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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.

References

REFERENCES

Andrews, D.W.K. (1999) Estimation when a parameter is on a boundary. Econometrica 67, 1341–1383.CrossRefGoogle Scholar
Andrews, D.W.K. (2000) Inconsistency of the bootstrap when a parameter is on the boundary of the parameter space. Econometrica 68, 399–405.CrossRefGoogle Scholar
Andrews, D.W.K. (2001) Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69, 683–734.CrossRefGoogle Scholar
Andrews, D.W.K. & Buchinsky, M. (2000) A three-step method for choosing the number of bootstrap repetitions. Econometrica 68, 23–51.CrossRefGoogle Scholar
Athreya, K.B. (1987) Bootstrap of the mean in the infinite variance case. Annals of Statistics 15, 724–731.CrossRefGoogle Scholar
Basawa, I.V., Mallik, A.K., McCornick, W.P., Reeves, J.H., & Taylor, R.L. (1991) Bootstrapping unstable first-order autoregressive processes. Annals of Statistics 19, 1098–1101.CrossRefGoogle Scholar
Borkovec, M. & Klüppelberg, C. (2001) The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Annals of Applied Probability 11, 1220–1241.Google Scholar
Boswijk, H.P., Cavaliere, G., Georgiev, I., & Rahbek, A. (2019) Bootstrapping Non-Stationary Stochastic Volatility. Working Paper, Tinbergen Institute, TI 2019-083/III.CrossRefGoogle Scholar
Boswijk, H.P., Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2016) Inference on co-integration parameters in heteroskedastic vector autoregressions. Journal of Econometrics 192, 64–85.CrossRefGoogle Scholar
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.CrossRefGoogle Scholar
Bühlmann, P. (1997) Sieve bootstrap for time series. Bernoulli 3, 123–148.CrossRefGoogle Scholar
Cavaliere, G. & Georgiev, I. (2019) Inference Under Random Limit Bootstrap Measures. arXiv:2930679.Google Scholar
Cavaliere, G., Nielsen, H.B., Pedersen, R.S., & Rahbek, A. (2019) Bootstrap inference on the boundary of the parameter space with application to conditional volatility models. Journal of Econometrics (forthcoming).Google Scholar
Cavaliere, G., Nielsen, H.B., & Rahbek, A. (2015) Bootstrap testing of hypotheses on co-integration relations in vector autoregressive models. Econometrica 83, 813–831.CrossRefGoogle Scholar
Cavaliere, G., Nielsen, H.B., & Rahbek, A. (2017) On the consistency of bootstrap testing for a parameter on the boundary of the parameter space. Journal of Time Series Analysis 38, 513–534.CrossRefGoogle Scholar
Cavaliere, G., Nielsen, H.B., & Rahbek, A. (2020) Bootstrapping noncausal autoregressions: With applications to explosive bubble modeling. Journal of Business & Economic Statistics 38, 55–67.CrossRefGoogle Scholar
Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2010a) Co-integration rank testing under conditional heteroskedasticity. Econometric Theory 26, 1719–1760.CrossRefGoogle Scholar
Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2010b) Testing for co-integration in vector autoregressions with non-stationary volatility. Journal of Econometrics 158, 7–24.CrossRefGoogle Scholar
Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2012) Bootstrap determination of the co-integration rank in vector autoregressive models. Econometrica 80, 1721–1740.Google Scholar
Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2014) Bootstrap determination of the co-integration rank in vector autoregressive heteroskedastic VAR models Econometric Reviews 33, 606–650CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2007) Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140, 919–947.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2008) Bootstrap unit root tests for time series with non-stationary volatility. Econometric Theory 24, 43–71.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2008) Heteroskedastic time series with a unit root. Econometric Theory 25, 1228–1276.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2009a) Bootstrap M unit root tests. Econometric Reviews 28, 393–421.CrossRefGoogle Scholar
Chen, M., Li, D., & Ling, S. (2013) Non-stationarity and quasi-maximum likelihood estimation on a double autoregressive model. Journal of Time Series Analysis 35, 189–202.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J.G. (2000) Bootstrap tests: How many bootstraps?. Econometric Reviews 19, 55–68.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J.G. (2002) Bootstrap J tests of nonnested linear regression models. Journal of Econometrics 109, 167–193.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J.G. (2002) The power of bootstrap and asymptotic tests. Journal of Econometrics 133, 421–441.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J.G. (2006) Bootstrap methods in econometrics, Chapter 23. In Mills, T.C. & Patterson, K.D. (eds.), Palgrave Handbooks of Econometrics, vol. 3. pp. 812–838. Palgrave Macmillan, Basingstoke.Google Scholar
