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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?

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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 $(y_1,y_2,y_3)$.

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  • $\begingroup$ Thanks for the response. I take your point with doing a Granger-causality test in order to determine the exogeneity of $y_3$. But can the AIC really not be used to compare models with different dependent variables? Or is the point here that we just have a different number of equations (3 vs. 2)? I would be grateful for additional details. $\endgroup$
    – Anthony
    Commented Aug 18, 2021 at 8:47
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    $\begingroup$ @Anthony, we have several threads discussing the prerequisites for model comparison with AIC, e.g. "Prerequisites for AIC model comparison" (but there are more). Having the exact same observations on which the model's likelihood is evaluated is key. If you add or remove dependent variables, this is violated, making AICs incomparable across models. $\endgroup$
    – Richard Hardy
    Commented Aug 18, 2021 at 9:04
  • $\begingroup$ This helps, thanks! $\endgroup$
    – Anthony
    Commented Aug 18, 2021 at 9:32
  • $\begingroup$ @Anthony, you are welcome! $\endgroup$
    – Richard Hardy
    Commented Aug 18, 2021 at 9:34

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