normal(loc=0, scale=2, size=1000)\n",
    "true_beta_0 = 1.1\n",
    "true_beta_1 = -8.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = true_beta_0 + true_beta_1*x_1 + true_error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  4.,  11.,  40., 136., 204., 252., 199.,  97.,  43.,  14.]),\n",
       " array([-233.16897238, -197.60977719, -162.050582  , -126.49138681,\n",
       "         -90.93219163,  -55.37299644,  -19.81380125,   15.74539394,\n",
       "          51.30458912,   86.86378431,  122.4229795 ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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EfLf7Fei5JG/qrpD4A2DYkd+KZKX3UQrbklyWZGOX9dFJZh1mGvfrBUz1fu2+qc+5FTh3NcXA3Cud7zxT+xEgSV6e5JXnpulduHCIXr7t3WLbmfzXY79h2abxte9Z6b/g0vuDw7PAwe720W7894HD9P7y/E3gd/vWmaX34j8N/BXdO2snlbWbd0+X5yn6rtyYYNZb6R1FnAZOAl+c4v06MOs07tfzcv8t8DjwGL1v6nUL5Z7kDbgJ+Pcu1z2TztOX6zXd1+O3uq/Ne7rxXwD2A9/p7q+YUL7P0Dud+b/d1+kdF8o2ja99VfnxA5LUoklfLSNJGgPLXZIaZLlLUoMsd0lqkOUuSQ2y3CWpQZa7JDXo/wBEnjei2R1sdAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y')"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x_1, y, color='red')\n",
    "plt.xlabel(\"x1\")\n",
    "plt.ylabel(\"y\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# fit linear regression\n",
    "\n",
    "X = sm.add_constant(x_1) # adding a constant\n",
    "model = sm.OLS(y, X).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.999</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.999</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>8.497e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th>  <td>  0.00</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:48:12</td>     <th>  Log-Likelihood:    </th> <td> -2071.8</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>   4148.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   998</td>      <th>  BIC:               </th> <td>   4158.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td>    0.9585</td> <td>    0.075</td> <td>   12.724</td> <td> 0.000</td> <td>    0.811</td> <td>    1.106</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>   -8.1773</td> <td>    0.009</td> <td> -921.790</td> <td> 0.000</td> <td>   -8.195</td> <td>   -8.160</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 0.734</td> <th>  Durbin-Watson:     </th> <td>   2.045</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.693</td> <th>  Jarque-Bera (JB):  </th> <td>   0.813</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.055</td> <th>  Prob(JB):          </th> <td>   0.666</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.914</td> <th>  Cond. No.          </th> <td>    10.6</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.999\n",
       "Model:                            OLS   Adj. R-squared:                  0.999\n",
       "Method:                 Least Squares   F-statistic:                 8.497e+05\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):               0.00\n",
       "Time:                        15:48:12   Log-Likelihood:                -2071.8\n",
       "No. Observations:                1000   AIC:                             4148.\n",
       "Df Residuals:                     998   BIC:                             4158.\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const          0.9585      0.075     12.724      0.000       0.811       1.106\n",
       "x1            -8.1773      0.009   -921.790      0.000      -8.195      -8.160\n",
       "==============================================================================\n",
       "Omnibus:                        0.734   Durbin-Watson:                   2.045\n",
       "Prob(Omnibus):                  0.693   Jarque-Bera (JB):                0.813\n",
       "Skew:                          -0.055   Prob(JB):                        0.666\n",
       "Kurtosis:                       2.914   Cond. No.                         10.6\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#true_beta_0 = 1.1\n",
    "#true_beta_1 = -8.2\n",
    "model.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "shape, scale = 2., 2.\n",
    "x_2 = np.random.gamma(shape, scale, size=1000)\n",
    "true_beta_2 = 15.5\n",
    "y = true_beta_0 + true_beta_1*x_1 + + true_beta_2*x_2 + true_error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([266., 326., 204., 110.,  58.,  19.,   8.,   5.,   1.,   3.]),\n",
       " array([ 0.07671512,  2.04852073,  4.02032635,  5.99213196,  7.96393757,\n",
       "         9.93574318, 11.90754879, 13.87935441, 15.85116002, 17.82296563,\n",
       "        19.79477124]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(1,2,figsize=(10,5))\n",
    "axes[0].scatter(x_2, y, color='red')\n",
    "axes[0].set_xlabel(\"x2\")\n",
    "axes[0].set_ylabel(\"y\")\n",
    "\n",
    "axes[1].scatter(x_1, y, color='red')\n",
    "axes[1].set_xlabel(\"x1\")\n",
    "axes[1].set_ylabel(\"y\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.DataFrame({'x1':x_1,'x2':x_2,'y':y})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.625</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.625</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   1665.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th> <td>6.68e-215</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:49:18</td>     <th>  Log-Likelihood:    </th> <td> -5214.2</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>1.043e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   998</td>      <th>  BIC:               </th> <td>1.044e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>   64.4706</td> <td>    1.744</td> <td>   36.959</td> <td> 0.000</td> <td>   61.047</td> <td>   67.894</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>        <td>   -8.3821</td> <td>    0.205</td> <td>  -40.802</td> <td> 0.000</td> <td>   -8.785</td> <td>   -7.979</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>278.390</td> <th>  Durbin-Watson:     </th> <td>   1.996</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td>  <th>  Jarque-Bera (JB):  </th> <td> 765.580</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 1.418</td>  <th>  Prob(JB):          </th> <td>5.71e-167</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 6.215</td>  <th>  Cond. No.          </th> <td>    10.6</td> \n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.625\n",
       "Model:                            OLS   Adj. R-squared:                  0.625\n",
       "Method:                 Least Squares   F-statistic:                     1665.\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):          6.68e-215\n",
       "Time:                        15:49:18   Log-Likelihood:                -5214.2\n",
