Convergence of Moments of Least Squares Estimators for the Coefficients of an Autoregressive Process of Unknown Order

Abstract

Given a realization of $T$ consecutive observations of a stationary autoregressive process of unknown, possibly infinite, order $m$, it is assumed that a process of arbitrary finite order $p$ is fitted by least squares. Under appropriate conditions it is known that the estimators of the autoregressive coefficients are asymptotically normal. The question considered here is whether the moments of the (scaled) estimators converge, as $T \rightarrow \infty$, to the moments of their asymptotic distribution. We establish a general result for stationary processes (valid, in particular, in the Gaussian case) which is sufficient to imply this convergence.