On the Strong Approximation of the Distributions of Estimators in Linear Stochastic Models, I and II: Stationary and Explosive AR Models

Abstract

It is shown that in the stationary autoregressive case (Part I [Sections 1-5)] the distributions of least squares estimators and their close relatives such as sample autocovariances and the recently introduced general $M$-estimators converge to suitable Gaussian distributions uniformly over all Borel sets and uniformly over suitable neighborhoods of the parameter. Specifically, the notion of strongly asymptotically shift equivariance is introduced and it is shown that the distributions of any estimators satisfying this asymptotic equivariance condition converge in the preceding strong sense whenever they converge weakly (in law), provided the likelihood function of the sample is appropriately smooth. This smoothness of the likelihood is verified under mild conditions. Then, restricted to a broad class of models which include autoregressive models, a more easily verifiable condition implying the aforementioned asymptotic equivariance is derived and is shown to be satisfied by the estimators mentioned earlier. The methods used in the present paper are different from the usual method of characteristic functions; some indications of their possible wider scope are given. In Part II (Sections 6-8) the explosive autoregressive model is considered and a simple extension of the preceding result is applied to show that the least squares estimators converge in the preceding strong sense under suitable random or nonrandom normalization.