Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-02-05T12:44:14.713Z Has data issue: false hasContentIssue false
  • English
  • Français

CONDITIONS FOR THE PROPAGATION OF MEMORY PARAMETER FROM DURATIONS TO COUNTS AND REALIZED VOLATILITY

Published online by Cambridge University Press:  01 June 2009

Rohit Deo ,
Clifford M. Hurvich ,
Philippe Soulier  and
Yi Wang
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish sufficient conditions on durations that are stationary with finite variance and memory parameter to ensure that the corresponding counting process N(t) satisfies Var N(t) ~ Ct2d+1 (C > 0) as t → ∞, with the same memory parameter that was assumed for the durations. Thus, these conditions ensure that the memory parameter in durations propagates to the same memory parameter in the counts. We then show that any autoregressive conditional duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, whereas any long memory stochastic duration model with d > 0 and all finite moments yields long memory in counts, with the same d. Finally, we provide some results about the propagation of long memory to the empirically relevant case of realized variance estimates affected by market microstructure noise contamination.

Type
Research Article
Information
Econometric Theory , Volume 25 , Issue 3 , June 2009 , pp. 764 - 792
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The authors thank the referees for invaluable suggestions that led to more elegant and shorter proofs and a better economic interpretation of the results. They also thank Jushan Bai, Xiaohong Chen, and Raymond Thomas for helpful comments and suggestions. Part of this research was conducted while Deo was at the University of Texas–Austin.

