On adaptive estimation in nonstationary ARMA models with GARCH errors

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfying a generalised autore-gressive conditional heteroscedastic process. The locally asymptotic quadratic form of the log-likelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. Adaptivity is discussed and it is found that the parameters in the model are generally not adaptively estimable if the density of the rescaled error is asymmetric. For the model with symmetric density of the rescaled error, a new efficiency crite-rion is established for a class of defined Mν-estimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to b