Forecasting Realized Volatility: A Review
Abstract
Modeling financial volatility is an important part of empirical finance. This paper provides a literature review of the most relevant volatility models, with a particular focus on forecasting models. We firstly discuss the empirical foundations of different kinds of volatility. The paper, then, analyses the non-parametric measure of volatility, named realized variance, and its empirical applications. A wide range of realized volatility models, both univariate and multivariate, is presented, such as time series models, MIDAS and GARCH-MIDAS models, Realized GARCH, and HEAVY models. We further discuss forecasting evaluation methods specifically suited for volatility models.
References
[1] Alexander, C., and A. M. Chibumba. 1996. Multivariate orthogonal factor GARCH, Working paper, University of Sussex.
[2] Amihud, Y., and H. Mendelson. 1987. Trading mechanisms and stock returns: an empirical investigation. Journal of Finance, 42: 533–553. DOI: http://dx.doi.org/10.1111/j.1540-6261.1987.tb04567.x
[3] Andersen, T. G. 2009. Stochastic volatility in Encyclopedia of Complexity and Systems Science. Springer Verlag.
[4] Andersen, T. G., and T. Bollerslev. 1998. Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts. International Economic Review, 39: 885–905.
[5] Andersen, T. G., T. Bollerslev, P. Christoffersen, and F. X. Diebold. 2006. Volatility and correlation forecasting in Handbook of Economic Forecasting, 778–878. Amsterdam: North-Holland.
[6] Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens. 2001. The distribution of realized stock return volatility. Journal of Financial Economics, 61: 43–76. DOI: https://doi.org/10.1016/S0304-405X(01)00055-1
[7] Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2000. Exchange rate returns standardized by realized volatility are (nearly) Gaussian. Multinational Finance Journal, 4: 159– 179.
[8] Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2001. The distribution of exchange rate volatility. Journal of American Statistical Association, 96: 42–55. DOI: https://doi.org/10.1198/016214501750332965
[9] Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2003. Modeling and Forecasting Realized Volatility. Econometrica, 71: 579–625. DOI: http://dx.doi.org/10.1111/1468-0262.00418
[10] Andersen, T. G., T. Bollerslev, and N. Meddahi .2005. Correcting the Errors: Volatility Forecast Evaluation Using High-Frequency Data and Realized Volatilities. Econometrica, 73(1): 279–296. DOI: http://dx.doi.org/10.1111/j.1468-0262.2005.00572.x
[11] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. 1999. Coherent measures of risk. Mathematical Finance, 9: 203–228. DOI: http://dx.doi.org/10.1111/1467-9965.00068
[12] Asai M., M. McAleer, and J. Yu. 2006. Multivariate stochastic volatility: A review. Econometric Reviews, 25: 145–175. DOI: https://doi.org/10.1080/07474930600713564
[13] Asgharian, H., C. Christiansen, and A. J. Hou. 2015. Macro-Finance Determinants of the Long-Run Stock-Bond Correlation: The DCC-MIDAS Specification. Journal of Financial Econometrics, 14(3): 617-642. DOI: https://doi.org/10.1093/jjfinec/nbv025
[14] Asgharian, H., A. J. Hou, AND F. Javed. 2013. The Importance of the Macroeconomic Variables in Forecasting Stock Return Variance: A GARCH-MIDAS Approach. Journal of Forecasting, 32: 600–612. DOI: https://doi.org/10.1002/for.2256
[15] Back, K. 1991. Asset pricing for general processes. Journal of Mathematical Economics, 20: 371–395. DOI: https://doi.org/10.1016/0304-4068(91)90037-T
[16] Baillie, R. T., Bollerslev, T., Mikkelsen, H.O. 1996. Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74(1): 3–30. DOI: https://doi.org/10.1016/S0304-4076(95)01749-6
[17] Bandi, F. M., and J. R. Russell. 2005. Realized covariation, realized beta and microstructure noise. Unpublished paper, Graduate School of Business, University of Chicago.
[18] Bandi, F. M., and J. R. Russell. (2006). Volatility. in Handbook of Financial Engineering. Elsevier.
[19] Bandi, F. M., and J. R. Russell. 2007. Microstructure noise, realized volatility, and optimal sampling. Unpublished paper, Graduate School of Business, University of Chicago.
[20] Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde, and N. Shephard. 2008. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica, 76(6): 1481–1536. DOI: https://doi.org/10.3982/ECTA6495
[21] Barndorff-Nielsen, O. E., and N. Shephard. 2002a. Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, 64(2): 253–280. DOI: https://doi.org/10.1111/1467-9868.00336
[22] Barndorff-Nielsen, O. E., and N. Shephard. 2002b. Estimating Quadratic Variation Using Realised Variance. Journal of Applied Econometrics, 17(5) 457–477. DOI: https://doi.org/10.1002/jae.691
[23] Barndorff-Nielsen, O. E., and N. Shephard. 2004. Power and Bipower Variation with Stochastic Volatility and Jumps. Journal of Financial Econometrics, 2(1): 1–37. DOI: https://doi.org/10.1093/jjfinec/nbh001
[24] Barr, D. G. 2013. Value at Risk. Bank of England, Centre for Central Banking Studies.
[25] Baruník, J., and F. Čech. 2016. On the modelling and forecasting multivariate realized volatility: Generalized Heterogeneous Autoregressive (GHAR) model. Journal of Forecasting, 36(2): 181-206. DOI: https://doi.org/10.1002/for.2423
[26] Bauer, G. H., and K. Vorkink. 2011. Forecasting multivariate realized stock market volatility. Journal of Econometrics, 160(1): 93–101. DOI: https://doi.org/10.1016/j.jeconom.2010.03.021
[27] Bauwens, L., C. Hafner, and S. Laurent. 2012. Handbook of Volatility Models and Their Applications. Wiley Online Library. DOI: https://doi.org/10.1002/9781118272039.ch1
[28] Becker, R., A. Clements, and R. O’Neill. 2010. A Cholesky-MIDAS model for predicting stock portfolio volatility. Working paper, Centre for Growth and Business Cycle Research Discussion Paper Series.
[29] Black, F. 1976. Noise. Journal of Finance, 41(3): 529–543. DOI: https://doi.org/10.1111/j.1540-6261.1986.tb04513.x
[30] Blanco, C., and G. Ihle. 1999. How Good is Your VaR? Using Backtesting to Assess System Performance. Financial Engineering News, 11(8): 1–2.
[31] Bollen, B. E., and B. Inder. 2002. Estimating daily volatility in financial markets utilizing intraday data. Journal of Empirical Finance, 9(5): 551–562. DOI: https://doi.org/10.1016/S0927-5398(02)00010-5
[32] Bollerslev, T. 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3): 307–327. DOI: https://doi.org/10.1016/0304-4076(86)90063-1
[33] Bollerslev, T. 1990. Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model,” The Review of Economics and Statistics, pp. 498–505.
