Skew exponential power stochastic volatility model for analysis of skewness, non-normal tails, quantiles and expectiles

Computational Statistics - Tập 31 - Trang 49-88 - 2015
Genya Kobayashi1
1Faculty of Law, Politics, and Economics, Chiba University, Chiba, Japan

Tóm tắt

This paper proposes a unified framework to analyse the skewness, tail heaviness, quantiles and expectiles of the return distribution based on a stochastic volatility model using a new parametrisation of the skew exponential power (SEP) distribution. The SEP distribution can express a wide range of distribution shapes through two shape parameters and one skewness parameter. Since the asymmetric Laplace and skew normal distributions are included as special cases, the proposed model is related to quantile regression and expectile regression. The efficient and simple Markov chain Monte Carlo estimation methods are also described. The proposed model is demonstrated using the simulated data and real data on daily return of foreign exchange rate.

Tài liệu tham khảo

Aas K, Haff IH (2006) The generalized hyperbolic skew student’s \(t\)-distribution. J Financ Econom 4:275–309 Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insur Math Econ 44:182–198 Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178 Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica 46:199–208 Balakrishnan N, Lai CD (2009) Continuous bivariate distributions. Springer, New York Bauwens L, Hafner CM, Laurent S (2012) Handbook of volatility models and their applications. Wiley, Hoboken Bedford T, Cooke RM (2002) Vines: a new graphical model for dependent random variables. Ann Stat 30:1031–1068 Bottazzi G, Secchi A (2011) A new class of asymmetric exponential power densities with applications to economics and finance. Ind Corp Change 20:991–1030 Cappuccio N, Lubian D, Raggi D (2004) MCMC Bayesian estimation of a skew-GED stochastic volatility model. Stud Nonlinear Dyn Econ 8:1–29 Chen CWS, Gerlach RH, Wei DCM (2009) Bayesian causal effects in quantiles: accounting for heteroscedasticity. Comput Stat Data Anal 53:1993–2007 Chen CWS, Liu FC, So MKP (2013) Threshold variable selection of asymmetric stochastic volatility models. Comput Stat 28:2415–2447 Chen Q, Gerlach RH, Lu Z (2012) Bayesian Value-at-Risk and expected shortfall forecasting via the asymmetric Laplace distribution. Comput Stat Data Anal 56:3498–3516 Chib S (1995) Marginal likelihood from the Gibbs output. J Am Stat Assoc 90:1313–1321 Chib S (2001) Markov chain Monte Carlo methods: computation and inference. In: Heckman JJ, Leamer E (eds) Handbook of econometrics. North Holland, Amsterdam, pp 3569–3649 Chib S, Jeliazkov I (2001) Marginal likelihood from the Metropolis–Hastings output. J Am Stat Assoc 97:270–291 Chib S, Nardari F, Shephard N (2002) Markov chain Monte Carlo methods for stochastic volatility models. J Econ 108:281–316 Chollete L, Heinen A, Valdesogo A (2009) Modeling international financial returns with a multivariate regime-switching copula. J Financ Econ 7:237–480 Choy STB, Wan WY, Chan CM (2008) Bayesian student-\(t\) stochastic volatility models via scale mixtures. Adv Econ 23:595–618 Christoffersen P (1998) Evaluating interval forecasts. Int Econ Rev 39:841–862 Christoffersen PF, Pelletier D (2004) Backtesting Value-at-Risk: a duration based approach. J Financ Econ 2:84–108 de Jong P, Shephard N (1995) The simulation smoother for time series models. Biometrika 82:339–350 De Rossi G, Harvey A (2009) Quantiles, expectiles, and splines. J Econ 152:179–185 Doornik J (2007) Ox: object oriented matrix programming. Timberlake Consultants Press, London Dufour J-M (2006) Monte Carlo tests with nuisance parameters: a general approach to finite sample inference and nonstandard asymptotics. J Econ 133:443–477 Durbin J, Koopman SJ (2002) A simple and efficient simulation smoother for state space time series analysis. Biometrika 89:603–615 Embrechts P, Kaufmann R, Patie P (2005) Strategic long-term financial risks: single risk factors. Comput Optim Appl 32:61–90 Engle RF, Russell JR (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66:1127–1162 Engle RF, Manganelli S (2004) CAViaR: conditional autoregressive value at risk by regression quantiles. J Bus Econ Stat 22:367–381 Fernández C, Steel MFJ (1995) On Bayesian modeling of fat tails and skewness. J Am Stat Assoc 93:359–371 Gerlach RH, Chen CWS, Chan NYC (2011) Bayesian time-varying quantile