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I have a set of multivariate time series observations that I am trying to model using VAR processes, using AIC to choose the best model. However, instead of determining the best model order for each individual sample (itself multivariate), I would like to figure out the best model order for fitting all the samples. However, each sample will still have unique parameters.

In other words, how can multiple AICs be combined?

More information:

All criteria [AIC, HQC, SWC] add a penalty to the one-step ahead MSE which depends on the sample size $T$, the number of variables $m$ and the number of lags $q$.

(F. Canova, Methods for Applied Macroeconomic Research)

I have $N$ samples, all with the same number of variables $m$, all cropped to identical length $T$. I would like to compute the pool information criterion for some lag $q$. I can easily sum or average the one-step ahead MSEs for each sample. But what is the best way to pool the penalty terms? The naive approaches would be to either sum (/average) the penalty terms or just apply a single penalty term. The latter seems to capture the situation where you are fitting the same VAR model to each sample. Does the former capture the situation where you are allowing the parameters to vary for each sample but keeping the number of lags $q$ constant?

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