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While tests for stationarity and cointegration have important econometric and economic implications, they do not always offer conclusive results. In this paper we suggest that exploiting the parametric structure of the multivariate correlated unobserved components framework can provide a more powerful way to test for stationarity and cointegration than the non-parametric, asymptotic tests currently available. The parametric test nests a partial or restricted unobserved components model within a more general unobserved components model. Then we estimate both the general and the restricted models and determine the likelihood ratio test statistic. The distribution of this likelihood ratio test statistic is nonstandard, but a Monte Carlo simulation provides proper error bands for use in inference. We then compare these results to the asymptotic, non-parametric KPSS test and the common trends test of Nyblom and Harvey (2000)