QADF

The module provides access to a quantile augmented Dickey-Fuller unit root procedure. The test performs unit root quantile autoregression inference following the Koenker and Xiao (2004) methodology.

Parts of the code/logic such as formulas for the bandwidth and critical values (Hansen, 1995) were written using the qradf function (by Saban Nazlioglu) in the Aptech GAUSS tspdlib library as a reference.

Usage

y = pd.read_csv('data/examples.csv', index_col=0, squeeze=True)
y.index = pd.date_range('2000-01-01', '2020-11-01', freq='MS')
qADF = QADF(y, model='c', pmax=5, ic='AIC')
qADF.fit(tau=0.42)
qADF.summary()

quantile: 0.42
Lags: 2
α₀(τ): 2.15
ρ₁(τ): 0.973
ρ₁(OLS): 0.968
δ²: 0.278
Half-lives: 25.461
tₙ(τ): -2.916
CV10%: -2.0271
CV5%: -2.3735
CV1%: -3.0307

Notebook example

import numpy as np
import pandas as pd
import time
import matplotlib.pyplot as plt
from quantileADF import QADF, bootstraps

import warnings
from statsmodels.tools.sm_exceptions import IterationLimitWarning
warnings.simplefilter('ignore', IterationLimitWarning)

y = pd.read_csv('data/examples.csv', index_col=0, squeeze=True)
y.index = pd.date_range('2000-01-01', '2020-11-01', freq='MS')

y.head()

2000-01-01    74.340017
2000-02-01    75.456058
2000-03-01    78.506961
2000-04-01    80.907992
2000-05-01    84.377459
Freq: MS, Name: Ukraine, dtype: float64

qADF = QADF(y, model='c', pmax=5, ic='AIC')
qADF.fit(tau=0.42)
qADF.summary()

quantile: 0.42
Lags: 2
α₀(τ): 2.15
ρ₁(τ): 0.973
ρ₁(OLS): 0.968
δ²: 0.278
Half-lives: 25.461
tₙ(τ): -2.916
CV10%: -2.0271
CV5%: -2.3735
CV1%: -3.0307

quantiles = np.arange(0.1, 1, 0.1)
CountryQADF = qADF.fitForQuantiles(quantiles)
CountryQADF

	Lags	α₀(τ)	ρ₁(τ)	ρ₁(OLS)	δ²	Half-lives	tₙ(τ)	CV10%	CV5%	CV1%	QKS	name
quantile
0.1	2	-2.304	1.002	0.968	0.328	∞	0.058	-2.0885	-2.4298	-3.0839	2.681	Ukraine
0.2	2	-0.096	0.991	0.968	0.307	73.523	-0.558	-2.0646	-2.4066	-3.0678	2.681	Ukraine
0.3	2	1.330	0.978	0.968	0.289	31.646	-2.061	-2.0415	-2.3858	-3.0460	2.681	Ukraine
0.4	2	2.021	0.974	0.968	0.280	26.503	-2.681	-2.0288	-2.3749	-3.0325	2.681	Ukraine
0.5	2	2.219	0.975	0.968	0.277	27.527	-2.512	-2.0251	-2.3718	-3.0285	2.681	Ukraine
0.6	2	2.597	0.976	0.968	0.266	28.12	-2.016	-2.0096	-2.3585	-3.0121	2.681	Ukraine
0.7	2	3.391	0.972	0.968	0.250	24.476	-2.058	-1.9867	-2.3388	-2.9878	2.681	Ukraine
0.8	2	2.856	0.986	0.968	0.228	48.461	-0.969	-1.9559	-2.3123	-2.9550	2.681	Ukraine
0.9	2	4.874	0.973	0.968	0.168	24.866	-1.232	-1.8653	-2.2276	-2.8731	2.681	Ukraine

repetitions = 1000
boots = bootstraps(y, lags=2, n_replications=repetitions)

pd.DataFrame(boots).plot(kind='line', legend=False, figsize=(10,5))
y.plot(legend=True, color='black')
plt.show()

%%time
results = pd.concat([qADF.setup(boots[yStar]).fitForQuantiles(quantiles) for yStar in boots])

CPU times: user 19min 33s, sys: 12min 33s, total: 32min 6s
Wall time: 18min 2s

cvQKS = results.groupby('name')['QKS'].mean()
cvQKS.plot(kind='hist')
t = CountryQADF['QKS'][0.1]
conf_int = np.percentile((cvQKS), [90, 95, 99])
pValue = (cvQKS >= t).mean()

plt.show()
print('tₙ(τ):', CountryQADF['QKS'][0.1])
print('Confidence interval:',conf_int)
print('p-value:', pValue)

tₙ(τ): 2.681
Confidence interval: [2.7542  3.1293  3.74112]
p-value: 0.115