Davison, A. C., Hinkley, D. V., & Young, G.A. (2003) Recent developments in bootstrap methodology. Statistical Science 18, 141–157.Google Scholar
Deo, R.S. (2000) Spectral tests of the martingale hypothesis under conditional heteroskedasticity. Journal of Econometrics 99, 291–315.CrossRefGoogle Scholar
Engle, R.F. & Granger, C.W.J. (1987) Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Fachin, S. (2000) Bootstrap and asymptotic tests of long-run relationships in cointegrated systems. Oxford Bulletin of Economics and Statistics 62, 543–551.CrossRefGoogle Scholar
Fachin, S. & Omtzigt, P. (2006) Boostrapping and Bartlett correction in the cointegrated VAR model. Econometric Reviews 25, 41–60.Google Scholar
Giné, E. & Zinn, J. (1990) Bootstrapping general empirical measures. Annals of Probability 18, 851–869.CrossRefGoogle Scholar
Gonçalves, S. & Kilian, L. (2004) Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 89–120.CrossRefGoogle Scholar
Gredenhoff, M. & Jacobson, T. (2001) Bootstrap testing linear restrictions on cointegrating vectors. Journal of Business & Economic Statistics 19, 63–72.CrossRefGoogle Scholar
Hall, P. (1992) The Bootstrap and Edgeworth Expansions. Springer-Verlag, New York, NY.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
Harvey, D., Leybourne, S.J., Sollis, R., & Taylor, A.M.R. (2016) Tests for explosive financial bubbles in the presence of non-stationary volatility. Journal of Empirical Finance 38, 548–574.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2004a) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641–646.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2004b) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203–1226.CrossRefGoogle Scholar
Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford, UK.Google Scholar
Klüppelberg, C., Maller, R.A., Van de Vyver, M., & Wee, D. (2002) Testing for reduction to random walk in autoregressive conditional heteroskedasticity models. Econometrics Journal 5, 387–416.CrossRefGoogle Scholar
Knight, K. (1989) On the bootstrap of the mean in the infinite variance case. Annals of Statistics 17, 1168–1175.CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2013) Testing and inference in nonlinear cointegrating vector error-correction models. Econometric Theory 29, 1238–1288CrossRefGoogle Scholar
Lange, T., Rahbek, A., & Jensen, S.T. (2011) Estimation and asymptotic inference in the AR-ARCH model. Econometric Reviews 30, 129–153.CrossRefGoogle Scholar
Ling, S. (2004) Estimation and testing stationarity for double-autoregressive models. Journal of the Royal Statistical Society B 66, 63–78.CrossRefGoogle Scholar
Ling, S. (2007a) A double AR(p) model: Structure and estimation. Statistica Sinica 17, 161–175.Google Scholar
Ling, S. (2007b) Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models. Journal of Econometrics 140, 849–873.CrossRefGoogle Scholar
Ling, S. & Li, W.K. (1998) Limiting distributions of maximum likelihood estimators for unstable ARMA models with GARCH errors. Annals of Statistics 26, 84–12.Google Scholar
Ling, S. & Mcaleer, M. (2003) Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280–310.CrossRefGoogle Scholar
Liu, R.Y. (1988) Bootstrap procedures under some non i.i.d. models. Annals of Statistics 16, 1696–1708.CrossRefGoogle Scholar
MacKinnon, J.G. (2009) Bootstrap hypothesis testing. In Belsley, D.A. & Kontoghiorghes, J.(eds.), Handbook of Computational Econometrics, pp. 183–213. Wiley, Chichester.CrossRefGoogle Scholar
Mammen, E. (1993) Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255–285.CrossRefGoogle Scholar
Nielsen, H.B. & Rahbek, A. (2014) Unit root vector autoregression with volatility induced stationarity. Journal of Empirical Finance 29, 144–167.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B., Wu, Y., & Yu, J. (2011) Explosive behavior in the 1990s Nasdaq: When did exuberance escalate stock values? International Economic Review 52, 201–226.CrossRefGoogle Scholar
Politis, D., Romano, J., & Wolf, M. (1999) Subsampling. Springer, Berlin, Germany.CrossRefGoogle Scholar
Swensen, A.R. (2003) Bootstrapping unit root tests for integrated processes. Journal of Time Series Analysis 24, 99–126.CrossRefGoogle Scholar
Swensen, A.R. (2006) Bootstrap algorithms for testing and determining the cointegration rank in VAR models. Econometrica 74, 1699–1714.CrossRefGoogle Scholar
Wu, C.F.J. (1986) Jackknife, bootstrap, and other resampling methods. Annals of Statistics 14, 1261–1295.CrossRefGoogle Scholar