Weak convergence of sample covariance matrices and testing for seasonal unit roots

Abstract

The paper has two main contributions. First, weak convergence results are derived from sampling moments of processes that contains a unit root at an arbitrary frequency, where, in contrast to the previous literature, the proofs are mainly based on algebraic manipulations and well known weak convergence results for martingale difference sequences. These convergence results are used to derive the limiting distribution of the ordinary least squares estimator for unit root autoregressions. As as second contribution, a Phillips-Perron type test for a unit root at an arbitrary frequency is introduced and its limiting distributions are derived. This test is further extended to a joint test for multiple unit roots and seasonal integration. The limiting distributions of these test statistics are asymptotically equivalent to various statistics presented earlier in the seasonal unit root literature.