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ECONOMETRIC ANALYSIS OF VOLATILITY COMPONENT MODELS

Published online by Cambridge University Press:  26 August 2014

Fangfang Wang  and
Eric Ghysels
Fangfang Wang*
Affiliation:
University of Illinois at Chicago
Eric Ghysels
Affiliation:
University of North Carolina at Chapel Hill
*
*Address correspondence to Fangfang Wang, Department of Information and Decision Sciences, University of Illinois at Chicago, 601 South Morgan Street, Chicago, IL 60607-7124; e-mail: [email protected].
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Abstract

Volatility component models have received considerable attention recently, not only because of their ability to capture complex dynamics via a parsimonious parameter structure, but also because it is believed that they can handle well structural breaks or nonstationarities in asset price volatility. This paper revisits component volatility models from a statistical perspective and attempts to explore the stationarity of the underlying processes. There is a clear need for such an analysis, since any discussion about nonstationarity presumes we know when component models are stationary. As it turns out, this is not the case and the purpose of the paper is to rectify this. We also look into the sampling behavior of the maximum likelihood estimates of recently proposed volatility component models and establish their consistency and asymptotic normality.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 2: Haavelmo Memorial Issue: Part Two , April 2015 , pp. 362 - 393
Copyright
Copyright © Cambridge University Press 2014 

