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Composite likelihood estimation of stationary Gaussian processes with a view toward stochastic volatility

Published 19 Mar 2024 in econ.EM and q-fin.MF | (2403.12653v1)

Abstract: We develop a framework for composite likelihood inference of parametric continuous-time stationary Gaussian processes. We derive the asymptotic theory of the associated maximum composite likelihood estimator. We implement our approach on a pair of models that has been proposed to describe the random log-spot variance of financial asset returns. A simulation study shows that it delivers good performance in these settings and improves upon a method-of-moments estimation. In an application, we inspect the dynamic of an intraday measure of spot variance computed with high-frequency data from the cryptocurrency market. The empirical evidence supports a mechanism, where the short- and long-term correlation structure of stochastic volatility are decoupled in order to capture its properties at different time scales.

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References (63)
  1. Alizadeh, S., M. W. Brandt, and F. X. Diebold, 2002, “Range-based estimation of stochastic volatility models,” Journal of Finance, 57(3), 1047–1092.
  2. Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys, 2000, “Great realizations,” Risk, 13(3), 105–108.
  3.   , 2003, “Modeling and forecasting realized volatility,” Econometrica, 71(2), 579–625.
  4. Arcones, M. A., 1994, “Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors,” Annals of Probability, 22(4), 2242–2274.
  5. Barndorff-Nielsen, O., J. M. Corcuera, and M. Podolskij, 2013, “Limit theorems for functionals of higher order differences of Brownian semi-stationary processes,” in Prokhorov and Contemporary Probability Theory, ed. by A. N. Shiryaev, S. R. S. Varadhan, and E. L. Presman. Springer, Heidelberg, pp. 69–96.
  6. Barndorff-Nielsen, O. E., and J. Schmiegel, 2009, “Brownian semistationary processes and volatility/intermittency,” in Radon Series on Computational and Applied Mathematics: Advanced Financial Modelling, ed. by H. Albrecher, W. J. Runggaldier, and W. Schachermayer. De Gruyter, Berlin, pp. 1–25.
  7. Barndorff-Nielsen, O. E., and N. Shephard, 2002, “Econometric analysis of realized volatility and its use in estimating stochastic volatility models,” Journal of the Royal Statistical Society: Series B, 64(2), 253–280.
  8. Bennedsen, M., 2020, “Semiparametric inference on the fractal index of Gaussian and conditionally Gaussian time series data,” Econometric Reviews, 39(9), 875–903.
  9. Bennedsen, M., A. Lunde, and M. S. Pakkanen, 2022, “Decoupling the short- and long-term behavior of stochastic volatility,” Journal of Financial Econometrics, 20(5), 961–1006.
  10. Bennedsen, M., A. Lunde, N. Shephard, and A. E. D. Veraart, 2023, “Inference and forecasting for continuous-time integer-valued trawl processes,” Journal of Econometrics, 236(2), 105476.
  11. Besag, J., 1974, “Spatial interaction and the statistical analysis of lattice systems,” Journal of the Royal Statistical Society: Series B, 36(2), 192–236.
  12. Bolko, A. E., K. Christensen, M. Pakkanen, and B. Veliyev, 2023, “A GMM approach to estimate the roughness of stochastic volatility,” Journal of Econometrics, 235(2), 745–778.
  13. Brent, R. P., F. G. Gustavson, and D. Y. Y. Yun, 1980, “Fast solution of Toeplitz systems of equations and computation of Padé approximants,” Journal of Algorithms, 1(3), 259–295.
  14. Breuer, P., and P. Major, 1983, “Central limit theorems for non-linear functionals of Gaussian fields,” Journal of Multivariate Analysis, 13(3), 425–441.
  15. Chentsov, N. N., 1956, “Weak convergence of stochastic processes whose trajectories have no discontinuities of the second kind and the “heuristic” approach to the Kolmogorov-Smirnov test,” Theory of Probability and Its Applications, 1(1), 140–144.
  16. Cheridito, P., H. Kawaguchi, and M. Maejima, 2003, “Fractional Ornstein-Uhlenbeck processes,” Electronic Journal of Probability, 8(3), 1–14.
