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Score-driven location plus scale models: asymptotic theory and an application to forecasting Dow Jones volatility

  • Szabolcs Blazsek , Alvaro Escribano EMAIL logo and Adrian Licht


We present the Beta-t-QVAR (quasi-vector autoregression) model for the joint modelling of score-driven location plus scale of strictly stationary and ergodic variables. Beta-t-QVAR is an extension of Beta-t-EGARCH (exponential generalized autoregressive conditional heteroscedasticity) and Beta-t-EGARCH-M (Beta-t-EGARCH-in-mean). We prove the asymptotic properties of the maximum likelihood (ML) estimator for correctly specified Beta-t-QVAR models. We use Dow Jones Industrial Average (DJIA) data for the period of 1985–2020. We find that the volatility forecasting accuracy of Beta-t-QVAR is superior to the volatility forecasting accuracies of Beta-t-EGARCH, Beta-t-EGARCH-M, A-PARCH (asymmetric power ARCH), and GARCH for the period of 2010–2020.

Corresponding author: Alvaro Escribano, Department of Economics, Universidad Carlos III de Madrid, Getafe 28903, Spain, E-mail:

Funding source: Universidad Francisco Marroquin

Funding source: Agencia Estatal de Investigacion

Award Identifier / Grant number: 2019/00419/001

Funding source: Comunidad de Madrid

Award Identifier / Grant number: MadEco-CM S2015/HUM-3444

Funding source: Ministerio de Economia, industria y Competitividad

Award Identifier / Grant number: ECO2016-00105-001 and MD 2014-0431


A previous version of this paper was presented at the MKE Annual Conference (December 2020, Budapest). We thank the helpful comments and suggestions of Robert Lieli, Gabor Korosi, and conference participants. The authors are also thankful for the comments of Matthew Copley and the anonymous reviewer. All remaining errors are our own.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: Blazsek and Licht acknowledge funding from Universidad Francisco Marroquin. Escribano acknowledges funding from Ministerio de Economia, Industria y Competitividad (ECO2016-00105-001 and MDM 2014-0431), Comunidad de Madrid (MadEco-CMS2015/HUM-3444), and Agencia Estatal de Investigacion (2019/00419/001).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


Black, F. 1976. “Studies of Stock Market Volatility Changes.” In 1976 Proceedings of the American Statistical Association Business and Economic Statistics Section.Search in Google Scholar

Blasques, F., P. Gorgi, S. J. Koopman, and O. Wintenberger. 2018. “Feasible Invertibility Conditions and Maximum Likelihood Estimation for Observation-Driven Models.” Electronic Journal of Statistics 12: 1019–52. in Google Scholar

Blasques, F., J. van Brummelen, S. J. Koopman, and A. Lucas. 2022. “Maximum Likelihood Estimation for Score-Driven Models.” Journal of Econometrics 227 (2): 325–46.10.1016/j.jeconom.2021.06.003Search in Google Scholar

Blazsek, S., A. Escribano, and A. Licht. 2020. “Identification of Seasonal Effects in Impulse Responses Using Score-Driven Multivariate Location Models.” Journal of Econometric Methods 10 (1): 53–66. in Google Scholar

Blazsek, S., A. Escribano, and A. Licht. 2021a. “Co-Integration with Score-Driven Models: An Application to US Real GDP Growth, US Inflation Rate, and Effective Federal Funds Rate.” Macroeconomic Dynamics. forthcoming. in Google Scholar

Blazsek, S., A. Escribano, and A. Licht. 2021b. “Multivariate Markov-Switching Score-Driven Models: An Application to the Global Crude Oil Market.” Studies in Nonlinear Dynamics & Econometrics. forthcoming. in Google Scholar

Blazsek, S., and A. Licht. 2020. “Dynamic Conditional Score Models: A Review of Their Applications.” Applied Economics 52 (11): 1181–99. in Google Scholar

Blazsek, S., and M. Villatoro. 2015. “Is Beta-t-EGARCH(1,1) Superior to GARCH(1,1)?” Applied Economics 47 (17): 1764–74. in Google Scholar

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31 (3): 307–27. in Google Scholar

Bollerslev, T. 1987. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics 69 (3): 542–7. in Google Scholar

Brandt, A. 1986. “The Stochastic Equation Yn+1 = AnYn + Bn with Stationary Coefficients.” Advances in Applied Probability 18 (1): 211–20. in Google Scholar

Chua, C. T., J. Goh, and Z. Zhang. 2010. “Expected Volatility, Unexpected Volatility, and the Cross-Section of Stock Returns.” Journal of Financial Research 33 (2): 103–23. in Google Scholar

Creal, D., S. J. Koopman, and A. Lucas. 2008. “A General Framework for Observation Driven Time-Varying Parameter Models.” In Tinbergen Institute Discussion Paper 08-108/4. (accessed August 30, 2021).10.2139/ssrn.1297183Search in Google Scholar

