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I'm doing some studies on variables (included their lags) that could influence the response variable (y1). The model that best fits the data is a VAR model (Vector Autoregressive Model) including three delays.

Header data:

date       y1       x1       x2       x3
2015-01-01 27.21140 25.09518 22.89878 23.91072
2015-02-01 27.17886 25.24704 22.95191 24.14761
2015-03-01 27.17886 25.47703 23.30820 24.32759
2015-04-01 27.33185 25.25882 23.06275 24.38328
2015-05-01 27.33185 25.70178 23.34213 24.52245
2015-06-01 27.50768 25.61009 23.50188 24.58411
2015-07-01 27.50768 25.26646 23.18284 24.34915
2015-08-01 27.47379 25.04455 23.02650 24.49288
2015-09-01 27.47379 25.27471 23.27283 24.46413
2015-10-01 27.29188 25.77447 23.20467 24.73335
2015-11-01 27.29188 25.46621 23.15823 24.37441
2015-12-01 27.28285 25.23452 23.17160 24.46506
2016-01-01 27.28285 25.53932 23.12888 24.57102
2016-02-01 27.24459 25.22238 23.11612 24.37433
2016-03-01 27.24459 24.93847 23.43555 24.40179
2016-04-01 27.33247 25.61391 23.59323 24.58246
2016-05-01 27.33247 25.20036 23.46443 24.40286
2016-06-01 27.54237 25.32496 23.69215 24.35019
2016-07-01 27.54237 25.16219 23.73906 25.28031
2016-08-01 27.56065 25.28213 23.87010 25.24637
2016-09-01 27.56065 25.24242 23.51359 25.19638
2016-10-01 27.34306 25.07353 23.38320 24.82677
2016-11-01 27.34306 25.33576 23.44870 24.55016

I was surprised so an autoregressive model, taking into account only variable y1 and their lags, did not fit well. However, choosing the variables x1, x2 and x3 and their delays for VAR model, fit perfectly to the original time series.

Causality or coincidence? I applied Granger Causality Test (page 9) and I reject the null hypothesis of no Granger causality.

Should I do something else? Test for residuals, for example? The model is well fitted, so I would like to make predictions.

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  • $\begingroup$ For forecasting, you do not necessarily need causality. To borrow the example of Hugh Perkins, you can predict rain from people carrying umbrellas even though umbrellas do not cause rain. $\endgroup$ – Richard Hardy Mar 23 '17 at 14:12
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You cannot establish causality without changing something, ideally randomly. Otherwise, there might be an underlying latent factor that correlates with both of your variables, and you'd never know. For example, you see people carrying umbrellas, and later on in the day it rains. Should we ban people carrying umbrellas, to stop it raining? :-P

To test the hypothesis "carrying umbrellas causes it to rain", you'd need to somehow arrange that umbrellas are arbitrarily banned on some days, and arbitrarily enforced on others.

Of course, this is expensive. And hard...

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