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Asymptotically Optimal Smoothing with ARCH Models

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  • Daniel B. Nelson
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    Abstract

    Suppose an observed time series is generated by a stochastic volatility model-i.e., there is an unobservable state variable controlling the volatility of the innovations in the series. As shown by Nelson (1992), and Nelson and Foster (1994), a misspecified ARCH model will often be able to consistently (as a continuous time limit is approached) estimate the unobserved volatility process, using information in the lagged residuals. This paper shows how to more efficiently estimate such a volatility process using information in both lagged and led residuals. In particular, this paper expands the optimal filtering results of Nelson and Foster (1994) and Nelson (1994) to smoothing.

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    Bibliographic Info

    Paper provided by National Bureau of Economic Research, Inc in its series NBER Technical Working Papers with number 0161.

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    Date of creation: Aug 1994
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    Publication status: published as Nelson, Daniel B. "Asymptotically Optimal Smoothing With ARCH Models," Econometrica, 1996, v64(3,May), 561-573.
    Handle: RePEc:nbr:nberte:0161

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    1. GHYSELS, Eric & HARVEY, Andrew & RENAULT, Eric, 1995. "Stochastic Volatility," CORE Discussion Papers 1995069, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    3. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 1994. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(4), pages 371-89, October.
    4. Daniel B. Nelson, 1994. "Asymptotic Filtering Theory for Multivariate ARCH Models," NBER Technical Working Papers 0162, National Bureau of Economic Research, Inc.
    5. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
    6. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    7. Ruiz, Esther, 1994. "Quasi-maximum likelihood estimation of stochastic volatility models," Journal of Econometrics, Elsevier, vol. 63(1), pages 289-306, July.
    8. Nelson, Daniel B & Foster, Dean P, 1994. "Asymptotic Filtering Theory for Univariate ARCH Models," Econometrica, Econometric Society, vol. 62(1), pages 1-41, January.
    9. Neil Shephard, 2005. "Stochastic volatility," Economics Series Working Papers 2005-W17, University of Oxford, Department of Economics.
    10. Foster, Dean P & Nelson, Daniel B, 1996. "Continuous Record Asymptotics for Rolling Sample Variance Estimators," Econometrica, Econometric Society, vol. 64(1), pages 139-74, January.
    11. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(04), pages 419-438, December.
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    Cited by:
    1. Guillermo Benavides & Carlos Capistrán, 2009. "Forecasting Exchange Rate Volatility: The Superior Performance of Conditional Combinations of Time Series and Option Implied Forecasts," Working Papers 2009-01, Banco de México.
    2. Jacobi, Frank, 2005. "ARCH-Prozesse und ihre Erweiterungen - Eine empirische Untersuchung für Finanzmarktzeitreihen -," Arbeitspapiere des Instituts für Statistik und Ökonometrie 31, Johannes Gutenberg-Universität Mainz, Institut für Statistik und Ökonometrie.
    3. Deschamps, Philippe J., 2012. "Bayesian estimation of generalized hyperbolic skewed student GARCH models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3035-3054.
    4. Christensen, Kim & Podolski, Mark, 2005. "Asymptotic theory for range-based estimation of integrated variance of a continuous semi-martingale," Technical Reports 2005,18, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    5. Goeij, P. C. de & Marquering, W., 2004. "Modeling the conditional covariance between stock and bond returns: A multivariate GARCH approach," Open Access publications from Tilburg University urn:nbn:nl:ui:12-194709, Tilburg University.
    6. Shirley J. Huang & Qianqiu Liu & Jun Yu, 2007. "Realized Daily Variance of S&P 500 Cash Index: A Revaluation of Stylized Facts," Annals of Economics and Finance, Society for AEF, vol. 8(1), pages 33-56, May.
    7. Matei, Marius, 2011. "Non-Linear Volatility Modeling of Economic and Financial Time Series Using High Frequency Data," Journal for Economic Forecasting, Institute for Economic Forecasting, vol. 0(2), pages 116-141, June.
    8. Shinichi Aihara, 2000. "Estimation of stochastic volatility in the Hull-White model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(3), pages 153-181.

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