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Mixed causal-noncausal AR processes and the modelling of explosive bubbles. (English) Zbl 1433.62258

Summary: 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 C. Gouriéroux and J.-M. Zakoïan [J. R. Stat. Soc., Ser. B, Stat. Methodol. 79, No. 3, 737–756 (2017; Zbl 1411.62252)], 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G30 Order statistics; empirical distribution functions

Citations:

Zbl 1411.62252
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