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WEAK DIFFUSION LIMITS OF DYNAMIC CONDITIONAL CORRELATION MODELS

Published online by Cambridge University Press:  13 June 2016

Christian M. Hafner ,
Sebastien Laurent  and
Francesco Violante
Christian M. Hafner*
Affiliation:
Université catholique de Louvain
Sebastien Laurent
Affiliation:
Aix-Marseille University
Francesco Violante
Affiliation:
Aarhus University
*
*Address correspondance to Christian M. Hafner, Université catholique de Louvain, ISBA and CORE, B-1348 Louvain-la-Neuve, Belgium, e-mail: Christian.hafner@uclouvain.be.
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Abstract

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The properties of dynamic conditional correlation (DCC) models, introduced more than a decade ago, are still not entirely known. This paper fills one of the gaps by deriving weak diffusion limits of a modified version of the classical DCC model. The limiting system of stochastic differential equations is characterized by a diffusion matrix of reduced rank. The degeneracy is due to perfect collinearity between the innovations of the volatility and correlation dynamics. For the special case of constant conditional correlations, a nondegenerate diffusion limit can be obtained. Alternative sets of conditions are considered for the rate of convergence of the parameters, obtaining time-varying but deterministic variances and/or correlations. A Monte Carlo experiment confirms that the often used quasi-approximate maximum likelihood (QAML) method to estimate the diffusion parameters is inconsistent for any fixed frequency, but that it may provide reasonable approximations for sufficiently large frequencies and sample sizes.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 3 , June 2017 , pp. 691 - 716
Copyright
Copyright © Cambridge University Press 2016 

Footnotes

This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program Investissements d’Avenir supervised by the Agence Nationale de la Recherche.

