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

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.     

Testing for the Cointegrating Rank of a VAR Process with Level Shift and Trend Break

Abstract. A test for the cointegrating rank of a vector autoregressive (VAR) process with a possible shift and broken linear trend is proposed. The break point is assumed to be known. Our test is not a likelihood ratio test but the deterministic terms including the broken trends are removed first by a generalized least squares procedure. Then, a likelihood ratio‐type test is applied to the adjusted series. The asymptotic null distribution of the test is derived and it is shown by a Monte Carlo experiment that the test has better small‐sample properties in many cases than a corresponding Gaussian likelihood ratio test for the cointegrating rank. Moreover, response surface techniques can be used to easily obtain p‐values of the test for any possible break date.     

The power of unit root tests against nonlinear local alternatives

This article extends the analysis of local power of unit root tests in a nonlinear direction by considering local nonlinear alternatives and tests built specifically against stationary nonlinear models. In particular, we focus on the popular test proposed by Kapetanios et al. (2003, Journal of Econometrics 112, 359–379) in comparison to the linear Dickey–Fuller test. To this end, we consider different adjustment schemes for deterministic terms. We provide asymptotic results which imply that the error variance has a severe impact on the behaviour of the tests in the nonlinear case; the reason for such behaviour is the interplay of non‐stationarity and nonlinearity. In particular, we show that nonlinearity of the data generating process can be asymptotically negligible when the error variance is moderate or large (compared to the ‘amount of nonlinearity’), rendering the linear test more powerful than the nonlinear one. Should however the error variance be small, the nonlinear test has better power against local alternatives. We illustrate this in an asymptotic framework of what we call persistent nonlinearity. The theoretical findings of this article explain previous results in the literature obtained by simulation. Furthermore, our own simulation results suggest that the user‐specified adjustment scheme for deterministic components (e.g. OLS, GLS, or recursive adjustment) has a much higher impact on the power of unit root tests than accounting for nonlinearity, at least under local (linear or nonlinear) alternatives.     

Weak identification in the ESTAR model and a new model

Determining good parameter estimates in (exponential smooth transition autoregressive) models is known to be difficult. We show that the phenomena of getting strongly biased estimators is a consequence of the so‐called identification problem, the problem of properly distinguishing the transition function in relation to extreme parameter combinations. This happens in particular for either very small or very large values of the error term variance. Furthermore, we introduce a new alternative model – the TSTAR model – which has similar properties as the ESTAR model but reduces the effects of the identification problem. We also derive a linearity and a unit root test for this model.