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

• Published in: Journal of time series analysis Vol. 38; no. 6; pp. 809 - 837
• Main Authors: Fermin, Lisandro Javier, Rios, Ricardo, Rodriguez, Luis Angel
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.11.2017
• Subjects: Algorithms; Autoregressive models; Autoregressive process; Autoregressive processes; Computer simulation; Consistency; Economic models; Markov analysis; Markov chains; Markov switching; Missing data; MOS subject classification: Primary: 60G17, Secondary:62G07; Nonparametric statistics; non‐parametric kernel estimation; Parameter estimation; Property; Robbins–Monro approximation; Switching
• ISSN: 0143-9782; 1467-9892
• DOI: 10.1111/jtsa.12237

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.