A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency

Abstract

We consider nonparametric estimation for autoregressive processes with Markov switching. In this context, the Nadaraya-Watson type estimator of regression funtions is interpreted as solution of a local weighted least-square problem, which does not closed-form solution in the case of hidden Markov switching. We introduce a nonparametric recursive algorithm to approximate the estimator. Our algorithm restores the missing data by means of a Monte-Carlo step and estimate the regression function via a Robbins-Monro step. Consistency of the estimator is proved using the strong α-mixing property of the model. Finally, we present some simulations illustrating the performances of our nonparametric estimation procedure.