       "No. Observations:                1000   AIC:                         1.043e+04\n",
       "Df Residuals:                     998   BIC:                         1.044e+04\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept     64.4706      1.744     36.959      0.000      61.047      67.894\n",
       "x1            -8.3821      0.205    -40.802      0.000      -8.785      -7.979\n",
       "==============================================================================\n",
       "Omnibus:                      278.390   Durbin-Watson:                   1.996\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):              765.580\n",
       "Skew:                           1.418   Prob(JB):                    5.71e-167\n",
       "Kurtosis:                       6.215   Cond. No.                         10.6\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = ols(\"y ~ x1\", data=df)\n",
    "results = model.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.405</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.404</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   678.9</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th> <td>1.37e-114</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:49:20</td>     <th>  Log-Likelihood:    </th> <td> -5445.4</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>1.089e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   998</td>      <th>  BIC:               </th> <td>1.090e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>  -42.4927</td> <td>    3.060</td> <td>  -13.885</td> <td> 0.000</td> <td>  -48.498</td> <td>  -36.487</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>        <td>   16.1138</td> <td>    0.618</td> <td>   26.056</td> <td> 0.000</td> <td>   14.900</td> <td>   17.327</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 0.303</td> <th>  Durbin-Watson:     </th> <td>   1.899</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.859</td> <th>  Jarque-Bera (JB):  </th> <td>   0.289</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.042</td> <th>  Prob(JB):          </th> <td>   0.865</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.998</td> <th>  Cond. No.          </th> <td>    8.77</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.405\n",
       "Model:                            OLS   Adj. R-squared:                  0.404\n",
       "Method:                 Least Squares   F-statistic:                     678.9\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):          1.37e-114\n",
       "Time:                        15:49:20   Log-Likelihood:                -5445.4\n",
       "No. Observations:                1000   AIC:                         1.089e+04\n",
       "Df Residuals:                     998   BIC:                         1.090e+04\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept    -42.4927      3.060    -13.885      0.000     -48.498     -36.487\n",
       "x2            16.1138      0.618     26.056      0.000      14.900      17.327\n",
       "==============================================================================\n",
       "Omnibus:                        0.303   Durbin-Watson:                   1.899\n",
       "Prob(Omnibus):                  0.859   Jarque-Bera (JB):                0.289\n",
       "Skew:                           0.042   Prob(JB):                        0.865\n",
       "Kurtosis:                       2.998   Cond. No.                         8.77\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = ols(\"y ~ x2\", data=df)\n",
    "results = model.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.999</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.999</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>7.128e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th>  <td>  0.00</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>15:49:23</td>     <th>  Log-Likelihood:    </th> <td> -2071.8</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>   4150.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   997</td>      <th>  BIC:               </th> <td>   4164.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>    0.9697</td> <td>    0.115</td> <td>    8.429</td> <td> 0.000</td> <td>    0.744</td> <td>    1.195</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>        <td>   -8.1774</td> <td>    0.009</td> <td> -920.881</td> <td> 0.000</td> <td>   -8.195</td> <td>   -8.160</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>        <td>   15.4973</td> <td>    0.021</td> <td>  730.525</td> <td> 0.000</td> <td>   15.456</td> <td>   15.539</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 0.735</td> <th>  Durbin-Watson:     </th> <td>   2.045</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.693</td> <th>  Jarque-Bera (JB):  </th> <td>   0.816</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.054</td> <th>  Prob(JB):          </th> <td>   0.665</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.911</td> <th>  Cond. No.          </th> <td>    17.0</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.999\n",
       "Model:                            OLS   Adj. R-squared:                  0.999\n",
       "Method:                 Least Squares   F-statistic:                 7.128e+05\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):               0.00\n",
       "Time:                        15:49:23   Log-Likelihood:                -2071.8\n",
       "No. Observations:                1000   AIC:                             4150.\n",
       "Df Residuals:                     997   BIC:                             4164.\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept      0.9697      0.115      8.429      0.000       0.744       1.195\n",
       "x1            -8.1774      0.009   -920.881      0.000      -8.195      -8.160\n",
       "x2            15.4973      0.021    730.525      0.000      15.456      15.539\n",
       "==============================================================================\n",
       "Omnibus:                        0.735   Durbin-Watson:                   2.045\n",
       "Prob(Omnibus):                  0.693   Jarque-Bera (JB):                0.816\n",
       "Skew:                          -0.054   Prob(JB):                        0.665\n",
       "Kurtosis:                       2.911   Cond. No.                         17.0\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#true_beta_0 = 1.1\n",
    "#true_beta_1 = -8.2\n",
    "#true_beta_2 = 15.5\n",
    "\n",
    "model = ols(\"y ~ x1 + x2\", data=df)\n",
    "results = model.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_pred = df[['x1','x2']]\n",
    "x_pred = sm.add_constant(x_pred)\n",
    "\n",
