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Published online by Cambridge University Press:  28 January 2019

Sébastien Fries
CREST and Paris-Saclay University
Jean-Michel Zakoian*
CREST and University of Lille
*Address correspondence to Jean-Michel Zakoian, CREST and University of Lille, CREST, 5 Avenue Henri Le Chatelier, 91120 Palaiseau, France; e-mail:


Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

Copyright © Cambridge University Press 2019 

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The authors are grateful to three anonymous referees, the Co-editor, and the Editor for insightful comments, suggestions, and useful criticisms. The second author acknowledges financial support from the Agence Nationale de la Recherche (ANR), which supported this work via the Project MultiRisk (ANR CE26 2016 - CR), the Labex ECODEC and the Chaire ACPR “Régulation et risques systémiques”



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