Generalized information criterion for the AR model

This paper studies the generalized information criterion (GIC) for the problem of subset selection in the autoregressive (AR) model under the condition that some of the parameters are irrelevant to the AR model. We prove that the resulting estimator obtained by minimizing the GIC is consistent in subset selection under the mild conditions. Here, the consistency means that the probability of obtaining true sub-model increases to one asymptotically. The results hold when the numbers of the relevant and irrelevant parameters increase to infinity and when the minimum of the absolute values of nonzero parameters decreases toward zero as the sample size increases. A practical method to reduce the computation time is introduced by applying the penalized estimation, which is useful when the order of AR model is very large. Simulation studies are presented to confirm the theoretical results for finite samples. As a real life example, we apply the GIC to annual total sunspot numbers.