On adaptive estimation in nonstationary ARMA Models with GARCH errors

Abstract

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfyinga generalized autoregressive 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. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined $M_{\nu}$-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 be asymptotically efficient in the class of $M_{\nu}$-estimators. The paper includes the results that the stationary ARMA-GARCH model is LAN, and that the parameters in the model with symmetric density of the rescaled error are adaptively estimable after a reparameterization of the GARCH process. This paper also establishes the locally asymptotic quadratic form of the log-likelihood ratio for nonlinear time series models with ARCH-type errors.