A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

Standard

A Primer on Bootstrap Testing of Hypotheses in Time Series Models : With an Application to Double Autoregressive Models. / Cavaliere, Giuseppe; Rahbek, Anders.

I: Econometric Theory, Bind 37, Nr. 1, 2021, s. 1-48.

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

Harvard

Cavaliere, G & Rahbek, A 2021, 'A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models', Econometric Theory, bind 37, nr. 1, s. 1-48. https://doi.org/10.1017/S0266466620000067

APA

Cavaliere, G., & Rahbek, A. (2021). A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models. Econometric Theory, 37(1), 1-48. https://doi.org/10.1017/S0266466620000067

Vancouver

Cavaliere G, Rahbek A. A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models. Econometric Theory. 2021;37(1):1-48. https://doi.org/10.1017/S0266466620000067

Author

Cavaliere, Giuseppe ; Rahbek, Anders. / A Primer on Bootstrap Testing of Hypotheses in Time Series Models : With an Application to Double Autoregressive Models. I: Econometric Theory. 2021 ; Bind 37, Nr. 1. s. 1-48.

Bibtex

@article{87e33a1e584f436eb625cc0f78e643b6,
title = "A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models",
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.",
author = "Giuseppe Cavaliere and Anders Rahbek",
year = "2021",
doi = "10.1017/S0266466620000067",
language = "English",
volume = "37",
pages = "1--48",
journal = "Econometric Theory",
issn = "0266-4666",
publisher = "Cambridge University Press",
number = "1",

}

RIS

TY - JOUR

T1 - A Primer on Bootstrap Testing of Hypotheses in Time Series Models

T2 - With an Application to Double Autoregressive Models

AU - Cavaliere, Giuseppe

AU - Rahbek, Anders

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

U2 - 10.1017/S0266466620000067

DO - 10.1017/S0266466620000067

M3 - Journal article

VL - 37

SP - 1

EP - 48

JO - Econometric Theory

JF - Econometric Theory

SN - 0266-4666

IS - 1

ER -

ID: 237363097