Semiparametric stochastic volatility modelling using penalized splines
Tóm tắt
Stochastic volatility (SV) models mimic many of the stylized facts attributed to time series of asset returns, while maintaining conceptual simplicity. The commonly made assumption of conditionally normally distributed or Student-t-distributed returns, given the volatility, has however been questioned. In this manuscript, we introduce a novel maximum penalized likelihood approach for estimating the conditional distribution in an SV model in a nonparametric way, thus avoiding any potentially critical assumptions on the shape. The considered framework exploits the strengths both of the hidden Markov model machinery and of penalized B-splines, and constitutes a powerful alternative to recently developed Bayesian approaches to semiparametric SV modelling. We demonstrate the feasibility of the approach in a simulation study before outlining its potential in applications to three series of returns on stocks and one series of stock index returns.
Tài liệu tham khảo
Abanto-Valle CA, Bandyopadhyay D, Lachos V, Enriquez I (2010) Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions. Comput Stat Data Anal 54:2883–2898
Abraham B, Balakrishna N, Sivakumar R (2006) Gamma stochastic volatility models. J Forecast 25:153–171
Arlot S, Celisse A (2010) Model selection. Stat Surv 4:40–79
Bartolucci F, De Luca G (2001) Maximum likelihood estimation of a latent variable time-series model. Appl Stoch Models Bus Ind 17:5–17
Bartolucci F, De Luca G (2003) Likelihood-based inference for asymmetric stochastic volatility models. Comput Stat Data Anal 42:445–449
Bergmeier C, Benítez JM (2012) On the use of cross-validation for time series predictor evaluation. Inf Sci 191:192–213
Chib S, Nardari F, Shephard N (2002) Markov chain Monte Carlo methods for stochastic volatility models. J Econom 108:281–316
Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Financ 1:223–236
de Boor C (1978) A practical guide to splines. Springer, Berlin
Delatola E-I, Griffin JE (2011) Bayesian nonparametric modelling of the return distribution with stochastic volatility. Bayesian Anal 6:901–926
Delatola E-I, Griffin JE (2013) A Bayesian semiparametric model for volatility with a leverage effect. Comput Stat Data Anal 60:97–110
Durham GB (2006) Monte Carlo methods for estimating, smoothing, and filtering one- and two-factor stochastic volatility models. J Econom 133:273–305
Eilers PHC, Marx BD (1996) Flexible smoothing with \(B\)-splines and penalties. Stati Sci 11:89–121
Fridman M, Harris L (1998) A maximum likelihood approach for non-Gaussian stochastic volatility models. J Bus Econ Stat 16:284–291
Fernandez C, Steel MFJ (1998) On Bayesian modeling of fat tails and skewness. J Am Stat Assoc 93:359–371
Gallant AR, Hsieh D, Tauchen GE (1997) Estimation of stochastic volatility models with diagnostics. J Econom 81:159–192
Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction, and estimation. J Am Stat Assoc 102:359–378
Harvey AC, Shephard N (1996) Estimation of an asymmetric stochastic volatility model for asset returns. J Bus Econ Stat 14:429–434
Harvey CR, Siddique A (2000) Conditional skewness in asset pricing tests. J Financ 55:1263–1295
Jacquier E, Polson NG, Rossi PE (2004) Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. J Econom 122:185–212
Jondeau E, Rockinger M (2003) Conditional volatility, skewness and kurtosis: existence, persistence and co-movements. J Econ Dyn Control 27:1699–1737
Jensen MJ, Maheu JM (2010) Bayesian semiparametric stochastic volatility modeling. J Econom 157:306–316
Kim S, Shephard N, Chib S (1998) Stochastic volatility: likelihood inference and comparison with ARCH models. Rev Econ Stud 65:361–393
Krivobokova T, Crainiceanu CM, Kauermann G (2008) Fast adaptive penalized splines. J Comput Graph Stat 17:1–20
Langrock R (2011) Some applications of nonlinear and non-Gaussian state-space modelling by means of hidden Markov models. J Appl Stat 38:2955–2970
Langrock R, King R (2013) Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates. Ann Appl Stat 7:1709–1732
Langrock R, Kneib T, Sohn A, DeRuiter SL (2014) Nonparametric inference in hidden Markov models using P-splines. Biometrics (to appear). arXiv:1309.0423v2
Langrock R, MacDonald IL, Zucchini W (2012) Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models. J Empir Financ 19:147–161
Nakajima J, Omori Y (2012) Stochastic volatility model with leverage and asymmetrically heavy-tailed error using GH skew Student’s -distribution. Comput Stat Data Anal 56:3690–3704
Racine J (2000) Consistent cross-validatory model-selection for dependent data: \(h\nu \)-block cross-validation. J Econom 99:39–61
Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472
Ruppert D (2002) Selecting the number of knots for penalized splines. J Comput Graph Stat 11:735–757
Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, Cambridge
Schellhase C, Kauermann G (2012) Density estimation and comparison with a penalized mixture approach. Comput Stat 27:757–777
Shephard N (1996) Statistical aspects of ARCH and stochastic volatility. In: Cox DR, Hinkley DV, Barndorff-Nielsen OE (eds) Time series models: in econometrics, finance and other fields. Chapman & Hall, London, pp 1–67
Zucchini W, MacDonald IL (2009) Hidden Markov Models for time series: an introduction using R. Chapman & Hall, London