Bootstrapping Noncausal Autoregressions: With Applications to Explosive Bubble Modeling

Abstract

In this article, we develop new bootstrap-based inference for noncausal autoregressions with heavy-tailed innovations. This class of models is widely used for modeling bubbles and explosive dynamics in economic and financial time series. In the noncausal, heavy-tail framework, a major drawback of asymptotic inference is that it is not feasible in practice as the relevant limiting distributions depend crucially on the (unknown) decay rate of the tails of the distribution of the innovations. In addition, even in the unrealistic case where the tail behavior is known, asymptotic inference may suffer from small-sample issues. To overcome these difficulties, we propose bootstrap inference procedures using parameter estimates obtained with the null hypothesis imposed (the so-called restricted bootstrap). We discuss three different choices of bootstrap innovations: wild bootstrap, based on Rademacher errors; permutation bootstrap; a combination of the two (“permutation wild bootstrap”). Crucially, implementation of these bootstraps do not require any a priori knowledge about the distribution of the innovations, such as the tail index or the convergence rates of the estimators. We establish sufficient conditions ensuring that, under the null hypothesis, the bootstrap statistics estimate consistently particular conditionaldistributions of the original statistics. In particular, we show that validity of the permutation bootstrap holds without any restrictions on the distribution of the innovations, while the permutation wild and the standard wild bootstraps require further assumptions such as symmetry of the innovation distribution. Extensive Monte Carlo simulations show that the finite sample performance of the proposed bootstrap tests is exceptionally good, both in terms of size and of empirical rejection probabilities under the alternative hypothesis. We conclude by applying the proposed bootstrap inference to Bitcoin/USD exchange rates and to crude oil price data. We find that indeed noncausal models with heavy-tailed innovations are able to fit the data, also in periods of bubble dynamics. Supplementary materials for this article are available online.

KEY WORDS:

• Bootstrap
• Bubble dynamics
• Heavy tails
• Noncausal autoregressions.

ACKNOWLEDGMENTS

This research was supported by Danish Council for Independent Research (DSF Grant 015-00028B). Rahbek is grateful for support from Center for Information and Bubble Studies (CIBS) at the University of Copenhagen. CIBS is sponsored by the Carlsberg Foundation. The authors thank the editor, one associate editor, three referees, Jean-Michel Zakoïan, Luca Fanelli, Alain Hecq, and Iliyan Georgiev as well as seminar participants at Hitotsubashi University, Kyoto University, Toulouse University, Heidelberg University, University of Messina, CREST, Paris, Cambridge University, and Oxford University for comments and discussions of previous draft of the article.

Funding

Danish Council for Independent Research. [015-00028B]