Tsionas, Mike (2012): Simple techniques for likelihood analysis of univariate and multivariate stable distributions: with extensions to multivariate stochastic volatility and dynamic factor models.
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In this paper we consider a variety of procedures for numerical statistical inference in the family of univariate and multivariate stable distributions. In connection with univariate distributions (i) we provide approximations by finite location-scale mixtures and (ii) versions of approximate Bayesian computation (ABC) using the characteristic function and the asymptotic form of the likelihood function. In the context of multivariate stable distributions we propose several ways to perform statistical inference and obtain the spectral measure associated with the distributions, a quantity that has been a major impediment in using them in applied work. We extend the techniques to handle univariate and multivariate stochastic volatility models, static and dynamic factor models with disturbances and factors from general stable distributions, a novel way to model multivariate stochastic volatility through time-varying spectral measures and a novel way to multivariate stable distributions through copulae. The new techniques are applied to artificial as well as real data (ten major currencies, SP100 and individual returns). In connection with ABC special attention is paid to crafting well-performing proposal distributions for MCMC and extensive numerical experiments are conducted to provide critical values of the “closeness” parameter that can be useful for further applied econometric work.
|Item Type:||MPRA Paper|
|Original Title:||Simple techniques for likelihood analysis of univariate and multivariate stable distributions: with extensions to multivariate stochastic volatility and dynamic factor models|
|Keywords:||Univariate and multivariate stable distributions, MCMC, Approximate Bayesian Computation, Characteristic function|
|Subjects:||C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General
C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General
|Depositing User:||Mike Tsionas|
|Date Deposited:||15. Sep 2012 21:04|
|Last Modified:||22. Aug 2015 04:40|
Abramowitz, M., and I.A. Stegun, 1965, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover.
Aguilar, O. and M. West, 2000, Bayesian dynamic factor models and variance matrix discounting for portfolio allocation, Journal of Business and Economic Statistics, 18, 338-357.
Bazant, Z.P., and B.H. Oh, 1986, Efficient numerical integration on the surface of a sphere, Zeitschrift für Angewandte Mathematik und Mechanik 1, 37-49.
Belisle, C., H. Romeijn, and R. Smith, R., 1993, Hit and run algorithms for generating multivariate distributions, Mathematics of Operations Research 18, 255-266.
Bernanke, B., J. Boivin and P.S. Eliasz, 2005, Measuring the effects of monetary policy: A factor-augmented vector autoregressive (FAVAR) approach, Quarterly Journal of Economics, 120, 387-422.
Broda, S.A., Haas, M., Krause, J., Paolella, M.S., and S.C. Steude, 2012, Stable mixture GARCH models, Journal of Econometrics, in press.
Buckle, D.J., 1995, Bayesian inference for stable distributions, Journal of the American Statistical Association 90, 605-613.
Byczowski, T., J.P. Nolan, and B. Rajput, 1993, Approximation of multidimensional stable distributions, Journal of Multivariate Analysis 46, 13-31.
Cambanis, S., and G. Miller, 1981, Linear problems in pth order and stable processes, SIAM Journal of Applied Mathematics 41, 43-69.
Carrasco, M., and J. Florens, 2000, Generalization of GMM to a continuum of moment conditions, Econometric Theory 16, 797–834.
Carrasco, M., and J. Florens, J., 2002, Efficient GMM estimation using the empirical characteristic function. Working Paper; Department of Economics: University of Rochester.
Chambers, J.M., C.L. Mallows, and B.W. Stuck, 1976, A method for simulating stable random variables, Journal of the American Statistical Association 71, 340-344.
Cheng, B.N., and S.T. Rachev, 1995, Multivariate stable futures prices, Mathematical Finance 5, 133-153.
Chib, S. and Greenberg, E. (1994): Bayes inference for regression models with ARMA(p, q) errors. Journal of Econometrics 64, 183–206.
Chib, S., Nardari, F. and Shephard, N. (2006): Analysis of high dimensional multivariate stochastic volatility models. Journal of Econometrics 134, 341–371.
Chib, S., Y. Omori, and M. Asai, 2009, Multivariate Stochastic Volatility, in T.G. Anderson et al (eds), Handbook of Financial Time Series, Springer Verlag, Heidelberg.
Demarta, S. and A.J. McNeil, 2005, The t-copula and related copulas, International Statistical Review, 73, 111-129.
