Consistent order selection with strongly dependent data and its application to efficient estimation

Order selection based on criteria by Akaike (IEEE Trans. Automat. Control AC-19 (1974) 716), AIC, Schwarz (Ann. Stat. (1978) 461), BIC or Hannan and Quinn's (J. R. Stat. Soc. Ser. B (1979) 190) HIC is often applied in empirical examples. They have been used in the context of order selection of weakly dependent ARMA models, AR models with unit or explosive roots and in the context of regression or distributed lag regression models for weakly dependent data. On the other hand, it has been observed that data exhibits the so-called strong dependence in many areas. Because the interest to this type of data, our main objective in this paper is to examine order selection for a distributed lag regression model that covers in a unified form weak and strong dependence. To that end, and because the possible adverse properties of the aforementioned criteria, we propose a criterion function based on the decomposition of the variance of the innovations of the model in terms of their frequency components. Assuming that the order of the model is finite, say p₀, we show that the proposed criterion consistently estimates p₀. In addition, we show that adaptive estimation for the parameters of the model is possible without knowledge of p₀. Finally, a small Monte-Carlo experiment is included to illustrate the finite sample performance of the proposed criterion.