This paper develops a linearity test that can be applied to cointegrating relations. We consider the widely used RESET speciﬁcation test and show that when this test is applied to nonstationary time series its asymptotic distribution involves a mixture of noncentral chi-squared distributions, which leads to severe size distortions in conventional testing based on the central chi-squared. Nonstationarity is shown to introduce two bias terms in the limit distribution, which are the source of the size distortion in testing. Appropriate corrections for this asymptotic bias leads to a modiﬁed version of the RESET test which has a central chi-squared limit distribution under linearity. The modiﬁed test has power not only against nonlinear cointegration but also against the absence of cointegration. Simulation results reveal that the modiﬁed test has good size inﬁnite samples and reasonable power against many nonlinear models as well as models with no cointegration, conﬁrming the analytic results. In an empirical illustration, the linear purchasing power parity (PPP) speciﬁcation is tested using US, Japan, and Canada monthly data after Bretton Woods. While commonly used ADF and PP cointegration tests give mixed results on the presence of linear cointegration in the series, the modiﬁed test rejects the null of linear PPP cointegration.
Hong, Seung Hyun and Phillips, Peter C.B., "Testing Linearity in Cointegrating Relations with an Application to Purchasing Power Parity" (2005). Cowles Foundation Discussion Papers. 1828.