Hostname: page-component-788cddb947-nxk7g Total loading time: 0 Render date: 2024-10-11T10:14:45.544Z Has data issue: false hasContentIssue false
  • English
  • Français

QML INFERENCE FOR VOLATILITY MODELS WITH COVARIATES

Published online by Cambridge University Press:  01 February 2018

Christian Francq  and
Le Quyen Thieu
Christian Francq*
Affiliation:
CREST and Université de Lille, RiskDesign and Université Pierre et Marie Curie
Le Quyen Thieu
Affiliation:
CREST and Université de Lille, RiskDesign and Université Pierre et Marie Curie
*
*Address correspondence to Christian Francq, 5 Avenue Henry Le Chatelier 91120 Plaiseau, France; e-mail: christian.francq@univ-lille3.fr.
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is obtained for a wide class of asymmetric GARCH models with exogenous covariates. The true value of the parameter is not restricted to belong to the interior of the parameter space, which allows us to derive tests for the significance of the parameters. In particular, the relevance of the exogenous variables can be assessed. The results are obtained without assuming that the innovations are independent, which allows conditioning on different information sets. Monte Carlo experiments and applications to financial series illustrate the asymptotic results. In particular, an empirical study demonstrates that the realized volatility can be a helpful covariate for predicting squared returns.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 1 , February 2019 , pp. 37 - 72
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We are grateful to the Editor, Associate Editor and anonymous reviewers for useful comments and suggestions. Christian Francq gratefully acknowledges the financial support from the labex ECODEC and the Agence Nationale de la Recherche (ANR) via the Project MultiRisk (ANR-16-CE26-0015-02).

References

REFERENCES

Ali, G. (2013) EGARCH, GJR-GARCH, TGARCH, AVGARCH, NGARCH, IGARCH and APARCH models for pathogens at marine recreational sites. Journal of Statistical and Econometric Methods 2, 57–73.Google Scholar
Andrews, D.W. (1999) Estimation when a parameter is on a boundary. Econometrica 67, 1341–1383.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
Billingsley, P. (1961) The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788–792.Google Scholar
Billingsley, P. (1995) Probability and Measure. John Wiley.Google Scholar
Bollerslev, T. (2008) Glossary to ARCH (GARCH). In Bollerslev, T., Russell, J.R., & Watson, M. (eds.), Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle. Oxford University Press, 137–163.Google Scholar
Bougerol, P. & Picard, N. (1992a) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992b) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714–1729.CrossRefGoogle Scholar
Brandt, A. (1986) The stochastic equation Y n+1 = A nY n + B n with stationary coefficients. Advance in Applied Probability 18, 221–254.Google Scholar
Ding, Z., Granger, C., & Engle, R.F. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.CrossRefGoogle Scholar
Engle, R.F., Hendry, D.F., & Richard, J.F. (1983) Exogeneity. Econometrica 51, 277–304.CrossRefGoogle Scholar
Engle, R.F. & Patton, A.J. (2001) What good is a volatility model. Quantitative Finance 1, 237–245.CrossRefGoogle Scholar
Escanciano, J.C. (2009) Quasi-maximum likelihood estimation of semi-strong GARCH models. Econometric Theory 25, 561–570.CrossRefGoogle Scholar
Fisher, T.J. & Gallagher, C.M. (2012) New weighted portmanteau statistics for time series goodness of fit testing. Journal of the American Statistical Association 107, 777–787.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.CrossRefGoogle Scholar
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, 1265–1284.CrossRefGoogle Scholar
Francq, C. & Zakoian, J.-M. (2009a) Bartlett’s formula for a general class of nonlinear processes. Journal of Time Series Analysis 30, 449–465.CrossRefGoogle Scholar
Francq, C. & Zakoian, J.-M. (2009b) 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. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley.CrossRefGoogle Scholar
Fuertes, A.M., Izzeldin, M., & Kalotychou, E. (2009) On forecasting daily stock volatility: The role of intraday information and market conditions. International Journal of Forecasting 25, 259–281.CrossRefGoogle Scholar
Glosten, L.R., Jaganathan, R., & Runkle, D. (1993) On the relation between the expected values and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
Guo, S., Ling, S., & Zhu, K. (2014) Factor double autoregressive models with application to simultaneous causality testing. Journal of Statistical Planning and Inference 148, 82–94.CrossRefGoogle Scholar
Hall, P. & Yao, Q. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317.CrossRefGoogle Scholar
Hamadeh, T. & Zakoïan, J.-M. (2011) Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes. Journal of Statistical Planning and Inference 141, 488–507.CrossRefGoogle Scholar
Han, H. & Kristensen, D. (2014) Asymptotic theory for the QMLE in GARCH-X models with stationary and nonstationary covariates. Journal of Business & Economic Statistics 32, 416–429.CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2005) Asymptotics of the QMLE for a class of ARCH(q) models. Econometric Theory 21, 946–961.CrossRefGoogle Scholar
Laurent, S., Lecourt, C., & Palm, F.C. (2016) Testing for jumps in conditionally Gaussian ARMA-GARCH models, a robust approach. Computational Statistics & Data Analysis 100, 383–400.CrossRefGoogle Scholar
Li, W.K. & Mak, T.K. (1994) On the squared residual autocorrelations in nonlinear time series with conditional heteroskedasticity. Journal of Time Series Analysis 15, 627–636.CrossRefGoogle Scholar
Ling, S. (2007) Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models. Journal of Econometrics 140, 849–873.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Nijman, T. & Sentana, E. (1996) Marginalization and contemporaneous aggregation in multivariate GARCH processes. Journal of Econometrics 71, 71–87.CrossRefGoogle Scholar
Pan, J., Wang, H., & Tong, H. (2008) Estimation and tests for power-transformed and threshold GARCH models. Journal of Econometrics 142, 352–378.CrossRefGoogle Scholar
Pedersen, R.S. (2017) Inference and testing on the boundary in extended constant conditional correlation GARCH models. Journal of Econometrics 196, 23–36.CrossRefGoogle Scholar
Sucarrat, G. & Escribano, A. (2010) The Power Log-GARCH Model. Working document, Economic Series 10–13, University Carlos III, Madrid.Google Scholar
Taylor, S.J. (1986) Modelling Financial Time Series. Wiley.Google Scholar
Wintenberger, O. (2013) Continuous invertibility and stable QML estimation of the EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846–867.CrossRefGoogle Scholar
Zakoïan, J.-M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.CrossRefGoogle Scholar