On nonlinear TAR processes and threshold estimation

TL;DR

This paper investigates the estimation of thresholds in nonlinear autoregressive time series with smooth innovations, deriving the asymptotic distribution of the Bayes estimator and demonstrating its efficiency.

Contribution

It introduces a method for threshold estimation in nonlinear TAR processes with non-Gaussian innovations and establishes the asymptotic properties of the Bayes estimator.

Findings

Bayes estimator's asymptotic distribution derived

Estimator attains minimax risk lower bound

Method applicable to nonlinear, non-Gaussian time series

Abstract

We consider the problem of threshold estimation for autoregressive time series with a "space switching" in the situation, when the regression is nonlinear and the innovations have a smooth, possibly non Gaussian, probability density. Assuming that the unknown threshold parameter is sampled from a continuous positive density, we find the asymptotic distribution of the Bayes estimator. As usually in the singular estimation problems, the sequence of Bayes estimators is found to be asymptotically efficient, attaining the minimax risk lower bound.