The Size and the power of the bootstrap tests for linear restrictions in misspecified 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 restrictions on cointegrating space to misspecification on the number of cointegrating relations. In addition, since all the distributional results within the maximum likelihood cointegration model rely on asymptotic considerations, it is important to consider the sensitivity of inference 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 find 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 converges to zero even for a sample size T=50. Moreover, it turns out that when the number of cointegrating vectors are correctly specified the bootstrap succeeds where the asymptotic approximation is not satisfactory, that is, for a sample size T<200. The only valid alternative to the resampling procedure is the F-type test\ proposed by Podivinsky (1994). However, when the number of cointegrating vectors is under-fitted or over-fitted relying on the asymptotic approximation is misleading, since the tests considered exhibit sizes sometimes very far away from the nominal size. In this situation the bootstrap test is much more robust to misspecifications. The analysis of the power reveals that the \ procedures have power. However, it is difficult to evaluate the power properties without\ investigating the asymptotic power, so further work is needed.