VAR - VARX model selection Suppose I have three stationary economic time series $y_i$ that are not cointegrated and I want to investigate the relationship between them. I happen to be unsure about the "endogenousness" of $y_3$ so I first fit a VAR model with all three variables, but I also fit a VARX model with $y_1$ and $y_2$ endogenous but $y_3$ exogenous.
Can I use an information criterion like the AIC to help me select the better model? Or if not, what are my other options?

You cannot use AIC because the dependent variables of the two models are not the same: $(y_1,y_2,y_3)$ vs. $(y_1,y_2)$. However, you could test whether the lags of $(y_1,y_2)$ have zero coefficients in the equation for $y_3$ using an $F$-test. If you do not reject $H_0$ of the coefficients jointly being zero, you may consider $y_3$ exogenous in the system of $(y_1,y_2,y_3)$.
