Asymptotics of Cholesky GARCH models and time-varying conditional betas

Darolles, Serges and Francq, Christian and Laurent, Sébastien (2018): Asymptotics of Cholesky GARCH models and time-varying conditional betas.

Abstract

This paper proposes a new model with time-varying slope coefficients. Our model, called CHAR, is a Cholesky-GARCH model, based on the Cholesky decomposition of the conditional variance matrix introduced by Pourahmadi (1999) in the context of longitudinal data. We derive stationarity and invertibility conditions and prove consistency and asymptotic normality of the Full and equation-by-equation QML estimators of this model. We then show that this class of models is useful to estimate conditional betas and compare it to the approach proposed by Engle (2016). Finally, we use real data in a portfolio and risk management exercise. We find that the CHAR model outperforms a model with constant betas as well as the dynamic conditional beta model of Engle (2016).

• Item Type: MPRA Paper
• Language: English
• Keywords: Multivariate-GARCH; conditional betas; covariance
• Subjects: C - Mathematical and Quantitative Methods > C5 - Econometric Modeling; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

Alexander, C. (2001) Orthogonal GARCH. In Mastering Risk. Mastering Risk, 2 21-38.

Alexander, C. and Chibumba, A. M. (1997) Multivariate orthogonal factor GARCH. Mimeo: University of Sussex, UK.

Amado, C. and Teräsvirta, T. (2014) Conditional correlation models of autoregressive conditional heteroscedasticity with nonstationary GARCH equations. Journal of Business and Economic Statistics 32, 69-87.

Aue, A., Hörmann, S., Horváth, L. and Reimherr, M. (2009) Break detection in the covariance structure of multivariate time series models. The Annals of Statistics 37, 4046-4087.

Avarucci, M., Beutner, E. and Za�aroni, P. (2013) On moment conditions for quasi-maximum likelihood estimation of multivariate ARCH models. Econometric Theory 29, 545-566.

Bauwens, L., Hafner, C. and Laurent, S. (2012) Volatility Models. Handbook of Volatility Models and Their Applications eds L. Bauwens, C. Hafner and S. Laurent, John Wiley & Sons, Inc., Hoboken, NJ, USA.

Bauwens, L., Laurent, S. and Rombouts, J.V.K. (2006) Multivariate GARCH models: a survey. Journal of Applied Econometrics 21, 79-109.

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

Billingsley, P. (1961) The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788-792.

Blasques, F., Gorgi, P., Koopman, S.J. and Wintenberger O. (2016) Feasible invertibility conditions and maximum likelihood estimation for observation-driven Models, Tinbergen Institute Discussion Paper 16-082/III

Bollerslev, T., Engle, R.F. and Wooldridge, J.M. (1988) A capital asset pricing model with time-varying covariances. Journal of Political Economy 96, 116-131.

Bollerslev, T. (1990) Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Review of Economics and Statistics 72, 498-505.

Boswijk, H. P. and van der Weide, R. (2011) Method of moments estimation of GO-GARCH models. Journal of Econometrics 163, 118-126.

Boudt, K., Laurent, S., Lunde, A., Quaedvlieg, R. and Sauri, O. (2017) Positive semidefinite integrated covariance estimation, factorizations and asynchronicity. Journal of Econometrics 196, 347-367.

Bougerol, P. and Picard, N. (1992a) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714-1729.

Bougerol, P. and Picard, N. (1992b) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115-127.

Chan, K.S. (1993) Consistency and limiting distribution of the LSE of a TAR. Ann. Statist. 21, 520-533.

Cox, D.R. (1981) Statistical analysis of time series: some recent developments. Scandinavian Journal of Statistics 8, 93-115.

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

Doornik, J. (2012) Object-oriented matrix programming using Ox. Timberlake Consultants Press.

Durbin J. and Koopman, S.J. (2012) Time series analysis by state space methods, Volume 38, Oxford Statistical Science Series, Oxford University Press.

Engle, R.F. (2016) Dynamic conditional beta. Journal of Financial Econometrics 14, 643-667.

Fama, E. and French, K. (1992) The cross-section of expected returns. Journal of Finance 47, 427-465.

Fama, E. and French, K. (2004) The capital asset pricing model: theory and evidences. Journal of Economic Perspectives 18, 25-46.

Fan, J., Wang, M. and Yao, Q. (2008) Modelling multivariate volatilities via conditionally uncorrelated components. Journal of the Royal Statistical Society, Series B, 70, 679-702.

Francq, C. and Thieu, L.Q. (2015) QML inference for volatility models with covariates. MPRA preprint No. 63198.

Francq, C. and Zakoïan, J-M. (2010) GARCH models: structure, statistical inference and financial applications. John Wiley.

Francq, C. and Zakoïan, J-M. (2012) QML estimation of a class of multivariate asymmetric GARCH models. Econometric Theory 28, 179-206.

Francq, C. and Zakoïan, J-M. (2016) Estimating multivariate volatility models equation by equation. Journal of the Royal Statistical Society B 78, 613-635.

Francq, C. and Zakoïan, J-M. (2017) Estimation risk for the VaR of portfolios driven by semi-parametric multivariate models. Preprint.

Glosten, L.R., Jaganathan, R. and 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.

Hamadeh, T. and 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.

Hansen, P.R., Lunde, A. and Nason, J.M. (2011) The model confidence set. Econometrica 79, 453-497.

Hansen, P.R., Lunde, A. and Voev, V. (2014) Realized beta GARCH: Multivariate GARCH model with realized measures of volatility and covolatility. Journal of Applied Econometrics 29, 774-799.

Harvey, A. (2013) Dynamic models for volatility and heavy tails: With applications to financial and economic time series. New York: Cambridge University Press.

He, C. and Teräsvirta, T. (2004) An extended constant conditional correlation GARCH model and its fourth-moment structure. Econometric Theory 20, 904-926.

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

Lanne, M. and P. Saikkonen (2007) A multivariate generalized orthogonal factor GARCH model. Journal of Business and Economic Statistics 25, 61-75.

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

Maheu, J.M and Shamsi, A.Z. (2016) Nonparametric dynamic conditional beta. MPRA Paper No. 73764.

McAleer, M., Hoti, S., and Chan, F. (2009) Structure and asymptotic theory for multivariate asymmetric conditional volatility. Econometric Reviews 28, 422-440.

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347-370.

Noureldin, D., Shephard, N. and Sheppard, K. (2014). Multivariate rotated ARCH models. Journal of Econometrics 179, 16-30.

Patton, A. and Verardo, M. (2012) Does beta move with news? Firm-specific information flows and learning about profitability. The Review of Financial Studies 25, 2789-2839.

Pedersen, R.S. (2017) Inference and testing on the boundary in extended constant conditional correlation GARCH models. Journal of Econometrics 196, 23-36.

Pourahmadi, M. (1999) Joint mean-covariance models with applications to longitudinal data: Unconstrained parametrization. Biometrika 86, 677-690.

Silvennoinen, A. and Teräsvirta, T. (2009) Multivariate GARCH models. Handbook of Financial Time Series T.G. Andersen, R.A. Davis, J-P. Kreiss and T. Mikosch, eds. New York: Springer.

Tsay, R. S. (2010) Analysis of financial time series, 3rd Edition, John Wiley & Sons.

van der Weide, R. (2002) GO-GARCH: a multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics 17, 549-564.

Wald, A. (1949) Note on the consistency of the maximum likelihood estimate. The Annals of Mathematical Statistics 20, 595-601.

Wintenberger, O. (2013). Continuous invertibility and stable QML estimation of the EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846-867.