Testing the null stationarity for multiple time series

This paper introduces various consistent tests for the null of stationarity against the alternative of nonstationarity applicable to multiple time series with and without the presence of time trends. The tests are based on the multivariate AR(1) model and derived by the principles of AR unit root tests. An important feature of the tests from the practical viewpoint is that no a priori knowledge about the data generating process of the series under study is required when the lag length for the long-run variance estimation is estimated by using Andrews (1991 Econometrica 59, 812-858) automatic lag selection methods along with a simple inequality restriction. This simple restriction also make the tests diverge at faster rates. The asymptotic distributions of these tests are complex and nonstandard but expressed in a unified manner by using the standard vector Brownian motion. The distributions are tabulated by simulation for some practical cases. The rates of divergence under the alternative are also reported. Further, the asymptotic effects of misspecifying the order of time trends in the regression model are analyzed. Using the regression model which does not detrend time series properly results in rejecting the null hypothesis in large samples even when the null is true. Extensive simulation illustrates the finite same performance of the tests introduced in this paper. The multivariate tests using Andrews' automatic lag selection methods with a restriction work reasonably well in finite samples. In particular, it is illustrated that using the multivariate tests introduced in this paper is a better testing strategy for detrended time series in terms of the finite sample size and power than applying univariate tests several times to each component of a multiple time series. The tests are applied to the real interest rates of the major industrialized nations studied in Kugler and Neusser (1993, Journal of Applied Econometrics 8, 163-174), The null of stationarity is not rejected for the real interest rates at conventional significance levels. Keywords: Null of stationery; LM test; Sargan-Bhargava-Durbin-Hausman test; Multiple time series