[34] Bollerslev, T. 2009. Glossary to ARCH (GARCH). Working paper, Duke University.
[35] Bollerslev, T., R. F. Engle, and D. B. Nelson. 1994. ARCH models in Handbook of Econometrics. Elsevier Science, Amsterdam.
[36] Bollerslev, T., R. F. Engle, and J. M. Wooldridge. 1988. A capital asset pricing model with time-varying covariances. The Journal of Political Economy, 96(1): 116–131. DOI: http://dx.doi.org/10.1086/261527
[37] Bollerslev, T., and J. M. Wooldridge. 1992. Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11(2): 143–172. DOI: https://doi.org/10.1080/07474939208800229
[38] Bonato, M. 2009. Estimating the degrees of freedom of the Realized Volatility Wishart Autoregressive model. Working paper.
[39] Bonato, M., Caporin, M., Ranaldo, A. 2009. Forecasting realized (co)variances with a block structure Wishart autoregressive model. Working papers, Swiss National Bank.
[40] Cai, J. 1994. A Markov Model of Switching-Regime ARCH. Journal of Business and Economic Statistics, 12(3): 309–316. DOI: https://doi.org/10.2307/1392087
[41] Chan, L. K., J. Karceski, and J. Lakonishok. 1999. On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model. The Review of Financial Studies, 12(5): 937–974. DOI: https://doi.org/10.1093/rfs/12.5.937
[42] Chen, L. (1996). Stochastic Mean and Stochastic Volatility: A Four-Dimensional Term Structure of Interest Rates and Its Application to the Pricing of Derivative Securities. Financial Markets, Institutions, and Instruments, 5: 1–88.
[43] Christensen, B., and N. Prabhala. 1998. The relation between implied and realized volatility. Journal of Financial Economics, 50(2): 125–150. DOI: https://doi.org/10.1016/S0304-405X(98)00034-8
[44] Christiansen, C., M. Schmeling, and A. Schrimpf. 2012. A comprehensive look at financial volatility prediction by economic variables. Journal of Applied Econometrics, 27(6): 956–977. DOI: https://doi.org/10.1002/jae.2298
[45] Christoffersen, P. F. 1998. Evaluating interval forecasts. International Economic Review, 39(4): 841–862. DOI: https://doi.org/10.2307/2527341
[46] Clark, T. E., and K. D. West. 2007. Approximately normal tests for equal predictive accuracy in nested models. Journal of Econometrics, 138(1): 291–311. DOI: https://doi.org/10.1016/j.jeconom.2006.05.023
[47] Colacito, R., R. F. Engle, and E. Ghysels. 2011. A Component Model for Dynamic Correlations. Journal of Econometrics, 164(1): 45–59. DOI: https://doi.org/10.1016/j.jeconom.2011.02.013
[48] Conrad, C., and K. Loch. 2014. Anticipating Long-Term Stock Market Volatility. Journal of Applied Econometrics, 30(7): 1090–1114. DOI: https://doi.org/10.1002/jae.2404
[49] Corsi, F. 2009. A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7: 174–196. DOI: http://dx.doi.org/10.2139/ssrn.626064
[50] De Pooter, M., M. Martens, and D. Van Dijk. 2008. Predicting the Daily Covariance Matrix for S&P 100 Stocks Using Intraday Data - But Which Frequency to Use? Econometric Reviews, 27: 199–229. DOI: https://doi.org/10.1080/07474930701873333
[51] Diebold, F. X., and R. S. Mariano. 1995. Comparing predictive accuracy. Journal of Business and Economic Statistics, 13, 253–263. DOI: https://doi.org/10.1080/07350015.1995.10524599
[52] Ebens, H. 1999. Realized Stock Index Volatility. Working Paper No. 420, Department of Economics, Johns Hopkins University, Baltimore.
[53] Engle, R. F. 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4): 987–1007. DOI: https://doi.org/10.2307/1912773
[54] Engle, R. F. 2002. Dynamic conditional correlation: a simple class of multivariate GARCH models. Journal of Business and Economic Statistics, 20(3): 339–350. DOI: https://doi.org/10.1198/073500102288618487
[55] Engle, R. F., AND G. M. Gallo. 2006. A multiple indicators model for volatility using intra-daily data. Journal of Econometrics, 131: 3–27. DOI: https://doi.org/10.1016/j.jeconom.2005.01.018
[56] Engle, R. F., E. Ghysels, and B. Sohn. 2013. Stock Market Volatility and Macroeconomic Fundamentals. Review of Economics and Statistics, 95(3) 776–797. DOI: https://doi.org/10.1162/REST_a_00300
[57] Engle, R. F., and K. F. Kroner. 1995. Multivariate simultaneous generalized ARCH. Econometric Theory, 11(1): 122–150. DOI: https://doi.org/10.1017/S0266466600009063
[58] Engle, R. F., D. M. Lilien, and R. P. Robins. 1987. Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica, 55(2): 391–407. DOI: https://doi.org/10.2307/1913242
[59] Engle, R. F., V. K. Ng, and M. Rothschild. 1990. Asset pricing with a factor ARCH covariance structure: empirical estimates for treasury bills. Journal of Econometrics, 45: 213–238. DOI: https://doi.org/10.1016/0304-4076(90)90099-F
[60] Engle, R. F., and J. G. Rangel. 2008. The Spline-GARCH Model for Low-Frequency Volatility and Its Global Macroeconomic Causes. Review of Financial Studies, 21: 1187–1222. DOI: http://dx.doi.org/10.2139/ssrn.939447
[61] Engle, R. F., and K. Sheppard. 2001. Theoretical and empirical properties of dynamical conditional correlation model multivariate GARCH, UCSD Discussion.
[62] Epps, T. W. 1979. Comovements in Stock Prices in the Very Short Run. Journal of the American Statistical Association, 74(366): 291–298. DOI: https://doi.org/10.2307/2286325
[63] Fleming, J., C. Kirby, and B. Ostdiek. 2003. The economic value of volatility timing using realized volatility. Journal of Financial Economics, 67(3): 473–509. DOI: https://doi.org/10.1016/S0304-405X(02)00259-3
[64] Ghysels, E., Harvey, A. and Renault, E. 1996. Stochastic volatility in Statistical models in finance, 119–191. Amsterdam: North-Holland.
[65] Ghysels, E., Rubia, A., Valkanov, R. 2009. Multi-Period Forecasts of Volatility: Direct, Iterated, and Mixed-Data Approaches. Working paper.
[66] Ghysels, E., P. Santa-Clara, and R. Valkanov. 2004. The MIDAS touch: Mixed data sampling regression models. Working paper, UNC and UCLA.