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References

REFERENCES

Adrian, T. & Rosenberg, J. (2008) Stock returns and volatility: Pricing the short-run and long-run components of market risk. Journal of Finance 63, 2997–3030.CrossRefGoogle Scholar
Asgharian, H., Hou, A. J., & Javed, F. (2013) The Importance of the Macroeconomic Variables in Forecasting Stock Return Variance: A GARCH-MIDAS Approach, Journal of Forecasting, 32(7), 600–612.CrossRefGoogle Scholar
Alizadeh, S., Brandt, M., & Diebold, F. (2002) Range-based estimation of stochastic volatility models. Journal of Finance 57, 1047–1091.CrossRefGoogle Scholar
Andreou, E. & Ghysels, E. (2002) Detecting multiple breaks in financial market volatility dynamics. Journal of Applied Econometrics 17, 579–600.CrossRefGoogle Scholar
Baele, L., Bekaert, G., & Inghelbrecht, K. (2010) The determinants of stock and bond return comovements. Review of Financial Studies 23, 2374–2428.CrossRefGoogle Scholar
Basrak, B., Davis, R., & Mikosch, T. (2001) Sample ACF of multivariate stochastic recurrence equations with application to GARCH. Preprint, Available atwww.math.ku.dk/mikosch.Google Scholar
Bauwens, L., Hafner, C. M., & Pierret, D. (2013) Multivariate volatility modeling of electricity futures, Journal of Applied Econometrics, 28(5), 743–761.CrossRefGoogle Scholar
Berkes, I., Gombay, E., Horváth, L., & Kokoszka, P. (2004) Sequential change-point detection in GARCH(p,q) models. Econometric Theory 20, 1140–1167.CrossRefGoogle Scholar
Berkes, I., Horváth, L., & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bollerslev, T. & Wooldridge, J. (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews 11, 143–172.CrossRefGoogle Scholar
Bougerol, P. (1987) Tightness of products of random matrices and stability of linear stochastic systems. Annals of Probability 15, 40–74.CrossRefGoogle 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, 1714–1730.CrossRefGoogle Scholar
Brandt, A. (1986) The stochastic equation Y n+1 = A nY n + B n with stationary coefficients. Advances in Applied Probability 18, 211–220.Google Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Chauvet, M., Senyuz, Z., & Yoldas, E. (2010) What Does Realized Volatility Tell Us About Macroeconomic Fluctuations? Discussion paper, Working paper, University of California, Riverside.Google Scholar
Chen, M. & An, H. (1998) A note on the stationarity and the existence of moments of the GARCH model. Statistica Sinica 8, 505–510.Google Scholar
Chen, W.-P., Choudhry, T., & Wu, C.-C. (2013) The extreme value in crude oil and US dollar markets. Journal of International Money and Finance 36, 191–210.CrossRefGoogle Scholar
Chen, J. & Gupta, A. (1997) Testing and locating variance changepoints with application to stock prices. Journal of the American Statistical Association 92, 739–747.CrossRefGoogle Scholar
Chernov, M., Gallant, A., Ghysels, E., & Tauchen, G. (2003) Alternative models for stock price dynamics. Journal of Econometrics 116, 225–257.CrossRefGoogle Scholar
Cohen, J. & Newman, C. (1984) The stability of large random matrices and their products. Annals of Probability 12, 283–310.CrossRefGoogle Scholar
Dahlhaus, R. & Rao, S. (2006) Statistical inference for time-varying ARCH processes. Annals of Statistics 34, 1075–1114.CrossRefGoogle Scholar
Ding, Z. & Granger, C. (1996) Modeling volatility persistence of speculative returns: A new approach. Journal of Econometrics 73, 185–215.CrossRefGoogle Scholar
Dorion, C. (2010) Business Conditions, Market Volatility and Option Prices. Discussion paper, HEC Montreal Working paper.Google Scholar
Duan, J. (1997) Augmented GARCH (p,q) process and its diffusion limit. Journal of Econometrics 79, 97–127.CrossRefGoogle Scholar
Engle, R. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Engle, R. & Lee, G. (1999) A long-run and short-run component model of stock return volatility. In Engle, R. & White, H. (eds.), Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive WJ Granger, pp. 475–497. Oxford University Press.CrossRefGoogle Scholar
Engle, R. & Rangel, J. (2008) The spline-GARCH model for low-frequency volatility and its global macroeconomic causes. Review of Financial Studies 21, 1187–1222.CrossRefGoogle Scholar
Engle, R. F., Ghysels, E., & Sohn, B. (2013) Stock market volatility and macroeconomic fundamentals, Review of Economics and Statistics, 95(3), 776–797.CrossRefGoogle Scholar
Francq, C. & Zakoian, J. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.CrossRefGoogle Scholar
Francq, C. & Zakoian, J. (2006) Mixing properties of a general class of GARCH (1, 1) models without moment assumptions on the observed process. Econometric Theory 22, 815–834.CrossRefGoogle Scholar
Francq, C. & Zakoian, J. (2009) A tour in the asymptotic theory of GARCH estimation. Handbook of Financial Time Series, pp. 85–111. Springer.CrossRefGoogle Scholar
Francq, C. & Zakoian, J. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley.CrossRefGoogle Scholar
Furstenberg, H. (1963) Noncommuting random products. Transactions of the American Mathematical Society 108, 377–428.CrossRefGoogle Scholar
Furstenberg, H. & Kesten, H. (1960) Products of random matrices. Annals of Mathematical Statistics 31, 457–469.CrossRefGoogle Scholar
Gallant, A. & White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Basil Blackwell.Google Scholar
Ghysels, E. (2013) Matlab toolbox for MIDAS regressions. Available athttp://www.unc.edu/~eghysels.Google Scholar
Ghysels, E., Santa-Clara, P., & Valkanov, R. (2004) The MIDAS Touch: Mixed Data Sampling Regression Models. CIRANO.Google Scholar
Ghysels, E., Santa-Clara, P., & Valkanov, R. (2005) There is a risk-return trade-off after all. Journal of Financial Economics 76, 509–548.CrossRefGoogle Scholar
