This paper considers an approximate dynamic factor model for a large panel of time series possibly sharing stochastic trends, with the aim of disentangling long-run from short-run co-movements. First, we propose a new Quasi Maximum Likelihood estimator of the model based on the Kalman filter and the Expectation Maximisation (EM) algorithm for non-stationary data. This estimator is shown to be more efficient than traditional estimators based on principal component analysis of first differences and their integration. Second, we show how to separate trends and cycles in the estimated factors by using a non-parametric decomposition based on eigenanalysis of a matrix similar to the long-run covariance matrix. Third, we employ our methodology to estimate aggregate real output, or Gross Domestic Output (GDO), and the output gap on a panel of US quarterly macroeconomic indicators. Specifically, we first derive an estimate of GDO as that part of GDP/GDI that is driven only by the common shocks, and then, by applying our trend-cycle decomposition to the factors driving GDO, we produce an estimate of the output gap.