# F-test and Wald test in time series

Assume we have the time series $$X_{1,t}$$ and $$X_{2,t}$$ where $$X_{i,t}=(x_{i,1},...,x_{i,T})$$. We build two models as (Assume the time series are weakly stationary): \begin{align} X_{1,t}=\sum_{\tau=1}^{p_1} a_{\tau} X_{1,t-\tau}+\epsilon_{1,t} \end{align}

\begin{align} X_{1,t}=\sum_{\tau=1}^{p_1} a_{\tau} X_{1,t-\tau}+\sum_{\tau=1}^{p_2} b_{\tau} X_{2,t-\tau}+\epsilon_t \end{align}

The first one is an $$AR(p_1)$$ model and the second one is also an $$AR(p_1)$$ model but with the exogenous variable $$X_j$$ (These are the two models that is built to check the Granger-causality). We obtain variances for first one, let's call it $$\sigma^2_{unexp}$$, meaning the unexplained variance and the second one $$\sigma^2_{exp}$$, the explained var.

1. Now, we want to see if $$\sigma^2_{unexp}$$ is smaller than $$\sigma^2_{exp}$$, meaning we have an improved model. The question is: Is there a difference in performing F-test or Wald test in this case? Is it the same as one way ANOVA? How is the sample variances calculated for the F-statistics?
2. Secondly, we can asses if there's improvement, also by looking at the coefficients $$b_{\tau}$$. If the coefficients $$b_{\tau}$$ are jointly significantly different from zero, then we have an improvement. What is the test for this (I think it's still F-test) and what are the steps?

I apologize if it's a lot to ask but I just want to get a clear image of the whole process, which seems to be found nowhere!