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Semiparametric Bayesian inference of long-memory stochastic volatility models. (English) Zbl 1062.62232
The author considers a long-memory stochastic volatility model defined as $y_{t}=\sigma\exp\{h_{t}\}\xi_{t},\quad (1-L)^{d}h_{t}=\sigma_{\eta}\eta_{t},\quad t=1,\dots,T,$ where at time $$t$$ the mean corrected return from holding a financial instrument is $$y_{t}$$, and $$h_{t}$$ is the unobservable log volatility that behaves as a fractionally integrated process. It is assumed that $$| d |<1/2$$ and that the innovations $$\xi_{t}$$ and $$\eta_{t}$$ are uncorrelated standard normal white noise processes. The fractional differencing operator is $$(1-L)^{d}$$, where $$L$$ is the lag operator, and $$x_{t-s}=L^{s}x_{t}$$ is defined by its binomial expansion.
It is described how to quickly and efficiently sample from the posterior distribution of long-memory stochastic volatility model parameters with Markov chain Monte Carlo simulators. The author proposes a Markov chain Monte Carlo simulator in the wavelet domain that converges quickly to the target density and produces a mix of draws from the desired posterior distribution. The proposed algorithm augments the latent volatility wavelet coefficients with the long-memory stochastic volatility parameters. Using simulated and empirical stock return data the author finds that the proposed algorithm produces uncorrelated draws of the posterior distribution and point estimates that rival existing long-memory stochastic volatility estimators.

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 65C40 Numerical analysis or methods applied to Markov chains 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F15 Bayesian inference 91B28 Finance etc. (MSC2000)
##### Software:
Ox; longmemo; sapa
Full Text:
##### References:
 [1] Adenstedt R. K., Annals of Statistics 2 pp 1095– (1974) · Zbl 0296.62081 [2] Andersen T., Journal of Finance 52 pp 975– (1997) [3] DOI: 10.1016/0304-4076(95)01732-1 · Zbl 0854.62099 [4] Beran J., Statistics for Long-memory Processes (1994) · Zbl 0869.60045 [5] Berger J. O., Bayesian Statistics 4 pp 35– (1992) [6] Bollerslev T., Journal of Business and Economic Statistics 17 pp 9– (1999) [7] DOI: 10.1016/S0304-4076(97)00072-9 · Zbl 0905.62116 [8] DOI: 10.1214/aos/1028144856 · Zbl 0929.62091 [9] D. Chan, R. Kohn, and C. Kirby (2003 ) Multivariate stochastic volatility with leverage . Working Paper, Australian Graduate School of Management, University of New South Wales. [10] Chib S., Handbook of Econometrics 5 pp 3569– (2001) [11] DOI: 10.1023/A:1008853808677 [12] Chib S., American Statistician 49 pp 327– (1995) [13] Chib S., Econometric Theory 12 pp 409– (1996) [14] Chib S., Biometrika 85 pp 347– (1998) [15] DOI: 10.1016/S0304-4076(02)00122-7 · Zbl 1030.62017 [16] DOI: 10.1016/S0304-4076(01)00137-3 · Zbl 1099.62539 [17] S. Chib, F. Nardari, and N. Shephard (2002 ) Analysis of high dimensional multivariate stochastic volatility models , unpublished manuscript, Olin School of Business, John M. Washington University. · Zbl 1418.62377 [18] Daubechies I., Ten Lectures on Wavelets (1992) · Zbl 0776.42018 [19] DOI: 10.1017/S0266466601174025 · Zbl 1018.62079 [20] DOI: 10.1109/18.333875 · Zbl 0810.60079 [21] Doornik D. F., Ox: An Object-Oriented Matrix Language (2001) [22] Escobar M. D., Journal of the American Statistical Association 89 pp 268– (1994) [23] Escobar M. D., Journal of the American Statistical Association 90 pp 577– (1995) [24] Ferguson T. S., Recent Advances in Statistics: Papers in