A General Test for the Cointegrating Rank in Vector Autoregressive Models

The article proposes a general test for the cointegrating rank in vector autoregressive models. The test is based on the eigenvalues of the companion matrix, more precisely on the sum of the real parts of those closest to one. The roots of the companion matrix are often inspected as a diagnostic tool. Here this practice is elevated to the level of a formal test. The asymptotic distribution of the test statistic is derived and tabulated by simulation. The new test is compared with the likelihood ratio test. Neither one of the tests is dominating the other over the entire parameter space. The new test safeguards the researcher against making spurious inferences on the cointegrating rank in the presence of explosive roots and of integration of higher order than one. A numerical illustration is given. <p> It is shown that the limiting distribution of the eigenvalues closest to one of the companion matrix is free of nuisance parameters, a useful result in its own right. When applied to a univariate autoregressive model, the augmented Dickey—Fuller test is obtained. The new test can be regarded as the multivariate version of the Dickey—Fuller coefficent test of a unit root. When there is a single common trend, the suggested test statistic has the same asymptotic distribution as the least squares estimate of the autoregressive coeficent in a univariate first order autoregressive model with a unit root. In the numerical application the test is applied to Swedish money demand data, and it is shown how to determine the cointegrating rank using the new test. The results from a small sample simulation experiment mimicking the actual data used in the application are also discussed. <p>