Near-optimal Unit Root Test with Stationary Covariate with Better Finite Sample Size

Numerous tests for integration and cointegration have been proposed in the literature. Since Elliott, Rothemberg, and Stock (1996), the search for tests with better power has moved in the direction of finding tests with some optimality properties both in univariate and multivariate models. Although the optimal tests constructed so far have asymptotic power that is indistinguishable from the power envelope, it is well known that they can have severe size distortions in finite samples. This paper proposes a simple and powerful test that can be used to test for unit root or for no cointegration when the cointegration vector is known. Although this test is not optimal in the sense of Elliott and Jansson (2003), it has better finite sample size properties while having asymptotic power curves that are indistinguishable from the power curves of optimal tests. Similarly to Hansen (1995), Elliott and Jansson (2003), Zivot (2000), and Elliott, Jansson and Pesavento (2005) the proposed test achieves higher power by using additional information contained in covariates correlated with the variable being tested. The test is constructed by applying Hansen's test to variables that are detrended under the alternative in a regression augmented with leads and lags of the stationary covariates. Using local to unity parametrization, the asymptotic distribution of the test under the null and the local alternative is analytically computed.