    "ypred = results.predict(x_pred)\n",
    "rmse_ = rmse(y, ypred)\n",
    "mse_ = mse(y, ypred)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# MSE by sample size"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "train_props = [random.uniform(0,0.95) for i in range(100)]\n",
    "train_props = sorted(train_props)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "# mse over sample sizes\n",
    "train_sizes = []\n",
    "train_mses = []\n",
    "test_mses = []\n",
    "\n",
    "for p in train_props:\n",
    "    X_train, X_test, y_train, y_test = train_test_split(df[['x1','x2']], df['y'],\n",
    "                                                        test_size=1 - p, random_state=42)\n",
    "    \n",
    "    X_train = sm.add_constant(X_train) # adding a constant\n",
    "    model = sm.OLS(y_train, X_train).fit()\n",
    "    \n",
    "    X_test = sm.add_constant(X_test)\n",
    "    ypred = model.predict(X_test)\n",
    "    \n",
    "    test_mse = mse(y_test, ypred)\n",
    "    train_mse = mse(y_train, model.predict(X_train))\n",
    "    \n",
    "    test_mses.append(test_mse)\n",
    "    train_mses.append(train_mse)\n",
    "    \n",
    "    train_sizes.append(len(X_train))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'mse')"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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A/E1U1br53cVjERHuO38U/5o7rdnjByRF946uIREJFZGvRWS+n/tERB4WkR0isl5EJgU7HvAkAk+LoKrWRVFFDemtJAKw1UNuAx9s6KZB45KDUFsBKcNg7KWQejy893PYtbh74lFKBWzRlhzeW3+In5xxHEPT4gJ6TGZyNAcLKzBBbvl3RYvgVmBzM/fNBoZ7LnOBx7sgnkZdQ95v9621CABG9ItneHoc73ZX9VC+p8og5TgIDYMffAihkbD2le6JRynVqqpaF/9YtIObX17N8PQ4bjptWMCPHZAUTWWNm/wgF6kENRGISCZwHvB0M4dcCLxgrOVAkohkBDMm8GxOU9U4EfibTObPnHH9+WpPAUe6eAo4UD9QnOL5Q4rpA+MuhS3vQUVh18ejlGrVb+Zv4k8fbuX049N54YYpRIQF/rE7wLN1brAHjIPdIvgbcCfgbub+AcD+Br9ne25rRETmisgqEVmVm5vb4aDiosKbJIK0uJYHi73mjM/AGLpnyYmCXRAWDfH962+bMheqS2HFE10fj1KqRcYYPthwhPPGZjDv6hPJSIxu/UENeLfOPdBTE4GIzAFyjDGrWzrMz21NOsOMMU8aYyYbYyanpaV1OLb4qPoxgpw2tgiGpcVxQkZCx6uH8nbAV89AxdHAH5O/w7YGQhr8s2WMg+Nnw/LHoLK4YzEppTrV7rwy8kqrmDk8tfWD/RjgTQSFwa0cCmaLYAZwgYjsAV4FzhSRl3yOyQYGNvg9EwhefWZNJbxzC/2ksG4eQW5JFSKQEhsR8NPMGZfBmn2FHRvNX/YwvHcbPDQaFtwJBbtbf0z+DugztOntp90BlYXwVXM9cEqp7rBydwEAU4b0adfjE6LCiY8K67ldQ8aYe4wxmcaYLOBy4FNjzFU+h70DXOOpHpoGFBljgtfnkrcN1rzAmNIvqaxxU+Nyk1NSRUpsRLN1vP7MGWeHMd7rSKug+AAkD4FRF8CqZ+GRSfDvq5tfUM5VC0f32IFiXwNOtHMLvnwUqsvaH5NSqlOt3F1AalwEQ1ObThgLVGZyzLHTNSQiM0Xkes/1NBEZ0p4XFJGbROQmz68LgF3ADuAp4Eftec6AGbsrWVqt/bAtrawlt6SK1FZmFfsanBLL2AGJHRsnKD4IfUfDxfPgf9fDjFth2wew9M/+jy/cC+5a/4kA4LQ7oTwfVv2z/TEppTrVit0FTBnSp9nlawLRFXMJAkoEIvIr4C7gHs9N4YBvN0+zjDGLjTFzPNfnGWPmea4bY8yPjTHDjDFjjTGr2hZ+G7ntmHVypR2fLq2qJbekMqDSUV9zxmWwLruIffnt7LsrOgAJnkHfhP5w1v3Qb1x9iaivutLRZkrPBk2DIafaLqea4E9AUUq1LPtoOQcKK5g6JKVDz5OZHM2Bo8GdSxBoi+Bi4AKgDMAYcxCID1ZQQWNsIkissImgxNMiaG15CX/O83QPzf+mHd1DVSVQVVSfCLz6DIGjzYwVFDSYQ9CcU++0i9KteaHpfbsWw7xToLKo7fEqpdqso+MDXpnJ0ZRU1VJcUdv6we0UaCKoNjYdGQARaX+HV3fydA3FlO5DcFNSWUNuaevLS/iTmRzDxEFJzF/Xju6hYs9jEnwqZZOHQFE21PqZPJK/A6ISIaaFbxdZM2HQdPj8b1DrswzG+tfh8HrY8Gbb41VKtdnK3QUkRIUxom/HvjN7S04PB3HuUqCJ4DUReQI74euHwMfYPv2exW0TQairknQKyT5aQY3LBLS8hD9zxvVn06FiduW2ceOIYs+AsG8i6DPEtlqKPFMrXLWQvRpKDnsqhoZBS32NInDqHXYpirUvN75v9xL7c82LbYtVqV7u/nc28vqq/a0f2EYrPeMDISHtHx8A6iag1biam47VcQElAmPMn4E3gDeBEcB9xphHghZVsHhaBABZcoTdebbCpj0tAoDzxmYgQts3rKlLBD5dQ8me8fcVT8Br18KfhsLTZ8Kb/wP5u1ruFvIadqatIvr8r+DyLLV9dA8U7bOL1R1cA0c2ti1epXqpr/YU8NyyPby/4TDLduR12odtTnElu/LKOtwtBOAtaLz99XVBW/040MHiWGz55x3YlkC0iIQHJaJgMvX/yFkhh9mVZ7/Jt7dF0C8xipMG92n7P06x5/h4n9U0Uo4DBFY+AftXwMjz4YQLYO8XtpUQSCIQsWMFhftg/b/tbd7WwJy/QmiEtgqU8vj7x9sB+HxHHlc8vYInl+zqlOdduceOD3R0oBggxNMLsOVwSdA2sw+0a2gJECkiA7DdQtcDzwUlomByN2wRHGbjQTsTt70tArCDxtuOlLLtSEngDyo+ADGpEO4zSB2XBtfNh5u/hNs2w0X/sF09xg2Y5iuGfB1/jq1AWvoX2720eynEptvxgxHn2gThO4aglMOs2lPA5zvyiAwLobrWfkl8cskuiitrWnlk61buLiAmIpTR/RM6/FyhDbqWvj2qb4efz59AE4EYY8qB7wCPGGMuBkYFJaJgatAimBxfyF5P6Wd6Qturhrxmj+1HSFu7h4oOQGKTJZWsrJnQd1T9WEC/sZA0yF4PNBGI2HkFBbvs4PDuJTDkFHv7pKuhogC2Lgg8XqV6ob9/sp2U2Ai+NzkTgIsnDqCoooZnPw9gln8rVu4u4MTByW2aqNqc0AbjgoNTglOnE3AiEJHpwJXAe57b/O/0fizztghi0zkxvoDTjk8jPT6S2IjQdj9lenwUU4ekMH/9wcDrfIsPNh0obo6I7R6SUDtYHKgR50H6KHjrRig9bOcYAAw9AxIytXtIOdrqvQUs3Z7H3FOHMnVICgP7RPPAhaM5Z3Rfnlm6m8Ly9i/7fLSsmi2HS5jaCeMDQN0mWicOTu6U5/Mn0ERwK3A38B9jzEbPrOJPgxZVsHhbBKnDCTm6m+eum8yi20/v0Kw/sCuS7sot47HFO3l77YHWE0LxgaYDxS05/W74wQcQ1YZmZkgInHo7dWv4ZZ3iuT0UJlwBOz+tL2NVymH+9vF2+sRGcPX0wZw/vj9L7zyThKhw/ves4ympquWppe0fK/hqj3f+QMfHBwAmDEyif2IUv75wdKc8nz+BJoJy7FLS3xeR9dg1gs4IWlTB4q0aSjkOasqRshxiIzvesJk9JoPo8FD+9OFWbn11Leuym5m0ZQx8/IBdIC5tZOAvEBkPA6e0PbBRF9Vfb7hY3dhLAANbmmwa1zZ525sfazAGPv0tHGhp8VkVVMbY7kFXx/u8e5M1+47WtQZiIhr//z8hI4HzxmXwzy/2kF/avnG0lbsLiAgLYfzAxM4Il74JUSy751uM7t85z+dPoJ+CLwO3Axtofm+BY5+3ayh1uP2ZvxPi+3X4afvERvDlPWeSX1bN7L8v5RdvfcOz151E34ZjD64aeOensO4VOPE6OPH6Dr9uq0JC7dIVEXGN5x+kjbCJaMmf7YzjyHh7TGR8g0sCRMb5/O45LjzalqQ+OtmOYfxwEYT6FJFtXQBL/mgvt6wJfHwjSIorazhaVs3R8hqOlldTWF7N8X3jg/qfq0tVFEJVsZ1wmL3Kbla05T0ozoZxl8EFj0JY4Cvs9mZ/97YGpg32e//PzhrOgm8O8Y9FO7nv/LYPha7cU8DEgUlEhrW/y7mrBZoIco0x7wY1kq7g7RpK8SSCgl2QNaNTnjopJoKkmAjmXTWJn7zyNRf/4wueue4kTshIsCuCvnYt7PgITr/XDuR2sDsqYDN/5v/20+6CFfPs8tdVJfZDpKqk0VyLZkkohHkqrQ5/A2//2O6UVnzIjn/UVsDW9+39kYnw9LfgspfsQHgXqXW5+Xp/IZ9szuHTLUfYdqTppL+IsBB+eMoQIsNCueTETPontW3TkGNCbbX9d/zsD3aDIq+wKLsibXmerRI7sAbO+R0cf3b3xXoM+HrfUT7blstds0Y22xtwXHo8l580kH8u283M4SmcOTLwSp2Syho2HCjiJ2cEUOp9DAk0EfxKRJ4GPgHq2kvGmP8EJapg8SaCZM83gaLsTn+JM0f25bUbp3PD81/xvXlf8uR3B3Py8pvh0Fo4/++2NXAsGPMde2nIGKit9CQGn0t1aX2yqCqBqlJIGmgXuFv02/o5C2AThXHBd5+BAZPglcvghYvsPIZJVwftLRVV1LB4aw6fbslh8dZciipqCAsRpg7tw4UTBpAeH0lyTATJseFEhoVy7bMr+cciu4bTvM928vOzR3DdyVmNyvWOWa5a2PY+fPJru7z68bNg+Nm2IiztBBh2BkTE2n/TbR/Ch/fCK9+D486yCSFtRHe/g0731JJdHCis4Lazjychquk0p8oaF39euJXkmHCume6/NeB135zRrNtfxK2vruWdn8xkSIDLSK/eexS36bzxga4SaCK4HhiJXXXU2zVkgJ6VCLxdQ6EREB7b+BtUJxozIJH//ngG9z4zn4w3L6Q29Chhl70MI88Nyut1GhHb7RMeDXHpgT3GGFuRZAwkZNhJcpXFkP0VjJhtn/OGj+D16+Cdn9gPrbPut91WnWj7kRIufeJLjpbXkBIbwVkn9OVbJ6Qzc3iq3w8FgBdumMLzy/ZwzfQs/rJwK7+Zv4n3vznEM9eeRGLMMTpfsmCXrfha+4qtBuszFK54zc4d8UcERsyyM85XPgmf/REemw5Tfgjfus8mi7Ywxs5P+fIftpswpo/tjmp0aeY23+7DTrThQBG/f38zbgMfbjzM778zlpTYSH759gZOH5HG8PR4/vThFvbkl3P/+aNaHRuMjgjliatP5PxHP+fGF1fx1o9mBDSeuHJ3AWEhwqTBSZ311rqEBFLyKCLfGGPGdkE8rZo8ebJZtaqdq1WvfQX+ezP8dC08e479FnXBw50X3KF18PH9cPkrkLcN90uXUF5RwTUVP+ekU2Zx16yRHV53pC2MMdz/zkaSYiL42bePD8prFJXXsGhrDicN6cOAlrpWXLXwwV12F7X4DJs8sk6xP5Nb/nbWkqpaF99kF3HTS2sQgYcvn8iUIX3a/K3eGMN/1x7gzjfWc3zfeF6/aXqTgUSMsV8eSnOgLNeu9NrwesVR+54mXNH2D9iW1FTC5ndhzfOwZylICAw/ByZdY1sBoW0oeCjLg0//H6x+DjLG2yQSX9/1sb+gnNKqWtul6au2Gub/DNa+ZLudYtPsHhjeS8VR22psTmRigyTRBxIzIXOKLYTwFjO0o8vUGMMl875kT14Zf71sAr+ev4kdOaWkxEZQUlVLjcttv6+kxvLrC0dzyvDAt7v9fHse1zy7gtljMnj0iomtVhhe8vgyXMbw1o86p8u5M4nIamPMZH/3BfoXtFxERhljNnViXF3P2zUUEmr/o3b2bl7v/NR2AS19CJY/Tkh0ElFz32H0chdPLNnFvoJy