Simple, Robust, and Accurate F and t Tests in Cointegrated Systems
This paper proposes new, simple, and more accurate statistical tests in a cointegrated system that allows for endogenous regressors and serially dependent errors. The approach involves first transforming the time series using orthonormal basis functions in L²[0,1], which has energy concentrated at low frequencies, and then running an augmented regression based on the transformed data and constructing the test statistics in the usual way. The approach is essentially the same as the trend instrumental variable approach of Phillips (2014), but we hold the number of orthonormal basis functions fixed in order to develop the standard F and t asymptotic theory. The tests are extremely simple to implement, as they can be carried out in exactly the same way as if the transformed regression is a classical linear normal regression. In particular, critical values are from the standard F or t distribution. The proposed F and t tests are robust in that they are asymptotically valid regardless of whether the number of basis functions is held fixed or allowed to grow with the sample size. The F and t tests have more accurate size in finite samples than existing tests such as the asymptotic chi-squared and normal tests based on the fully modified OLS estimator of Phillips and Hansen (1990) and can be made as powerful as the latter test.