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EXPONENTIAL REALIZED GARCH-ITÔ VOLATILITY MODELS

Published online by Cambridge University Press:  10 November 2022

Donggyu Kim Open the ORCID record for Donggyu Kim [Opens in a new window]
Donggyu Kim*
Affiliation:
College of Business, Korea Advanced Institute of Science and Technology (KAIST)
*
Address correspondence to Donggyu Kim, College of Business, Korea Advanced Institute of Science and Technology (KAIST), Seoul, South Korea; e-mail: donggyukim@kaist.ac.kr.
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Abstract

This paper introduces a novel Itô diffusion process to model high-frequency financial data that can accommodate low-frequency volatility dynamics by embedding the discrete-time nonlinear exponential generalized autoregressive conditional heteroskedasticity (GARCH) structure with log-integrated volatility in a continuous instantaneous volatility process. The key feature of the proposed model is that, unlike existing GARCH-Itô models, the instantaneous volatility process has a nonlinear structure, which ensures that the log-integrated volatilities have the realized GARCH structure. We call this the exponential realized GARCH-Itô model. Given the autoregressive structure of the log-integrated volatility, we propose a quasi-likelihood estimation procedure for parameter estimation and establish its asymptotic properties. We conduct a simulation study to check the finite-sample performance of the proposed model and an empirical study with 50 assets among the S&P 500 compositions. Numerical studies show the advantages of the proposed model.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 37
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

The author thanks the Editor (Professor Peter C.B. Phillips), the Co-Editor (Professor D. Kristensen), and anonymous two referees for their careful reading of this paper and valuable comments. The research of the author was supported in part by the National Research Foundation of Korea (NRF) (2021R1C1C1003216).

