Financial and economic time series can feature locally explosive behaviour when bubbles are formed. We develop a time-varying parameter model that is capable of describing this behaviour in time series data. Our proposed dynamic model can be used to predict the emergence, existence and burst of bubbles. We adopt a flexible observation driven model specification that allows for different bubble shapes and behaviour. We establish stationarity, ergodicity, and bounded moments of the data generated by our model. Furthermore, we obtain the consistency and asymptotic normality of the maximum likelihood estimator. Given the parameter estimates in the model, the implied filter is capable of extracting the unobserved bubble process from the observed data. We study finite-sample properties of our estimator through a Monte Carlo simulation study. Finally, we show that our model compares well with existing noncausal models in a financial application concerning the Bitcoin/US dollar exchange rate.