Using Granger Causality Test to Know If One Time Series Is Impacting in Predicting Another?

Granger causality test is used to determine if one time series will be useful to forecast another variable by investigating causality between two variables in a time series. The method is a probabilistic account of causality; it uses observed data sets to find patterns of correlation. One good thing about time series vector autoregression (VAR) is that we could test ‘causality’ in some sense. This test is first proposed by Granger (1969), and therefore we refer to it as the Granger causality.

Simple Mechanism to define Granger Causality:

It is based on the idea that if X causes Y, then the forecast of Y based on previous values of Y AND the previous values of X should best result in the forecast of Y based on previous values of Y alone.

Granger causality should not be used to test if a lag of Y causes Y. Instead, it is generally used on exogenous (not Y lag) variables only. In simple terms ‘X is said to Granger-cause Y if Y can be better predicted using the histories of both X and Y than it can by using the history of Y alone’

When performing Granger Causality Test we need to consider two assumptions:

1. Future values cannot cause the past values.
2. A notably distinct information is contained in cause about effect which will not be available elsewhere

Python implementation of statsmodel package for the Granger test.

Lets do the code:

import pandas as pd
import numpy as np
import matplotlib 
import seaborn as sns
import random
import matplotlib.pyplot as plt
from dateutil.parser import parse
from scipy import signal
from scipy.interpolate import interp1d
from scipy import stats
from statsmodels.tsa.stattools import adfuller, kpss, acf, pacf, grangercausalitytests
from statsmodels.nonparametric.smoothers_lowess import lowess
from statsmodels.tsa.seasonal import seasonal_decompose
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
from sklearn.metrics import mean_squared_error
%matplotlib inline
# Upload Data in a DataFrame
df = pd.read_csv('IxR_Data.csv', parse_dates=['Value_Date'])
df.set_index(['Value_Date'])
# Get few records
df.head()

We will do some EDA …. Not relevent at this to show here. But Just wanted to show you that how the series are behaving in data time series

ax = sns.lineplot(x="Value_Date", y="IxD", data=df)
ax1 = sns.lineplot(x="Value_Date", y='RxB', data=df)

We can see from the above chart that both Time Series move (not so) simultaneously, and lead to the inter relationship can be use for the prediction of any variable.

Now we are going to use Gregner Test on the data series

res = grangercausalitytests(df[[‘IxD’, ‘RxB’]], maxlag=15)

Results:

Granger Causality
 number of lags (no zero) 1
 ssr based F test: F=13.3271 , p=0.0003 , df_denom=519, df_num=1
 ssr based chi2 test: chi2=13.4041 , p=0.0003 , df=1
 likelihood ratio test: chi2=13.2349 , p=0.0003 , df=1
 parameter F test: F=13.3271 , p=0.0003 , df_denom=519, df_num=1
 
 Granger Causality
 number of lags (no zero) 2
 ssr based F test: F=13.4540 , p=0.0000 , df_denom=516, df_num=2
 ssr based chi2 test: chi2=27.1687 , p=0.0000 , df=2
 likelihood ratio test: chi2=26.4840 , p=0.0000 , df=2
 parameter F test: F=13.4540 , p=0.0000 , df_denom=516, df_num=2
 
 Granger Causality
 number of lags (no zero) 3
 ssr based F test: F=8.9401 , p=0.0000 , df_denom=513, df_num=3
 ssr based chi2 test: chi2=27.1863 , p=0.0000 , df=3
 likelihood ratio test: chi2=26.4994 , p=0.0000 , df=3
 parameter F test: F=8.9401 , p=0.0000 , df_denom=513, df_num=3
 
 Granger Causality
 number of lags (no zero) 4
 ssr based F test: F=6.7321 , p=0.0000 , df_denom=510, df_num=4
 ssr based chi2 test: chi2=27.4037 , p=0.0000 , df=4
 likelihood ratio test: chi2=26.7047 , p=0.0000 , df=4
 parameter F test: F=6.7321 , p=0.0000 , df_denom=510, df_num=4
 
 Granger Causality
 number of lags (no zero) 5
 ssr based F test: F=5.8029 , p=0.0000 , df_denom=507, df_num=5
 ssr based chi2 test: chi2=29.6440 , p=0.0000 , df=5
 likelihood ratio test: chi2=28.8268 , p=0.0000 , df=5
 parameter F test: F=5.8029 , p=0.0000 , df_denom=507, df_num=5
 
 Granger Causality
 number of lags (no zero) 6
 ssr based F test: F=5.0143 , p=0.0001 , df_denom=504, df_num=6
 ssr based chi2 test: chi2=30.8620 , p=0.0000 , df=6
 likelihood ratio test: chi2=29.9760 , p=0.0000 , df=6
 parameter F test: F=5.0143 , p=0.0001 , df_denom=504, df_num=6
 
 Granger Causality
 number of lags (no zero) 7
 ssr based F test: F=4.3764 , p=0.0001 , df_denom=501, df_num=7
 ssr based chi2 test: chi2=31.5520 , p=0.0000 , df=7
 likelihood ratio test: chi2=30.6250 , p=0.0001 , df=7
 parameter F test: F=4.3764 , p=0.0001 , df_denom=501, df_num=7
 
