Residual Based Tests for Cointegration with GLS Detrended Data

Abstract

We propose residual based tests for cointegration using local GLS detrending (Elliott, Rothemberg and Stock, 1996, ERS) applied separately to each variable of the system. We consider two cases, one where only a constant is included and one where a constant and a time trend are included. We derive the asymptotic distribution of a feasible point optimal test which allows us to derive the power envelope. The quasi-differencing parameter c is selected such as the asymptotic power of the feasible point optimal test is 50.0%. Different c exists because the limiting distribution of the feasible point optimal test depends of the number of right-hand regressors and the specification of the deterministic components. The limiting distributions of various residuals based tests are derived for a general quasi-differencing parameter c and critical values are tabulated for values of c = 0 irrespective of the nature of the deterministic components and the values suggested by the power envelope. Simulations show that using GLS detrending allows tests with higher power and that using c selected by the power envelope, as the quasi-differencing parameter, according to the two cases analyzed, is preferable. An explanation for this feature is provided.