forecasting for Value-at-Risk in financial markets. J Bus Econ Stat 29:481–492 Gerlach RH, Chen CWS, Lin L (2012) Bayesian semi-parametric expected shortfall forecasting in financial markets. The University of Sydney Business School, BA Working Paper 01/2012 Hafner CM, Manner H (2012) Dynamic stochastic copula models: estimation, inference and applications. J Appl Econ 27:269–295 Ishihara T, Omori Y (2012) Efficient Bayesian estimation of a multivariate stochastic volatility model with cross leverage and heavy-tailed errors. Comput Stat Data Anal 56:3674–3689 Jones MC (1994) Expectiles and M-quantiles are quantiles. Stat Probab Lett 20:149–153 Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46:33–50 Koenker R (2005) Quantile regression. Cambridge University Press, New York Kozumi H, Kobayashi G (2011) Gibbs sampling methods for Bayesian quantile regression. J Stat Comput Simul 81:1565–1578 Kupiec P (1995) Techniques for verifying the accuracy of risk measurement models. J Deriv 3:73–84 Li S (2011) Three essays on econometrics: asymmetric exponential power distribution, econometric computation, and multifactor model. Ph.D. Thesis, Rutgers The State University of New Jersey, New Brunswick Nakajima J, Omori Y (2009) Leverage, heavy-tails and correlated jumps in stochastic volatility models. Comput Stat Data Anal 53:2535–2553 Nakajima J, Omori Y (2012) Stochastic volatility model with leverage and asymmetrically heavy-tailed error using GH skew student’s-\(t\) distribution. Comput Stat Data Anal 56:3690–3704 Nakajima J (2013) Stochastic volatility model with regime-switching skewness in heavy-tailed errors for exchange rate returns. Stud Nonlinear Dyn Econ 17:499–520 Nakajima J (2014) Bayesian analysis of multivariate stochastic volatility with skew distribution. Econ Rev. doi:10.1080/07474938.2014.977093 Naranjo L, Pérez CJ, Martín J (2012) Bayesian analysis of a skewed exponential power distribution. In: Proceedings of COMPSTAT 2012, 20th international conference on computational statistics, pp 641–652 Naranjo L, Pérez CJ, Martín J (2015) Bayesian analysis of some models that use the asymmetric exponential power distribution. Stat Comput 25:497–514 Nelsen RB (2006) An introduction to copulas. Springer, Yew York Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55:819–847 Omori Y, Chib S, Shephard N, Nakajima J (2007) Stochastic volatility with leverage: fast likelihood inference. J Econ 140:425–449 Omori Y, Watanabe T (2008) Block sampler and posterior mode estimation for asymmetric stochastic volatility models. Comput Stat Data Anal 52:2892–2910 Park T, van Dyk D (2008) Partially collapsed Gibbs samplers: theory and methods. J Am Stat Assoc 103:790–796 Pitt MK, Shephard N (1999) Filtering via simulation: auxiliary particle filter. J Am Stat Assoc 94:590–599 Rubio FJ, Steel MFJ (2013) Bayesian modelling of skewness and kurtosis with two-piece scale and shape transformations. CRiSM working paper 13–10, University of Warwick Shephard N, Pitt MK (1997) Likelihood analysis of non-Gaussian measurement time series. Biometrika 84:653–667 Steel MFJ (1998) Bayesian analysis of stochastic volatility models with flexible tails. Econ Rev 17:109–143 Takahashi M, Omori Y, Watanabe T (2009) Estimating stochastic volatility models using daily returns and realized volatility simultaneously. Comput Stat Data Anal 53:2404–2426 Takahashi M, Watanabe T, Omori Y (2014) Volatility and quantile forecasts by realized stochastic volatility models with generalized hyperbolic distribution. CIRJE discussion papers 949 Tsiotas G (2012) On generalised asymmetric stochastic volatility models. Comput Stat Data Anal 56:151–172 Watanabe T, Omori Y (2004) A multi-move sampler for estimating non-Gaussian time series models: comments on Shephard and Pitt (1997). Biometrika 91:246–248 Wichitaksorn N,Wang JJJ, ChoySTB, Gerlach R (2014) Analyzing return asymmetry and quantiles through stochastic volatility models using asymmetric Laplace error via uniform scale mixtures. Appl Stoch Models Bus Ind. doi: 10.1002/asmb/2062 Yu K, Moyeed RA (2001) Bayesian quantile regression. Stat Probab Lett 54:437–447 Zhu D, Zinde-Walsh V (2009) Properties and estimation of asymmetric exponential power distribution. J Econ 148:86–99 Zhu D, Galbraith JW (2011) Modeling and forecasting expected shortfall with the generalized asymmetric student-\(t\) and asymmetric exponential power distributions. J Empir Financ 18:765–778