Tests for Linearity in Star Models: Supwald and Lm‐Type Tests

Abstract

This article investigates approximation and supremum approaches for testing linearity in smooth transition autoregressive (STAR) models. We show that since the approximation of STAR models by Taylor series expansions may not accurately describe the specific transition dynamic when the process is away from the null, LM‐type tests may fail to detect the form of nonlinearity for which they are designed for. Investigating a supremum approach, the article provides the asymptotic distribution of a SupWald test that is obtained by taking the supremum of a Wald statistic over the Cartesian product of the spaces for the transition and threshold parameters. Simulated asymptotic critical values for the resulting tests are provided for a wide range of autoregressive orders and shown to differ across exponential and logistic STAR (ESTAR and LSTAR) models. Monte Carlo experiments show that SupWald tests for ESTAR and LSTAR models outperform LM‐type tests, compares well relative to the recently developed score‐based tests and each SupWald statistic performs the best against the true alternative for which it is formed. SupWald tests also provide results that are consistent with the findings from (independently) estimating and diagnostic testing of STAR models in real exchange rate data.