Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown

Sucarrat, Genaro and Grønneberg, Steffen and Escribano, Alvaro (2013): Estimation and Inference in Univariate and Multivariate Log-GARCH-X Models When the Conditional Density is Unknown.

Abstract

Exponential models of Autoregressive Conditional Heteroscedasticity (ARCH) enable richer dynamics (e.g. contrarian or cyclical), provide greater robustness to jumps and outliers, and guarantee the positivity of volatility. The latter is not guaranteed in ordinary ARCH models, in particular when additional exogenous or predetermined variables ("X") are included in the volatility specification. Here, we propose estimation and inference methods for univariate and multivariate Generalised log-ARCH-X (i.e. log-GARCH-X) models when the conditional density is not known via (V)ARMA-X representations. The multivariate specification allows for volatility feedback across equations, and time-varying correlations can be fitted in a subsequent step. Finally, our empirical applications on electricity prices show that the model-class is particularly useful when the X-vector is high-dimensional.

• Item Type: MPRA Paper
• Language: English
• Keywords: ARCH, exponential GARCH, log-GARCH, ARMA-X, Multivariate GARCH
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection

References

Aielli, G. P. (2009). Dynamic Conditional Correlations: On Properties and Eestimation. http://ssrn.com/abstract=1507743.

Bai, J. (1994). Weak convergence of the sequential empirical process of residuals in arma models. The Annals of Statistics 22, 2051-2061.

Bardet, J.-M. and O. Wintenberger (2009). Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes. Unpublished working paper.

Bauwens, L., S. Laurent, and J. Rombouts (2006). Multivariate GARCH Models: A Survey. Journal of Applied Econometrics 21, 79-109.

Bauwens, L. and G. Sucarrat (2010). General to Specific Modelling of Exchange Rate Volatility: A Forecast Evaluation. International Journal of Forecasting 26, 885-907.

Berkes, I., L. Horvath, and P. Kokoszka (2003). GARCH processes: structure and estimation. Bernoulli 9, 201-227. 22

Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307{327.

Brockwell, P. J. and R. A. Davis (2006). Time Series: Theory and Methods. New York: Springer. 2nd. Edition.

Carnero, M. A., D. Pena, and E. Ruiz (2007). Effects of outliers on the identification and estimation of GARCH models. Journal of Time Series Analysis 28, pp. 471- 497.

Comte, F. and O. Lieberman (2003). Asymptotic Theory for Multivariate GARCH Processes. Journal of Multivariate Analysis 84, 61-84.

Engle, R. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflations. Econometrica 50, 987-1008.

Engle, R. (2002). Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models. Journal of Business and Economic Statistics 20, 339-350.

Engle, R. and J. Maricucci (2006). A long-run Pure Variance Common Features model for the common volatilities of the Dow Jones. Journal of Econometrics 132, 7-42.

Engle, R. F. and T. Bollerslev (1986). Modelling the persistence of conditional variances. Econometric Reviews 5, 1-50.

Escribano, A., , J. I. Pena, and P. Villaplana (2011). Modelling Electricity Prices: International Evidence. Oxford Bulletin of Economics and Statistics 73, 622-650.

Francq, C., O. Wintenberger, and J.-M. Zakoian (2012). GARCH Models Without Positivity Constraints: Exponential or Log-GARCH? To appear in Journal of Econometrics, http//dx.doi.org/10.1016/j.jeconom.2013.05.004.

Francq, C. and J.-M. Zakoian (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605-637.

Francq, C. and J.-M. Zakoian (2006). Linear-representation Based Estimation of Stochastic Volatility Models. Scandinavian Journal of Statistics 33, 785-806.

Francq, C. and J.-M. Zakoian (2010a). GARCH Models. New York: Marcel Dekker.

Francq, C. and J.-M. Zakoian (2010b). QML estimation of a class of multivariate GARCH models without moment conditions on the observed process. Unpublished working paper.

Franses, P. H., J. Neele, and D. Van Dijk (2001). Modelling asymmetric volatility in weekly Dutch temperature data. Environmental Modeling and Software 16, 131-137.

Geweke, J. (1986). Modelling the Persistence of Conditional Variance: A Comment. Econometric Reviews 5, 57-61.

Gradshteyn, I. S. and I. M. Ryzhik (2007). Table of Integrals, Series and Products. New York: Academic Press. Seventh edition. Available via http://books. google.com/.

Grønneberg, S. and G. Sucarrat (2013). Asymptotic Theory for Univariate and Multivariate Log-GARCH-X Models. Work in progress.

Hafner, C. and A. Preminger (2009). Asymptotic theory for a factor GARCH model. Econometric Theory 25, 336-363.

