Time series analysis # 6 Spurious regression and cointegration

1. Overview

2. Spurious regression

Definition

There seems to be a significant relationship between $ x_t $ and $ y_t $ when regressing $ y_t = \ alpha + \ beta x_t + \ epsilon_t $ for two unrelated unit root processes $ x_t $ and $ y_t $. The phenomenon that looks like is called spurious regression.

Verification

#Data generation
sigma_x, sigma_y = 1, 2
T = 10000
xt = np.cumsum(np.random.randn(T) * sigma_x).reshape(-1, 1)
yt = np.cumsum(np.random.randn(T) * sigma_y).reshape(-1, 1)

save.png

from sklearn.linear_model import LinearRegression
reg = LinearRegression().fit(xt,yt)
print('R-squared : ',reg.score(xt,yt))
print('coef : ',reg.coef_, 'intercept', reg.intercept_)

R-squared : 0.4794854506874714 coef : [[-0.62353254]] intercept [-24.27600549]

import statsmodels.api as sm
reg = sm.OLS(yt,sm.add_constant(xt,prepend=False)).fit()
reg.summary()
Dep. Variable: y R-squared: 0.479
Model: OLS Adj. R-squared: 0.479
Method: Least Squares F-statistic: 9210.
Date: Tue, 07 Jan 2020 Prob (F-statistic): 0.00
Time: 22:36:57 Log-Likelihood: -51058.
No. Observations: 10000 AIC: 1.021e+05
Df Residuals: 9998 BIC: 1.021e+05
Df Model: 1
Covariance Type: nonrobust
coef std err t P>abs(t) [0.025 0.975]
const -24.2760 0.930 -26.113 0.000 -26.098 -22.454
x1 -0.6235 0.006 -95.968 0.000 -0.636 -0.611

How to avoid

Include lag variables in the model

x_t, y_t, y_t_1 = xt[1:], yt[1:], yt[:-1]
X = np.column_stack((x_t, y_t_1))
reg = sm.OLS(y_t,sm.add_constant(X)).fit()
reg.summary()
Dep. Variable: y R-squared: 0.999
Model: OLS Adj. R-squared: 0.999
Method: Least Squares F-statistic: 3.712e+06
Date: Thu, 09 Jan 2020 Prob (F-statistic): 0.00
Time: 22:12:59 Log-Likelihood: -21261.
No. Observations: 9999 AIC: 4.253e+04
Df Residuals: 9996 BIC: 4.255e+04
Df Model: 2
Covariance Type: nonrobust
coef std err t P>abs(t) [0.025 0.975]
const -0.0815 0.049 -1.668 0.095 -0.177 0.014
x1 -0.0004 0.000 -0.876 0.381 -0.001 0.000
x2 0.9989 0.001 1964.916 0.000 0.998 1.000

Regression after taking the difference of the unit root process and making it a stationary process

x_t, y_t = np.diff(xt.flatten()).reshape(-1,1), np.diff(yt.flatten()).reshape(-1,1)
reg = sm.OLS(y_t,sm.add_constant(x_t)).fit()
reg.summary()
Dep. Variable: y R-squared: 0.000
Model: OLS Adj. R-squared: 0.000
Method: Least Squares F-statistic: 3.297
Date: Thu, 09 Jan 2020 Prob (F-statistic): 0.0694
Time: 22:33:26 Log-Likelihood: -21262.
No. Observations: 9999 AIC: 4.253e+04
Df Residuals: 9997 BIC: 4.254e+04
Df Model: 1
Covariance Type: nonrobust
coef std err t P>abs(t) [0.025 0.975]
const -0.0138 0.020 -0.681 0.496 -0.054
x1 -0.0374 0.021 -1.816 0.069 -0.078

3. Cointegration

Definition

Implication

Granger Representation theorem

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