Threshold model with a time‐varying threshold based on Fourier approximation

Abstract

Classical threshold models assume that threshold values are constant and stable, which appears overly restrictive and unrealistic. In this article, we extend Hansen's (2000) constant threshold regression model by allowing for a time‐varying threshold which is approximated by a Fourier function. Least‐square estimation of regression slopes and the time‐varying threshold is proposed, and test statistics for the existence of threshold effect and threshold constancy are constructed. We also develop the asymptotic distribution theory for the time‐varying threshold estimator. Through Monte Carlo simulations, we show that the proposed estimation and testing procedures work reasonably well in finite samples, and there is little efficiency loss by the allowance for Fourier approximation in the estimation procedure even when there is no time‐varying feature in the threshold. On the contrary, the slope estimates are seriously biased when the threshold is time‐varying but being treated as a constant. The model is illustrated with an empirical application to a nonlinear Taylor rule for the United States.