DiCiccio, T. J., R.E. Kass, A. Raftery, and L. Wasserman, 1997, Computing Bayes factors by combining simulation and asymptotic approximation, Journal of the American Statistical Association 92, 903-915.
Dominicy, Y., and D. Veredas, 2012, The method of simulated quantiles, Journal of Econometrics, in press.
Durbin, J. and S. J. Koopman, 2000, Time series analysis of non-Gaussian observations based on state space models from both classical and Bayesian perspectives (with discussion), Journal of the Royal Statistical Society B 62, 3-56.
Engle, R., and R. Watson, 1981, A one-factor multivariate time series model of metropolitan wage rates, Journal of the American Statistical Association 76, 774-781.
Feuerverger, A., and P. McDunnough, 1981a, On efficient inference in symmetric stable laws and processes, in M. Csorgo, ed., Statistics and Related Topics. New York: North-Holland, pp. 109–122.
Feuerverger, A., and P. McDunnough, 1981b, On some Fourier methods for inference, Journal of the American Statistical Association 76, 379–387.
Feuerverger, A. and R. A. Mureika , 1977, The empirical characteristic function and its applications, Annals of Statistics 5, 88-97.
Forni, M., and M. Lippi, 2001, The generalized dynamic factor model: Representation theory, Econometric Theory 17, 1113-1141.
Forni, M., M. Hallin, M. Lippi and L. Reichlin, 2005, The generalized dynamic factor model: Identification and estimation, Review of Economics and Statistics 82, 540-554.
Forni, M., M. Hallin, M. Lippi and L. Reichlin, 2005, The generalized dynamic factor model: One-sided estimation and forecasting, Journal of the American Statistical Association 100, 830-840.
Frees, E.W. and E.A. Valdez, 1998), Understanding Relationships Using Copulas, North American Actuarial Journal 2, 1- 25.
Garcia, R., E. Renault, and D. Veredas, 2011, Estimation of stable distributions by indirect inference, Journal of Econometrics 161, 325-337.
Gautier, E., and Y. Kitamura, 2011, Nonparametric estimation in random coefficient binary choice models, http://arxiv.org/pdf/0907.2451.pdf
Georgiadis, A.T., and B. Mulgrew, 2001, Adaptive Bayesian decision feedback equaliser for alpha-stable noise environments, Signal Processing 81, 1603-1623.
Geweke, J.F. and K.J. Singleton, 1980, Interpreting the likelihood ratio statistic in factor models when sample size is small, Journal of the American Statistical Association, 75, 133-137.
Geweke, J. F., 1977, The Dynamic Factor Analysis of Economic Time Series Models," in Latent Variables in Socioeconomic Models, ed. by D. Aigner, and A. Goldberger, pp. 365-383. North-Holland.
Geweke, J.F. and G. Zhou, G., 1996, Measuring the pricing error of the arbitrage pricing theory, The Review of Financial Studies, 9, 557-587.
Hallin, M., Swan, Y., Verdebout, T., and D. Veredas, 2012, One-step R-estimation in linear models with stable errors, Journal of Econometrics, in press.
Harvey, A. C., E. Ruiz, and N. Shephard, N., 1994, Multivariate stochastic variance models. Review of Economic Studies 61, 247–264.
Joe, H., 1997, Multivariate Models and Dependence Concepts, Chapman and Hall.
Kim, S., N. Shephard and S. Chib, 1998, Stochastic volatility: Likelihood inference and comparison with ARCH models, Review of Economic Studies, 65, 361-393.
Knight, J. L., S.E. Satchell, and J. Yu, 2002, Estimation of the stochastic volatility model by the empirical characteristic function method, Australian & New Zealand Journal of Statistics 44, 319–335.
Knight, J. L., and J. Yu, J., 2002, The empirical characteristic function in time series estimation, Econometric Theory 18:691–721.
Koutrouvelis, I. A., 1980, Regression-type estimation of the parameters of stable laws, Journal of the American Statistical Association 75, 918–928.
Koutrouvelis, I. A., 1981, An iterative procedure for estimation of the parameters of stable laws, Communications in Statististcs - Simulation and Computation 10, 17–28.
Li, F., M. Villani, and R. Kohn, 2010, Flexible Modeling of Conditional Distributions Using Smooth Mixtures of Asymmetric Student T Densities, Journal of Statistical Planning and Inference 140, 3638-3654.
Liu, S.-M., and B.W. Brorsen, 1995, Maximum likelihood estimation of a GARCH – stable model, Journal of Applied Econometrics 10, 273-285.
