On the Least Squares Estimation of Multiple-Threshold-Variable Autoregressive Models

Abstract

Most threshold models to-date contain a single threshold variable. However, in many empirical applications, models with multiple threshold variables may be needed and are the focus of this article. For the sake of readability, we start with the Two-Threshold-Variable Autoregressive (2-TAR) model and study its Least Squares Estimation (LSE). Among others, we show that the respective estimated thresholds are asymptotically independent. We propose a new method, namely the weighted Nadaraya-Watson method, to construct confidence intervals for the threshold parameters, that turns out to be, as far as we know, the only method to-date that enjoys good probability coverage, regardless of whether the threshold variables are endogenous or exogenous. Finally, we describe in some detail how our results can be extended to the K-Threshold-Variable Autoregressive (K-TAR) model, K > 2. We assess the finite-sample performance of the LSE by simulation and present two real examples to illustrate the efficacy of our modeling.

Keywords:

• Compound Poisson process
• Degeneracy of a spatial process
• Multiple threshold variables
• TAR model
• Weighted Nadaraya-Watson method

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The authors are very grateful to the two referees, the associate editor, and the coeditor for their constructive suggestions and comments, which have led to a substantial improvement of our article. Zhang acknowledges that her work was mainly carried out during studying in Tsinghua University and is from Chapter 2 of her PhD thesis.

Additional information

Funding

Li’s research is supported in part by the NSFC (No.71973077) and the Tsinghua University Initiative Scientific Research Program (No.2019Z07L01009).