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Noncausal, or anticipative, heavy-tailed processes generate trajectoriesfeaturing locally explosive episodes akin to speculative bubbles in financialtime series data. For $(X_t)$ a two-sided infinite $\alpha$-stable movingaverage (MA), conditional moments up to integer order four are shown to existprovided $(X_t)$ is anticipative enough, despite the process featuring infinitemarginal variance. Formulae of these moments at any forecast horizon under anyadmissible parameterisation are provided. Under the assumption of errors withregularly varying tails, closed-form formulae of the predictive distributionduring explosive bubble episodes are obtained and expressions of the ex antecrash odds at any horizon are available. It is found that the noncausalautoregression of order 1 (AR(1)) with AR coefficient $\rho$ and tail exponent$\alpha$ generates bubbles whose survival distributions are geometric withparameter $\rho^{\alpha}$. This property extends to bubbles witharbitrarily-shaped collapse after the peak, provided the inflation phase isnoncausal AR(1)-like. It appears that mixed causal-noncausal processes generateexplosive episodes with dynamics \`a la Blanchard and Watson (1982) which couldreconcile rational bubbles with tail exponents greater than 1. Applications ofthe conditional moments to bubble modelling by noncausal processes arediscussed and the use of the closed-form crash odds is illustrated on theNasdaq and S&P500 series.

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