/nrZBKLCgz+ItOFAEb9bsJllO/MBuGra4IBmUNe43BSUVZNXWkVeaTX5pVXklVaRX1pNruen9/fy6lqqXW4qa9yEhwrfmZjJzacPI8tfMzo0DM77ix0n2PSOLV/1diclDYKsUzFDTqG433SWHgnnpeV7GZgcw/iBSUwYmERmcjS788rYkVPKjtxSduaUsiOnlH0F5bgNpMZF8q8fTuO49Lh2nS8R4eKJmcRGhDH3xdW8smIf/5O0Br55A8q8H/a5dvyj6aPth1t4DGx6237QTv4BTJlrW0keRRU1VNa4SI+PDKxk+chGu6T4uldtpVnSYDjz/2DClW0rP24oNhXO/5v9EvTG9fD0WXDl61QmD+fxxTt5fPFOql1uvj2qL3fNGmnPpzG2+uuj++xyJ6fdBaff4/9Du7badk81TBDl+VDuc1vpEdi3wu7OB57uRLct3kjOsu81eTBEJ9tChdAI+zcUGmGTRsrwugT41tcHWL33KH/87jhOPT6N+bfM5O+fbOeJz3Zy77knMHtsBhsPFHHaiLQ2D+DOHJ7KnbNG8uD7Wxi7JJGbTvNT9GAMuGqocIeyLruQH8xs155d3SrQFsFmYBiwGztGINh9ZcYFN7ymOtQiWP08vPtT+NlG+Nf37X+mK/7d+uMCNe8Uu9QzQPpouOoNSOiPMYZnPt/NbxdsZnxmEk9dM7nNy1rUutwBzVI8XFTJnz7cyn++ziYpOpwLxvfn+S/38rOzjufWs4bXHUzrFNoAAB3ySURBVLd6bwHvrD1IXlk1eSVV5Hs+/AvL/ZcaRoSGkBoXQWp8JCmxESRGh/PVnqNMGpzM5ScN5MONh3n1q/3Uej5EhqfHc9Ppw4jzNKdrXW5yS6s4XFTpuVRQe2QzyUeWM6hoFSOr1pOA7arb6c7gQddVrI6aSkFZ04k94aHCkNRYjkuP47i0OIalx3HysNQOLRXS0OVPLOOyw3/mYvMJJA605cZx6fYbcGyavR6XbpftiEu3y4WEhtkPhP0r7JahW96zH25jvgNpI/ky5WJue3sXh4oqSY2LZMyABEb3T2BM/0RG909kYGwNkrMFcjZCzmbYv9J+qQiNgBPOt9/+s06180M6y8Gv4ZXLcFdX8DO5g7eLhnHB+P4MT4/jiSW7SKjJ5VeD1vOtqk8IO7rDdqde8LCn/LgTuN2QuwX2L7drY4WE2YKDo3vsjnxF2dTNg/EVGgnpJ1CTPobbNwxmT58ZvHXzyY1a3EUVNSREhXV4npAxhp+88jXvbzjE8z+YwimDY+y527/SdoHuXwmVRRwYeQ2z1kzn79ed2qYB5q7SUosg0ETgt+1ujNnbwdjarEOJYNWztml72xZ48wZA4Pr3Wn1YwB6bDjmbYPBMuPxliG7cT/jBhsP877+/JjUukhdvmBrQAFRheTX3vvUNy3bm8/L/TG223LGsqpYnPtvJk0t34XbD9TOy+NEZx5EYHc51/1zJxoPFfHHXmUSEhfDm6mzu/s96wkND6JcQRWpcJClxEXU/U+IiSfP89N4WH9n6f6ickkqeWbqbt9ceJKekEreB4elxFFfWkFtShdvnTy0iNIS+iZH0S4iiX3w448OzGVu9jjGH3iS2bB/mvL9ScmAzFYe2UFtdQdGQ84iZ+D0y+/fvlKn7zVm/9B3GfWIHtT85eyGlMZkUlFVTUFZNUUUNqXGRDE6JYWCfGAb3iaFPbESjc1NUXsPmzeuIWPUUkw69CkClCefh8Btwjbuc8KLd1B7aQGLJdoazn5Eh+8mUvLrH14TFUpt6AlHjv4uMv9x2owRJdd4eDj9+PhmuAxQNOpvUKZeCcVO9+iXC9nxGCG5WmZEUDr+E6RfcQGxC8GJpwlXrKVAotuXX7lpbnJC33X7hOrKByn1riKotprj/TBLO+QX0n2T/Dx5aZ48bcoqtoKoutbP5+wyxrYxAGQNHd1O5ezkLP3yX4dWbGSn7kIZzkjKngHFh1r9GvoknZvYDxEy5tnPHwapKbS9GB5JahxPBsaRDiWDlU7Dgdrh9uy15LD0CNy7pnMCMgd9n2kXdLp7X7B/Buv2FXPfPlWQkRvPfH8+oW2sc7NT0u/+znp25ZRRX1FBUUUNVrZvQECEpOhwR4c2bpzdab8TlNry2aj9/WbiNvNIqzh/fnzvPGcHAPjF1x3y2LZdrn13Jny4Zx8aDxTy3bA/Th6Yw7+oTSYwOzgDehxsPc+OLqxmQFM3Jw1LolxhlLwn1P30/QOt4x3IAwqLtfzZ3jf32GBoJI8+z3SPDzuj0QWcA89x51Oxdyfcr72G1qa+uCRGIiwyjuLLxTlFxkWEM7BND/8Qo9uSXsTO3rO74rNhazqn6kKv7bKJ/0deNXyckjMrEYRyJGso2M5CV5Rl8WpDCrtoUQDghI4Erpw7iookD6lpWne038zfx+ucbeGfMErIOfWC7wAASB8H4y9k/6EJ+v6KKBd8cpm9CJG/cdHKjv63uUOtys3DTEZ5auosN+/K4v+8Srqx928burViDxtcbypwC026yS7f4G8CuLrPdlts/st/4PefEHR7Lypoh7IkazcUXXExk1tRGSfoX/3iRKwoeY7Rrk51fM+sPnVOevnMRvHiRbR2efjec8vN2PY0mAq8VT8L7d8Adu2xCOLwebumkma/Fh+ChkTD7TzB1bouHLtx4mLkvrua8sRlkJkfbvviyKrYfKSW3tIpvjUwnMTqcxOhwEqLDOWV4KjERoVwy70sSo8N5/abpVFa7eW7ZHhZtzWF3XhknDk7m/847gYmDmn7bMcZw1kOfsTuvDLeBG2YO4Z7ZI4P6rRogr7SK5JiItpdjul22Tzo+w36LCwmxifbQOpskvnnNDkzGZ9jJUhOuhLROGgxf96pdn2nWg+SP+QE7c8voExtR1x0WEiJU1rjYX1DO3vxy9hXYy978Mg4UVpCZHMOkQUlMGpTMuIFJ9R/gxsA3r9tun/RRdmHBlOFNJnnVutzszC1j5e58Xlm5n82HiomNCOXCiQO4aupgRnXCapZeH206wg9fWMW10wfzwIVj7DfwA6vth+fAaY26oVbvLeDKp1dw7tgMHrp0QqfF0JqqWhcfb8qhuLKGWrehsKyaf6/aT/bRCganxPCDGUP43uRMYqi2mzGVHLKD4BnjbTfens/tRMjwGNvVlLsZvn7ZbgmbMABOugFSj7f/7hOvssd//aLd0rXPUBg4FTJPsgPa6aP4dFseNzy/iosmDOChS8fXfZGprnUz7oEP+f5JA/nV0K2w8D47mW/URXD2b+oXjmyLyiJbIfbJr8FVBaO/A6MvglEXtutcaiLwWv44fHA33LkbPv4VbFsIt2/tnMB2L4Hnz4er37Kleq24/52NPLdsDxFhIaTG2m6YlLgIrj05izNG+C/d/HrfUa54agUVNS5E7KqEA5Kj+ckZx3HJiZktdt289XU2//fWBv7fxWO4eGJmu9/mMaG2ytbGr33ZfmszLph4NVz4aMee19tiHDTd/juGd+8EM2MMa/cX8vKKfby77iBVtW4mDkriyqmDmTMuo0NFBwcKKzj370vJTI7mzZtPDui5fvveJp75fDcf3XYaw9LaNygfCGMMuaVVrN5zlD98YEs+GzopK5kbZg7l26P6tm/Oh9sN2xfCisftzPqGJBRGXQBTb7Yf/n7+Tz38yXYe+mgbvzp/FNfPsAPDq/cW8N3Hv2TeVZOYNSYDqsth2SN2gygMDD0dEJv4o/vYlkR0sm3lVhbbsaTKYsjdapNVzha70yDYXobzH7bL1HeAJgKvZY/Cwl/A3ftg8R9sOd69BzonsK+egfdug//dYCdaBaCi2kVUeEibBrM+25bL7a+v48Lx/fnhqUMbL2PRCpfb9IzJUm1RcsTOql31DHz/VTt3oT2KD8IjJ8Lgk235b1jnDDx3lqLyGt5Yk83LK/ayK7eMhKgwLjlxINednMWglLZ11ZRX13L1MyvZeriE+bfM9F/l5UdeaRWn/nERU4f04dnrTurwICzA1sMl/PWjbazPLiQjKZqjZdUcKKygyrM/wHHpcdx77khGZSQSGiJEhIV0bndmzmZbRDLgRNvNOHCKLWttgdttmPviahZtzeHl/5nKtKEpPLZ4B3/8YCur/+8sUuIa/O0UZdsqsoNrbUFBbZWtoKo46r/bKiy6fgmY9JGQMcEmkU44151RPto7NOw7jPBMKHO7O6cSI3+H/UcMdHlp7KSVtjrt+DS++sVZbX4c0PuSANga+FkP2rLGBXfCkNMgoh192B/fb7ukzv3zMZcEABJjwrlh5hB+MCOL5bsKeHnFXl5cvodXVu7lF+eewFXTBgf0wbxiVz53vLGefQXlPPL9iQEnAbAlurefPYJfz9/E66uzuWRSJtUuN9UuNzW1bmpchhqXm6paNzWu+kt1rWlwjOd4l2Hp9lzeWXeQuIgwThuRRm5JFSMz4jlrVF8yk6MZ2CeGmcelEh7MLsz0E2D2g216SEiI8NBl47no0S/4yStrePeWmazcXcBx6XGNkwDYpHLxvKZPYowdBK8us1VnB1bb0tmkwZ1bGRYghyWCBvMIIj1N25ry+usdkbfdDmp2wz+i44VF2DkKz50HL1xg11cafk7rE61qKmHfMoiIt4ODM2+zVSXHMBFh+rAUpg9L4XBRJXe9uZ5fvr2RjzfncOtZwxmaGktSTNPF5cqra/njB1t5btkeBvWJ4dW505g2tO3LIFx3chYfbjzMnW+s58431nfovUSFh3DjqcO48dShJMf2rAXxEqLCefKaE7nw0S+46aU17Mwp5YIJbZjbIQJRifYCMHh6cAINkLMSgXeJCQmx0+PBtgo6IxHkb4f+Ezv+PKp9smbC7D/C53+DV6+wm+9Mvg4mXdv8chkf3GVn2ALE9YVTbuuqaDtFv8Qonrv+JF5cvpffLdjMdx6z1S0nD0vhrlkjGT/Qli+v3F3AHW+sY29+OddOH8xds0c2nTUdoJAQ4dErJvHqyn24jV24LzxUPD/tJSIshIhQqfvd3iZEhIYSHmZvjwgNITk2ImjVUF3huPR4/nLpBG56yRacdNZGNN2h5/4rtIe3RSChtpIAbH1ufAeft/iQnRAz9tIOPpHqkKk3wuQb7GJsXz1t+2YX/8EO/k24ApKy7MzaqES71MI6W+NPVBJc8Ej930QPIiJcMz2Ls0f1Y+3+QrYfKeGfy/Zw4T++4LxxGYSHCG+vO0hmcjT/+uE0pg/r+GJoafGR3PKt4a0f6ACzxvTjp2cex7wlu5jejhbWscKZicC7xARAdRs2nfentgpeu9qOD4z9XseeS3VcaJidiXvC+ba7btWztlxww5v1x4SE23JCVzXc9LndArQ94wrHkH6JUcxK7MesMf24bkYWTy3dzZNLdlJZ42bKkD7887qTOmUTJtXUbWeP4H9OHdrs4oY9gbP+Muq6hqRB11AH1hsyBt77uZ10cukLnVfLrjpH6nCY9Xs485d2GYOyPDs5yPuz31h76WXio8K57dvHc9W0Qby+KpsLxvfXJBBkPTkJgNMSgXHZbiGoHxeo6sBS1KuesZNPTrm93ZM8VBeIiAlobkdvkx4fxY972AYpqns4q8TF7apfkiDC0x/c3j0J9i6D9++y1Sln3Ns58SmlVDcIWiIQkSgRWSki60Rko4g84OeY00WkSETWei73BSsewI4ReFsE3jGCqnaMERRlw2vX2OVyv/NkUNa7UUqprhLMrqEq4ExjTKmIhAOfi8j7xpjlPsctNcbMCWIc9Yzblo5CfddQW8cIairg1SttDfp17zVZYVQppXqaoCUCY9eu8Pa7hHsu3bueRaOuoQbzCAJljF3G+tBauPxfvXLfV6WU8wR1jEBEQkVkLZADfGSMWeHnsOme7qP3RWR0M88zV0RWiciq3Nzc9gdkXPUtgpBQW0LYlq6hFfNg3b/g9HuP/f2HlVIqQEFNBMYYlzFmApAJTBGRMT6HrAEGG2PGA48A/23meZ40xkw2xkxOS+vACnzG3bg/37veUCB2fQYf/gJGzoFT72h/DEopdYzpkqohY0whsBiY5XN7sTGm1HN9ARAuIqlBC8TdoEUAtnsokDGCo3vh9etsXfrF83Q9IaVUrxLMqqE0EUnyXI8GzgK2+BzTTzxLJorIFE88+cGKqdE8ArADxq3NI6gut4PDbpddnrgHLkOglFItCWbVUAbwvIiEYj/gXzPGzBeRmwCMMfOAS4CbRaQWqAAuN8HcIMHt2zUU33LXkDHwzk/gyAa48nVIGRa00JRSqrsEs2poPdBkOU5PAvBefxTo4LZSbQnK3bhFEBFbv0erP189bdeo+davYPi3gx+fUkp1A2d1dhtX451+IlsZI9jxsd3PdObPgh+bUkp1E2clgobzCMAzWNxC11B5vt0gvRO2iVNKqWOVsxKBb9dQZHzLg8XlBRDTc9cYV0qpQDgsEfiUj0b3sfsR1FT6P76iAGJ67q5DSikVCGclAt+uoYQM+7PkkP9jKwq1RaCU6vWclQh8u4YSPJtNFx9semxFIWBsq0EppXox5yWChrOC4z2JwF+LoNwzr027hpRSvZyzEoHvEhPeriG/LYIC+1MTgVKql3NWImiyxEQChMc20yLwJALtGlJK9XLOSgS+g8Uidpyg+EDTY+u6hnSwWCnVuzkrEfgOFoPtHir20yLQriGllEM4MBH4vOX4/s0PFodG1O9kppRSvZSzEoHb1XQvgYQMmwjc7sa3lxfY8QFdXkIp1cs5KxH4DhYDJAwAd23TVUgrjmq3kFLKERyWCHz2IwC7qBxAiU8JaXm+DhQrpRzBWYnAdx4BNJhL4DNOUF4A0cldE5dSSnUjZyUCf11DdbOLtUWglHImhyUC07RrKC7dJoeGLQJjdIxAKeUYzkoEblfTKqCQUIjv13iZicoi23rQWcVKKQdwViLw1zUEdsC4YdeQzipWSjmIsxKB7xITXr6ziyuO2p/aNaSUcgBnJQJ/S0yAnUvQcHaxtgiUUg7isETgp3wUbNdQVTFUldjf61Ye1fJRpVTv56xE4PYzoQwa7FTmaRXognNKKQdxViJoqUUA9QPGRzZBVJK9KKVUL+ewRBBgi2DvFzB4hi44p5RyBGclAn9LTECDRHDAzic4uhuyZnRtbEop1U2clQiam0cQHm27gUoOwZ4v7G2DT+7a2JRSqps4KxE0N1gMni0rD9luocgE6Deua2NTSqluErREICJRIrJSRNaJyEYRecDPMSIiD4vIDhFZLyKTghUP0Pw8ArCJoOSgTQSDpjWfMJRSqpcJZougCjjTGDMemADMEpFpPsfMBoZ7LnOBx4MYj6drqJkB4PgMOLQO8rbZgWKllHKIoCUCY5V6fg33XIzPYRcCL3iOXQ4kiUhGsGJqdokJgKhE22IATQRKKUcJ6hiBiISKyFogB/jIGLPC55ABwP4Gv2d7bvN9nrkiskpEVuXm5vreHbiWuoYiYu3PsGjoP6H9r6GUUj1MUBOBMcZljJkAZAJTRGSMzyH++ml8Ww0YY540xkw2xkxOS0vrQEDNlI+CrRwCyBgPoeHtfw2llOphuqRqyBhTCCwGZvnclQ0MbPB7JuCzVVinBdH8hDKA2mr7U1sDSimHCWbVUJqIJHmuRwNnAVt8DnsHuMZTPTQNKDLG+Gwe3Em8/f/NdQ1VexacGzg1KC+vlFLHqrAgPncG8LyIhGITzmvGmPkichOAMWYesAA4F9gBlAPXBy0abyIIaSb3nfJzSBwIoy4KWghKKXUsCloiMMasByb6uX1eg+sG+HGwYmjE7bI/mxsjiE6GqTd2SShKKXUscc7MYuNNBDpRTCmlGnJOIvC2CHTGsFJKNeKcRNDaYLFSSjmUAxOBc96yUkoFwjmfito1pJRSfjknEZhWqoaUUsqhnPOpWDePQFsESinVkHMSQWvzCJRSyqGc86mo8wiUUsovByUC7RpSSil/nJMI3Fo+qpRS/jjnU1GrhpRSyi/nfCrqPAKllPLLOYlAl5hQSim/HJQItGtIKaX8cc6nonYNKaWUX85JBNo1pJRSfjkvETS3VaVSSjmUcz4VdYkJpZTyyzmfirrEhFJK+eWcRKCDxUop5ZdzEoEOFiullF8OSgQ6RqCUUv4451PRrauPKqWUP85JBLp5vVJK+eWcT0XtGlJKKb+c86moVUNKKeWXcxKBziNQSim/nJMI4vvDqAshKrG7I1FKqWNK0BKBiAwUkUUisllENorIrX6OOV1EikRkredyX7DiYdBUuPQFSBwQtJdQSqmeKCyIz10L/NwYs0ZE4oHVIvKRMWaTz3FLjTFzghiHUkqpFgStRWCMOWSMWeO5XgJsBvTruFJKHWO6ZIxARLKAicAKP3dPF5F1IvK+iIzuiniUUkrVC2bXEAAiEge8CfyvMabY5+41wGBjTKmInAv8Fxju5znmAnMBBg0aFOSIlVLKWYLaIhCRcGwSeNkY8x/f+40xxcaYUs/1BUC4iKT6Oe5JY8xkY8zktLS0YIaslFKOE8yqIQGeATYbYx5q5ph+nuMQkSmeePKDFZNSSqmmgtk1NAO4GvhGRNZ6brsXGARgjJkHXALcLCK1QAVwuTHGBDEmpZRSPoKWCIwxnwPSyjGPAo8GKwallFKtk572BVxEcoG9AR6eCuQFMZyeQs+DpedBz4GXE8/DYGOM30HWHpcI2kJEVhljJnd3HN1Nz4Ol50HPgZeeh8acs9aQUkopvzQRKKWUw/X2RPBkdwdwjNDzYOl50HPgpeehgV49RqCUUqp1vb1FoJRSqhWaCJRSyuF6RSIQkVkislVEdojI3X7uFxF52HP/ehGZ1B1xBlsA5+FKz/tfLyLLRGR8d8QZTK2dgwbHnSQiLhG5pCvj6yqBnAfPxlBrPRtHfdbVMXaFAP5PJIrIu54VkDeKyPXdEWe3M8b06AsQCuwEhgIRwDpglM8x5wLvY2c6TwNWdHfc3XQeTgaSPddn97bzEMg5aHDcp8AC4JLujrub/haSgE3AIM/v6d0ddzedh3uBP3iupwEFQER3x97Vl97QIpgC7DDG7DLGVAOvAhf6HHMh8IKxlgNJIpLR1YEGWavnwRizzBhz1PPrciCzi2MMtkD+FgBuwa6Km9OVwXWhQM7DFcB/jDH7AIwxvfFcBHIeDBDvWfwyDpsIars2zO7XGxLBAGB/g9+zaboTWiDH9HRtfY83YFtJvUmr50BEBgAXA/O6MK6uFsjfwvFAsogsFpHVInJNl0XXdQI5D48CJwAHgW+AW40x7q4J79gR9I1puoC/he18a2IDOaanC/g9isgZ2EQwM6gRdb1AzsHfgLuMMS7PCui9USDnIQw4EfgWEA18KSLLjTHbgh1cFwrkPJwDrAXOBIYBH4nIUtN0E61erTckgmxgYIPfM7HZva3H9HQBvUcRGQc8Dcw2xvS2vR8COQeTgVc9SSAVOFdEao0x/+2aELtEoP8n8owxZUCZiCwBxgO9KREEch6uBx40dpBgh4jsBkYCK7smxGNDb+ga+goYLiJDRCQCuBx4x+eYd4BrPNVD04AiY8yhrg40yFo9DyIyCPgPcHUv++bn1eo5MMYMMcZkGWOygDeAH/WyJACB/Z94GzhFRMJEJAaYCmzu4jiDLZDzsA/bKkJE+gIjgF1dGuUxoMe3CIwxtSLyE+BDbJXAs8aYjSJyk+f+edjqkHOBHUA59ltArxLgebgPSAEe83wjrjW9aAXGAM9BrxfIeTDGbBaRD4D1gBt42hizofui7nwB/j38BnhORL7BdiXdZYxx2vLUusSEUko5XW/oGlJKKdUBmgiUUsrhNBEopZTDaSJQSimH00SglFIOp4lA9QgikiQiP2rnYxeISFJnxxQsvu9VRPqLyBvdGZPq3bR8VPUIIpIFzDfGjPFzX6gxxtXF8YQZYzp9cTIRCcXOhvX7XpUKBm0RqJ7iQWCYZ/38P3nW0l8kIq9gFwtDRP7rWUBto4jM9T5QRPaISKqIZInIZhF5ynPMQhGJ9n0hEXlOROaJyFIR2SYiczy3Xycir4vIu8BCEenjec31IrLcs3wHInK/iLwoIp+KyHYR+aHndvHEvkFEvhGRyzy3+74X3/eaJSIbPMdGicg/PY//2rNulDe2/4jIB57X/KPn9lDP+/G+5s+C9Q+keq4eP7NYOcbdwBhjzASwH57YZYbHGGN2e475gTGmwPPh/pWIvOlnPaXhwPeNMT8UkdeA7wIv+Xm9LOA07EJki0TkOM/t04Fxntd5BPjaGHORiJwJvABM8Bw3Drv3RSzwtYi853nsBOyaPqmeGJd4jq97L57WT8P3mtUgrh8DGGPGishIbEI63nPfBGAiUAVs9cSXDgzwti56UheZ6jraIlA92coGSQDgpyKyDrvXwkDsh76v3caYtZ7rq7Ef+P68ZoxxG2O2Y9eeGem5/SNjTIHn+kzgRQBjzKdAiogkeu572xhT4VmuYBH2g34m8C9jjMsYcwT4DDipmffSnIavuQXYi11SGuATY0yRMaYSu+nMYE/sQ0XkERGZBThqVU0VGE0Eqicr817xtBDOAqYbY8YDXwNRfh5T1eC6i+Zbxb6DZ97fyxrc1tIyx/4e39K612Ut3NdQS8/R5L15NiIaDyzGtiaeDvB1lINoIlA9RQkQ38L9icBRY0y5p8tkWgdf73siEiIiw7BbHW71c8wS4EqoS0R5Ddaxv9DTn58CnI5dCXMJcJmn3z4NOBX/yx239F4bvubxwKBmYsNzTCoQYox5E/gl0Cv361Ydo2MEqkcwxuSLyBeeQdP3gfd8DvkAuElE1mM/GJd38CW3Yrtu+gI3GWMqpelGNvcD//S8ZjlwbYP7VnpiHAT8xhhzUETewo4TrMO2EO40xhz2JK6W3us/Gtz9GDDPs1pmLXCdMabKT2xeAzwxer/03RPwGVCOoeWjSvkQkeew5Zvtqt0XkfuBUmPMnzszLqWCRbuGlFLK4bRFoJRSDqctAqWUcjhNBEop5XCaCJRSyuE0ESillMNpIlBKKYf7/5YMhaIRTgjLAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# mse stables out once train proportion is above 10% and below 90%\n",
    "plt.plot(train_props, test_mses, label='test')\n",
    "plt.plot(train_props, train_mses, label='train')\n",
    "plt.legend()\n",
    "plt.xlabel('train proportions')\n",
    "plt.ylabel('mse')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Play around with parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 319,
   "metadata": {},
   "outputs": [],
   "source": [
    "# when the betas are close to 0, most of the points are generated by noise\n",
    "# doesn't seem to matter what the distribution of error. The model is still able to \n",
    "# find the linear relationship\n",
    "\n",
    "\"\"\"\n",
    "quoted from duke.edu about the normality assumption of the error term:\n",
    "\n",
    "Technically, the normal distribution assumption is not necessary if you are willing to assume \n",
    "the model equation is correct and your only goal is to estimate its coefficients and\n",
    "generate predictions in such a way as to minimize mean squared error.  The formulas\n",
    "for estimating coefficients require no more than that, and some references on regression\n",
    "analysis do not list normally distributed errors among the key assumptions.\n",
    "But generally we are interested in making inferences about the model and/or estimating the\n",
    "probability that a given forecast error will exceed some threshold in a particular direction,\n",
    "in which case distributional assumptions are important.  Also, a significant violation of\n",
    "the normal distribution assumption is often a \"red flag\" indicating that there is some \n",
    "other problem with the model assumptions and/or that there are a few unusual data points \n",
    "that should be studied closely and/or that a better model is still waiting out there somewhere.\n",
    "\"\"\"\n",
    "\n",
    "true_error = np.random.normal(loc=0, scale=2, size=1000)\n",
    "#true_error = np.random.gamma(2, 50, size=1000)\n",
    "#true_error = np.random.binomial(10, 0.2, size=1000)\n",
    "true_beta_0 = 100\n",
    "true_beta_1 = 10\n",
    "true_beta_2 = 40\n",
    "\n",
    "x_1 = np.random.normal(loc=5, scale=7, size=1000)\n",
    "x_2 = np.random.normal(loc=10, scale=2, size=1000)\n",
    "y = true_beta_0 + true_beta_1*x_1 + true_beta_2*x_2 + true_error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 320,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  1.,   8.,  30., 102., 216., 257., 216., 121.,  43.,   6.]),\n",
