Understanding output from statsmodels grangercausalitytests

Question

I'm new to Granger Causality and would appreciate any advice on understanding/interpreting the results of the python statsmodels output. I've constructed two data sets (sine functions shifted in time with noise added)

and put them in a "data" matrix with signal 1 as the first column and signal 2 as the second. I then ran the tests using:

granger_test_result = sm.tsa.stattools.grangercausalitytests(data, maxlag=40, verbose=True)`

The results showed that the optimal lag (in terms of the highest F test value) were for a lag of 1.

Granger Causality
('number of lags (no zero)', 1)
ssr based F test:         F=96.6366 , p=0.0000  , df_denom=995, df_num=1
ssr based chi2 test:   chi2=96.9280 , p=0.0000  , df=1
likelihood ratio test: chi2=92.5052 , p=0.0000  , df=1
parameter F test:         F=96.6366 , p=0.0000  , df_denom=995, df_num=1

However, the lag that seems to best describe the optimal overlap of the data is around 25 (in the figure below, signal 1 has been shifted to the right by 25 points):

Granger Causality
('number of lags (no zero)', 25)
ssr based F test:         F=4.1891  , p=0.0000  , df_denom=923, df_num=25
ssr based chi2 test:   chi2=110.5149, p=0.0000  , df=25
likelihood ratio test: chi2=104.6823, p=0.0000  , df=25
parameter F test:         F=4.1891  , p=0.0000  , df_denom=923, df_num=25

I'm clearly misinterpreting something here. Why wouldn't the predicted lag match up with the shift in the data?

Also, can anyone explain to me why the p-values are so small as to be negligible for most lag values? They only begin to show up as non-zero for lags greater than 30.

Thanks for any help you can give.

Answer 1

As stated here, in order to run a Granger Causality test, the time series' you are using must be stationary. A common way to achieve this is to transform both series by taking the first difference of each:

x = np.diff(x)[1:]
y = np.diff(y)[1:]

Here is the comparison of Granger Causality results at lag 1 and lag 25 for the similar dataset I generated:

Unchanged

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=19.8998 , p=0.0000  , df_denom=221, df_num=1
ssr based chi2 test:   chi2=20.1700 , p=0.0000  , df=1
likelihood ratio test: chi2=19.3129 , p=0.0000  , df=1
parameter F test:         F=19.8998 , p=0.0000  , df_denom=221, df_num=1

Granger Causality
number of lags (no zero) 25
ssr based F test:         F=6.9970  , p=0.0000  , df_denom=149, df_num=25
ssr based chi2 test:   chi2=234.7975, p=0.0000  , df=25
likelihood ratio test: chi2=155.3126, p=0.0000  , df=25
parameter F test:         F=6.9970  , p=0.0000  , df_denom=149, df_num=25

1st Difference

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=0.1279  , p=0.7210  , df_denom=219, df_num=1
ssr based chi2 test:   chi2=0.1297  , p=0.7188  , df=1
likelihood ratio test: chi2=0.1296  , p=0.7188  , df=1
parameter F test:         F=0.1279  , p=0.7210  , df_denom=219, df_num=1

Granger Causality
number of lags (no zero) 25
ssr based F test:         F=6.2471  , p=0.0000  , df_denom=147, df_num=25
ssr based chi2 test:   chi2=210.3621, p=0.0000  , df=25
likelihood ratio test: chi2=143.3297, p=0.0000  , df=25
parameter F test:         F=6.2471  , p=0.0000  , df_denom=147, df_num=25

I'll try to explain what is happening conceptually. Due to the series you are using having a clear trend in mean, the early lags at 1, 2, ... etc. all give significant predictive models in the F test. This is because you can negatively correlate the x values 1 lag a way with the y values very easily, due to the long term trend. Additionally (this one is more of an educated guess), I think the reason you see the F statistic for lag 25 very low compared to the early lags is that a lot of the variance explained by the x series is contained in the auto-correlation of y from lags 1-25, since the non-stationarity gives the auto correlation more predictive power.