[67] Ghysels, E. 2006. Predicting volatility: getting the most out of return data sampled at different frequencies. Journal of Econometrics, 131: 59–95. DOI: https://doi.org/10.1016/j.jeconom.2005.01.004
[68] Giacomini, R., White, H. 2006. Tests of conditional predictive ability. Econometrica, 74(6): 1545–1578. DOI: https://doi.org/10.1111/j.1468-0262.2006.00718.x
[69] Giot, P., and Laurent, S. 2004. Modelling daily Value-at-Risk using realized volatility and ARCH type models. Journal of Empirical Finance, 11(3): 379–398. DOI: https://doi.org/10.1016/j.jempfin.2003.04.003
[70] Glosten, L. R., R. Jagannathan, and D. E. Runkle. 1993. On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. Journal of Finance, 48(5): 1779–1801. DOI: https://doi.org/10.1111/j.1540-6261.1993.tb05128.x
[71] Gonzalo-Rivera, G. 1998. Smooth transition GARCH models. Studies in Nonlinear Dynamics and Econometrics, 3: 61–78.
[72] Gourieroux, C. S., Jasiak, J., Sufana, R. 2009. The Wishart Autoregressive process of multivariate stochastic volatility. Journal of Econometrics, 150(2): 167–181. DOI: https://doi.org/10.1016/j.jeconom.2008.12.016
[73] Hagerud, G. 1997. A New Non-Linear GARCH Model. Ph.D. Thesis, Stockholm School of Economics.
[74] Halbleib-Chiriac, R. 2007. Nonstationary Wishart Autoregressive Model. Working Paper, CoFE, University of Konstanz.
[75] Halbleib-Chiriac, R., AND V. Voev. 2011. Modelling and Forecasting Multivariate Realized Volatility. Journal of Applied Econometrics, 26(6): 922–947. DOI: https://doi.org/10.1002/jae.1152
[76] Hamilton, J. D., and R. Susmel. 1994. Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics, 64: 307–333. DOI: https://doi.org/10.1016/0304-4076(94)90067-1
[77] Hanoch, G., and H. Levy. 1969. The Efficiency Analysis of Choices Involving Risk. The Review of Economic Studies, 36(3): 335–346. DOI: https://doi.org/10.2307/2296431
[78] Hansen, P.R., Huang, Z., Shek, H.H. 2011. Realized GARCH: A Joint Model for Returns and Realized Measures of volatility. Journal of Applied Econometrics, 27(6): 877–906. DOI:https://doi.org/ 10.1002/jae.1234
[79] Hansen, P. R., Lunde, A. 2004. An unbiased measure of realized variance. Unpublished manuscript, Stanford University.
[80] Hansen, P. R., and A. Lunde. 2005. A realized variance for the whole day based on intermittent data. Journal of Financial Econometrics, 3(4): 525–554. DOI: https://doi.org/10.1093/jjfinec/nbi028
[81] Hansen, P. R., and A. Lunde. 2006. Consistent ranking of volatility models. Journal of Econometrics, 131(1-2): 97–121. DOI: https://doi.org/10.1016/j.jeconom.2005.01.005
[82] Hansen, P. R., A. Lunde, and J. M. Nason. 2011. The Model Confidence Set. Econometrica, 79(2): 435–497. DOI: https://doi.org/10.3982/ECTA5771
[83] Hansen, P. R., A. Lunde, and V. Voev. 2014. Realized Beta GARCH: A Multivariate GARCH Model With Realized Measures of Volatility. Journal of Applied Econometrics, 29(5): 774–799. DOI: https://doi.org/10.1002/jae.2389
[84] Harris, L. 1990. Estimation of stock variance and serial covariance from discrete observations. Journal of Financial and Quantitative Analysis, 25(3): 291–306. DOI: https://doi.org/10.2307/2330697
[85] Harris, L. 1991. Stock price clustering and discreteness. Review of Financial Studies, 4(3): 389–415. DOI: https://doi.org/10.1093/rfs/4.3.389
[86] Harvey, A., E. Ruiz, and N. Shephard. 1994. Multivariate stochastic variance models. Review of Economic Studies, 61(2): 247–264. DOI: https://doi.org/10.2307/2297980
[87] Hautsch, N., L. M. Kyj, and P. Malec. 2015. Do High-Frequency Data Improve High Dimensional Portfolio Allocations? Journal of Applied Econometrics, 30(2): 263–290. DOI: https://doi.org/10.1002/jae.2361
[88] Heston, S. L. 1993. A closed form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6: 327–343. DOI: https://doi.org/10.1093/rfs/6.2.327
[89] Higgins, M. L., and A. K. Bera. 1992. A Class of Nonlinear ARCH Models. International Economic Review, 33(1): 137–158. DOI: https://doi.org/10.2307/2526988
[90] Jacod, J., and P. Protter. 1998. Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability, 26(1): 267–307.
[91] Jagannathan, R., Ma, T. 2003. Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. The Journal of Finance, 58(4): 1651–1683. DOI: https://doi.org/10.1111/1540-6261.00580
[92] Jin, X., and J. M. Maheu. 2013. Modelling Realized Covariances and Returns. Journal of Financial Econometrics, 11(2): 335–369. DOI: https://doi.org/10.1093/jjfinec/nbs022
[93] Kuester, K., Mittnik, S., Paolella, M.S. 2006. Value-at-risk prediction: a comparison of alternative strategies. Journal of Financial Econometrics, 4(1): 53–89.
[94] Kupiec, P. 1995. Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2): 73–84. DOI: https://doi.org/10.3905/jod.1995.407942
[95] Kyj, L., Ostdiek, B., Ensor, K. 2009. Realized Covariance Estimation in Dynamic Portfolio Optimization. Working Paper.
[96] Laurent, S., J. V. Rombouts, and F. Violante. 2013. On Loss Functions and Ranking Forecasting Performances of Multivariate Volatility Models. Journal of Econometrics, 173(1): 1–10. DOI: https://doi.org/10.1016/j.jeconom.2012.08.004
[97] Laurent, S., and F. Violante. 2012. Volatility forecasts evaluation and comparison. Wiley Interdisciplinary Reviews: Computational Statistics, 4(1): 1–12. DOI: https://doi.org/10.1002/wics.190
[98] Lopez, J. A. 1998. Regulatory evaluation of Value-at-Risk models. Journal of Risk, 1(2): 37-64.
[99] Maheu, J. M., McCurdy, T.H. 2002. Nonlinear features of realized volatility. Review of Economics and Statistics, 84(4): 668–681. DOI: https://doi.org/10.1162/003465302760556486
[100] Markowitz, H. 1952. Portfolio selection. Journal of Finance, 7(1): 77–91. DOI: https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
[101] Martens, M., M. De Pooter, and D. J. Van Dijk. 2004. Modeling and Forecasting S&P 500 Volatility: Long Memory, Structural Breaks and Nonlinearity. Tinbergen Institute Discussion Paper No. 04-067/4.
