Model selection for integrated autoregressive processes of infinite order

Choosing good predictive models is an important ingredient in a great deal of statistical research. When the true model is relatively simple and can be parameterized by a prescribed finite set of parameters whose values are unknown, it is natural to ask whether a model selection criterion can exclude all redundant parameters, thereby achieving prediction efficiency through the most parsimonious correct model. A model selection criterion is said to be consistent if it can identify this ideal model with probability approaching 1 as the number of observations, n, goes to ∞. In the case of finite-order stationary autoregressive (AR) processes, Hannan and Quinn (1979) showed that BIC (Schwarz, 1978) and HQIC (Hannan and Quinn, 1979) are consistent. Tsay (1984) subsequently verified that the consistency of BIC and HQIC carries over to nonstationary AR process of finite order.