Consistent boundaries for the one-step-ahead forecast error criterion and the AIC in vector autoregressions

We propose an upper bound for the asymptotic approximation of the one-step-ahead forecast mean squared error (MSE) in infinite-order vector autoregression (VAR) settings, i.e., VAR(infinity). Once minimized over a truncation-lag of small order o(T^(1/3)), where T is the sample size, it yields a consistent truncation of the autoregression associated with the efficient one-step forecast error covariance matrix. When the infinite-order process degenerates to a finite-order VAR, we show that the resulting truncation is strongly consistent (eventually asymptotically), given a parameter epsilon >= 2. We particularly note that when epsilon tends to infinity, our order-selection criterion (upper bound) becomes inconsistent, with a variant of it reducing to Akaike information criterion (AIC). Thus, unlike the final prediction error (FPE) criterion and AIC, our criteria have the good sampling property of being consistent, like those by Hannan and Quinn, and Schwarz, respectively. Compared to conventional criteria, our model-selection procedures not only better handle the multivariate dynamic structure of the time series data, through a compound penalty term that we specify, but also tend to avoid model overfitting in large samples, hence the singularity problems encountered in practice. Variants of our primary criterion, which are in small samples less parsimonious than AIC in large systems, are also proposed. Besides being strongly consistent asymptotically, they tend to select the actual data-generating process (DGP) most of the time in small samples, as shown with Monte Carlo (MC) simulations.