[102] McAleer, M., and M. Medeiros. 2008a. A multiple regime smooth transition Heterogeneous Autoregressive model for long memory and asymmetries. Journal of Econometrics, 147(1): 104–119. DOI: https://doi.org/10.1016/j.jeconom.2008.09.032
[103] McAleer, M., and M. Medeiros. 2008b. Realized Volatility: a Review. Econometric Reviews, 27: 10–45. DOI: https://doi.org/10.1080/07474930701853509
[104] Merton, R. C. 1980. On estimating the expected return on the market: An explanatory investigation. Journal of Financial Economics, 8: 323–361.
[105] Mincer, J. A., and V. Zarnowitz. 1969. The Evaluation of Economic Forecasts in Economic Forecasts and Expectations: Analysis of Forecasting Behavior and Performance, NBER Chapters, 3–46. National Bureau of Economic Research, Inc.
[106] Nelson, D. B. 1990. Stationarity and persistence in the GARCH (1,1) model. Econometric Theory, 6(3): 318–334. DOI: https://doi.org/10.1017/S0266466600005296
[107] Noureldin, D., N. Shephard, and K. Sheppard. 2011. Multivariate high-frequency-based volatility (HEAVY) models. Journal of Applied Econometrics, 27(6): 907–933. DOI: https://doi.org/10.1002/jae.1260
[108] Oomen, R. C. 2001. Using High Frequency Stock Market Index Data to Calculate, Model & Forecast Realized Return Variance. Working paper.
[109] Oomen, R. C. 2005a. Properties of bias-corrected realized variance under alternative sampling schemes. Journal of Financial Econometrics, 3(4): 555–577. DOI: https://doi.org/10.1093/jjfinec/nbi027
[110] Oomen, R. C. 2005b. Properties of realized variance under alternative sampling schemes. Journal of Business and Economic Statistics, 24(2): 219–237. DOI: https://doi.org/10.1198/073500106000000044
[111] Patton, A. J. 2011. Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics, 160(1): 246 – 256. DOI: https://doi.org/10.1016/j.jeconom.2010.03.034
[112] Patton, A. J., and K. Sheppard. 2009. Evaluating Volatility and Correlation Forecasts. Handbook of Financial Time Series, 801-838. Springer, Berlin, Heidelberg. DOI: https://doi.org/10.1007/978-3-540-71297-8_36
[113] Paye, B. S. 2012. Dèjà vol: Predictive regressions for aggregate stock market volatility using macroeconomic variables. Journal of Financial Economics, 106(3): 527–546. DOI: https://doi.org/10.1016/j.jfineco.2012.06.005
[114] Poon, S.H., and C. W. Granger. 2003. Forecasting Volatility in Financial Markets: A Review. Journal of Economic Literature, 41(2): 478–539. DOI: https://doi.org/10.1257/002205103765762743
[115] Poterba, J. M., and L. H. Summers. 1986. The persistence of volatility and stock market fluctuations. The American Economic Review, 76(5): 1142–1151.
[116] Protter, P. E. 1992. Stochastic Integration and Differential Equations: A New Approach. New York: Springer-Verlag.
[117] Richardson, M., Stock, J.H. 1989. Drawing inference from statistics based on multiyear asset returns. Journal of Financial Economics, 25(2): 323–348. DOI: https://doi.org/10.1016/0304-405X(89)90086-X
[118] Ross, S. A. 1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3): 341–360. DOI: https://doi.org/10.1016/0022-0531(76)90046-6
[119] Schwert, W. G. 1989. Why does stock market volatility change over time? Journal of Finance, 44(5): 1115–1153. DOI: https://doi.org/10.1111/j.1540-6261.1989.tb02647.x
[120] Schwert, W. G. 1990. Indexes of US stock prices from 1802 to 1987. Journal of Business, 63: 399–426.
[121] Sharpe, W. F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19(3): 425–442. DOI: https://doi.org/10.2307/2977928
[122] Shephard, N. 1996. Statistical aspects of ARCH and stochastic volatility in Time Series Models in Econometrics, Finance and Other Fields.
[123] Shephard, N., and K. Sheppard. 2010. Realising the future: forecasting with high-frequency-based volatility (HEAVY) models. Journal of Applied Econometrics, 25(2): 197–231. DOI: https://doi.org/10.1002/jae.1158
[124] Shiryaev, A. N. 1999. Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific.
[125] Silvennoinen, A., and T. Teräsvirta. 2008. Multivariate GARCH Models. Working paper.
[126] Sowell, F. 1992. Maximum likelihood estimation of fractionally integrated time series models. Journal of Econometrics, 53: 165–188. DOI: https://doi.org/10.1016/0304-4076(92)90084-5
[127] Taylor, S. J. 1986. Modeling Financial Time Series. John Wiley and Sons.
[128] Taylor, S.J. 1994. Modeling Stochastic Volatility: A Review and Comparative Study. Mathematical Finance, 4(2): 183–204. DOI: https://doi.org/10.1111/j.1467-9965.1994.tb00057.x
[129] Taylor, S. J., X. Xu. 1997. The incremental volatility information in one million foreign exchange quotations. Journal of Empirical Finance, 4(4): 317–340. DOI: https://doi.org/10.1016/S0927-5398(97)00010-8
[130] Theil, H., and J. C. G. Boot. 1962. The final form of econometric equation systems. Review of International Statistical Institute, 30(2): 136–152. DOI: https://doi.org/10.2307/1401895
[131] Van der Weide, R. 2002. GO-GARCH: a multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics, 17(5): 549–564. DOI: https://doi.org/10.1002/jae.688
[132] von Neumann, J., and O. Morgenstern. 1947. Theory of Games and Economic Behavior. Princeton University Press, Princeton, New Jersey, 2nd Edition.
[133] West, K. 1996. Asymptotic inference about predictive ability. Econometrica, 64(5): 1067–1084. DOI: https://doi.org/10.2307/2171956
[134] Wold, S. 1976. Spline Functions in Data Analysis. Technometrics, 16(1): 1–11. DOI: https://doi.org/10.2307/1267485
[135] Zakoian, J. M. 1994. Threshold heteroskedastic. Journal of Economics Dynamics and Control, 18(5): 931–955. DOI: https://doi.org/10.1016/0165-1889(94)90039-6
[136] Zellner, A. 1962. An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias. Journal of the American Statistical Association, 57(298): 348–368. DOI: https://doi.org/10.2307/2281644
[137] Zhang, L. 2006. Estimating Covariation: Epps Effect, Microstructure Noise. Working paper.
