Testing for Cointegration in a System of Equations

This paper introduces various consistent tests for the null of cointegration against the alternative of noncointegration that can be applied to a system of equations as well as to a single equation. The tests are analogs of Choi and Ahn's (1993, Testing the Null of Stationarity for Multiple Time Series, working paper, The Ohio State University) multivariate tests for the null of stationarity and use Park's (1992, <italic>Econometrica</italic> 60, 119–143) canonical cointegrating regression (CCR) residuals to make the tests free of nuisance parameters in the limit. The asymptotic distributions of the tests are complex but expressed in unified manner by using standard vector Brownian motion. These distributions are tabulated by simulation for some practical cases. Furthermore, the rates of divergence of the tests are reported. Because there are methods for estimating cointegrating matrices other than CCR, it is illustrated for a model without time trends that the tests we introduce work exactly the same way in the limit when Phillips and Hansen's (1990, <italic>Review of Economic Studies</italic> 57, 99–125) fully modified ordinary least-squares (OLS) procedure is used. Also, is shown that difficulties arise when OLS residuals are used to formulate the tests. Small-scale simulation results are reported to examine the finite sample performance of the tests. The tests are shown to work reasonably wellin finite samples. In particular, it is illustrated that using the multivariate tests introduced in this paper can be a better testing strategy in terms of the finite sample size and power than applying univariate tests several times to each equation in a system of equations.