Semiparametrically optimal cointegration test

This paper aims to address the issue of semiparametric efficiency for cointegration rank testing in finite-order vector autoregressive models, where the innovation distribution is considered an infinite-dimensional nuisance parameter. Our asymptotic analysis relies on Le Cam’s theory of limit experiment, which in this context is of the Locally Asymptotically Brownian Functional (LABF) type likelihood ratios. By exploiting the structural representation of LABF, an Ornstein–Uhlenbeck experiment, we develop the asymptotic power envelopes of asymptotically invariant tests for both cases with and without time trends. We propose feasible tests based on a nonparametrically estimated density and demonstrate that their power can achieve the semiparametric power envelopes, making them semiparametrically optimal. We validate the theoretical results through large-sample simulations and illustrate satisfactory size control and excellent power performance of our tests under small samples. In both cases with and without time trends, we show that a remarkable amount of additional power can be obtained from non-Gaussian distributions.