       " array([-7.82010702, -6.41818551, -5.01626401, -3.6143425 , -2.21242099,\n",
       "        -0.81049949,  0.59142202,  1.99334353,  3.39526503,  4.79718654,\n",
       "         6.19910805]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 320,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAD4CAYAAAAXUaZHAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+WH4yJAAAO60lEQVR4nO3dbYxcZ32G8etuDKnESwnyJqS26QZkWpIWDFpcqog2NC1JSYXhA8iRiqwW1VAFlFRUxQ5qoR8sXF6L1IJkSEo+pKRWA01EKBBcKEJqEjZpSuKYFJe48WI3XvoCqVCDbP79sMfNst71jHd2PDsP109azTnPnDPnXml9z/GzZ86mqpAkteUnRh1AkrTyLHdJapDlLkkNstwlqUGWuyQ1aM2oAwCsXbu2JicnRx1DksbKvffe+52qmljsuVVR7pOTk0xPT486hiSNlST/ttRzTstIUoMsd0lqkOUuSQ2y3CWpQZa7JDXIcpekBlnuktQgy12SGmS5S1KDVsUnVKXVbHLHHSM57qHdV43kuGqDZ+6S1KCe5Z5kQ5IvJTmQZH+Sa7vxdyf5dpL7u69Xz9tnZ5KDSR5OcsUwvwFJ0qn6mZY5Dry9qu5L8gzg3iR3ds99qKreP3/jJBcDW4FLgJ8GvpjkBVV1YiWDS5KW1vPMvaqOVtV93fLjwAFg3Wl22QLcUlVPVNUjwEFg80qElST154zm3JNMAi8B7u6G3prk60luTHJeN7YOODxvtxkWeTNIsj3JdJLp2dnZMw4uSVpa3+We5OnArcB1VfU94KPA84FNwFHgAyc3XWT3OmWgak9VTVXV1MTEovealyQtU1/lnuQpzBX7zVX1KYCqeqyqTlTVD4GP8eTUywywYd7u64EjKxdZktRLP1fLBLgBOFBVH5w3fuG8zV4HPNgt3w5sTXJukouAjcA9KxdZktRLP1fLXAq8EXggyf3d2PXA1Uk2MTflcgh4M0BV7U+yF3iIuSttrvFKGUk6u3qWe1V9lcXn0T97mn12AbsGyCVJGoCfUJWkBlnuktQgy12SGmS5S1KDLHdJapDlLkkNstwlqUGWuyQ1yHKXpAZZ7pLUIMtdkhpkuUtSgyx3SWqQ5S5JDbLcJalBlrskNchyl6QGWe6S1CDLXZIaZLlLUoMsd0lqkOUuSQ1aM+oAUj8md9wx6ghn3Si/50O7rxrZsbUyPHOXpAZZ7pLUIMtdkhpkuUtSgyx3SWqQ5S5JDbLcJalBlrskNahnuSfZkORLSQ4k2Z/k2m782UnuTPLN7vG8efvsTHIwycNJrhjmNyBJOlU/Z+7HgbdX1QuBlwPXJLkY2AHsq6qNwL5une65rcAlwJXAR5KcM4zwkqTF9Sz3qjpaVfd1y48DB4B1wBbgpm6zm4DXdstbgFuq6omqegQ4CGxe6eCSpKWd0Zx7kkngJcDdwAVVdRTm3gCA87vN1gGH5+02041Jks6Svss9ydOBW4Hrqup7p9t0kbFa5PW2J5lOMj07O9tvDElSH/oq9yRPYa7Yb66qT3XDjyW5sHv+QuBYNz4DbJi3+3rgyMLXrKo9VTVVVVMTExPLzS9JWkQ/V8sEuAE4UFUfnPfU7cC2bnkbcNu88a1Jzk1yEbARuGflIkuSeunnfu6XAm8EHkhyfzd2PbAb2JvkTcCjwOsBqmp/kr3AQ8xdaXNNVZ1Y8eSSpCX1LPeq+iqLz6MDXL7EPruAXQPkkiQNwE+oSlKDLHdJapDlLkkNstwlqUGWuyQ1yHKXpAZZ7pLUIMtdkhpkuUtSgyx3SWqQ5S5JDbLcJalBlrskNchyl6QGWe6S1CDLXZIaZLlLUoMsd0lqkOUuSQ2y3CWpQZa7JDXIcpekBlnuktQgy12SGmS5S1KDLHdJapDlLkkNstwlqUGWuyQ1yHKXpAZZ7pLUIMtdkhrUs9yT3JjkWJIH5429O8m3k9zffb163nM7kxxM8nCSK4YVXJK0tH7O3D8BXLnI+IeqalP39VmAJBcDW4FLun0+kuSclQorSepPz3Kvqq8A/9nn620BbqmqJ6rqEeAgsHmAfJKkZRhkzv2tSb7eTduc142tAw7P22amGztFku1JppNMz87ODhBDkrTQcsv9o8DzgU3AUeAD3XgW2bYWe4Gq2lNVU1U1NTExscwYkqTFLKvcq+qxqjpRVT8EPsaTUy8zwIZ5m64HjgwWUZJ0ppZV7kkunLf6OuDklTS3A1uTnJvkImAjcM9gESVJZ2pNrw2SfBK4DFibZAZ4F3BZkk3MTbkcAt4MUFX7k+wFHgKOA9dU1YnhRJckLaVnuVfV1YsM33Ca7XcBuwYJJUkajJ9QlaQGWe6S1CDLXZIaZLlLUoMsd0lqkOUuSQ2y3CWpQZa7JDXIcpekBlnuktQgy12SGmS5S1KDLHdJapDlLkkNstwlqUGWuyQ1yHKXpAb1/EtMkn78TO64YyTHPbT7qpEct0WeuUtSgyx3SWqQ5S5JDbLcJalBlrskNchyl6QGWe6S1CDLXZIaZLlLUoMsd0lqkOUuSQ2y3CWpQZa7JDWoZ7knuTHJsSQPzht7dpI7k3yzezxv3nM7kxxM8nCSK4YVXJK0tH7O3D8BXLlgbAewr6o2Avu6dZJcDGwFLun2+UiSc1YsrSSpLz3v515VX0kyuWB4C3BZt3wT8GXgHd34LVX1BPBIkoPAZuAfVyauRm1U9/mWdGaWO+d+QVUdBegez+/G1wGH5203042dIsn2JNNJpmdnZ5cZQ5K0mJX+hWoWGavFNqyqPVU1VVVTExMTKxxDkn68LbfcH0tyIUD3eKwbnwE2zNtuPXBk+fEkScux3HK/HdjWLW8Dbps3vjXJuUkuAjYC9wwWUZJ0pnr+QjXJJ5n75enaJDPAu4DdwN4kbwIeBV4PUFX7k+wFHgKOA9dU1YkhZZckLaGfq2WuXuKpy5fYfhewa5BQkqTB+AlVSWqQ5S5JDbLcJalBlrskNchyl6QGWe6S1CDLXZIaZLlLUoMsd0lqkOUuSQ2y3CWpQZa7JDXIcpekBlnuktQgy12SGmS5S1KDLHdJapDlLkkNstwlqUGWuyQ1yHKXpAZZ7pLUIMtdkhpkuUtSgyx3SWqQ5S5JDbLcJalBlrskNchyl6QGWe6S1CDLXZIatGaQnZMcAh4HTgDHq2oqybOBvwYmgUPAG6rqvwaLKUk6Eytx5v7KqtpUVVPd+g5gX1VtBPZ165Kks2gY0zJbgJu65ZuA1w7hGJKk0xi03Av4QpJ7k2zvxi6oqqMA3eP5Ax5DknSGBppzBy6tqiNJzgfuTPKNfnfs3gy2Azz3uc8dMIYkab6Bztyr6kj3eAz4NLAZeCzJhQDd47El9t1TVVNVNTUxMTFIDEnSAssu9yRPS/KMk8vAq4AHgduBbd1m24DbBg0pSTozg0zLXAB8OsnJ1/mrqvpckq8Be5O8CXgUeP3gMSVJZ2LZ5V5V3wJevMj4fwCXDxJKkjSYQX+hKkkrZnLHHSM57qHdV43kuMPk7QckqUGWuyQ1yHKXpAZZ7pLUIMtdkhpkuUtSgyx3SWqQ5S5JDbLcJalBlrskNcjbD4yhUX1EW9L48MxdkhpkuUtSgyx3SWqQ5S5JDbLcJalBlrskNchyl6QGWe6S1CDLXZIaZLlLUoMsd0lqkOUuSQ2y3CWpQZa7JDXIcpekBlnuktQg/1jHAPyjGZJWK8/cJalBnrlL+rE3yv+FH9p91VBe1zN3SWrQ0Mo9yZVJHk5yMMmOYR1HknSqoZR7knOAvwB+A7gYuDrJxcM4liTpVMOac98MHKyqbwEkuQXYAjw0jIN51Yok/ahhlfs64PC89RngF+dvkGQ7sL1b/Z8kD5/B668FvjNQwrPHrMMzTnnHKSuMV95xygoL8uZPB3qtn1nqiWGVexYZqx9ZqdoD7FnWiyfTVTW1nH3PNrMOzzjlHaesMF55xykrnL28w/qF6gywYd76euDIkI4lSVpgWOX+NWBjkouSPBXYCtw+pGNJkhYYyrRMVR1P8lbg88A5wI1VtX8FD7Gs6ZwRMevwjFPeccoK45V3nLLCWcqbquq9lSRprPgJVUlqkOUuSQ0ay3JPsinJXUnuTzKdZPOoM/WS5G3d7Rj2J3nvqPP0kuQPklSStaPOcjpJ3pfkG0m+nuTTSZ416kwLjcutOJJsSPKlJAe6n9NrR52plyTnJPmnJJ8ZdZZekjwryd90P68HkvzSMI83luUOvBf4k6raBPxxt75qJXklc5/QfVFVXQK8f8SRTivJBuDXgUdHnaUPdwI/X1UvAv4F2DniPD9izG7FcRx4e1W9EHg5cM0qznrStcCBUYfo04eBz1XVzwEvZsi5x7XcC3hmt/xTrP5r6H8P2F1VTwBU1bER5+nlQ8AfsuCDZ6tRVX2hqo53q3cx95mK1eT/b8VRVT8ATt6KY9WpqqNVdV+3/Dhz5bNutKmWlmQ9cBXw8VFn6SXJM4FfBm4AqKofVNV/D/OY41ru1wHvS3KYubPgVXW2togXAK9IcneSf0jyslEHWkqS1wDfrqp/HnWWZfgd4O9GHWKBxW7FsWoL86Qkk8BLgLtHm+S0/oy5k5AfjjpIH54HzAJ/2U0jfTzJ04Z5wFX7xzqSfBF4ziJPvRO4HPj9qro1yRuYezf8tbOZb6EeedcA5zH3X92XAXuTPK9GdB1qj6zXA686u4lO73R5q+q2bpt3MjetcPPZzNaHnrfiWG2SPB24Fbiuqr436jyLSfKbwLGqujfJZaPO04c1wEuBt1XV3Uk+DOwA/mhYBxzL69yTfBd4VlVVkgDfrapn9tpvVJJ8jrlpmS936/8KvLyqZkcabIEkvwDsA77fDZ28bcTmqvr3kQXrIck24C3A5VX1/V7bn03dL83eXVVXdOs7AarqPSMNtoQkTwE+A3y+qj446jxLSfIe4I3MvaH/JHPTtJ+qqt8aabAlJHkOcFdVTXbrrwB2VNVw/gwT4zstcwT4lW75V4FvjjBLP/6WuZwkeQHwVFbhXeyq6oGqOr+qJrsfwhngpau82K8E3gG8ZrUVe2dsbsXRnSjdABxYzcUOUFU7q2p993O6Ffj71VrsAN2/ocNJfrYbupwh3QL9pFU7LdPD7wIfTrIG+F+evHXwanUjcGOSB4EfANtGNSXToD8HzgXunOsm7qqqt4w20pPOwq04VtKlzJ0NP5Dk/m7s+qr67AgzteRtwM3dm/y3gN8e5sHGclpGknR64zotI0k6DctdkhpkuUtSgyx3SWqQ5S5JDbLcJalBlrskNej/ANdhvuSF2VoDAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(true_error)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 321,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f5f9232e070>"
      ]
     },
     "execution_count": 321,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x_2, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 322,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f5f92280ac0>"
      ]
     },
