Interpreting why a VAR produces lower error than VARMA? I trained various VARMA models on the same dataset consisting of different number of AR and MA terms, from $VARMA(0,1)$ and $VARMA(1,0)$ to $VARMA(6,6)$ and all the combinations in-between.
After evaluating all the model configurations, the top five best performing models based on $AIC$ and $BIC$ criterions had no MA terms, i.e. $VARMA(p, 0)$ for several different p.
How should one interpret why MA terms did not bring any performance gain and were in fact detrimental to performance? $VAR$ models tended to perform better than $VARMA$ on my dataset. The dataset consisted of commodity prices.
 A: Generally, VARMA is more parsimonious than VAR, so theoretically it should beat VAR in terms of AIC and BIC in some if not most instances. Why is this not happening?
Features of the data
Perhaps the data generating process is close to a very simple process such as a random walk. Some commodity prices are not very far from that. Then a model as parsimonious as a VAR(1) (or a VAR(0) for first differences of the data) might be all you need.
Estimation of VAR and VARMA
VAR and VARMA are usually estimated in different ways. It may be that the estimation precision of VAR is better than of VARMA for your sample size.
Comparability of AIC and BIC values
Are the AIC and BIC values comparable between VAR and VARMA? AIC and BIC involve likelihoods, and likelihoods often involve constants that get treated differently in different software implementations. This is because they do not matter when comparing models estimated using the same implementation. However, that might cause problems across implementations.
Moreover, some implementations report average likelihood (averaged over the sample) while other report regular likelihood.
Also, even comparing VAR(1) with VAR(2) can be problematic depending on which data points exactly the likelihoods are evaluated on. For a VAR(1), the likelihood may be estimated on all but the first data point. For a VAR(2), it may be the first two points that are dropped. Such treatment would make the likelihoods and the AIC and BIC based on them incomparable.