  17. Chong, C. H., and V. Todorov, 2023, “The Fine Structure of Volatility Dynamics,” Working paper, Hong Kong University of Science and Technology (HKUST).
  18. Christensen, K., M. Thyrsgaard, and B. Veliyev, 2019, “The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing,” Journal of Econometrics, 212(2), 556–583.
  19. Christoffersen, P. F., K. Jacobs, and K. Mimouni, 2010, “Volatility dynamics for the S&P500: Evidence from realized volatility, daily returns, and option prices,” Review of Financial Studies, 23(8), 3141–3189.
  20. Comte, F., and E. Renault, 1998, “Long memory in continuous-time stochastic volatility models,” Mathematical Finance, 8(4), 291–323.
  21. Corsi, F., 2009, “A simple approximate long-memory model of realized volatility,” Journal of Financial Econometrics, 7(2), 174–196.
  22. Cox, D. R., and N. Reid, 2004, “A note on pseudolikelihood constructed from marginal densities,” Biometrika, 91(3), 729–737.
  23. Davis, R. A., and C. Y. Yau, 2011, “Comments on pairwise likelihood in time series models,” Statistica Sinica, 21(1), 255–277.
  24. Durbin, J., 1960, “The fitting of time-series models,” Review of the International Statistical Institute, 28(3), 233–244.
  25. Fukasawa, M., T. Takabatake, and R. Westphal, 2022, “Consistent estimation for fractional stochastic volatility model under high-frequency asymptotics,” Mathematical Finance, 32(4), 1086–1132.
  26. Garnier, J., and K. Sølna, 2018, “Option pricing under fast-varying and rough stochastic volatility,” Annals of Finance, 14(4), 489–516.
  27. Gatheral, J., T. Jaisson, and M. Rosenbaum, 2018, “Volatility is rough,” Quantitative Finance, 18(6), 933–949.
  28. Gneiting, T., and M. Schlather, 2004, “Stochastic models that separate fractal dimension and the Hurst effect,” SIAM Review, 46(2), 269–282.
  29. Godambe, V. P., 1960, “An optimum property of regular maximum likelihood estimation,” Annals of Mathematical Statistics, 31(4), 1208–1211.
  30. Hansen, P. R., C. Kim, and W. Kimbrough, 2022, “Periodicity in cryptocurrency volatility and liquidity,” Journal of Financial Econometrics, Forthcoming, https://doi.org/10.1093/jjfinec/nbac034.
  31. Hansen, P. R., and A. Lunde, 2006, “Consistent ranking of volatility models,” Journal of Econometrics, 131(1–2), 97–121.
  32. Hosking, J. R. M., 1996, “Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series,” Journal of Econometrics, 73(1), 261–284.
  33. Jacod, J., Y. Li, P. A. Mykland, M. Podolskij, and M. Vetter, 2009, “Microstructure noise in the continuous case: The pre-averaging approach,” Stochastic Processes and their Applications, 119(7), 2249–2276.
  34. Lang, G., and F. Roueff, 2001, “Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates,” Statistical Inference for Stochastic Processes, 4(3), 283–306.
  35. Levinson, N., 1946, “The Wiener (root mean square) error criterion in filter design and prediction,” Journal of Mathematics and Physics, 25(1–4), 261–278.
  36. Li, J., P. C. B. Phillips, S. Shi, and J. Yu, 2023, “Weak identification of long memory with implications for inference,” Working paper, Singapore Management University.
  37. Li, J., V. Todorov, and G. Tauchen, 2013, “Volatility occupation times,” Annals of Statistics, 41(4), 1865–1891.
  38. Li, M., and S. C. Lim, 2008, “Modeling network traffic using generalized Cauchy process,” Physica A: Statistical Mechanics and its Applications, 387(11), 2584–2594.
  39. Lindsay, B. G., 1988, “Composite likelihood methods,” Contemporary Mathematics, 80(1), 220–239.
  40. Lindsay, B. G., G. Y. Yi, and J. Sun, 2011, “Issues and strategies in the selection of composite likelihoods,” Statistica Sinica, 21(1), 71–105.
