Incorporating lag order selection uncertainty in parameter inference for AR models

Parameter inference on autoregressive models is usually carried out conditionally on a previously selected lag order. In the majority of cases the lag order selection is carried out using information criteria and in particular the Akaike [2nd International Symposium on Information Theory (1973) 267–281], Schwarz [Annuls of Statistics (1978) 461–464] or Hannan and Quinn [Journal of the Royal Statistical Society (Series B), 41 (1979) 190–195] criteria. It is well known that the latter two criteria are consistent in lag order selection in the sense of picking the true order of the system with probability one asymptotically. On the other hand, Akaike’s criterion is known to overestimate the lag order in this sense. In this note we discuss the asymptotic distribution, of the parameter estimates without conditioning on the lag order selected.