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

Abstract

Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For $(X_t)$ a two-sided infinite α-stable moving average (MA), conditional moments up to integer order four are shown to exist provided $(X_t)$ 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 ${\rho }^{\alpha }$. 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.

Keywords:

• Conditional moments
• Crashes
• Noncausal process
• Prediction
• Rational expectation
• Speculative bubble

Supplementary Materials

The supplementary materials contains all mathematical proofs, additional examples and simulation experiments referenced in the article, and codes for replicating the main figures and tables.

Acknowledgments

The author is thankful to the editor, the associate editor and two referees, whose comments have greatly improved the article. The author is extraordinarily indebted to Jean-Michel Zakoïan, and further thanks Denisa-Georgiana Banulescu, Jean-Marc Bardet, Frédérique Bec, Francisco Blasques, Ophélie Couperier, Gilles De Truchis, Elena Dumitrescu, Christian Francq, Christian Gouriéroux, Alain Hecq, Siem Jan Koopman, Jérémy Leymarie, Yang Lu, Andre Lucas, Anders Rahbek, Li Sun, Sean Telg, Arthur Thomas and Elisa Voisin for insightful discussions.

Additional information

Funding

The author gratefully acknowledges the support of the Groupe des Écoles Nationales d’Économie et Statistique (GENES), the Agence Nationale de la Recherche (via the Project MultiRisk ANR CE26 2016 - CR), and the support of the European Commission (via the Project NONCAUSALBubble H2020-MSCA-IF-2019 896504).