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A PRIMER ON BOOTSTRAP TESTING OF HYPOTHESES IN TIME SERIES MODELS: WITH AN APPLICATION TO DOUBLE AUTOREGRESSIVE MODELS

Published online by Cambridge University Press:  20 March 2020

Giuseppe Cavaliere  and
Anders Rahbek
Giuseppe Cavaliere
Affiliation:
University of Bologna Exeter Business School
Anders Rahbek*
Affiliation:
University of Copenhagen
*
Address correspondence to Anders Rahbek, Department of Economics, University of Copenhagen, Øster Farimagsgade 5, 1353 Copenhagen K, Denmark; e-mail: [email protected].
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Abstract

In this article, we discuss the bootstrap as a tool for statistical inference in econometric time series models. Importantly, in the context of testing, properties of the bootstrap under the null (size) as well as under the alternative (power) are discussed. Although properties under the alternative are crucial to ensure the consistency of bootstrap-based tests, it is often the case in the literature that only validity under the null is discussed. We provide new results on bootstrap inference for the class of double-autoregressive (DAR) models. In addition, we review key examples from the bootstrap time series literature in order to emphasize the importance of properly defining and analyzing the bootstrap generating process and associated bootstrap statistics, while also providing an up-to-date review of existing approaches. DAR models are particularly interesting for bootstrap inference: first, standard asymptotic inference is usually difficult to implement due to the presence of nuisance parameters; second, inference involves testing whether one or more parameters are on the boundary of the parameter space; third, even second-order moments may not exist. In most of these cases, the bootstrap is not considered an appropriate tool for inference. Conversely, and taking testing nonstationarity to illustrate, we show that although a standard bootstrap based on unrestricted parameter estimation is invalid, a correct implementation of the bootstrap based on restricted parameter estimation (restricted bootstrap) is first-order valid. That is, it is able to replicate, under the null hypothesis, the correct limiting distribution. Importantly, we also show that the behavior of this bootstrap under the alternative hypothesis may be more involved because of possible lack of finite second-order moments of the bootstrap innovations. This feature makes for some parameter configurations, the restricted bootstrap unable to replicate the null asymptotic distribution when the null is false. We show that this possible drawback can be fixed by using a novel bootstrap in this framework. For this “hybrid bootstrap,” the parameter estimates used to construct the bootstrap data are obtained with the null imposed, while the bootstrap innovations are sampled with replacement from unrestricted residuals. We show that the hybrid bootstrap mimics the correct asymptotic null distribution, irrespective of the null being true or false. Monte Carlo simulations illustrate the behavior of both the restricted and the hybrid bootstrap, and we find that both perform very well even for small sample sizes.

Type
ET LECTURE
Information
Econometric Theory , Volume 37 , Issue 1 , February 2021 , pp. 1 - 48
Copyright
© Cambridge University Press 2020

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Footnotes

The article is based on the Econometric Theory Lecture given by Anders Rahbek at the 8th Italian Congress of Econometrics and Empirical Economics (ICEEE), Lecce, 23–25 January 2019. We thank Peter Phillips, three anonymous referees, conference participants and, importantly, James MacKinnon, Ye Lu, and David Harris for important feedback on previous versions of the article. Part of the article was written when Anders Rahbek visited the Department of Economics of the University of Bologna, whose hospitality is gratefully acknowledged. This research was supported by the Danish Council for Independent Research (DSF Grant 015-00028B) and by an ALMA IDEA grant from the University of Bologna.

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