Determining the Number and Values of Thresholds for Multi-regime Threshold Ornstein–Uhlenbeck Processes

Abstract

The threshold Ornstein–Uhlenbeck process is a stochastic process that satisfies a stochastic differential equation with a drift term modeled as a piecewise linear function and a diffusion term characterized by a positive constant. This paper addresses the challenge of determining both the number and values of thresholds based on the continuously observed process. We present three testing algorithms aimed at determining the unknown number and values of thresholds, in conjunction with least squares estimators for drift parameters. The limiting distribution of the proposed test statistic is derived. Additionally, we employ sequential and global methods to determine the values of thresholds, and prove their weak convergence. Monte Carlo simulation results are provided to illustrate and support our theoretical findings. We utilize the model to estimate the term structure of US treasury rates and currency foreign exchange rates.