     "execution_count": 322,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x_1, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 323,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   1.000</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   1.000</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>1.366e+06</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th>  <td>  0.00</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>17:10:36</td>     <th>  Log-Likelihood:    </th> <td> -2127.3</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>   4261.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   997</td>      <th>  BIC:               </th> <td>   4275.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>  100.3183</td> <td>    0.325</td> <td>  308.886</td> <td> 0.000</td> <td>   99.681</td> <td>  100.956</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>        <td>    9.9951</td> <td>    0.009</td> <td> 1095.536</td> <td> 0.000</td> <td>    9.977</td> <td>   10.013</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>        <td>   39.9697</td> <td>    0.031</td> <td> 1283.303</td> <td> 0.000</td> <td>   39.909</td> <td>   40.031</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 0.905</td> <th>  Durbin-Watson:     </th> <td>   2.073</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.636</td> <th>  Jarque-Bera (JB):  </th> <td>   0.980</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.065</td> <th>  Prob(JB):          </th> <td>   0.613</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.918</td> <th>  Cond. No.          </th> <td>    61.5</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       1.000\n",
       "Model:                            OLS   Adj. R-squared:                  1.000\n",
       "Method:                 Least Squares   F-statistic:                 1.366e+06\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):               0.00\n",
       "Time:                        17:10:36   Log-Likelihood:                -2127.3\n",
       "No. Observations:                1000   AIC:                             4261.\n",
       "Df Residuals:                     997   BIC:                             4275.\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept    100.3183      0.325    308.886      0.000      99.681     100.956\n",
       "x1             9.9951      0.009   1095.536      0.000       9.977      10.013\n",
       "x2            39.9697      0.031   1283.303      0.000      39.909      40.031\n",
       "==============================================================================\n",
       "Omnibus:                        0.905   Durbin-Watson:                   2.073\n",
       "Prob(Omnibus):                  0.636   Jarque-Bera (JB):                0.980\n",
       "Skew:                          -0.065   Prob(JB):                        0.613\n",
       "Kurtosis:                       2.918   Cond. No.                         61.5\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 323,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.DataFrame({'x1':x_1,'x2':x_2,'y':y})\n",
    "model = ols(\"y ~ x1 + x2\", data=df)\n",
    "results = model.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 318,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.472</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.472</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   893.6</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th> <td>9.72e-141</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>17:08:45</td>     <th>  Log-Likelihood:    </th> <td> -5979.3</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>1.196e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   998</td>      <th>  BIC:               </th> <td>1.197e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>  215.7054</td> <td>   14.867</td> <td>   14.509</td> <td> 0.000</td> <td>  186.530</td> <td>  244.880</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>        <td>   43.4299</td> <td>    1.453</td> <td>   29.894</td> <td> 0.000</td> <td>   40.579</td> <td>   46.281</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>33.830</td> <th>  Durbin-Watson:     </th> <td>   1.985</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td>  37.185</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.427</td> <th>  Prob(JB):          </th> <td>8.42e-09</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.404</td> <th>  Cond. No.          </th> <td>    50.7</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.472\n",
       "Model:                            OLS   Adj. R-squared:                  0.472\n",
       "Method:                 Least Squares   F-statistic:                     893.6\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):          9.72e-141\n",
       "Time:                        17:08:45   Log-Likelihood:                -5979.3\n",
       "No. Observations:                1000   AIC:                         1.196e+04\n",
       "Df Residuals:                     998   BIC:                         1.197e+04\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept    215.7054     14.867     14.509      0.000     186.530     244.880\n",
       "x2            43.4299      1.453     29.894      0.000      40.579      46.281\n",
       "==============================================================================\n",
       "Omnibus:                       33.830   Durbin-Watson:                   1.985\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               37.185\n",
       "Skew:                           0.427   Prob(JB):                     8.42e-09\n",
       "Kurtosis:                       3.404   Cond. No.                         50.7\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 318,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = ols(\"y ~ x2\", data=df)\n",
    "results = model.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 293,
   "metadata": {},
   "outputs": [],
   "source": [
    "# even tho x_1 and x_2 are independent, if they have similar distribution and \n",
    "# their true_beta are of the same magnitude then\n",
    "# the r2 for just x_2 is much smaller since x_1 already explains\n",
    "# most of those variance\n",
    "\n",
    "# changing x2 to be a completely diff distribution like gamma gives x2 a much higher r2\n",
    "true_error = np.random.normal(loc=0, scale=2, size=1000)\n",
    "\n",
    "x_1 = np.random.normal(loc=5, scale=7, size=1000)\n",
    "#x_2 = np.random.normal(loc=10, scale=2, size=1000)\n",
    "x_2 = np.random.gamma(4, 4, size=1000)\n",
    "x_3 = np.random.binomial(n=30, p=0.5, size=1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {},
   "outputs": [],
   "source": [
    "true_beta_0 = 5\n",
    "true_beta_1 = 100\n",
    "true_beta_2 = -100\n",
    "true_beta_3 = 250\n",
    "\n",
    "y = true_beta_0 + true_beta_1*x_1 + true_beta_2*x_2 + true_beta_3*x_3 + true_error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 62., 196., 263., 222., 119.,  64.,  39.,  23.,  10.,   2.]),\n",
       " array([ 0.92005978,  5.70930757, 10.49855536, 15.28780316, 20.07705095,\n",
       "        24.86629874, 29.65554653, 34.44479432, 39.23404212, 44.02328991,\n",