Liu, Z.-Y., K.-C. Chiu, and L. Xu, 2004, Investigations on non-Gaussian factor analysis, IEEE Signal Processing Letters 11, 597-600. Lombardi, M. J., and G. Calzolari, 2009, Indirect Estimation of -Stable Stochastic Volatility Models, Computational Statistics and Data Analysis 53, 2298-2308.
Lombardi, M. J., and G. Calzolari, 2008, Indirect Estimation of -Stable Distributions and Processes, Econometrics Journal 11, 193-208.
Lombardi, M. J., and D. Veredas, 2007, Indirect estimation of elliptical stable distributions, Computational Statistics and Data Analysis 53, 2309-2324.
Lopes, H.F., and M. West, 2004, Bayesian Model assessment in factor analysis, Statistica Sinica 14, 41-67.
Madan, D.B., and E. Seneta, 1987, Simulation of Estimates Using the Empirical Characteristic Function, International Statistical Review 55, 153-161.
Marjoram, P., J. Molitor, V. Plagnol, and S. Tavare, 2003, Markov Chain Monte Carlo without likelihoods, Proceedings of the National Academy of Sciences 100, 15324-15328.
Matsui, M., and A. Takemura, 2006, Some improvements in numerical evaluation of symmetric stable density and its derivatives, Communications in Statistics – Theory and Methods 35, 149-172.
McCulloch, J. H., 1986, Simple consistent estimators of stable distribution parameters, Communications in Statistics - Simulation and Computation 15, 1109-1136.
McCulloch, J. H. , 1994, Estimation of bivariate stable spectral densities, Technical Report, Department of Economics, Ohio State University.
McCulloch, J. H., 1998, Numerical Approximation of the Symmetric Stable Distribution and Density, in R. Adler, R. Feldman, and M. Taqqu (eds), A practical guide to heavy tails: Statistical techniques for analyzing heavy tailed data, Boston, Birkhauser, 489-500.
Meerschaert, M.M., and H.-P. Scheffler, 1999, Moment estimator for random vectors with heavy tails, Journal of Multivariate Analysis 71, 145-159.
Meerschaert, M.M., and H.P. Scheffler, 2001, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley & Sons, New York, 2001.
Meerschaert, M.M., and H.P. Scheffler, 2003, Portfolio modeling with heavy tailed random vectors, Handbook of Heavy-Tailed Distributions in Finance, pp. 595-640, S. T. Rachev, Ed., Elsevier North-Holland, New York.
Meintanis, S.G., and E. Taufer, 2012, Inference procedures for stable-Paretian stochastic volatility models, Mathematical and Computer Modelling 25, 1199-1212.
Mittnik, S., T. Doganoglu, and D. Chenyao, 1999, Computing the probability density function of the stable Paretian distribution, Mathematical and Computer Modelling 29, 235-240.
Mittnik, S., S.T. Rachev, T. Doganoglu, and D. Chenyao, 1999, Maximum likelihood estimation of stable Paretian models, Mathematical and Computer Modelling 29, 275-293.
Modarres, R. and J. P. Nolan, 1994, A method for simulating stable random vectors. Computational Statistics 9, 11-19. Nagaev, A., 2000, On non-parametric estimation of the Poisson spectral measure of a stable law, Journal of Mathematical Sciences 106, 2854-2859.
Nelsen, R., 2006, An Introduction to Copulas, 2nd ed., New York: NY, Springer.
Nolan, J., 1997, Numerical calculation of stable densities and distribution functions, Communications in Statistics – Stochastic Models 13, 759-774.
Nolan, J. P., 1998, Multivariate stable distributions: approximation, estimation, simulation and identification. In R. J. Adler, R. E. Feldman, and M. S. Taqqu (Eds.), A Practical Guide to Heavy Tails: Statistical techniques for analyzing heavy tailed data, pp. 509-526. Boston: Birkhauser.
Nolan, J. P., A. Panorska, and J. H. McCulloch, 2001, Estimation of stable spectral measures, Mathematical and Computer Modelling 34, 1113-1122.
Nolan, J. P. and B. Rajput, 1995, Calculation of multidimensional stable densities, Communications in Statistics - Simulation 24, 551-556.
Ogata, H., 2012, Estimation for multivariate stable distributions with generalized empirical likelihood, Journal of Econometrics, in press.
