Should the RMSE of the unrestricted (VAR) model for a time series that is being Granger caused by another be lesser than its restricted counterpart?

Question

I have a couple of time series, say, T1 and T2. I have established (using the grangercausalitytest library of Statsmodels in Python) that T1 Granger-causes T2 at 10% significance for lags 4 and 5. I now want to know whether I can validate the result by using VAR (unrestricted model).

Going by the definition of Granger causality mentioned in the book "Introduction to Modern Time Series Analysis", for 2 time series, "x" and "y",

...if future values of y can be better predicted, i.e. with a smaller forecast error variance, if current and past values of x are used,

I should get less RMSE for the VAR model as compared to the Autoregression model with only T2. (I'm using ARIMA(p,0,0) for autoregression with p = 4,5 as the Granger - cause occurs at these lags)

My first question is my interpretation correct?

If not, why not? And if that is the case, the RMSE of what should I be comparing? I ask this question because T1 Granger-causes T2 at lags 4 and 5. Should I take the RMSE (for both VAR and Autoregression models) after lags of 4?

Also, should there be any other caveats?

Answer 1

Let me denote the first time series $\{Y_1\}$ and the second one $\{Y_2\}$.

Interpretation
Your interpretation is correct.

RMSE of what?
You may compare the RMSE of one-step ahead forecasts from the AR(p) model for $\{Y_2\}$ with one of the ARDL(p,p) model for $\{Y_2\}$ for either $p=4$ or $p=5$. The ARDL(p,p) model corresponds to the equation for $\{Y_2\}$ from the VAR(p) model that includes both $\{Y_1\}$ and $\{Y_2\}$. I think RMSE of $h$-step ahead forecast for $h>1$ would also work but may be a less common choice.

For more details on the logic of comparing AR and ARDL models in the context of Granger causality, see this.