[138] 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. DOI: https://doi.org/10.1198/016214505000000169
[139] Zhou, B. 1996. High frequency data and volatility in foreign-exchange rates. Journal of Business and Economic Statistics, 14(1): 45–52. DOI: https://doi.org/10.2307/1392098
[2] Amihud, Y., and H. Mendelson. 1987. Trading mechanisms and stock returns: an empirical investigation. Journal of Finance, 42: 533–553. DOI: http://dx.doi.org/10.1111/j.1540-6261.1987.tb04567.x
[3] Andersen, T. G. 2009. Stochastic volatility in Encyclopedia of Complexity and Systems Science. Springer Verlag.
[4] Andersen, T. G., and T. Bollerslev. 1998. Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts. International Economic Review, 39: 885–905.
[5] Andersen, T. G., T. Bollerslev, P. Christoffersen, and F. X. Diebold. 2006. Volatility and correlation forecasting in Handbook of Economic Forecasting, 778–878. Amsterdam: North-Holland.
[6] Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens. 2001. The distribution of realized stock return volatility. Journal of Financial Economics, 61: 43–76. DOI: https://doi.org/10.1016/S0304-405X(01)00055-1
[7] Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2000. Exchange rate returns standardized by realized volatility are (nearly) Gaussian. Multinational Finance Journal, 4: 159– 179.
[8] Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2001. The distribution of exchange rate volatility. Journal of American Statistical Association, 96: 42–55. DOI: https://doi.org/10.1198/016214501750332965
[9] Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys. 2003. Modeling and Forecasting Realized Volatility. Econometrica, 71: 579–625. DOI: http://dx.doi.org/10.1111/1468-0262.00418
[10] Andersen, T. G., T. Bollerslev, and N. Meddahi .2005. Correcting the Errors: Volatility Forecast Evaluation Using High-Frequency Data and Realized Volatilities. Econometrica, 73(1): 279–296. DOI: http://dx.doi.org/10.1111/j.1468-0262.2005.00572.x
[11] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. 1999. Coherent measures of risk. Mathematical Finance, 9: 203–228. DOI: http://dx.doi.org/10.1111/1467-9965.00068
[12] Asai M., M. McAleer, and J. Yu. 2006. Multivariate stochastic volatility: A review. Econometric Reviews, 25: 145–175. DOI: https://doi.org/10.1080/07474930600713564
[13] Asgharian, H., C. Christiansen, and A. J. Hou. 2015. Macro-Finance Determinants of the Long-Run Stock-Bond Correlation: The DCC-MIDAS Specification. Journal of Financial Econometrics, 14(3): 617-642. DOI: https://doi.org/10.1093/jjfinec/nbv025
[14] Asgharian, H., A. J. Hou, AND F. Javed. 2013. The Importance of the Macroeconomic Variables in Forecasting Stock Return Variance: A GARCH-MIDAS Approach. Journal of Forecasting, 32: 600–612. DOI: https://doi.org/10.1002/for.2256
[15] Back, K. 1991. Asset pricing for general processes. Journal of Mathematical Economics, 20: 371–395. DOI: https://doi.org/10.1016/0304-4068(91)90037-T
[16] Baillie, R. T., Bollerslev, T., Mikkelsen, H.O. 1996. Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74(1): 3–30. DOI: https://doi.org/10.1016/S0304-4076(95)01749-6
[17] Bandi, F. M., and J. R. Russell. 2005. Realized covariation, realized beta and microstructure noise. Unpublished paper, Graduate School of Business, University of Chicago.
[18] Bandi, F. M., and J. R. Russell. (2006). Volatility. in Handbook of Financial Engineering. Elsevier.
[19] Bandi, F. M., and J. R. Russell. 2007. Microstructure noise, realized volatility, and optimal sampling. Unpublished paper, Graduate School of Business, University of Chicago.
[20] Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde, and N. Shephard. 2008. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica, 76(6): 1481–1536. DOI: https://doi.org/10.3982/ECTA6495
[21] Barndorff-Nielsen, O. E., and N. Shephard. 2002a. Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, 64(2): 253–280. DOI: https://doi.org/10.1111/1467-9868.00336
[22] Barndorff-Nielsen, O. E., and N. Shephard. 2002b. Estimating Quadratic Variation Using Realised Variance. Journal of Applied Econometrics, 17(5) 457–477. DOI: https://doi.org/10.1002/jae.691
[23] Barndorff-Nielsen, O. E., and N. Shephard. 2004. Power and Bipower Variation with Stochastic Volatility and Jumps. Journal of Financial Econometrics, 2(1): 1–37. DOI: https://doi.org/10.1093/jjfinec/nbh001
[24] Barr, D. G. 2013. Value at Risk. Bank of England, Centre for Central Banking Studies.
[25] Baruník, J., and F. Čech. 2016. On the modelling and forecasting multivariate realized volatility: Generalized Heterogeneous Autoregressive (GHAR) model. Journal of Forecasting, 36(2): 181-206. DOI: https://doi.org/10.1002/for.2423
[26] Bauer, G. H., and K. Vorkink. 2011. Forecasting multivariate realized stock market volatility. Journal of Econometrics, 160(1): 93–101. DOI: https://doi.org/10.1016/j.jeconom.2010.03.021
[27] Bauwens, L., C. Hafner, and S. Laurent. 2012. Handbook of Volatility Models and Their Applications. Wiley Online Library. DOI: https://doi.org/10.1002/9781118272039.ch1
[28] Becker, R., A. Clements, and R. O’Neill. 2010. A Cholesky-MIDAS model for predicting stock portfolio volatility. Working paper, Centre for Growth and Business Cycle Research Discussion Paper Series.
[29] Black, F. 1976. Noise. Journal of Finance, 41(3): 529–543. DOI: https://doi.org/10.1111/j.1540-6261.1986.tb04513.x
[30] Blanco, C., and G. Ihle. 1999. How Good is Your VaR? Using Backtesting to Assess System Performance. Financial Engineering News, 11(8): 1–2.
[31] Bollen, B. E., and B. Inder. 2002. Estimating daily volatility in financial markets utilizing intraday data. Journal of Empirical Finance, 9(5): 551–562. DOI: https://doi.org/10.1016/S0927-5398(02)00010-5
[32] Bollerslev, T. 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3): 307–327. DOI: https://doi.org/10.1016/0304-4076(86)90063-1
[33] Bollerslev, T. 1990. Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model,” The Review of Economics and Statistics, pp. 498–505.
[34] Bollerslev, T. 2009. Glossary to ARCH (GARCH). Working paper, Duke University.
[35] Bollerslev, T., R. F. Engle, and D. B. Nelson. 1994. ARCH models in Handbook of Econometrics. Elsevier Science, Amsterdam.
