A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency - Fermin, Lisandro Javier; Rios, Ricardo; Rodriguez, Luis Angel (Journal of time series analysis, 2017)
We approach the problem of non‐parametric estimation for autoregressive Markov switching processes. In this context, the Nadaraya–Watson‐type regression functions estimator is interpreted as a solution of a local weighted least‐square problem, which does not admit a closed‐form solution in the case of hidden Markov switching. We introduce a non‐parametric recursive algorithm to approximate the estimator. Our algorithm restores the missing data by means of a Monte Carlo step and estimates the regression function via a Robbins–Monro step. We prove that non‐parametric autoregressive models with Markov switching are identifiable when the hidden Markov process has a finite state space. 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 non‐parametric estimation procedure.