Ghysels, E., Santa-Clara, P., & Valkanov, R. (2006) Predicting volatility: Getting the most out of return data sampled at different frequencies. Journal of Econometrics 131, 59–95.CrossRefGoogle Scholar
Ghysels, E., Sinko, A., & Valkanov, R. (2006) MIDAS regressions: Further results and new directions. Econometric Reviews 26, 53–90.CrossRefGoogle Scholar
Glasserman, P. & Yao, D. (1995) Stochastic vector difference equations with stationary coefficients. Journal of Applied Probability 32, 851–866.CrossRefGoogle Scholar
Goldsheid, I. (1991) Lyapunov exponents and asymptotic behaviour of the product of random matrices. Lyapunov Exponents, pp. 23–37. Springer.CrossRefGoogle Scholar
Golosnoy, V., Gribisch, B., & Liesenfeld, R. (2012) The conditional autoregressive Wishart model for multivariate stock market volatility. Journal of Econometrics 167, 211–223.CrossRefGoogle Scholar
Hafner, C. & Linton, O. (2010) Efficient estimation of a multivariate multiplicative volatility model. Journal of Econometrics 159, 55–73.CrossRefGoogle Scholar
He, C. & Terasvirta, T. (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173–192.CrossRefGoogle Scholar
Horvath, L., Kokoszka, P., & Teyssière, G. (2001) Empirical process of the squared residuals of an ARCH sequence. Annals of Statistics 29, 445–469.CrossRefGoogle Scholar
Horvath, L., Kokoszka, P., & Zhang, A. (2006) Monitoring constancy of variance in conditionally heteroskedastic time series. Econometric Theory 22, 373–402.CrossRefGoogle Scholar
Inclan, C. & Tiao, G. (1994) Use of cumulative sums of squares for retrospective detection of changes of variance. Journal of the American Statistical Association 89, 913–923.Google Scholar
Kesten, H. & Spitzer, F. (1984) Convergence in distribution of products of random matrices. Probability Theory and Related Fields 67, 363–386.Google Scholar
Kingman, J. (1973) Subadditive ergodic theory. Annals of Probability 1, 883–899.CrossRefGoogle Scholar
Kokoszka, P. & Leipus, R. (2000) Change-point estimation in ARCH models. Bernoulli 6, 513–540.CrossRefGoogle Scholar
Kristensen, D. (2009) On stationarity and ergodicity of the bilinear model with applications to GARCH models. Journal of Time Series Analysis 30, 125–144.CrossRefGoogle Scholar
Kulperger, R. & Yu, H. (2005) High moment partial sum processes of residuals in GARCH models and their applications. Annals of Statistics 33, 2395–2422.CrossRefGoogle Scholar
Lee, S. & Hansen, B. (1994) Asymptotic theory for the GARCH (1, 1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–29.CrossRefGoogle Scholar
Lindner, A. (2009) Stationarity, mixing, distributional properties and moments of GARCH (p, q)–processes. Handbook of Financial Time Series, pp. 43–69. Springer.CrossRefGoogle Scholar
Lu, Z. & Smith, R. (1997) Estimating local Lyapunov exponents. Fields Institute Communications 11, 135–151.Google Scholar
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.CrossRefGoogle Scholar
Maheu, J. & McCurdy, T. (2007) Components of market risk and return. Journal of Financial Econometrics 5, 560–590.CrossRefGoogle Scholar
Meitz, M. & Saikkonen, P. (2008) Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models. Econometric Theory 24, 1291–1320.CrossRefGoogle Scholar
Merton, R.C. (1980) On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics 8, 323–361.CrossRefGoogle Scholar
Meyn, S.P. & Caines, P.E. (1991) Asymptotic behavior of stochastic systems possessing Markovian realizations. SIAM Journal on Control and Optimization 29, 535–561.CrossRefGoogle Scholar
Mokkadem, A. (1990) Propriétés de mélange des processus autorégressifs polynomiaux. Annales de l’Institut Henri Poincaré (B) Probabilités et statistiques 26, 219–260.Google Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH (l, l) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Newman, C. (1986) The distribution of Lyapunov exponents: Exact results for random matrices. Communications in Mathematical Physics 103, 121–126.CrossRefGoogle Scholar
Peres, Y. (1992) Domains of analytic continuation for the top Lyapunov exponent. Annales de l’Institut Henri Poincaré (B) Probabilités et statistiques 28, 131–148.Google Scholar
Pfanzagl, J. (1969) On the measurability and consistency of minimum contrast estimates. Metrika 14, 249–272.CrossRefGoogle Scholar
Pham, D. (1985) Bilinear markovian representation and bilinear models. Stochastic Processes and their Applications 20, 295–306.CrossRefGoogle Scholar
Pham, D. (1986) The mixing property of bilinear and generalised random coefficient autoregressive models. Stochastic Processes and their Applications 23, 291–300.Google Scholar
Rangel, J. (2011) Macroeconomic news, announcements, and stock market jump intensity dynamics. Journal of Banking and Finance 35, 1263–1276.CrossRefGoogle Scholar
Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 44, 1207–1239.CrossRefGoogle Scholar
Seneta, E. (1981) Non-negative Matrices and Markov Chains. Springer Verlag.CrossRefGoogle Scholar
Straumann, D. & Mikosch, T. (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495.CrossRefGoogle Scholar
Vanneste, J. (2010) Estimating generalized Lyapunov exponents for products of random matrices. Physical Review E 81, 036701.CrossRefGoogle ScholarPubMed
Weiss, A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.CrossRefGoogle Scholar
Whang, Y. & Linton, O. (1999) The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series. Journal of Econometrics 91, 1–42.CrossRefGoogle Scholar
Yao, J. (2001) On square-integrability of an AR process with Markov switching. Statistics and Probability Letters 52, 265–270.CrossRefGoogle Scholar