[36] Bollerslev, T., R. F. Engle, and J. M. Wooldridge. 1988. A capital asset pricing model with time-varying covariances. The Journal of Political Economy, 96(1): 116–131. DOI: http://dx.doi.org/10.1086/261527
[37] Bollerslev, T., and J. M. Wooldridge. 1992. Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11(2): 143–172. DOI: https://doi.org/10.1080/07474939208800229
[38] Bonato, M. 2009. Estimating the degrees of freedom of the Realized Volatility Wishart Autoregressive model. Working paper.
[39] Bonato, M., Caporin, M., Ranaldo, A. 2009. Forecasting realized (co)variances with a block structure Wishart autoregressive model. Working papers, Swiss National Bank.
[40] Cai, J. 1994. A Markov Model of Switching-Regime ARCH. Journal of Business and Economic Statistics, 12(3): 309–316. DOI: https://doi.org/10.2307/1392087
[41] Chan, L. K., J. Karceski, and J. Lakonishok. 1999. On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model. The Review of Financial Studies, 12(5): 937–974. DOI: https://doi.org/10.1093/rfs/12.5.937
[42] Chen, L. (1996). Stochastic Mean and Stochastic Volatility: A Four-Dimensional Term Structure of Interest Rates and Its Application to the Pricing of Derivative Securities. Financial Markets, Institutions, and Instruments, 5: 1–88.
[43] Christensen, B., and N. Prabhala. 1998. The relation between implied and realized volatility. Journal of Financial Economics, 50(2): 125–150. DOI: https://doi.org/10.1016/S0304-405X(98)00034-8
[44] Christiansen, C., M. Schmeling, and A. Schrimpf. 2012. A comprehensive look at financial volatility prediction by economic variables. Journal of Applied Econometrics, 27(6): 956–977. DOI: https://doi.org/10.1002/jae.2298
[45] Christoffersen, P. F. 1998. Evaluating interval forecasts. International Economic Review, 39(4): 841–862. DOI: https://doi.org/10.2307/2527341
[46] Clark, T. E., and K. D. West. 2007. Approximately normal tests for equal predictive accuracy in nested models. Journal of Econometrics, 138(1): 291–311. DOI: https://doi.org/10.1016/j.jeconom.2006.05.023
[47] Colacito, R., R. F. Engle, and E. Ghysels. 2011. A Component Model for Dynamic Correlations. Journal of Econometrics, 164(1): 45–59. DOI: https://doi.org/10.1016/j.jeconom.2011.02.013
[48] Conrad, C., and K. Loch. 2014. Anticipating Long-Term Stock Market Volatility. Journal of Applied Econometrics, 30(7): 1090–1114. DOI: https://doi.org/10.1002/jae.2404
[49] Corsi, F. 2009. A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7: 174–196. DOI: http://dx.doi.org/10.2139/ssrn.626064
[50] De Pooter, M., M. Martens, and D. Van Dijk. 2008. Predicting the Daily Covariance Matrix for S&P 100 Stocks Using Intraday Data - But Which Frequency to Use? Econometric Reviews, 27: 199–229. DOI: https://doi.org/10.1080/07474930701873333
[51] Diebold, F. X., and R. S. Mariano. 1995. Comparing predictive accuracy. Journal of Business and Economic Statistics, 13, 253–263. DOI: https://doi.org/10.1080/07350015.1995.10524599
[52] Ebens, H. 1999. Realized Stock Index Volatility. Working Paper No. 420, Department of Economics, Johns Hopkins University, Baltimore.
[53] Engle, R. F. 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4): 987–1007. DOI: https://doi.org/10.2307/1912773
[54] Engle, R. F. 2002. Dynamic conditional correlation: a simple class of multivariate GARCH models. Journal of Business and Economic Statistics, 20(3): 339–350. DOI: https://doi.org/10.1198/073500102288618487
[55] Engle, R. F., AND G. M. Gallo. 2006. A multiple indicators model for volatility using intra-daily data. Journal of Econometrics, 131: 3–27. DOI: https://doi.org/10.1016/j.jeconom.2005.01.018
[56] Engle, R. F., E. Ghysels, and B. Sohn. 2013. Stock Market Volatility and Macroeconomic Fundamentals. Review of Economics and Statistics, 95(3) 776–797. DOI: https://doi.org/10.1162/REST_a_00300
[57] Engle, R. F., and K. F. Kroner. 1995. Multivariate simultaneous generalized ARCH. Econometric Theory, 11(1): 122–150. DOI: https://doi.org/10.1017/S0266466600009063
[58] Engle, R. F., D. M. Lilien, and R. P. Robins. 1987. Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica, 55(2): 391–407. DOI: https://doi.org/10.2307/1913242
[59] Engle, R. F., V. K. Ng, and M. Rothschild. 1990. Asset pricing with a factor ARCH covariance structure: empirical estimates for treasury bills. Journal of Econometrics, 45: 213–238. DOI: https://doi.org/10.1016/0304-4076(90)90099-F
[60] Engle, R. F., and J. G. Rangel. 2008. The Spline-GARCH Model for Low-Frequency Volatility and Its Global Macroeconomic Causes. Review of Financial Studies, 21: 1187–1222. DOI: http://dx.doi.org/10.2139/ssrn.939447
[61] Engle, R. F., and K. Sheppard. 2001. Theoretical and empirical properties of dynamical conditional correlation model multivariate GARCH, UCSD Discussion.
[62] Epps, T. W. 1979. Comovements in Stock Prices in the Very Short Run. Journal of the American Statistical Association, 74(366): 291–298. DOI: https://doi.org/10.2307/2286325
[63] Fleming, J., C. Kirby, and B. Ostdiek. 2003. The economic value of volatility timing using realized volatility. Journal of Financial Economics, 67(3): 473–509. DOI: https://doi.org/10.1016/S0304-405X(02)00259-3
[64] Ghysels, E., Harvey, A. and Renault, E. 1996. Stochastic volatility in Statistical models in finance, 119–191. Amsterdam: North-Holland.
[65] Ghysels, E., Rubia, A., Valkanov, R. 2009. Multi-Period Forecasts of Volatility: Direct, Iterated, and Mixed-Data Approaches. Working paper.
[66] Ghysels, E., P. Santa-Clara, and R. Valkanov. 2004. The MIDAS touch: Mixed data sampling regression models. Working paper, UNC and UCLA.
[67] Ghysels, E. 2006. Predicting volatility: getting the most out of return data sampled at different frequencies. Journal of Econometrics, 131: 59–95. DOI: https://doi.org/10.1016/j.jeconom.2005.01.004
[68] Giacomini, R., White, H. 2006. Tests of conditional predictive ability. Econometrica, 74(6): 1545–1578. DOI: https://doi.org/10.1111/j.1468-0262.2006.00718.x
[69] Giot, P., and Laurent, S. 2004. Modelling daily Value-at-Risk using realized volatility and ARCH type models. Journal of Empirical Finance, 11(3): 379–398. DOI: https://doi.org/10.1016/j.jempfin.2003.04.003
[70] Glosten, L. R., R. Jagannathan, and D. E. Runkle. 1993. On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. Journal of Finance, 48(5): 1779–1801. DOI: https://doi.org/10.1111/j.1540-6261.1993.tb05128.x
[71] Gonzalo-Rivera, G. 1998. Smooth transition GARCH models. Studies in Nonlinear Dynamics and Econometrics, 3: 61–78.
