The Size and the Power of the Bootstrap Tests for Linear Restrictions in Misspeci…ed Cointegrating Relationships

This paper considers computer intensive methods for inference on cointegrating vectors in maximum likelihood analysis. It investigates the robustness of LR, Wald tests and an F¡type test for linear re-strictions on cointegrating space to misspeci…cation of the number of cointegrating relations. In addition, since all the distributional results within the maximum likelihood cointegration model rely on asymp-totic considerations, it is important to consider the sensitivity of in-ference procedures to the sample size. In this paper we use bootstrap hypothesis testing as a way to improve inference for linear restriction on the cointegrating space. We …nd that the resampling procedure is a very useful device for tests that lack the invariance property such as the Wald test, where the size distortion of the bootstrap test is small even for a sample size T = 50. Moreover, it turns out that when the number of cointegrating vectors are correctly speci…ed the bootstrap succeeds where the asymptotic approximation is not satisfactory, that is, for a sample size T < 200. The only valid alternative to the re-sampling procedure is the F-type test proposed by Podivinsky (1992). However, when the number of cointegrating vectors is over-…tted re-lying on the asymptotic approximation is misleading, since the tests considered exhibit sizes very far from the nominal size. In this sit-uation the bootstrap test is much more robust to misspeci…cations. The analysis of the power reveals that the procedures have power. However, it is di¢cult to evaluate the power properties without inves-tigating the asymptotic power, so further work is needed