References

REFERENCES

Aït-Sahalia, Y., Fan, J., & Xiu, D. (2010) High-frequency covariance estimates with noisy and asynchronous financial data. Journal of the American Statistical Association 105(492), 1504–1517.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Xiu, D. (2016) Increased correlation among asset classes: Are volatility or jumps to blame, or both? Journal of Econometrics 194(2), 205–219.CrossRefGoogle Scholar
Andersen, T.G. & Bollerslev, T. (1997a) Heterogeneous information arrivals and return volatility dynamics: Uncovering the long-run in high frequency returns. The Journal of Finance 52(3), 975–1005.CrossRefGoogle Scholar
Andersen, T.G. & Bollerslev, T. (1997b) Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance 4(2–3), 115–158.CrossRefGoogle Scholar
Andersen, T.G. & Bollerslev, T. (1998a) Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39(4), 885–905.CrossRefGoogle Scholar
Andersen, T.G. & Bollerslev, T. (1998b) Deutsche mark-dollar volatility: Intraday activity patterns, macroeconomic announcements, and longer run dependencies. The Journal of Finance 53(1), 219–265.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2003) Modeling and forecasting realized volatility. Econometrica 71(2), 579–625.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., & Huang, X. (2011) A reduced form framework for modeling volatility of speculative prices based on realized variation measures. Journal of Econometrics 160(1), 176–189.CrossRefGoogle Scholar
Andrews, D.W. (1992) Generic uniform convergence. Econometric Theory 8(2), 241–257.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2008) Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica 76(6), 1481–1536.Google Scholar
Bernardi, M. & Catania, L. (2018) The model confidence set package for R. International Journal of Computational Economics and Econometrics 8(2), 144–158.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31(3), 307–327.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992) Strict stationarity of generalized autoregressive processes. The Annals of Probability 20(4), 1714–1730.CrossRefGoogle Scholar
Campbell, J.Y. & Thompson, S.B. (2008) Predicting excess stock returns out of sample: Can anything beat the historical average? The Review of Financial Studies 21(4), 1509–1531.CrossRefGoogle Scholar
Corsi, F. (2009) A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7(2), 174–196.CrossRefGoogle Scholar
Diebold, F.X. & Mariano, R.S. (2002) Comparing predictive accuracy. Journal of Business & Economic Statistics 20(1), 134–144.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 987–1007.CrossRefGoogle Scholar
Fan, J. & Kim, D. (2018) Robust high-dimensional volatility matrix estimation for high-frequency factor model. Journal of the American Statistical Association 113(523), 1268–1283.CrossRefGoogle ScholarPubMed
Fan, J. & Wang, Y. (2007) Multi-scale jump and volatility analysis for high-frequency financial data. Journal of the American Statistical Association 102(480), 1349–1362.CrossRefGoogle Scholar
Francq, C., Wintenberger, O., & Zakoian, J.-M. (2013) GARCH models without positivity constraints: Exponential or log GARCH? Journal of Econometrics 177(1), 34–46.CrossRefGoogle Scholar
Hansen, P.R. & Huang, Z. (2016) Exponential GARCH modeling with realized measures of volatility. Journal of Business & Economic Statistics 34(2), 269–287.CrossRefGoogle Scholar
Hansen, P.R., Huang, Z., & Shek, H.H. (2012) Realized GARCH: A joint model for returns and realized measures of volatility. Journal of Applied Econometrics 27(6), 877–906.CrossRefGoogle Scholar
Hansen, P.R. & Lunde, A. (2005) A realized variance for the whole day based on intermittent high-frequency data. Journal of Financial Econometrics 3(4), 525–554.CrossRefGoogle Scholar
Hansen, P.R., Lunde, A., & Nason, J.M. (2011) The model confidence set. Econometrica 79(2), 453–497.Google Scholar
Haug, S. & Czado, C. (2007) An exponential continuous-time GARCH process. Journal of Applied Probability 44(4), 960–976.CrossRefGoogle Scholar
Jacod, J., Li, Y., Mykland, P.A., Podolskij, M., & Vetter, M. (2009) Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Processes and their Applications 119(7), 2249–2276.CrossRefGoogle Scholar
Kawakatsu, H. (2006) Matrix exponential GARCH. Journal of Econometrics 134(1), 95–128.CrossRefGoogle Scholar
Kim, D. & Fan, J. (2019) Factor GARCH–Itô models for high-frequency data with application to large volatility matrix prediction. Journal of Econometrics 208(2), 395–417.CrossRefGoogle Scholar
Kim, D., Shin, M., & Wang, Y. (2022) Overnight GARCH–Itô volatility models. Journal of Business & Economic Statistics, to appear.CrossRefGoogle Scholar
Kim, D. & Wang, Y. (2016) Unified discrete-time and continuous-time models and statistical inferences for merged low-frequency and high-frequency financial data. Journal of Econometrics 194, 220–230.CrossRefGoogle Scholar
Kim, D., Wang, Y., & Zou, J. (2016) Asymptotic theory for large volatility matrix estimation based on high-frequency financial data. Stochastic Processes and their Applications 126, 3527–3577.CrossRefGoogle Scholar
Klüppelberg, C., Lindner, A., & Maller, R. (2004) A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour. Journal of Applied Probability 41(3), 601–622.CrossRefGoogle Scholar
Martens, M. (2002) Measuring and forecasting S&P 500 index-futures volatility using high-frequency data. Journal of Futures Markets: Futures, Options, and Other Derivative Products 22(6), 497–518.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the Econometric Society 59, 347–370.CrossRefGoogle Scholar
Patton, A.J. (2011) Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160(1), 246–256.CrossRefGoogle Scholar
Shephard, N. & Sheppard, K. (2010) Realising the future: Forecasting with high-frequency-based volatility (HEAVY) models. Journal of Applied Econometrics 25(2), 197–231.CrossRefGoogle Scholar
Shin, M., Kim, D., & Fan, J. (2021) Adaptive robust large volatility matrix estimation based on high-frequency financial data. Available at SSRN 3793394.CrossRefGoogle Scholar
Song, X., Kim, D., Yuan, H., Cui, X., Lu, Z., Zhou, Y., & Wang, Y. (2021) Volatility analysis with realized GARCH–Itô models. Journal of Econometrics 222(1), 393–410.CrossRefGoogle Scholar
Tao, M., Wang, Y., Yao, Q., & Zou, J. (2011) Large volatility matrix inference via combining low-frequency and high-frequency approaches. Journal of the American Statistical Association 106(495), 1025–1040.CrossRefGoogle Scholar
Taylor, N. (2007) A note on the importance of overnight information in risk management models. Journal of Banking & Finance 31(1), 161–180.CrossRefGoogle Scholar
Todorova, N. & Souček, M. (2014) Overnight information flow and realized volatility forecasting. Finance Research Letters 11(4), 420–428.CrossRefGoogle Scholar
Tseng, T.-C., Lai, H.-C., & Lin, C.-F. (2012) The impact of overnight returns on realized volatility. Applied Financial Economics 22(5), 357–364.CrossRefGoogle Scholar
Xiu, D. (2010) Quasi-maximum likelihood estimation of volatility with high frequency data. Journal of Econometrics 159(1), 235–250.CrossRefGoogle Scholar
Zhang, L. (2006) Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12(6), 1019–1043.CrossRefGoogle 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(472), 1394–1411.CrossRefGoogle Scholar
Zhang, X., Kim, D., & Wang, Y. (2016) Jump variation estimation with noisy high frequency financial data via wavelets. Econometrics 4(3), 34.CrossRefGoogle Scholar