[72] Gourieroux, C. S., Jasiak, J., Sufana, R. 2009. The Wishart Autoregressive process of multivariate stochastic volatility. Journal of Econometrics, 150(2): 167–181. DOI: https://doi.org/10.1016/j.jeconom.2008.12.016
[73] Hagerud, G. 1997. A New Non-Linear GARCH Model. Ph.D. Thesis, Stockholm School of Economics.
[74] Halbleib-Chiriac, R. 2007. Nonstationary Wishart Autoregressive Model. Working Paper, CoFE, University of Konstanz.
[75] Halbleib-Chiriac, R., AND V. Voev. 2011. Modelling and Forecasting Multivariate Realized Volatility. Journal of Applied Econometrics, 26(6): 922–947. DOI: https://doi.org/10.1002/jae.1152
[76] Hamilton, J. D., and R. Susmel. 1994. Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics, 64: 307–333. DOI: https://doi.org/10.1016/0304-4076(94)90067-1
[77] Hanoch, G., and H. Levy. 1969. The Efficiency Analysis of Choices Involving Risk. The Review of Economic Studies, 36(3): 335–346. DOI: https://doi.org/10.2307/2296431
[78] Hansen, P.R., Huang, Z., Shek, H.H. 2011. Realized GARCH: A Joint Model for Returns and Realized Measures of volatility. Journal of Applied Econometrics, 27(6): 877–906. DOI:https://doi.org/ 10.1002/jae.1234
[79] Hansen, P. R., Lunde, A. 2004. An unbiased measure of realized variance. Unpublished manuscript, Stanford University.
[80] Hansen, P. R., and A. Lunde. 2005. A realized variance for the whole day based on intermittent data. Journal of Financial Econometrics, 3(4): 525–554. DOI: https://doi.org/10.1093/jjfinec/nbi028
[81] Hansen, P. R., and A. Lunde. 2006. Consistent ranking of volatility models. Journal of Econometrics, 131(1-2): 97–121. DOI: https://doi.org/10.1016/j.jeconom.2005.01.005
[82] Hansen, P. R., A. Lunde, and J. M. Nason. 2011. The Model Confidence Set. Econometrica, 79(2): 435–497. DOI: https://doi.org/10.3982/ECTA5771
[83] Hansen, P. R., A. Lunde, and V. Voev. 2014. Realized Beta GARCH: A Multivariate GARCH Model With Realized Measures of Volatility. Journal of Applied Econometrics, 29(5): 774–799. DOI: https://doi.org/10.1002/jae.2389
[84] Harris, L. 1990. Estimation of stock variance and serial covariance from discrete observations. Journal of Financial and Quantitative Analysis, 25(3): 291–306. DOI: https://doi.org/10.2307/2330697
[85] Harris, L. 1991. Stock price clustering and discreteness. Review of Financial Studies, 4(3): 389–415. DOI: https://doi.org/10.1093/rfs/4.3.389
[86] Harvey, A., E. Ruiz, and N. Shephard. 1994. Multivariate stochastic variance models. Review of Economic Studies, 61(2): 247–264. DOI: https://doi.org/10.2307/2297980
[87] Hautsch, N., L. M. Kyj, and P. Malec. 2015. Do High-Frequency Data Improve High Dimensional Portfolio Allocations? Journal of Applied Econometrics, 30(2): 263–290. DOI: https://doi.org/10.1002/jae.2361
[88] Heston, S. L. 1993. A closed form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6: 327–343. DOI: https://doi.org/10.1093/rfs/6.2.327
[89] Higgins, M. L., and A. K. Bera. 1992. A Class of Nonlinear ARCH Models. International Economic Review, 33(1): 137–158. DOI: https://doi.org/10.2307/2526988
[90] Jacod, J., and P. Protter. 1998. Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability, 26(1): 267–307.
[91] Jagannathan, R., Ma, T. 2003. Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. The Journal of Finance, 58(4): 1651–1683. DOI: https://doi.org/10.1111/1540-6261.00580
[92] Jin, X., and J. M. Maheu. 2013. Modelling Realized Covariances and Returns. Journal of Financial Econometrics, 11(2): 335–369. DOI: https://doi.org/10.1093/jjfinec/nbs022
[93] Kuester, K., Mittnik, S., Paolella, M.S. 2006. Value-at-risk prediction: a comparison of alternative strategies. Journal of Financial Econometrics, 4(1): 53–89.
[94] Kupiec, P. 1995. Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2): 73–84. DOI: https://doi.org/10.3905/jod.1995.407942
[95] Kyj, L., Ostdiek, B., Ensor, K. 2009. Realized Covariance Estimation in Dynamic Portfolio Optimization. Working Paper.
[96] Laurent, S., J. V. Rombouts, and F. Violante. 2013. On Loss Functions and Ranking Forecasting Performances of Multivariate Volatility Models. Journal of Econometrics, 173(1): 1–10. DOI: https://doi.org/10.1016/j.jeconom.2012.08.004
[97] Laurent, S., and F. Violante. 2012. Volatility forecasts evaluation and comparison. Wiley Interdisciplinary Reviews: Computational Statistics, 4(1): 1–12. DOI: https://doi.org/10.1002/wics.190
[98] Lopez, J. A. 1998. Regulatory evaluation of Value-at-Risk models. Journal of Risk, 1(2): 37-64.
[99] Maheu, J. M., McCurdy, T.H. 2002. Nonlinear features of realized volatility. Review of Economics and Statistics, 84(4): 668–681. DOI: https://doi.org/10.1162/003465302760556486
[100] Markowitz, H. 1952. Portfolio selection. Journal of Finance, 7(1): 77–91. DOI: https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
[101] Martens, M., M. De Pooter, and D. J. Van Dijk. 2004. Modeling and Forecasting S&P 500 Volatility: Long Memory, Structural Breaks and Nonlinearity. Tinbergen Institute Discussion Paper No. 04-067/4.
[102] McAleer, M., and M. Medeiros. 2008a. A multiple regime smooth transition Heterogeneous Autoregressive model for long memory and asymmetries. Journal of Econometrics, 147(1): 104–119. DOI: https://doi.org/10.1016/j.jeconom.2008.09.032
[103] McAleer, M., and M. Medeiros. 2008b. Realized Volatility: a Review. Econometric Reviews, 27: 10–45. DOI: https://doi.org/10.1080/07474930701853509
[104] Merton, R. C. 1980. On estimating the expected return on the market: An explanatory investigation. Journal of Financial Economics, 8: 323–361.
