What happens if a VAR model includes a series that is not stationary? What effects does it have on its results and impulse response functions? I am writing VAR models and can't seem to figure out why a VAR model need to be stationary.
A modest answer:
Stability implies stationarity. So if a model is not stationary, it's not stable, which implies it might eventually blow-up. What would be the interest in models with such characteristic? If a VAR is unstable, it might happen that your impulse-response functions will not converge to zero, implying the shocks would have permanent effects, which is unreasonable in most cases.
Specifically, [weak] stationarity implies both the mean and covariance of the process do not change over time, which is a basic assumption for the estimation of the classical VAR model. Finally, for a VAR model to be stationary, it's a necessary condition that all of your series are.
Hint: A way to relax such hypothesis is, for example, switch to cointegration (VECM) models.