The thesis considers time series and econometric models with stochastic trend components. Locally Best Invariant tests for the presence of stochastic trends are constructed and their asymptotic distributions derived. Particular attention is paid to models with structural breaks, as the tests have high power also against alternative hypotheses in which the trends of the series contain a small number of breaks but are otherwise deterministic. Asymptotic critical values of the tests are tabulated for series with a single breakpoint. A modification of the LBI statistic is then proposed, for which the asymptotic distribution depends only on the number of the breaks and not on their location. Common stochastic trends imply cointegration and thus testing the number of common trends can also be regarded as testing the dimension of the cointegration space. A test for common trends recently proposed in the literature is extended to series which contain structural breaks. Testing for the presence of a nonstationary seasonal component is then examined. The LBI test, adjusted for serial correlation by means of a nonparametric correction, is extended in various directions and its performance is compared with that of a parametric test. Representation, estimation and tests of cointegrated structural time series models form the subject of one chapter, where numerous links with the literature on vector autoregressions are established. Panel data regression models where the individual effects take the form of individual specific random walks are considered in the last chapter. Imposing the constraint of a common signal-to-noise ratio across individuals makes the maximum likelihood estimator computationally feasible also when the number of units in the cross section is large. For these models an average LBI test for stationarity and for the presence of fixed effects is proposed.