[105] Mincer, J. A., and V. Zarnowitz. 1969. The Evaluation of Economic Forecasts in Economic Forecasts and Expectations: Analysis of Forecasting Behavior and Performance, NBER Chapters, 3–46. National Bureau of Economic Research, Inc.
[106] Nelson, D. B. 1990. Stationarity and persistence in the GARCH (1,1) model. Econometric Theory, 6(3): 318–334. DOI: https://doi.org/10.1017/S0266466600005296
[107] Noureldin, D., N. Shephard, and K. Sheppard. 2011. Multivariate high-frequency-based volatility (HEAVY) models. Journal of Applied Econometrics, 27(6): 907–933. DOI: https://doi.org/10.1002/jae.1260
[108] Oomen, R. C. 2001. Using High Frequency Stock Market Index Data to Calculate, Model & Forecast Realized Return Variance. Working paper.
[109] Oomen, R. C. 2005a. Properties of bias-corrected realized variance under alternative sampling schemes. Journal of Financial Econometrics, 3(4): 555–577. DOI: https://doi.org/10.1093/jjfinec/nbi027
[110] Oomen, R. C. 2005b. Properties of realized variance under alternative sampling schemes. Journal of Business and Economic Statistics, 24(2): 219–237. DOI: https://doi.org/10.1198/073500106000000044
[111] Patton, A. J. 2011. Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics, 160(1): 246 – 256. DOI: https://doi.org/10.1016/j.jeconom.2010.03.034
[112] Patton, A. J., and K. Sheppard. 2009. Evaluating Volatility and Correlation Forecasts. Handbook of Financial Time Series, 801-838. Springer, Berlin, Heidelberg. DOI: https://doi.org/10.1007/978-3-540-71297-8_36
[113] Paye, B. S. 2012. Dèjà vol: Predictive regressions for aggregate stock market volatility using macroeconomic variables. Journal of Financial Economics, 106(3): 527–546. DOI: https://doi.org/10.1016/j.jfineco.2012.06.005
[114] Poon, S.H., and C. W. Granger. 2003. Forecasting Volatility in Financial Markets: A Review. Journal of Economic Literature, 41(2): 478–539. DOI: https://doi.org/10.1257/002205103765762743
[115] Poterba, J. M., and L. H. Summers. 1986. The persistence of volatility and stock market fluctuations. The American Economic Review, 76(5): 1142–1151.
[116] Protter, P. E. 1992. Stochastic Integration and Differential Equations: A New Approach. New York: Springer-Verlag.
[117] Richardson, M., Stock, J.H. 1989. Drawing inference from statistics based on multiyear asset returns. Journal of Financial Economics, 25(2): 323–348. DOI: https://doi.org/10.1016/0304-405X(89)90086-X
[118] Ross, S. A. 1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3): 341–360. DOI: https://doi.org/10.1016/0022-0531(76)90046-6
[119] Schwert, W. G. 1989. Why does stock market volatility change over time? Journal of Finance, 44(5): 1115–1153. DOI: https://doi.org/10.1111/j.1540-6261.1989.tb02647.x
[120] Schwert, W. G. 1990. Indexes of US stock prices from 1802 to 1987. Journal of Business, 63: 399–426.
[121] Sharpe, W. F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19(3): 425–442. DOI: https://doi.org/10.2307/2977928
[122] Shephard, N. 1996. Statistical aspects of ARCH and stochastic volatility in Time Series Models in Econometrics, Finance and Other Fields.
[123] Shephard, N., and K. Sheppard. 2010. Realising the future: forecasting with high-frequency-based volatility (HEAVY) models. Journal of Applied Econometrics, 25(2): 197–231. DOI: https://doi.org/10.1002/jae.1158
[124] Shiryaev, A. N. 1999. Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific.
[125] Silvennoinen, A., and T. Teräsvirta. 2008. Multivariate GARCH Models. Working paper.
[126] Sowell, F. 1992. Maximum likelihood estimation of fractionally integrated time series models. Journal of Econometrics, 53: 165–188. DOI: https://doi.org/10.1016/0304-4076(92)90084-5
[127] Taylor, S. J. 1986. Modeling Financial Time Series. John Wiley and Sons.
[128] Taylor, S.J. 1994. Modeling Stochastic Volatility: A Review and Comparative Study. Mathematical Finance, 4(2): 183–204. DOI: https://doi.org/10.1111/j.1467-9965.1994.tb00057.x
[129] Taylor, S. J., X. Xu. 1997. The incremental volatility information in one million foreign exchange quotations. Journal of Empirical Finance, 4(4): 317–340. DOI: https://doi.org/10.1016/S0927-5398(97)00010-8
[130] Theil, H., and J. C. G. Boot. 1962. The final form of econometric equation systems. Review of International Statistical Institute, 30(2): 136–152. DOI: https://doi.org/10.2307/1401895
[131] Van der Weide, R. 2002. GO-GARCH: a multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics, 17(5): 549–564. DOI: https://doi.org/10.1002/jae.688
[132] von Neumann, J., and O. Morgenstern. 1947. Theory of Games and Economic Behavior. Princeton University Press, Princeton, New Jersey, 2nd Edition.
[133] West, K. 1996. Asymptotic inference about predictive ability. Econometrica, 64(5): 1067–1084. DOI: https://doi.org/10.2307/2171956
[134] Wold, S. 1976. Spline Functions in Data Analysis. Technometrics, 16(1): 1–11. DOI: https://doi.org/10.2307/1267485
[135] Zakoian, J. M. 1994. Threshold heteroskedastic. Journal of Economics Dynamics and Control, 18(5): 931–955. DOI: https://doi.org/10.1016/0165-1889(94)90039-6
[136] Zellner, A. 1962. An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias. Journal of the American Statistical Association, 57(298): 348–368. DOI: https://doi.org/10.2307/2281644
[137] Zhang, L. 2006. Estimating Covariation: Epps Effect, Microstructure Noise. Working paper.
[138] 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. DOI: https://doi.org/10.1198/016214505000000169
[139] Zhou, B. 1996. High frequency data and volatility in foreign-exchange rates. Journal of Business and Economic Statistics, 14(1): 45–52. DOI: https://doi.org/10.2307/1392098
Published
2018-02-25
How to Cite
BUCCI, Andrea.
Forecasting Realized Volatility: A Review.
Journal of Advanced Studies in Finance, [S.l.], v. 8, n. 2, p. 94-138, feb. 2018.
ISSN 2068-8393.
Available at: <https://journals.aserspublishing.eu/jasf/article/view/1782>. Date accessed: 11 sep. 2024.
doi: https://doi.org/10.14505//jasf.v8.2(16).02.
Section
Journal of Advanced Studies in Finance
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