Conditional Moments of Noncausal Alpha-Stable Processes and the Prediction of Bubble Crash Odds

Sebastien Fries

Research output: Contribution to JournalArticleAcademicpeer-review


Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For (Xt) a two-sided infinite α-stable moving average (MA), conditional moments up to integer order four are shown to exist provided (Xt) is anticipative enough, despite the process featuring infinite marginal variance. Formulas of these moments at any forecast horizon under any admissible parameterization are provided. Under the assumption of errors with regularly varying tails, closed-form formulas of the predictive distribution during explosive bubble episodes are obtained and expressions of the ex ante crash odds at any horizon are available. It is found that the noncausal autoregression of order 1 (AR(1)) with AR coefficient ρ and tail exponent α generates bubbles whose survival distributions are geometric with parameter ρ^α. This property extends to bubbles with arbitrarily shaped collapse after the peak, provided the inflation phase is noncausal AR(1)-like. It appears that mixed causal–noncausal processes generate explosive episodes with dynamics à la Blanchard and Watson which could reconcile rational bubbles with tail exponents greater than 1.
Original languageEnglish
Pages (from-to)1596-1616
Number of pages21
JournalJournal of Business and Economic Statistics
Issue number4
Publication statusPublished - 2022


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