Skip to main content
×
×
Home

TESTING GARCH-X TYPE MODELS

  • Rasmus Søndergaard Pedersen (a1) and Anders Rahbek (a1)
Abstract

We present novel theory for testing for reduction of GARCH-X type models with an exogenous (X) covariate to standard GARCH type models. To deal with the problems of potential nuisance parameters on the boundary of the parameter space as well as lack of identification under the null, we exploit a noticeable property of specific zero-entries in the inverse information of the GARCH-X type models. Specifically, we consider sequential testing based on two likelihood ratio tests and as demonstrated the structure of the inverse information implies that the proposed test neither depends on whether the nuisance parameters lie on the boundary of the parameter space, nor on lack of identification. Asymptotic theory is derived essentially under stationarity and ergodicity, coupled with a regularity assumption on the exogenous covariate X. Our general results on GARCH-X type models are applied to Gaussian based GARCH-X models, GARCH-X models with Student’s t-distributed innovations as well as integer-valued GARCH-X (PAR-X) models.

Copyright
Corresponding author
*Address correspondence to Rasmus Søndergaard Pedersen, Department of Economics, University of Copenhagen, Øster Farimagsgade 5, 1353 Copenhagen K, Denmark; e-mail: rsp@econ.ku.dk.
Footnotes
Hide All

Department of Economics, University of Copenhagen, Denmark. This research was supported by the Danish Council for Independent Research (DSF Grant 015-00028B). Pedersen is grateful for support from the Carlsberg Foundation. We thank the Co-Editor (Dennis Kristensen) and two referees as well as seminar participants at Cambridge, Helsinki, and Oxford Universities. We also thank participants at (EC)2 2017, Heikametrics 2017, Toulouse Financial Econometrics Conference 2018, and Brunel Conference 2018 for comments and discussions of a previous draft of the article.

Footnotes
References
Hide All
Agosto, A., Cavaliere, G., Kristensen, D., & Rahbek, A. (2016) Modeling corporate defaults: Poisson autoregressions with exogenous covariates (PARX). Journal of Empirical Finance 38, 640663.
Ahmad, A. & Francq, C. (2016) Poisson QMLE of count time series models. Journal of Time Series Analysis 37, 291314.
Andrews, D.W.K. (1999) Estimation when a parameter is on a boundary. Econometrica 67, 13411383.
Andrews, D.W.K. (2001) Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69, 683734.
Andrews, D.W.K. & Cheng, X. (2012) Estimation and inference with weak, semi-strong, and strong identification. Econometrica 80, 21532211.
Berkes, I. & Horváth, L. (2004) The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics 32, 633655.
Bierens, H.J. & Ploberger, W. (1997) Asymptotic theory of integrated conditional moment tests. Econometrica 65, 11291151.
Brown, B.M. (1971) Martingale central limit theorems. The Annals of Mathematical Statistics 42, 5966.
Chernoff, A. (1954) On the distribution of the likelihood ratio. The Annals of Mathematical Statistics 25, 573578.
Demos, A. & Sentana, E. (1998) Testing for GARCH effects: A one-sided approach. Journal of Econometrics 86, 97127.
Fokianos, K., Rahbek, A., & Tjostheim, D. (2009) Poisson autoregression. Journal of the American Statistical Association 104, 14301439.
Francq, C. & Thieu, L.Q. (2018) QML inference for volatility models with covariates. Econometric Theory, first published online 01 February 2018.
Francq, C. & Zakoïan, J.M. (2007) Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero. Stochastic Processes and Their Applications 117, 12651284.
Francq, C. & Zakoïan, J.M. (2009) Testing the nullity of GARCH coefficients: Correction of the standard tests and relative efficiency comparisons. Journal of the American Statistical Association 104, 313324.
Francq, C. & Zakoïan, J.M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications, Wiley.
Han, H. (2015) Asymptotic properties of GARCH-X processes. Journal of Financial Econometrics 13, 188221.
Han, H. & Kristensen, D. (2014) Asymptotic for the QMLE in GARCH-X models with stationary and nonstationary covariates. Journal of Business & Economic Statistics 32, 416429.
Han, H. & Park, J.Y. (2012) ARCH/GARCH with persistent covariate: Asymptotic theory of MLE. Journal of Econometrics 167, 95112.
Jensen, S.T. & Rahbek, A. (2004a) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 12031226.
Jensen, S.T. & Rahbek, A. (2004b) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641646.
Ketz, P. (2018) Subvector inference when the true parameter vector may be near or at the boundary. Journal of Econometrics, first published online 05 September 2018. doi: 10.1016/j.jeconom.2018.08.003.
Leeb, H. & Pötscher, B.M. (2005) Model selection and inference: Facts and fiction. Econometric Theory 21, 2159.
McCloskey, A. (2017) Bonferroni-based size-correction for nonstandard testing problems. Journal of Econometrics 200, 1735.
Mikosch, T. & Starica, C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. The Annals of Statistics 28, 14271451.
Pedersen, R.S. (2017) Inference and testing on the boundary in extended constant conditional correlation GARCH models. Journal of Econometrics 196, 2336.
Pedersen, R.S. & Rahbek, A. (2016) Nonstationary GARCH with 𝑡-distributed innovations. Economics Letters 138, 1921.
Ranga Rao, R. (1962) Relations between weak and uniform convergence of measures with applications. The Annals of Mathematical Statistics 33, 659680.
Silvapulle, M.J. & Silvapulle, P. (1995) A score test against one-sided alternatives. Journal of the American Statistical Association 90, 342349.
Straumann, D. (2005) Estimation in conditionally heteroscedastic time series models. Lecture Notes in Statistics. Springer.
White, H. (2001) Asymptotic Theory for Econometricians: Revised Edition. Emerald Group Publishing Ltd.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed