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.

当泡沫形成时，金融和经济时间序列可以具有局部爆炸性行为。我们开发了一个时变参数模型，能够在时间序列数据中描述这种行为。我们提出的动态模型可用于预测气泡的出现、存在和破裂。我们采用灵活的观察驱动模型规范，允许不同的气泡形状和行为。我们建立了模型生成的数据的平稳性、遍历性和有界矩。此外，我们获得了最大似然估计量的一致性和渐近正态性。给定模型中的参数估计，隐含过滤器能够从观察到的数据中提取未观察到的气泡过程。我们通过蒙特卡罗模拟研究来研究估计器的有限样本特性。最后，