Should the RMSE of an unrestricted VAR model decrease as compared to a restricted Autoregression model when there is Granger Causality

Question

I have 2 time series, say for instance, T1 and T2. T1 granger causes T2 at lag 2. Should this mean that if I make a VAR model with these two time series, and an autoregression model with just T2, the RMSE from the former should necessarily be lesser than the latter?

Answer 1

Consider two time series $\{Y_1\}$ and $\{Y_2\}$ where $\{Y_1\}$ might be Granger-causing $\{Y_2\}$. To examine whether that is the case, you can compare two models for $\{Y_2\}$, one being a pure autoregression (AR) and another augmented with appropriate lags of $\{Y_1\}$ (ARDL). The latter model can also be seen as a single equation from a VAR model for both $\{Y_1\}$ and $\{Y_2\}$.

The augmented model will never have a higher in-sample RMSE (lower $R^2$), because in-sample RMSE can only decrease ($R^2$ can only increase) when we add regressors, and we got ARDL by adding regressors to AR. However, if there is no Granger causality from $\{Y_1\}$ to $\{Y_2\}$, then the difference in in-sample RMSEs ($R^2$s) between the two models will tend to be small. On the other hand, if Granger causality is present, the difference will tend to be larger. This insight leads to an $F$-test which can employ the two $R^2$ values for testing for presence of Granger causality.

Regarding out-of-sample RMSEs (root mean square forecast errors), the ARDL will still tend to deliver lower values than the AR in presence of Granger causality – unless the effect of Granger causality is estimated with substantial error relative to the effect size (which may happen when the effect is small and/or the estimation sample is short or unlucky). Due to randomness in future realizations of $\{Y_2\}$, there may be instances where the opposite is the case. Meanwhile, in absence of Granger causality, the ARDL will tend to deliver higher out-of-sample RMSEs. Again, due to randomness, there may be instances where the opposite is the case.