Prediction of bubbles in presence of α-stable aggregates moving averages

Financial markets frequently exhibit dramatic episodes where asset prices undergo rapid growth followed by abrupt collapses, that are incompatible with standard linear time series models. While anticipative heavytailed linear processes offer a promising alternative for modeling such phenomena, they impose uniform bubble patterns across different episodes, contradicting empirical evidence. This paper introduces a new model, based on α-stable moving average aggregates, that accommodates heterogeneous bubble dynamics.We establish the theoretical properties of this model, demonstrating that it admits a semi-norm representation on a unit cylinder, thereby enabling the prediction of extreme trajectories with varying growth dynamics. We develop a minimum distance estimation procedure based on the joint characteristic function and establish its asymptotic properties. Monte Carlo simulations confirm the estimator's good finite-sample performance across various specifications, and we implement a subsampling methodology to empirically verify the convergence to asymptotic normality. Our empirical application to the CBOE Crude Oil ETF Volatility Index successfully decomposes observed volatility dynamics into distinct components with different persistence properties, revealing that what appears as a single bubble episode actually consists of multiple superimposed processes with heterogeneous growth rates and crash probabilities.