Tests of AR(2) unit roots and cointegration with the GPH statistic

Time domain tests of unit roots and cointegration are often made after a model search procedure has been made in order to reduce erroneus inference caused by model misspecification. An alternative to this is to use a semi parametric method like the GPH (Geweke & Porter-Hudak, 1983) test where the properties of the spectral density close to the zero frequency are exploited, see Cheung & Lai (1993) who use this to test for cointegration in a bivariate system. In this paper we further investigate the size and power properties of this test. The results provide evidence that the test has not only good power properties against AR(1) and ARFIMA(0,d,0) alternatives, which was shown by Cheung and Lai (1993) but also against AR(2) alternatives. We show that the power of the test is not as good when applied to a raw time series as when applied to residuals from a cointegrating regression. The power can, in the AR(1) case, be significantly improved by using a larger bandwith than is usually done for the GPH estimator without imposing any size distortions. For the AR(2) alternative, however, severe size distortions appear when this is done.