Bootstrapping ARMA time series models after model selection

Inference after model selection is a very important problem. This article derives the asymptotic distribution of some model selection estimators for autoregressive moving average time series models. Under strong regularity conditions, the model selection estimators are asymptotically normal, but generally the asymptotic distribution is a non normal mixture distribution. Hence bootstrap confidence regions that can handle this complicated distribution were used for hypothesis testing. A bootstrap technique to eliminate selection bias is to fit the model selection estimator β̂MS∗ to a bootstrap sample to find a submodel, then draw another bootstrap sample, and fit the same submodel to get the bootstrap estimator β̂MIX∗.