Multivariate Unit Root Tests and Testing for Convergence

We examine the properties of a multivariate Dickey-Fuller t-statistic designed to test for a unit root in a panel while taking account of cross-sectional dependence. The asymptotic distribution is presented and critical values provided. When intercepts are present, a modification along the lines of Elliot, Rothenberg and Stock (1996) can be implemented. The tests have invariance properties and can be carried out even if the number of series exceeds the number of time periods. Non-zero initial conditions actually boost the power of the (unmodified) Dickey-Fuller tests confirming that they are useful for testing the hypothesis that the series are in the process of converging. Typical applications are for a moderate number of series observed over a reasonably long period of time. The example given is for the per capital incomes of six US regions observed annually from 1950 to 1999.