References

REFERENCES

Andersen, T.G. & Bollerslev, T. (1997) Heterogeneous information arrivals and return volatility dynamics: Uncovering the long-run in high frequency returns. Journal of Finance 52, 975–1005.10.1111/j.1540-6261.1997.tb02722.xGoogle Scholar
Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2001) The distribution of exchange rate volatility. Journal of the American Statistical Association 96, 42–55.10.1198/016214501750332965CrossRefGoogle Scholar
Andrasfai, B. (1977) Introductory Graph Theory. Adam Hilger.Google Scholar
Baccelli, F. & Brémaud, P. (2003) Elements of Queueing Theory. Springer-Verlag.10.1007/978-3-662-11657-9CrossRefGoogle Scholar
Baillie, R., Bollerslev, T., & Mikkelsen, H. (1996) Fractionally integrated generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 74, 3–30.10.1016/S0304-4076(95)01749-6CrossRefGoogle Scholar
Bandi, F.M. & Russell, J.R. (2006) Separating microstructure noise from volatility. Journal of Financial Economics 79, 655–692.10.1016/j.jfineco.2005.01.005CrossRefGoogle Scholar
Barndorff-Nielsen, O. & Shephard, N. (2006) Impact of jumps on returns and realised variances: Econometric analysis of time-deformed Lévy processes. Journal of Econometrics 131, 217–252.10.1016/j.jeconom.2005.01.009CrossRefGoogle Scholar
Bauwens, L. & Veredas, D. (2004) The stochastic conditional duration model: A latent variable model for the analysis of financial durations. Journal of Econometrics 119, 381–412.10.1016/S0304-4076(03)00201-XCrossRefGoogle Scholar
Billingsley, P. (1986) Probability and Measure, 2nd ed. Wiley.Google Scholar
Bollerslev, T. & Jubinski, D. (1999) Equity trading volume and volatility: Latent information arrivals and common long-run dependencies. Journal of Business & Economic Statistics 17, 9–21.Google Scholar
Bollerslev, T. & Mikkelsen, H.O. (1996) Modeling and pricing long memory in stock market volatility. Journal of Econometrics 73, 151–184.10.1016/0304-4076(95)01736-4CrossRefGoogle Scholar
Bradley, R. (2005) Basic properties of strong mixing conditions. A survey and some open questions. Probability Surveys 2, 107–144.10.1214/154957805100000104CrossRefGoogle Scholar
Breidt, F.J., Crato, N., & de Lima, P. (1998) The detection and estimation of long memory in stochastic volatility. Journal of Econometrics 83, 325–348.10.1016/S0304-4076(97)00072-9CrossRefGoogle Scholar
Burkholder, D.L. (1966) Martingale transforms. Annals of Mathematical Statistics 37, 1494–1505.10.1214/aoms/1177699141CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.10.1017/S0266466602181023CrossRefGoogle Scholar
Chung, K.L. (1974) A Course in Probability Theory, 2nd ed. Academic Press.Google Scholar
Clark, P.K. (1973) A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–155.10.2307/1913889CrossRefGoogle Scholar
Comte, F. & Renault, E. (1998) Long memory in continuous-time stochastic volatility models. Mathematical Finance 8, 291–323.10.1111/1467-9965.00057CrossRefGoogle Scholar
Daley, D.J., Rolski, T., & Vesilo, R. (2000) Long-range dependent point processes and their Palm-Khinchin distributions. Advances in Applied Probability 32, 1051–1063.10.1017/S0001867800010454CrossRefGoogle Scholar
Daley, D.J. & Vere-Jones, D. (2003) An Introduction to the Theory of Point Processes, 2nd ed. Springer-Verlag.Google Scholar
Deo, R., Hsieh, M., & Hurvich, C.M. (2007) Long Memory in Intertrade Durations, Counts and Realized Volatility of NYSE Stocks. Preprint, Stern School of Business, New York University.Google Scholar
Deo, R.S. & Hurvich, C.M. (2001) On the log periodogram regression estimator of the memory parameter in long memory stochastic volatility models. Econometric Theory 17, 686–710.10.1017/S0266466601174025CrossRefGoogle Scholar
Deo, R., Hurvich, C., & Lu, Y. (2006) Forecasting realized volatility using a long memory stochastic volatility model: Estimation, prediction and seasonal adjustment. Journal of Econometrics 131, 29–58.10.1016/j.jeconom.2005.01.003CrossRefGoogle Scholar
Dittmann, I. & Granger, C.W.J. (2002) Properties of nonlinear transformations of fractionally integrated processes. Journal of Econometrics 110, 113–133.10.1016/S0304-4076(02)00089-1CrossRefGoogle Scholar
Douc, R., Roueff, F., & Soulier, P. (2008) On the existence of some ARCH(∞) processes. Stochastic Processes and Their Applications 118, 755–761.10.1016/j.spa.2007.06.002CrossRefGoogle Scholar
Doukhan, P. (1994) Mixing. Lecture Notes in Statistics 85. Springer-Verlag.10.1007/978-1-4612-2642-0CrossRefGoogle Scholar
Engle, R. & Russell, J. (1998) Autoregressive conditional duration: A new model for irregularly spaced transaction data. Econometrica 66, 1127–1162.10.2307/2999632CrossRefGoogle Scholar
Epps, T.W. & Epps, M.L. (1976) The stochastic dependence of security price changes and transaction volumes: Implications for the mixture-of-distribution hypothesis. Econometrica 44, 305–321.10.2307/1912726CrossRefGoogle Scholar
Hansen, P.R. & Lunde, A. (2006) Realized variance and market microstructure noise. Journal of Business & Economic Statistics 24, 127–161.10.1198/073500106000000071CrossRefGoogle Scholar
Harvey, A.C. (1998) Long memory in stochastic volatility. In Knight, J. & Satchell, S. (eds.), Forecasting Volatility in Financial Markets, pp. 307–320. Butterworth-Heinemann.Google Scholar
Hurvich, C.M., Moulines, E., & Soulier, P. (2005) Estimating long memory in volatility. Econometrica 73, 1283–1328.10.1111/j.1468-0262.2005.00616.xCrossRefGoogle Scholar
Ibragimov, I.A. & Linnik, Y.V. (1971) Independent and Stationary Sequences of Random Variables, ed. Kingman, J.F.C.. Wolters-Noordhoff.Google Scholar
Iglehart, D.L. & Whitt, W. (1971) The equivalence of functional central limit theorems for counting processes and associated partial sums. Annals of Mathematical Statistics 42, 1372–1378.10.1214/aoms/1177693249CrossRefGoogle Scholar
Jasiak, J. (1999) Persistence in Intertrade Durations. Working paper, York University.10.2139/ssrn.162008CrossRefGoogle Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.10.1017/S0266466600005296CrossRefGoogle Scholar
Nieuwenhuis, G. (1989) Equivalence of functional limit theorems for stationary point processes and their palm distributions. Probability Theory and Related Fields 81, 593–608.10.1007/BF00367306CrossRefGoogle Scholar
Oomen, R.C.A. (2006) Properties of realized variance under alternative sampling schemes. Journal of Business & Economic Statistics 24, 219–237.10.1198/073500106000000044CrossRefGoogle Scholar
Press, S.J. (1967) A compound events model for security prices. Journal of Business 40, 317–335.10.1086/294980CrossRefGoogle Scholar
Resnick, S.I. (1997) Heavy tail modeling and teletraffic data. With discussion and a rejoinder by the author. Annals of Statistics 25, 1805–1869.Google Scholar
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84.10.1016/0304-4076(91)90078-RCrossRefGoogle Scholar
Robinson, P.M. & Henry, M. (1999) Long and short memory conditional heteroskedasticity in estimating the memory parameter of levels. Econometric Theory 15, 299–336.10.1017/S0266466699153027CrossRefGoogle Scholar
Roll, R. (1984) A simple implicit measure of the effective bid-ask spread in an efficient market. Journal of Finance 39, 1127–1139.10.1111/j.1540-6261.1984.tb03897.xGoogle Scholar
Rydberg, T.H. & Shephard, N. (2003) Dynamics of trade-by-trade price movements: Decomposition and models. Journal of Financial Econometrics 1, 2–25.10.1093/jjfinec/nbg002CrossRefGoogle Scholar
Surgailis, D. & Viano, M.-C. (2002) Long memory properties and covariance structure of the EGARCH model. European Series in Applied and Industrial Mathematics: Probability and Statistics 6, 311–329.Google Scholar
Tauchen, G. & Pitts, M. (1983) The price variability-volume relationship on speculative markets. Econometrica 51, 485–505.10.2307/1912002CrossRefGoogle Scholar
Tsay, R. (2002) Analysis of Financial Time Series. Wiley.10.1002/0471264105CrossRefGoogle Scholar
Viano, M.-C., Deniau, C., & Oppenheim, G. (1995) Long-range dependence and mixing for discrete time fractional processes. Journal of Time Series Analysis 16, 323–338.10.1111/j.1467-9892.1995.tb00237.xCrossRefGoogle Scholar
Yokoyama, R. (1980) Moment bounds for stationary mixing sequences. Zeitschrift Für Wahrscheinlichskeitstheorie und Verwandte Gebiete 52, 45–57.10.1007/BF00534186CrossRefGoogle Scholar
Zhang, L., Mykland, P.A., & Aït-Sahalia, Y. (2005) A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association 100, 1394–1411.10.1198/016214505000000169CrossRefGoogle Scholar