Hamilton, J. D. (2010). Macroeconomics and ARCH. In T. Bollerslev, J. R. Russell, and M. Watson (Eds.), Festschrift in Honor of Robert F. Engle. Oxford: Oxford University Press.

Hannan, E. and M. Deistler (2012). The statistical theory of linear systems. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Originally published in 1988 by Wiley, New York.

Hansen, P. R., Z. Huan, and H. H. Shek (2012). Realized GARCH: A Joint Model for Returns and Realized Measures of Volatility. Journal of Applied Econometrics 27, 877-906.

Harvey, A. C. (1976). Estimating Regression Models with Multiplicative Heteroscedasticity. Econometrica 44, 461-465.

Harvey, A. C. (2013). Dynamic Models for Volatility and Heavy Tails. New York: Cambridge University Press.

Harvey, A. C. and T. Chakravarty (2008). Beta-t-(E)GARCH. Cambridge Working Papers in Economics 0840, Faculty of Economics, University of Cambridge.

Harvey, A. C. and N. Shephard (1996). Estimation of an Asymmetric Stochastic Volatility Model for Asset Returns. Journal of Business and Economic Statistics 14, 429-434.

Jeantheau, T. (1998). Strong consistency of estimators for multivariate arch models. Econometric Theory 14, pp. 70-86.

Kawakatsu, H. (2006). Matrix exponential GARCH. Journal of Econometrics 134, 95-128.

Koopman, S. J., M. Ooms, and M. A. Carnero (2007). Periodic Seasonal REGARFIMA-GARCH Models for Daily Electricity Spot Prices. Journal of the American Statistical Association 102, 16-27.

Kristensen, D. and A. Rahbek (2009). Asymptotics of the QMLE for Non-Linear ARCH Models. Journal of Time Series Econometrics 1, Issue 1, Article 2. Available at: http://www.bepress.com/jtse/vol1/iss1/art2.

Ling, S. and M. McAleer (2003). Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280-310.

Ljung, G. and G. Box (1979). On a Measure of Lack of Fit in Time Series Models. Biometrika 66, 265-270.

Lumsdaine, R. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575-596.

Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Berlin: Springer-Verlag.

Milhøj, A. (1987). A Multiplicative Parametrization of ARCH Models. Research Report 101, University of Copenhagen: Institute of Statistics.

Nelson, D. B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica 59, 347-370.

Pantula, S. (1986). Modelling the Persistence of Conditional Variance: A Comment. Econometric Reviews 5, 71-73.

Psaradakis, Z. and E. Tzavalis (1999). On regression-based tests for persistence in logarithmic volatility models. Econometric Reviews 18, 441-448.

SAS Institute Inc. (2013). JMP Version 10. Cary, NC: SAS Institute Inc.

Straumann, D. and T. Mikosch (2006). Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach. The Annals of Statistics 34, 2449-2495.

Sucarrat, G. (2012). AutoSEARCH: General-to-Specific (GETS) Model Selection. R package version 1.2.

Sucarrat, G. (2013). Models of Zero-Augmented Financial Return with Time-Varying Zero-Probabilities. Work in progress.

Sucarrat, G. and A. Escribano (2010). The Power Log-GARCH Model. Universidad Carlos III de Madrid Working Paper 10-13 in the Economic Series, June 2010. http://e-archivo.uc3m.es/bitstream/10016/8793/1/we1013.pdf.

Sucarrat, G. and A. Escribano (2012). Automated Model Selection in Finance: General-to-Specific Modelling of the Mean and Volatility Specifications. Oxford Bulletin of Economics and Statistics 74, 716-735.

Sucarrat, G. and A. Escribano (2013). Unbiased QML Estimation of Log-GARCH Models in the Presence of Zero Errors. Work in progress.

Terasvirta, T. (2009). An introduction to univariate GARCH models. In T. Mikosch, T. Kreiss, J.-P. Davis, R. Andersen, and T. Gustav (Eds.), Handbook of Financial Time Series. Berlin: Springer.

Terasvirta, T. (2012). Nonlinear Models for Autoregressive Conditional Heteroskedasticity. In L. Bauwens, C. Hafner, and S. Laurent (Eds.), Handbook of Volatility Models and Their Applications. New Jersey: Wiley.

Weiss, A. (1986). Asymptotic Theory for ARCH Models: Estimation and Testing. Econometric Theory 2, 107-131.

White, H. (1980). A Heteroskedasticity-Consistent Covariance Matrix and a Direct Test for Heteroskedasticity. Econometrica 48, 817-838.

Wintenberger, O. (2012). QML estimation of the continuously invertible EGARCH(1,1) model. Unpublished working paper.

Yu, H. (2007). High moment partial sum processes of residuals in ARMA models and their applications. Journal of Time Series Analysis 28, 72-91.