       "        48.8125377 ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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G/S+SrdMzyOFPKgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   1.000</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   1.000</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>1.381e+08</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th>  <td>  0.00</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>16:43:35</td>     <th>  Log-Likelihood:    </th> <td> -2108.0</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>   4224.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   996</td>      <th>  BIC:               </th> <td>   4244.</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     3</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>    4.8343</td> <td>    0.386</td> <td>   12.509</td> <td> 0.000</td> <td>    4.076</td> <td>    5.593</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>        <td>  100.0083</td> <td>    0.009</td> <td> 1.14e+04</td> <td> 0.000</td> <td>   99.991</td> <td>  100.026</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x2</th>        <td> -100.0038</td> <td>    0.008</td> <td>-1.28e+04</td> <td> 0.000</td> <td> -100.019</td> <td>  -99.988</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th>        <td>  250.0176</td> <td>    0.024</td> <td> 1.06e+04</td> <td> 0.000</td> <td>  249.971</td> <td>  250.064</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 2.306</td> <th>  Durbin-Watson:     </th> <td>   2.016</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.316</td> <th>  Jarque-Bera (JB):  </th> <td>   2.162</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.099</td> <th>  Prob(JB):          </th> <td>   0.339</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.112</td> <th>  Cond. No.          </th> <td>    144.</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       1.000\n",
       "Model:                            OLS   Adj. R-squared:                  1.000\n",
       "Method:                 Least Squares   F-statistic:                 1.381e+08\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):               0.00\n",
       "Time:                        16:43:35   Log-Likelihood:                -2108.0\n",
       "No. Observations:                1000   AIC:                             4224.\n",
       "Df Residuals:                     996   BIC:                             4244.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept      4.8343      0.386     12.509      0.000       4.076       5.593\n",
       "x1           100.0083      0.009   1.14e+04      0.000      99.991     100.026\n",
       "x2          -100.0038      0.008  -1.28e+04      0.000    -100.019     -99.988\n",
       "x3           250.0176      0.024   1.06e+04      0.000     249.971     250.064\n",
       "==============================================================================\n",
       "Omnibus:                        2.306   Durbin-Watson:                   2.016\n",
       "Prob(Omnibus):                  0.316   Jarque-Bera (JB):                2.162\n",
       "Skew:                           0.099   Prob(JB):                        0.339\n",
       "Kurtosis:                       3.112   Cond. No.                         144.\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.DataFrame({'x1':x_1,'x2':x_2,'x3':x_3,'y':y})\n",
    "model = ols(\"y ~ x1 + x2 + x3\", data=df)\n",
    "results = model.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.287</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.286</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   401.8</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 16 Dec 2020</td> <th>  Prob (F-statistic):</th> <td>2.28e-75</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>16:43:43</td>     <th>  Log-Likelihood:    </th> <td> -8408.1</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>  1000</td>      <th>  AIC:               </th> <td>1.682e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   998</td>      <th>  BIC:               </th> <td>1.683e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>    \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>-1238.1806</td> <td>  196.940</td> <td>   -6.287</td> <td> 0.000</td> <td>-1624.645</td> <td> -851.717</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x3</th>        <td>  257.6221</td> <td>   12.853</td> <td>   20.044</td> <td> 0.000</td> <td>  232.400</td> <td>  282.844</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>15.743</td> <th>  Durbin-Watson:     </th> <td>   1.927</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td>  16.049</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.304</td> <th>  Prob(JB):          </th> <td>0.000327</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.120</td> <th>  Cond. No.          </th> <td>    88.2</td>\n",
       "</tr>\n",
       "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.287\n",
       "Model:                            OLS   Adj. R-squared:                  0.286\n",
       "Method:                 Least Squares   F-statistic:                     401.8\n",
       "Date:                Wed, 16 Dec 2020   Prob (F-statistic):           2.28e-75\n",
       "Time:                        16:43:43   Log-Likelihood:                -8408.1\n",
       "No. Observations:                1000   AIC:                         1.682e+04\n",
       "Df Residuals:                     998   BIC:                         1.683e+04\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept  -1238.1806    196.940     -6.287      0.000   -1624.645    -851.717\n",
       "x3           257.6221     12.853     20.044      0.000     232.400     282.844\n",
       "==============================================================================\n",
       "Omnibus:                       15.743   Durbin-Watson:                   1.927\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               16.049\n",
       "Skew:                          -0.304   Prob(JB):                     0.000327\n",
       "Kurtosis:                       3.120   Cond. No.                         88.2\n",
       "==============================================================================\n",
       "\n",
       "Notes:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = ols(\"y ~ x3\", data=df)\n",
    "results = model.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f5f932b6eb0>"
      ]
     },
     "execution_count": 244,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x_2, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f5f9320ef10>"
      ]
     },
     "execution_count": 245,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x_1, x_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "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": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}