Testing for Cointegration in Nonlinear Star Error Correction Models

In this paper we propose a new testing procedure to detect the presence of a cointegrating relationship that follows a globally stationary smooth transition autoregressive (STAR) process. We start from a general VAR model, embed the STAR error correction mechanism (ECM) and then derive the generalised nonlinear STAR error correction model. We provide two operational versions of the tests. Firstly, we obtain the associated nonlinear ECM-based test. Secondly, we generalise the well-known residual-based test for cointegration in linear models by Engle and Granger (1987) and obtain its nonlinear analogue. We derive the relevant asymptotic distributions of the proposed tests. We find via Monte Carlo simulation exercises that our proposed tests have much better power than the Engle and Granger test against the alternative of a globally stationary STAR cointegrating process. In an application to the price-dividend relationship, we also find that our test is able to find cointegration, whereas the linear-based tests fail to do so. Further analysis of impulse response functions of error correction terms (under the alternative) shows that the time taken to recover one half of a one standard deviation shock varies between five and twenty years, whereas the time taken to recover one half of a large shock varies between just 4 to 18 months. This clearly implies that data periods dominated by extreme volatility may display substantial mean reversion of the price-dividend relationship. By contrast this relationship may well look like a unit root when the underlying shocks take on smaller values