Conditional moments of noncausal alpha-stable processes and the prediction of bubble crash odds

Noncausal, or anticipative, alpha-stable processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For (X_t) a two-sided infinite alpha-stable moving average (MA), conditional moments up to integer order four are shown to exist provided (X_t) is anticipative enough. The functional forms of these moments at any forecast horizon under any admissible parameterisation are obtained by extending the literature on arbitrary bivariate alpha-stable random vectors. The dynamics of noncausal processes simplifies during explosive episodes and allows to express ex ante crash odds at any horizon in terms of the MA coefficients and of the tail index alpha. The results are illustrated in a synthetic portfolio allocation framework and an application to the Nasdaq and S&P500 series is provided.