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INFERENCE ON GARCH-MIDAS MODELS WITHOUT ANY SMALL-ORDER MOMENT

Published online by Cambridge University Press:  12 May 2023

Christian FrancqOpen the ORCID record for Christian Francq[Opens in a new window],
Baye Matar KandjiOpen the ORCID record for Baye Matar Kandji[Opens in a new window] and
Jean-Michel ZakoianOpen the ORCID record for Jean-Michel Zakoian[Opens in a new window]
Christian Francq
Affiliation:
CREST–ENSAE and University of Lille
Baye Matar Kandji
Affiliation:
CREST–ENSAE
Jean-Michel Zakoian*
Affiliation:
CREST-ENSAE and University of Lille
*
Address correspondence to Jean-Michel Zakoïan, CREST–ENSAE, Palaiseau, France; e-mail: zakoian@ensae.fr
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Abstract

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In GARCH-mixed-data sampling models, the volatility is decomposed into the product of two factors which are often interpreted as “short-run” (high-frequency) and “long-run” (low-frequency) components. While two-component volatility models are widely used in applied works, some of their theoretical properties remain unexplored. We show that the strictly stationary solutions of such models do not admit any small-order finite moment, contrary to classical GARCH. It is shown that the strong consistency and the asymptotic normality of the quasi-maximum likelihood estimator hold despite the absence of moments. Tests for the presence of a long-run volatility relying on the asymptotic theory and a bootstrap procedure are proposed. Our results are illustrated via Monte Carlo experiments and real financial data.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 6 , December 2024, pp. 1422 - 1455
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Footnotes

The authors are grateful to two anonymous referees, the Co-Editor, and the Editor for insightful comments, suggestions, and useful criticisms. The authors are also grateful to the Agence Nationale de la Recherche (ANR), which supported this work via the Project MLforRisk (ANR-21-CE26-0007).

References

REFERENCES

Amado, C. & Teräsvirta, T. (2017) Specification and testing of multiplicative time-varying GARCH models with applications. Econometric Reviews 36, 421–446.CrossRefGoogle Scholar
Andrews, D.W. (2001) Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69, 683–734.CrossRefGoogle Scholar
Berkes, I., Horváth, L., & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.CrossRefGoogle Scholar
Beutner, E., Heinemann, A., & Smeekes, S. (2020) A residual bootstrap for conditional value-at-risk. Preprint, arXiv:1808.0912. https://EconPapers.repec.org/RePEc:arx:papers:1808.09125.Google Scholar
Billingsley, P. (1961) The Lindeberg-Lévy theorem for martingales. Proceedings of the American Mathematical Society 12, 788–792.Google Scholar
Brandt, A. (1986) The stochastic equation ${Y}_{n+1}={A}_n{Y}_n+{B}_n$ with stationary coefficients. Advances in Applied Probability 18, 211–220.Google Scholar
Cavaliere, G., Nielsen, H.B., Pedersen, R.S., & Rahbek, A. (2022) Bootstrap inference on the boundary of the parameter space with application to conditional volatility models. Journal of Econometrics 227, 241–263.CrossRefGoogle Scholar
Conrad, C., Custovic, A., & Ghysels, E. (2018) Long- and short-term cryptocurrency volatility components: A GARCH-MIDAS analysis. Journal of Risk and Financial Management 11, 23.CrossRefGoogle Scholar
Conrad, C. & Engle, R. (2021) Modelling volatility cycles: The (MF) ${}^2$ GARCH model. NYU Stern School of Business, Forthcoming.CrossRefGoogle Scholar
Conrad, C. & Loch, K. (2015) Anticipating long-term stock market volatility. Journal of Applied Econometrics 30, 1090–1114.CrossRefGoogle Scholar
Conrad, C. & Schienle, M. (2020) Testing for an omitted multiplicative long-term component in GARCH models. Journal of Business & Economic Statistics 38, 229–242.CrossRefGoogle Scholar
Davis, R. & Resnick, S. (1986) Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14, 533–558.CrossRefGoogle Scholar
Ding, Z. & Granger, C. (1996) Modeling volatility persistence of speculative returns: A new approach. Journal of Econometrics 73, 185–215.CrossRefGoogle Scholar
Engle, R.F., Ghysels, E., & Sohn, B. (2013) Stock market volatility and macroeconomic fundamentals. Review of Economics and Statistics 95, 776–797.CrossRefGoogle Scholar
Engle, R.F. & Lee, G. (1999) A long-run and short-run component model of stock return volatility. In Engle, R. & White, H. (eds), Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive WJ Granger , pp. 475–497. Oxford University Press.CrossRefGoogle Scholar
Francq, C., Horvàth, L., & Zakoïan, J.-M. (2010) Sup-tests for linearity in a general nonlinear AR (1) model. Econometric Theory 26, 965–993.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH. Bernoulli 10, 605–637.CrossRefGoogle Scholar
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, 313–324.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2019) GARCH Models: Structure, Statistical Inference and Financial Applications , 2nd Edition . Wiley.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2022) Testing the existence of moments for GARCH processes. Journal of Econometrics 227, 47–64.CrossRefGoogle Scholar
Giacomini, R., Politis, D.N., & White, H. (2013) A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric Theory 29, 567–589.CrossRefGoogle Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Ibragimov, R., Kim, J., & Skrobotov, A. (2023) New robust inference for predictive regressions. Econometric Theory. Preprint, arXiv:2006.01191v4.CrossRefGoogle Scholar
Ibragimov, R., Pedersen, R.S., & Skrobotov, A. (2021) New approaches to robust inference on market (non-)efficiency volatility clustering and non linear dependence. Preprint, arXiv:2006.01212v3.Google Scholar
Kandji, B.M. (2023) Iterated function systems driven by non independent sequences: Structure and inference. Journal of Applied Probability, forthcoming.Google Scholar
Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131, 207–248.CrossRefGoogle Scholar
Kreiss, J.P., Paparoditis, E., & Politis, D.N. (2011) On the range of validity of the autoregressive sieve bootstrap. Annals of Statistics 39, 2103–2130.CrossRefGoogle Scholar
Leucht, A., Kreiss, J.P., & Neumann, M.H. (2015) A model specification test for GARCH (1, 1) processes. Scandinavian Journal of Statistics 42, 1167–1193.CrossRefGoogle Scholar
Mikosch, T. & Stărică, C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Annals of Statistics 28, 1427–1451.CrossRefGoogle Scholar
Patton, A.J. (2011) Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160, 246–256.CrossRefGoogle Scholar
Shimizu, K. (2013) The bootstrap does not always work for heteroscedastic models. Statistics & Risk Modeling 30, 189–204.CrossRefGoogle Scholar
Tanny, D. (1974) A zero-one law for stationary sequences. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 30, 139–148.CrossRefGoogle Scholar
van der Vaart, A.W. (2000) Asymptotic Statistics . Cambridge University Press.Google Scholar
Wang, F. & Ghysels, E. (2015) Econometric analysis of volatility component models. Econometric Theory 31, 362–393.CrossRefGoogle Scholar
White, H. (1982) Maximum likelihood estimation of misspecified models. Econometrica 50, 1–25.CrossRefGoogle Scholar