Creal, D., S. J. Koopman, and A. Lucas. 2011. “A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations.” Journal of Business & Economic Statistics 29 (4): 552–63. in Google Scholar

Creal, D., S. J. Koopman, and A. Lucas. 2013. “Generalized Autoregressive Score Models with Applications.” Journal of Applied Econometrics 28 (5): 777–95. in Google Scholar

Creal, D., B. Schwaab, S. J. Koopman, and A. Lucas. 2014. “Observation-Driven Mixed-Measurement Dynamic Factor Models with an Application to Credit Risk.” The Review of Economics and Statistics 96 (5): 898–915. in Google Scholar

Cox, D. R. 1981. “Statistical Analysis of Time Series: Some Recent Developments.” Scandinavian Journal of Statistics 8: 93–115.Search in Google Scholar

Ding, Z., C. W. J. Granger, and R. F. Engle. 1993. “A Long Memory Property of Stock Market Returns and a New Model.” Journal of Empirical Finance 1 (1): 83–106. in Google Scholar

Elton, J. H. 1990. “A Multiplicative Ergodic Theorem for Lipschitz Maps.” Stochastic Processes and their Applications 34: 39–47. in Google Scholar

Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50 (4): 987–1007. in Google Scholar

Engle, R. F., D. M. Lilien, and R. P. Robins. 1987. “Estimating Time-Varying Risk Premia in the Term Structure: The ARCH-M Model.” Econometrica 55 (2): 391–407. in Google Scholar

Giacomini, R., and H. White. 2006. “Tests of Conditional Predicitve Ability.” Econometrica 74 (6): 1545–78. in Google Scholar

Glosten, L. R., R. Jagannathan, and D. E. Runkle. 1993. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance 48 (5): 1779–801. in Google Scholar

Hansen, P. R., and A. Lunde. 2005. “A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH(1,1)?” Journal of Applied Econometrics 20 (7): 873–89. in Google Scholar

Harvey, A. C. 2013. “Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series.” In Econometric Society Monographs. Cambridge: Cambridge University Press.10.1017/CBO9781139540933Search in Google Scholar

Harvey, A. C., and T. Chakravarty. 2008. “Beta-t-(E)GARCH.” In Cambridge Working Papers in Economics 0840. Cambridge: Faculty of Economics, University of Cambridge. (accessed August 30, 2021).Search in Google Scholar

Harvey, A., and R. J. Lange. 2017. “Volatility Modeling with a Generalized t Distribution.” Journal of Time Series Analysis 38 (2): 175–90. in Google Scholar

Harvey, A. C., and R. J. Lange. 2018. “Modeling the Interactions between Volatility and Returns Using EGARCH-M.” Journal of Time Series Analysis 39 (6): 909–19. in Google Scholar

Herwartz, H., and H. Lütkepohl. 2000. “Multivariate Volatility Analysis of VW Stock Prices.” International Journal of Intelligent Systems in Accounting, Finance & Management 9: 35–54.<35::aid-isaf176>;2-v.10.1002/(SICI)1099-1174(200003)9:1<35::AID-ISAF176>3.0.CO;2-VSearch in Google Scholar

Liu, L. Y., A. J. Patton, and K. Sheppard. 2015. “Does Anything Beat 5-Minute RV? A Comparison of Realized Measures across Multiple Asset Classes.” Journal of Econometrics 187 (1): 293–311. in Google Scholar

Lütkepohl, H. 2005. New Introduction to Multivariate Time Series Analysis. Berlin: Springer-Verlag.10.1007/978-3-540-27752-1Search in Google Scholar

Nelson, D. B. 1991. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica 59 (2): 347–70. in Google Scholar

Patton, A. J. 2011. “Data-Based Ranking of Realised Volatility Estimators.” Journal of Econometrics 161 (2): 284–303. in Google Scholar

Shapiro, S. S., and M. B. Wilk. 1965. “An Analysis of Variance Test for Normality (Complete Samples).” Biometrika 52 (3/4): 591–611. in Google Scholar

Straumann, D., and T. Mikosch. 2006. “Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach.” Annals of Statistics 34 (5): 2449–95. in Google Scholar

Veronesi, P. 1999. “Stock Market Overreaction to Bad News in Good Times: A Rational Expectations Equilibrium Model.” Review of Financial Studies 12 (5): 975–1007. in Google Scholar

White, H. 2001. Asymptotic Theory for Econometricians, Revised edition. San Diego: Academic Press.Search in Google Scholar

Wooldridge, J. M. 1994. “Estimation and Inference for Dependent Processes.” In Handbook of Econometrics, vol. 4, edited by R. F. Engle and D. L. McFadden, 2639–738. Amsterdam: North-Holland.10.1016/S1573-4412(05)80014-5Search in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (

Received: 2021-09-07
Accepted: 2022-02-16
Published Online: 2022-03-07

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