TY - JOUR

T1 - The validity of bootstrap testing for threshold autoregression

AU - Giannerini, Simone

AU - Goracci, Greta

AU - Rahbek, Anders

N1 - Funding Information: This work was supported by a STSM Grant from COST Action CA17120 ( European Cooperation in Science and Technology ). Greta Goracci acknowledges the support of Libera Università di Bolzano, Grant WW202G (TARMAECON) . Simone Giannerini acknowledges the support of University of Bologna, ALMArie CURIE 2021 project . Anders Rahbek acknowledges financial support from the Independent Research Fund Denmark (Grant no. 0133-00162B ) Publisher Copyright: © 2023 Elsevier B.V.

PY - 2024

Y1 - 2024

N2 - We consider bootstrap-based testing for threshold effects in non-linear threshold autoregressive (TAR) models. It is well-known that classic tests based on asymptotic theory tend to be biased in case of small, or even moderate sample sizes, especially when the estimated parameters indicate non-stationarity, or in presence of heteroskedasticity, as often witnessed in the analysis of financial or climate data. To address the issue we propose a supremum Lagrange Multiplier test statistic (sLM), where the null hypothesis specifies a linear autoregressive (AR) model against the alternative of a TAR model. We consider both the classical recursive residual i.i.d. bootstrap (sLMi) and a wild bootstrap (sLMw), applied to the sLM statistic, and establish their validity under the null hypothesis. The framework is new, and requires the proof of non-standard results for bootstrap analysis in time series models; this includes a uniform bootstrap law of large numbers and a bootstrap functional central limit theorem. The Monte Carlo evidence shows that the bootstrap tests have correct empirical size even for small samples; the wild bootstrap version (sLMw) is also robust against the presence of heteroskedasticity. Moreover, there is no loss of empirical power when compared to the asymptotic test and the size of the tests is not affected if the order of the tested model is selected through AIC. Finally, we use our results to analyse the time series of the Greenland ice sheet mass balance. We find a significant threshold effect and an appropriate specification that manages to reproduce the main non-linear features of the series, such as the asymmetric seasonal cycle, the main periodicities, and the multimodality of the probability density function.

AB - We consider bootstrap-based testing for threshold effects in non-linear threshold autoregressive (TAR) models. It is well-known that classic tests based on asymptotic theory tend to be biased in case of small, or even moderate sample sizes, especially when the estimated parameters indicate non-stationarity, or in presence of heteroskedasticity, as often witnessed in the analysis of financial or climate data. To address the issue we propose a supremum Lagrange Multiplier test statistic (sLM), where the null hypothesis specifies a linear autoregressive (AR) model against the alternative of a TAR model. We consider both the classical recursive residual i.i.d. bootstrap (sLMi) and a wild bootstrap (sLMw), applied to the sLM statistic, and establish their validity under the null hypothesis. The framework is new, and requires the proof of non-standard results for bootstrap analysis in time series models; this includes a uniform bootstrap law of large numbers and a bootstrap functional central limit theorem. The Monte Carlo evidence shows that the bootstrap tests have correct empirical size even for small samples; the wild bootstrap version (sLMw) is also robust against the presence of heteroskedasticity. Moreover, there is no loss of empirical power when compared to the asymptotic test and the size of the tests is not affected if the order of the tested model is selected through AIC. Finally, we use our results to analyse the time series of the Greenland ice sheet mass balance. We find a significant threshold effect and an appropriate specification that manages to reproduce the main non-linear features of the series, such as the asymmetric seasonal cycle, the main periodicities, and the multimodality of the probability density function.

KW - Bootstrap test

KW - Greenland ice sheet

KW - Heteroskedasticity

KW - Law of large numbers

KW - Threshold autoregressive models

UR - http://www.scopus.com/inward/record.url?scp=85147225387&partnerID=8YFLogxK

U2 - 10.1016/j.jeconom.2023.01.004

DO - 10.1016/j.jeconom.2023.01.004

M3 - Journal article

AN - SCOPUS:85147225387

JO - Journal of Econometrics

JF - Journal of Econometrics

SN - 0304-4076

ER -