Unit Root Tests in Three-Regime Setar Models

Abstract

This paper proposes a simple testing procedure to distinguish a unit root process from a globally stationary three-regime self-exciting threshold autoregressive process. Following the threshold cointegration literature we assume that the process follows the random walk in the corridor regime, and therefore we propose that the null of a unit root be tested by the Wald statistic for the joint significance of autoregressive parameters in both lower and upper regimes. We establish that when threshold parameters are known, the suggested Wald test has a well-defined asymptotic null distribution free of nuisance parameters. In the general case where threshold parameters are unknown a priori, we consider the three most commonly used summary statistics based on their average, exponential average and supremum. Assuming that the grid set for thresholds can be selected such that the corridor regime be of finite width both under the null and under the alternative, we can establish both stochastic equicontinuity and uniform convergence of the aforementioned summary statistics. Monte Carlo evidence indicates that the proposed tests are more powerful than the Dickey-Fuller test that ignores the threshold nature under the alternative. We illustrate the usefulness of our proposed tests by examining stationarity of real exchange rates for the G7 countries.