Recursive estimation of possibly misspecified MA(1) models: Convergence of a general algorithm

Abstract

We introduce a recursive algorithm of conveniently general form for estimating the coefficient of a moving average model of order one and obtain convergence results for both correct and misspecified MA(1) models. The algorithm encompasses Pseudolinear Regression (PLR--also referred to as AML and $\mbox{RML}_1$) and Recursive Maximum Likelihood ($\mbox{RML}_2$) without monitoring. Stimulated by the approach of Hannan (Hannan, E. J. (1980), Recursive estimation based on ARMA models, Ann. Statist.8 (4) 762–777, MR572620), our convergence results are obtained indirectly by showing that the recursive sequence can be approximated by a sequence satisfying a recursion of simpler (Robbins-Monro) form for which convergence results applicable to our situation have recently been obtained.