Paulauskas, V. I., 1976, Some remarks on multivariate stable distributions. Journal of Multivariate Analysis 6, 356-368. Pitt, M., D. Chan, and R. Kohn, 2006, Efficient Bayesian inference for Gaussian copula regression models, Biometrika 93, 537-554.
Pivato, M., 2001, Analytical methods for multivariate stable probability distributions. Ph. D. thesis, University of Toronto, Department of Mathematics.
Pivato, M. and L. Seco, 2003, Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis, Journal of Multivariate Analysis 87, 219-240.
Plagnol V., and S. Tavare, 2004, Approximate Bayesian Computation and MCMC, in H. Niederreiter (ed) Proceedings of the MCQMC2002, Springer.
Pourahmadi, M., 1987, Some properties of empirical characteristic functions viewed as harmonizable processes, Journal of Statistical Planning and Inference 17, 345-359.
Press, S. J., 1972, Estimation in univariate and multivariate stable distributions, Journal of the American Statistical Association 67, 842-846.
Ravishanker, N., and Z. Qiou, 1999, Monte Carlo EM estimation for multivariate stable distributions, Statistics & Probability Letters 45, 335-340.
Roose, D., and E. De Doncker, 1981, Automatic integration over a sphere, Journal of Computational and Applied Mathematics 7, 203-224.
Samorodnitsky, G., and M. S. Taqqu, 1994, Stable non-Gaussian random processes, Chapman and Hall, New York.
Sargent, T. J., and C. Sims, 1977, Business Cycle Modelling without Pretending to have too much a-priori Economic Theory," in New Methods in Business Cycle Research, ed. by C. Sims. Federal Reserve Bank of Minneapolis.
Smith, M. and Pitts, A. (2006): Foreign exchange intervention by the Bank of Japan: Bayesian analysis using a bivariate stochastic volatility model. Econometric Reviews 25, 425–451.
Song, P., 2000, Multivariate Dispersion Models Generated from Gaussian Copula, Scandinavian Journal of Statistics, 27, 305-320.
Stock J.H., and M.W. Watson, 2005, Implications of dynamic factor models for VAR analysis, NBER Working Paper, no. 11467
Tanner, M.A., and W.H. Wong, 1987, The calculation of posterior distributions by data augmentation, Journal of the American Statistical Association 82, 528-540.
Toni, T., D. Welch, N. Strelkowa, A. Ipsen, and M.P.H. Stumpf, 2009, Approximate Bayesian Computation scheme for parameter inference and model selection in dynamical systems, Journal of the Royal Statistical Society Interface 31, 187-202.
Tran, K. C., 1998, Estimating mixtures of normal distributions via empirical characteristic function, Econometric Reviews 17, 167–83.
Tsionas, E.G., 1999, Monte Carlo inference in econometric models with symmetric stable disturbances, Journal of Econometrics 88, 365-401.
Tsionas, E.G., 2012a, Maximum likelihood estimation of stochastic frontier models by the Fourier transform, Journal of Econometrics 170, 234-248.
Tsionas, E.G., 2012b, Estimating tail indices and principal directions easily, Statistics & Probability Letters, in print.
Zolotarev, V.M., 1986, One-dimensional stable distributions, AMS Translations of Mathematical Monographs, vol. 65, AMS, Providence, Long Island.
Wegmann, D., C. Leuenberger, and L. Excoffier, 2009, Efficient Approximate Bayesian Computation Coupled With Markov Chain Monte Carlo Without Likelihood, Genetics 182, 1207-1218.
Weron, R., 1995, Performance of the estimators of stable law parameters, Hugo Steinhaus Center for Stochastic Methods, Research Report HSC /95/1.
Weron, R., 1996, On the Chambers-Mallows-Stuck method for simulating skewed stable random variables, Statistics & Probability Letters 28, 165-171.
Wilkinson, R.D., 2008, Approximate Bayesian Computation (ABC) gives exact results under the assumption of model error, Biometrika, 1-13.
Wong, F., Carter, C. and Kohn, R. (2003): Efficient estimation of covariance matrix selection models, Biometrika 90, 809–830.
Viroli, C., 2009, Bayesian inference in non-Gaussian factor analysis, Statistics and Computing 19, 451-463. de Vries, C., 1991, On the relation between GARCH and stable processes, Journal of Econometrics 48, 313-324.
Xu, D., and J. Knight, 2010, Continuous empirical characteristic function estimation of mixtures of normal parameters, Econometric Reviews 30, 25-50.
Yu, J., 2007, Empirical characteristic function and its applications, Econometric Reviews 23, 93-123.