Honor of Herman Chernoff on His Sixtieth Birthday pp 287– (1983) [25] Fletcher R., Practical Methods of Optimization (1987) · Zbl 0905.65002 [26] Fuller W. A., Introduction to Statistical Time Series (1996) · Zbl 0851.62057 [27] Gelfand A. E., Journal of the American Statistical Association 85 pp 398– (1990) [28] Geweke J., Bayesian Statistics 4 pp 169– (1992) · Zbl 1093.62107 [29] Geweke J., Journal of Time Series Analysis 4 pp 221– (1983) [30] Ghysels E., Handbook of Statistics 14 pp 119– (1996) [31] Granger C. W. J., Journal of Time Series Analysis 4 pp 221– (1980) · Zbl 0541.62106 [32] Harvey A. C., Forecasting Volatility in the Financial Market (1998) [33] Harvey A. C., Journal of Business and Economic Statistics 14 pp 429– (1996) [34] Hastings W. K., Biometrika 57 pp 97– (1970) [35] Hernandez E., A First Course on Wavelets (1996) [36] K. Hirano (1998 ) Essay on the econometric analysis of panel data . Ph.D. Thesis , Harvard University. [37] DOI: 10.1111/1468-0262.00305 · Zbl 1121.62557 [38] Hosking J. R. M., Biometrika 68 pp 165– (1981) [39] Hosking J. R. M., Water Resources Research 20 pp 1898– (1984) [40] DOI: 10.1002/(SICI)1099-131X(199901)18:1<17::AID-FOR686>3.3.CO;2-D [41] Jensen M. J., Studies in Nonlinear Dynamics and Econometrics 3 pp 239– (1999) [42] DOI: 10.1016/S0165-1889(99)00010-X · Zbl 0953.91058 [43] Jong P., Biometrika 82 pp 339– (1995) [44] DOI: 10.1111/1467-937X.00050 · Zbl 0910.90067 [45] Liu J. S., Biometrika 81 pp 27– (1994) [46] S. M. MacEachern (1992 ) Estimating normal means with a conjugate style Dirichlet process prior , Technical report no. 487, Dept. of Statistics, Ohio State University. · Zbl 0825.62053 [47] DOI: 10.1002/(SICI)1099-1255(199807/08)13:4<333::AID-JAE479>3.0.CO;2-I [48] DOI: 10.1109/34.192463 · Zbl 0709.94650 [49] Mallat S., A Wavelet Tour of Signal Processing (1999) · Zbl 0945.68537 [50] Metropolis N., Journal of Chemical Physics 21 pp 1087– (1953) [51] DOI: 10.1111/1368-423X.00046 · Zbl 0970.91060 [52] McCoy E. J., Journal of Computational and Graphical Statistics 5 pp 26– (1996) [53] McLeod A. I., Water Resource Research 14 pp 491– (1978) [54] Nelson D. B., Econometrica 59 pp 347– (1991) [55] Percival D. B., Spectral Analysis for Physical Applications (1998) [56] Percival D. B., Wavelet Methods for Time Series Analysis (2000) · Zbl 0963.62079 [57] Ray B. K., Journal of Business and Economic Statistics 18 pp 254– (2000) [58] Robinson P. M., Annals of Statistics 23 pp 1048– (1995) [59] Scott L. O., Journal of Financial and Quantitative Analysis 22 pp 419– (1987) [60] Sethuraman J., Statistical Decision Theory and Related Topics 2 pp 305– (1982) [61] Shephard N., Time Series Models (1996) [62] So M. K. P., Sankhya: The Indian Journal of Statistics 64 pp 1– (2002) [63] Tanner M. A., Journal of the American Statistical Association 82 pp 528– (1987) [64] Taylor S. J., Mathematical Finance 4 pp 183– (1994) [65] Taylor S. J., Modeling Financial Times Series (1986) [66] DOI: 10.1109/18.119750 · Zbl 0743.60079 [67] Tierney L., Annals of Statistics 21 pp 1701– (1994) [68] Tiwari R. C., Empirical Economics 13 pp 209– (1988) [69] B. Whitcher (1998 ) Assessing nonstationary time series using wavelets . Ph.D. dissertation , Department of Statistics, University of Washington. [70] Welch B. L., Journal of the Royal Statistical Society, Series B 25 pp 318– (1963) [71] West M., Aspects of Uncertainty:A Tribute to D.V. Lindley pp 363– (1994) [72] DOI: 10.1016/S0165-1765(99)00046-4 · Zbl 0922.90032 [73] DOI: 10.1109/78.120804
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