Analytical Evaluation of the Power of Tests for the Absence of Cointegration

Abstract

This paper proposes a theoretical explanation to the common empirical results in which different tests for cointegration give different answers. Using local to unity parametrization I compute the analytical power of some tests for the null of no cointegration: The ADF test on the residuals of the cointegration regression, Johansen's maximum eigenvalue test, the t-test on the Error Correction term and Boswijk (1994) Wald test. The tests are shown to be functions of Brownian Motions and Ornstein-Uhlenbeck processes and to depend on a single nuisance parameter, which is, in turn determined by the correlation at frequency zero of the independent variables with the errors of the cointegration regression. Monte Carlo experiments show that the tests can have significantly different performances for different values of the nuisance parameter. An application to the money demand equation is presented.JEL classification: C32

Combining non-cointegration tests

“…Also, as pointed out above, Pesavento (2004) demonstrates that the tests differ in their power in different parts of the (c-R 2 )-parameter space. In particular, any test is the most powerful one in some part of the parameter space.…”

“…1 Proposition 1 summarizes the local asymptotic distribution derived by Pesavento (2004). Proposition 1.…”

“…We consider the residual-based test of Engle and Granger (1987), a system-based test of Johansen (1988), as well as the error-correction-based tests of Boswijk (1994) and Banerjee et al (1998). Pesavento (2004) derives the local asymptotic power of these tests. She shows that, under model (1), their power only depends on the local-to-unity parameter c := T (ρ − 1) and the squared correlations of the elements of v 1t with v 2t .…”

“…This also has implications for their (local) asymptotic power under the alternative. Specifically, Pesavento (2004) shows that the relative power of cointegration tests depends crucially on the squared long-run correlations of error terms driving the variables of the analyzed system. That is, the power ranking of the tests varies by the value of that unknown nuisance parameter.…”

“…In particular, we propose to combine test statistics in the spirit of Fisher's (1932) famous test. We derive the asymptotic null distribution of our Fisher-type combination test for correlated cointegration test statistics and its local asymptotic power, exploiting Pesavento's (2004) results. Besides successfully tackling the above-mentioned multiple testing nature inherent in combining different test statistics, the combined test indeed enjoys a robust power performance over the range of the squared long-run error correlation.…”