Sucarrat, Genaro and Grønneberg, Steffen and Escribano, Alvaro (2013): Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown.
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Abstract
Exponential models of Autoregressive Conditional Heteroscedasticity (ARCH) are of special interest, since they 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 and/or predetermined variables (``X") are included in the volatility specification. We propose a general framework for the estimation and inference in univariate and multivariate Generalised logARCHX (i.e. logGARCHX) models when the conditional density is not known. The framework employs (V)ARMAX representations and relies on a biasadjustment in the logvolatility intercept. The bias is induced by (V)ARMA estimators, but the remaining parameters are consistently estimated by (V)ARMA methods. We derive a simple formula for the biasadjustment, and a closedform expression for its asymptotic variance. Next, we show that adding exogenous or predetermined variables and/or increasing the dimension of the model does not change the structure of the problem. Accordingly, the univariate biasadjustment is applicable not only in univariate logGARCHX models, but also in multivariate logGARCHX models. An empirical application illustrates the usefulness of the methods.
Item Type:  MPRA Paper 

Original Title:  Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown 
English Title:  Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown 
Language:  English 
Keywords:  ARCH, exponential GARCH, LogGARCHX, ARMAX, multivariate logGARCHX, VARMAX 
Subjects:  C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C32  TimeSeries 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 
Item ID:  62352 
Depositing User:  Dr. Genaro Sucarrat 
Date Deposited:  25. Feb 2015 15:39 
Last Modified:  25. Feb 2015 16:31 
References:  Bardet, J.M. and O. Wintenberger (2009). Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes. Unpublished working paper. Bauwens, L. and G. Sucarrat (2010). General to Specific Modelling of Exchange Rate Volatility: A Forecast Evaluation. International Journal of Forecasting 26, 885907. Bauwens, L., C. Hafner, and D. Pierret (2013). Multivariate Volatility Modelling of Electricity Futures. Journal of Applied Econometrics 28, 743761. Berkes, I., L. Horvath, and P. Kokoszka (2003). GARCH processes: structure and estimation. Bernoulli 9, 201227. 22 Brockwell, P. J. and R. A. Davis (2006). Time Series: Theory and Methods. New York: Springer. 2nd. Edition. Brownlees, C., F. Cipollini, and G. Gallo (2012). Multiplicative Error Models. In L. Bauwens, C. Hafner, and S. Laurent (Eds.), Handbook of Volatility Models and Their Applications, pp. 223247. New Jersey: Wiley. Comte, F. and O. Lieberman (2003). Asymptotic Theory for Multivariate GARCH Processes. Journal of Multivariate Analysis 84, 6184. Duan, N. (1983). Smearing Estimate: A Nonparametric Retransformation Method. Journal of the Americal Statistical Association 78, pp. 605610. Engle, R. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflations. Econometrica 50, 9871008. Engle, R. (2002). Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models. Journal of Business and Economic Statistics 20, 339350. Engle, R. and Kroner. (1995). Multivariate simultaneous GARCH. Econometric Theory 11, 122150. Engle, R. F. and T. Bollerslev (1986). Modelling the persistence of conditional variances. Econometric Reviews 5, 150. Escribano, A., J. I. Pena, and P. Villaplana (2011). Modelling Electricity Prices: International Evidence. Oxford Bulletin of Economics and Statistics 73, 622650. Francq, C. and G. Sucarrat (2013). An Exponential ChiSquared QMLE for LogGARCH Models Via the ARMA Representation. http://mpra.ub.unimuenchen.de/51783/. Francq, C., O. Wintenberger, and J.M. Zakoian (2013). GARCH Models Without Positivity Constraints: Exponential or LogGARCH? Forthcoming 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 ARMAGARCH processes. Bernoulli 10, 605637. Francq, C. and J.M. Zakoian (2006). Linearrepresentation Based Estimation of Stochastic Volatility Models. Scandinavian Journal of Statistics 33, 785806. Francq, C. and J.M. Zakoian (2010). QML estimation of a class of multivariate GARCH models without moment conditions on the observed process. Unpublished working paper. Francq, C. and J.M. Zakoian (2014). Estimating multivariate GARCH and stochastic correlation models equation by equation. MPRA Paper No. 54250. Online at http: //mpra.ub.unimuenchen.de/54250/. Franses, P. H., J. Neele, and D. Van Dijk (2001). Modelling asymmetric volatility in weekly Dutch temperature data. Environmental Modeling and Software 16, 131137. Geweke, J. (1986). Modelling the Persistence of Conditional Variance: A Comment. Econometric Reviews 5, 5761. Hafner, C. and A. Preminger (2009). Asymptotic theory for a factor GARCH model. Econometric Theory 25, 336363. 