In the aftermath of the global financial crisis, competing measures of the trend in macroeconomic variables such as US real GDP have featured prominently in policy debates. A key question is whether the large shocks to macroeconomic variables will have permanent effects — i.e., in econometric terms, do the data contain stochastic trends? Unobserved components models provide a convenient way to estimate stochastic trends for time series data, with their existence typically motivated by stationarity tests that allow for at most a deterministic trend under the null hypothesis. However, given the small sample sizes available for most macroeconomic variables, standard Lagrange multiplier tests of stationarity will perform poorly when the data are highly persistent. To address this problem, we propose the use of a likelihood ratio test of stationarity based directly on the unobserved components models used in estimation of stochastic trends. We demonstrate that a bootstrap version of this test has far better small-sample properties for empirically-relevant data generating processes than bootstrap versions of the standard Lagrange multiplier tests. An application to US real GDP produces stronger support for the presence of large permanent shocks when using the likelihood ratio test as compared to the standard tests.