 Granger Causality
 number of lags (no zero) 8
 ssr based F test: F=3.8112 , p=0.0002 , df_denom=498, df_num=8
 ssr based chi2 test: chi2=31.5303 , p=0.0001 , df=8
 likelihood ratio test: chi2=30.6027 , p=0.0002 , df=8
 parameter F test: F=3.8112 , p=0.0002 , df_denom=498, df_num=8
 
 Granger Causality
 number of lags (no zero) 9
 ssr based F test: F=3.4022 , p=0.0005 , df_denom=495, df_num=9
 ssr based chi2 test: chi2=31.7947 , p=0.0002 , df=9
 likelihood ratio test: chi2=30.8501 , p=0.0003 , df=9
 parameter F test: F=3.4022 , p=0.0005 , df_denom=495, df_num=9
 
 Granger Causality
 number of lags (no zero) 10
 ssr based F test: F=3.0085 , p=0.0011 , df_denom=492, df_num=10
 ssr based chi2 test: chi2=31.3696 , p=0.0005 , df=10
 likelihood ratio test: chi2=30.4478 , p=0.0007 , df=10
 parameter F test: F=3.0085 , p=0.0011 , df_denom=492, df_num=10
 
 Granger Causality
 number of lags (no zero) 11
 ssr based F test: F=2.9947 , p=0.0007 , df_denom=489, df_num=11
 ssr based chi2 test: chi2=34.4912 , p=0.0003 , df=11
 likelihood ratio test: chi2=33.3791 , p=0.0005 , df=11
 parameter F test: F=2.9947 , p=0.0007 , df_denom=489, df_num=11
 
 Granger Causality
 number of lags (no zero) 12
 ssr based F test: F=2.7394 , p=0.0013 , df_denom=486, df_num=12
 ssr based chi2 test: chi2=34.5641 , p=0.0005 , df=12
 likelihood ratio test: chi2=33.4453 , p=0.0008 , df=12
 parameter F test: F=2.7394 , p=0.0013 , df_denom=486, df_num=12
 
 Granger Causality
 number of lags (no zero) 13
 ssr based F test: F=2.6577 , p=0.0013 , df_denom=483, df_num=13
 ssr based chi2 test: chi2=36.4818 , p=0.0005 , df=13
 likelihood ratio test: chi2=35.2360 , p=0.0008 , df=13
 parameter F test: F=2.6577 , p=0.0013 , df_denom=483, df_num=13
 
 Granger Causality
 number of lags (no zero) 14
 ssr based F test: F=2.4680 , p=0.0022 , df_denom=480, df_num=14
 ssr based chi2 test: chi2=36.6399 , p=0.0008 , df=14
 likelihood ratio test: chi2=35.3812 , p=0.0013 , df=14
 parameter F test: F=2.4680 , p=0.0022 , df_denom=480, df_num=14
 
 Granger Causality
 number of lags (no zero) 15
 ssr based F test: F=2.3488 , p=0.0029 , df_denom=477, df_num=15
 ssr based chi2 test: chi2=37.5225 , p=0.0011 , df=15
 likelihood ratio test: chi2=36.2014 , p=0.0017 , df=15
 parameter F test: F=2.3488 , p=0.0029 , df_denom=477, df_num=15

Lets analyze the results:

The Test was performed with 15 Lags!

Simple notion:

X(t) granger causes Y(t) , if the past values of X(t) helps in predicting the future values of Y(t).

Y(t) is not just the function of Y( t -1 ) lag of 1 but also the function of X( t - 1)

number of lags (no zero) [xx]

Shows the Lags used in finding causility.

p values of all (lag 1 - 15)

Both p-values of every lag is same, show both-directional Granger-cause.

Granger Causality
 number of lags (no zero) 2
 ssr based F test: F=13.4540

Lag 2 show the highest F test value out of all the lags. F-test checks that the lagged values of X jointly improve the forecast of Y (or vice versa). It can conclude that for predicting Y with two predictors X1 and X2 where X2 is same X1 with small noised values.

Few Quick Points

• Testing for Granger-causality using F-statistics when one or both time series are non-stationary can lead to nearly false causality. If both the time series are NOT stationary then differencing, de-trending or other techniques must first be employed before using the Granger Causality test.
• F statistic can be used only when variables are cointegrated.
• We say that x Granger-causes y when the null hypothesis is rejected.
• The null hypothesis for the test is that lagged X-values do not explain the variation in Y. Put simple, it assumes that X(t) doesn’t Granger-cause Y(t).
• The Toda–Yamamoto approach is used to test causality for co-integrated time-series data.
• Cointegration analysis is a useful tool in order to examine if there exists a long run equilibrium relationship between two or more time series. (For example: nominal interest rate is connected with an increase on expected inflation in the long run. )
• Causation is can be One-Direction, Both-Direction or NO-Direction