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. Harvey, A. C. (1976). Estimating Regression Models with Multiplicative Heteroscedasticity. Econometrica 44, 461465. Harvey, A. C. (2013). Dynamic Models for Volatility and Heavy Tails. New York: Cambridge University Press. Harvey, A. C. and T. Chakravarty (2008). Betat(E)GARCH. Cambridge Working Papers in Economics 0840, Faculty of Economics, University of Cambridge. Ibragimov, R. and P. C. Phillips (2008). Regression asymptotics using martingale convergence methods. Econometric Theory 24 (4), 888947. Jeantheau, T. (1998). Strong consistency of estimators for multivariate arch models. Econometric Theory 14, pp. 7086. Kawakatsu, H. (2006). Matrix exponential GARCH. Journal of Econometrics 134, 95128. Koopman, S. J., M. Ooms, and M. A. Carnero (2007). Periodic Seasonal REGARFIMAGARCH Models for Daily Electricity Spot Prices. Journal of the American Statistical Association 102, 1627. Lee, S. (1997). A note on the residual empirical process in autoregressive models. Statistics and Probability Letters 32 (4), 405411. Ling, S. and M. McAleer (2003). Asymptotic theory for a vector ARMAGARCH model. Econometric Theory 19, 280310. Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Berlin: SpringerVerlag. 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, 347370. Pantula, S. (1986). Modelling the Persistence of Conditional Variance: A Comment. Econometric Reviews 5, 7173. Phillips, P. C. and V. Solo (1992). Asymptotics for linear processes. The Annals of Statistics, 9711001. Psaradakis, Z. and E. Tzavalis (1999). On regressionbased tests for persistence in logarithmic volatility models. Econometric Reviews 18, 441448. R Core Team (2014). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. Straumann, D. and T. Mikosch (2006). QuasiMaximumLikelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach. The Annals of Statistics 34, 24492495. Sucarrat, G. (2012). AutoSEARCH: GeneraltoSpecific (GETS) Model Selection. R package version 1.2. Sucarrat, G. (2013). betategarch: Simulation, estimation and forecasting of BetaSkewtEGARCH models. R package version 3.1. Sucarrat, G. (2014). lgarch: Simulation and estimation of logGARCH models. R package version 0.2. Sucarrat, G. and A. Escribano (2010). The Power LogGARCH Model. Universidad Carlos III de Madrid Working Paper 1013 in the Economic Series, June 2010. http://earchivo.uc3m.es/bitstream/10016/8793/1/we1013.pdf. Sucarrat, G. and A. Escribano (2012). Automated Model Selection in Finance: GeneraltoSpecific Modelling of the Mean and Volatility Specifications. Oxford Bulletin of Economics and Statistics 74, 716735. Sucarrat, G. and A. Escribano (2013). Unbiased QML Estimation of LogGARCH Models in the Presence of Zero Returns. MPRA Paper No. 50699. Online at http: //mpra.ub.unimuenchen.de/50699/. Wintenberger, O. (2013). Continuous Invertibility and Stable QML Estimation of the EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846867. Yu, H. (2007). High moment partial sum processes of residuals in ARMA models and their applications. Journal of Time Series Analysis 28, 7291. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/62352 
Available Versions of this Item

Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown. (deposited 29. Aug 2013 14:30)

Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown. (deposited 10. Jul 2014 20:12)

Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown. (deposited 11. Jul 2014 03:37)
 Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown. (deposited 25. Feb 2015 15:39) [Currently Displayed]

Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown. (deposited 11. Jul 2014 03:37)

Estimation and Inference in Univariate and Multivariate LogGARCHX Models When the Conditional Density is Unknown. (deposited 10. Jul 2014 20:12)