References

REFERENCES

Aielli, G.P. (2006) Consistent estimation of large scale dynamic conditional correlations. Department of Statistics, University of Florence.Google Scholar
Aielli, G.P. (2013) Dynamic conditional correlations: On properties and estimation. Journal of Business and Economic Statistics 31, 282–299.CrossRefGoogle Scholar
Alexander, C. & Lazar, E. (2005) On the continuous limit of GARCH. Discussion Papers in Finance 2005–13. ICMA Centre Discussion Paper No. DP2005-13. University of Reading, UK.CrossRefGoogle Scholar
Barone-Adesi, G., Rasmussen, H., & Ravanelli, C. (2005) An option pricing formula for the GARCH diffusion model. Computational Statistics and Data Analysis 49(2), 287–310.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bollerslev, T. (1990) Modeling the coherence on short-run nominal exchange rates: a multivariate generalized ARCH model. Review of Economics and Statistics 72, 498–505.CrossRefGoogle Scholar
Bollerslev, T., Engle, R., & Nelson, D. (1994) Arch Models. Handbook of Econometrics, vol. IV. North-Holland.Google Scholar
Broze, L., Scaillet, O., & Zakoian, J. (1998) Quasi indirect inference for diffusion processes. Econometric Theory 14, 161–186.CrossRefGoogle Scholar
Corradi, V. (2000) Reconsidering the continuous time limit of the GARCH(1,1) process, Journal of Econometrics 96, 145–153.CrossRefGoogle Scholar
Ding, Z., Granger, C., & Engle, R. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.Google Scholar
Drost, C. & Nijman, T. (1993) Temporal aggregation of GARCH processes. Econometrica 61, 909–927.CrossRefGoogle Scholar
Drost, F. & Werker, J. (1996) Closing the gap: Continuous time GARCH modeling. Journal of Econometrics 74, 31–57.CrossRefGoogle Scholar
Duan, J. (1997) Augmented p,q process and its diffusion limit. Journal of Econometrics 79, 97–127.CrossRefGoogle Scholar
Duffie, D. & Singleton, K. (1993) Simulated moments estimation of Markov models of asset prices. Econometrica 61, 929–952.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Engle, R. & Lee, G. (1996) Estimating diffusion models of stochastic volatility. In Rossi, P. (ed.), Modeling Stock Market Volatility: Bridging the Gap to Continuous Time, pp. 333–384. Academic Press.CrossRefGoogle Scholar
Ethier, S. & Kurtz, T. (1986) Markov Processes: Characterization and Convergence. Wiley.CrossRefGoogle Scholar
Fornari, F. & Mele, A. (1997) Weak convergence and distributional assumptions for a general class of nonlinear ARCH models. Econometric Reviews 16, 205–229.CrossRefGoogle Scholar
Fornari, F. & Mele, A. (2006) Approximating volatility diffusions with CEV-ARCH models. Journal of Economic Dynamics & Control 30(6), 931–966.CrossRefGoogle Scholar
Ghysels, E., Harvey, A., & Renault, E. (1996) Stochastic volatility. In Maddala, G. (ed.), Handbook of Statistics, vol. 14, pp. 119–191. North-Holland.Google Scholar
Hafner, C. (2003) Fourth moment structure of multivariate GARCH models. Journal of Financial Econometrics 1, 26–54.CrossRefGoogle Scholar
Hafner, C. (2008) Temporal aggregation of multivariate GARCH processes. Journal of Econometrics 142, 467–483.CrossRefGoogle Scholar
Hafner, C., Laurent, S., & Violante, F. (2016) Weak diffusion limits of dynamic conditional correlation models, CORE discussion paper 2016/09, Université catholique de Louvain, Belgium.Google Scholar
Hafner, C. & Rombouts, J. (2007) Estimation of temporally aggregated multivariate GARCH models. Journal of Statistical Computation and Simulation 77, 629–650.CrossRefGoogle Scholar
Hobson, D., & Rogers, L. (1998) Complete models with stochastic volatility. Mathematical Finance 8, 27–48.CrossRefGoogle Scholar
Jeantheau, T. (2004) A link between complete models with stochastic volatility and ARCH models. Finance and Stochastics 8, 111–131.CrossRefGoogle Scholar
Jones, C. (2003) The dynamics of stochastic volatility: evidence from underlying and options markets. Journal of Econometrics 116, 181–224.CrossRefGoogle Scholar
Kleppe, T., Yu, J., & Skaug, H. (2014) Maximum likelihood estimation of partially observed diffusion models. Journal of Econometrics 180, 73–80.CrossRefGoogle Scholar
Kushner, H. (1984) Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory. The MIT Press.Google Scholar
Lewis, A. (2000) Option Valuation under Stochastic Volatility. Finance Press.Google Scholar
Li, C. (2013) Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41, 1350–1380.CrossRefGoogle Scholar
Lütkepohl, H. (1996) Handbook of Matrices. Wiley.Google Scholar
Magnus, J. (1988) Linear Structures. Griffin.Google Scholar
Nelson, D. (1990) ARCH models as diffusion approximations. Journal of Econometrics 45, 7–38.Google Scholar
Nelson, D. (1996) Asymptotic filtering theory for multivariate ARCH models. Journal of Econometrics 71, 1–47.CrossRefGoogle Scholar
Nelson, D. & Foster, D. (1994) Asymptotic filtering theory for univariate ARCH models. Econometrica 62, 1–41.CrossRefGoogle Scholar
Phillips, P. & Yu, J. (2009) Maximum likelihood and Gaussian estimation of continuous time models in finance. In Anderson, T.G. et al. (eds.), Handbook of Financial Time Series, pp. 497–530. Springer.CrossRefGoogle Scholar
Steele, J. (2001) Stochastic Calculus and Financial Applications. Springer.CrossRefGoogle Scholar
Stentoft, L. (2011) American option pricing with discrete and continuous time models: An empirical comparison. Journal of Empirical Finance 18, 880–902.CrossRefGoogle Scholar
Stroock, D. & Varadhan, S. (1979) Multidimensional Diffusion Processes. Springer-Verlag.Google Scholar
Trifi, A. (2006) Issues of aggregation over time of conditional heteroscedastic volatility models: What kind of diffusion do we recover? Studies in Nonlinear Dynamics & Econometrics 10(4), 1314–1323.Google Scholar
Wang, Y. (2002) Asymptotic nonequivalence of GARCH models and diffusions. Annals of Statistics 30, 754–783.Google Scholar