This paper proposes a novel time-series model with a non-stationary stochastic trend, locally explosive mixed causal non-causal dynamics and fat-tailed innovations. The model allows for a description of financial time-series that is consistent with financial theory, for a decomposition of the time-series in trend and bubble components, and for meaningful real-time forecasts during bubble episodes. We provide sufficient conditions for strong consistency and asymptotic normality of the maximum likelihood estimator. The model-based filter for extracting the trend and bubbles is shown to be invertible and the extracted components converge to the true trend and bubble paths. A Monte Carlo simulation study confirms the good finite sample properties. Finally, we consider an empirical study of Nickel monthly price series and global mean sea level data. We document the forecasting accuracy against competitive alternative methods and conclude that our model-based forecasts outperform all these alternatives.