Testing for observation-dependent regime switching in mixture autoregressive models

TL;DR

This paper develops a likelihood ratio test for detecting observation-dependent regime switching in mixture autoregressive models, addressing complex nonstandard statistical challenges and demonstrating good finite sample performance.

Contribution

It introduces a novel testing framework for observation-dependent regime switching with a nonstandard asymptotic distribution, applicable to various dependence structures.

Findings

The test's asymptotic null distribution is derived and can be simulated.

Monte Carlo studies confirm the test's size and power are satisfactory.

The methodology applies to general mixture autoregressive models with dependent switching probabilities.

Abstract

Testing for regime switching when the regime switching probabilities are specified either as constants (`mixture models') or are governed by a finite-state Markov chain (`Markov switching models') are long-standing problems that have also attracted recent interest. This paper considers testing for regime switching when the regime switching probabilities are time-varying and depend on observed data (`observation-dependent regime switching'). Specifically, we consider the likelihood ratio test for observation-dependent regime switching in mixture autoregressive models. The testing problem is highly nonstandard, involving unidentified nuisance parameters under the null, parameters on the boundary, singular information matrices, and higher-order approximations of the log-likelihood. We derive the asymptotic null distribution of the likelihood ratio test statistic in a general mixture autoregressive setting using high-level conditions that allow for various forms of dependence of the regime switching probabilities on past observations, and we illustrate the theory using two particular mixture autoregressive models. The likelihood ratio test has a nonstandard asymptotic distribution that can easily be simulated, and Monte Carlo studies show the test to have satisfactory finite sample size and power properties.