TESTING FOR COINTEGRATION IN A SYSTEM OF EQUATIONS

This paper introduces various consistent tests for the null of cointeg ration against the alternative of noncointegration that can be applied to a system of equations as well as to a single equation. The tests a re analogs of Choi and Ahn's (1993, Testing the Null of Stationarity f or Multiple Time Series, working paper, The Ohio State University) mul tivariate tests for the null of stationarity and use Park's (1992, Eco nometrica 60, 119-143) canonical cointegrating regression (CCR) residu als to make the tests free of nuisance parameters in the limit. The as ymptotic distributions of the tests are complex but expressed in a uni fied manner by using standard vector Brownian motion. These distributi ons 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 il lustrated for a model without time trends that the tests we introduce work exactly the same way in the limit when Phillips and Hansen's (199 0, Review of Economic Studies 57, 99-125) fully modified ordinary leas t-squares (OLS) procedure is used. Also, it 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 performa nce of the tests. The tests are shown to work reasonably well in finit e samples. In particular, it is illustrated that using the multivariat e tests introduced in this paper can be a better testing strategy in t erms of the finite sample size and power than applying univariate rest s several times to each equation in a system of equations.