  41. Liu, H., W. Song, and E. Zio, 2021, “Generalized Cauchy difference iterative forecasting model for wind speed based on fractal time series,” Nonlinear Dynamics, 103(1), 759–773.
  42. Liu, X., S. Shi, and J. Yu, 2020, “Persistent and rough volatility,” Working paper, Singapore Management University.
  43. Mancini, C., 2009, “Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps,” Scandinavian Journal of Statistics, 36(2), 270–296.
  44. Maruyama, G., 1949, “The harmonic analysis of stationary stochastic processes,” Memoirs of the Faculty of Science, Kyūshū University, 4(1), 45–106.
  45. Meyer, S., and L. Held, 2014, “Power-law models for infectious disease spread,” Annals of Applied Statistics, 8(3), 1612–1639.
  46. Mykland, P. A., and L. Zhang, 2016, “Between data cleaning and inference: Pre-averaging and robust estimators of the efficient price,” Journal of Econometrics, 194(2), 242–262.
  47. Newey, W. K., and D. McFadden, 1994, “Large sample estimation and hypothesis,” in Handbook of Econometrics: Volume IV, ed. by R. F. Engle, and D. McFadden. North-Holland, Amsterdam, pp. 2112–2245.
  48. Pakel, C., N. Shephard, K. Sheppard, and R. F. Engle, 2020, “Fitting vast dimensional time-varying covariance models,” Journal of Business and Economic Statistics, 39(3), 652–668.
  49. Podolskij, M., and M. Vetter, 2009a, “Bipower-type estimation in a noisy diffusion setting,” Stochastic Processes and their Applications, 119(9), 2803–2831.
  50.   , 2009b, “Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps,” Bernoulli, 15(3), 634–658.
  51. Rogers, L. C. G., 1997, “Arbitrage with fractional Brownian motion,” Mathematical Finance, 7(1), 95–105.
  52. Rosenbaum, M., and J. Zhang, 2022, “On the universality of the volatility formation process: When machine learning and rough volatility agree,” preprint arXiv:2206.14114.
  53. Rosenblatt, M., 1979, “Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 49(2), 125–132.
  54. Shi, S., and J. Yu, 2022, “Volatility puzzle: Long memory or antipersistency,” Management Science, 69(7), 3759–4361.
  55. Shi, S., J. Yu, and C. Zhang, 2023, “On the spectral density of fractional Ornstein-Uhlenbeck processes,” Working paper, Singapore Management University.
  56. Stewart, M., 2003, “A superfast Toeplitz solver with improved numerical stability,” SIAM Journal on Matrix Analysis and Applications, 25(3), 669–693.
  57. Taylor, S. J., and X. Xu, 1997, “The incremental volatility information in one million foreign exchange quotations,” Journal of Empirical Finance, 4(4), 317–340.
  58. Trench, W. F., 1964, “An algorithm for the inversion of finite Toeplitz matrices,” Journal of the Society for Industrial and Applied Mathematics, 12(3), 515–522.
  59. Varin, C., N. Reid, and D. Firth, 2011, “An overview of composite likelihood methods,” Statistica Sinica, 21(1), 5–42.
  60. Wang, X., W. Xiao, and J. Yu, 2023, “Modeling and forecasting realized volatility with the fractional Ornstein-Uhlenbeck process,” Journal of Econometrics, 232(2), 389–415.
  61. Wang, X., W. Xiao, J. Yu, and C. Zhang, 2023, “Maximum likelihood estimation of fractional Ornstein-Uhlenbeck process with discretely sampled data,” Working paper, Singapore Management University.
  62. White, H., 1982, “Maximum likelihood estimation of misspecified models,” Econometrica, 50(1), 1–25.
  63. Wooldridge, J. M., 1994, “Estimation and inference for dependent processes,” in Handbook of Econometrics: Volume IV, ed. by R. F. Engle, and D. McFadden. North-Holland, Amsterdam, pp. 2639–2738.

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