We document how sampling from a conditional Student's t distribution is implemented in stochvol. Moreover, a simple example using EUR/CHF exchange rates illustrates how to use the augmented sampler. We conclude with results and implications. (author's abstract

Kastner, Gregor

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ePubWU Institutional RepositoryGregor KastnerHeavy-Tailed Innovations in the R Package stochvolPaperOriginal Citation:Kastner, Gregor (2015) Heavy-Tailed Innovations in the R Package stochvol. WU Vienna Universityof Economics and Business, Vienna.This version is available at: http://epub.wu.ac.at/4918/Available in ePubWU: March 2016ePubWU, the institutional repository of the WU Vienna University of Economics and Business, isprovided by the University Library and the IT-Services. The aim is to enable open access to thescholarly output of the WU.http://epub.wu.ac.at/Heavy-Tailed Innovations in the R Package stochvolGregor KastnerWU Vienna University of Economics and BusinessFirst version, May 15, 2015AbstractWe document how sampling from a conditional Student’s t distribution is implementedin stochvol. Moreover, a simple example using EUR/CHF exchange rates illustrates howto use the augmented sampler. We conclude with results and implications.Keywords: Student’s t distribution, data augmentation, EUR/CHF exchange rates.PrefaceThis note serves as a preliminary add-on to the more elaborate article“Dealing with StochasticVolatility in Time Series using the R package stochvol” (Kastner 2016a). It discusses andrelaxes the restriction to conditionally normal errors in the vanilla stochastic volatility (SV)model.1. The SV model with Student’s t errorsSeveral authors have suggested to use non-normal conditional innovation distributions forstochastic volatility modeling. Examples include the Student’s t distribution (Harvey, Ruiz,and Shephard 1994), the extended Generalized Inverse Gaussian (Silva, Lopes, and Migon2006), (semi-)parametric innovations (Jensen and Maheu 2010; Delatola and Griffin 2011),or the GH skew Student’s t distribution (Nakajima and Omori 2012). In the following, wedescribe how the estimation of the SV model with Student’s t errors is implemented in the R(R Core Team 2016) package stochvol (Kastner 2016b).Let y = (y1, y2, . . . , yn)> be a vector of returns with mean zero. The SV model with Student’st errors (in short SV-t) is given throughyt|ht, ν ∼ tν(0, expht) , (1)ht|ht−1, µ, φ, ση ∼ N(µ+ φ(ht−1 − µ), σ2η), (2)h0|µ, φ, ση ∼ N(µ, σ2η/(1− φ2)), (3)i.e., conditionally on ht, the data is assumed to follow a zero-mean non-standardized Student’st distribution with ν degrees of freedom and variance (ν expht)/(ν − 2) for ν > 2. FollowingChib, Nardari, and Shephard (2002), we assume that a priori the degrees of freedom parameterν ∼ U(a, b), i.e., follows a uniform distribution with support on the real interval (a, b). Allother prior components are chosen as in Kastner (2016a).2 Heavy-Tailed Innovations in the R Package stochvol2. UsageEstimating a stochastic volatility model with conditional t errors via stochvol is very similar toestimating a model with standard Gaussian errors, differing only through specifying a non-NAargument priornu. This triggers the sampler specified in Section 3. To provide an example,we investigate the historical daily EUR/CHF exchange rates and display these in Figure 1.R> library(stochvol)R> data(exrates)R> par(mfrow = c(2, 1), mar = c(1.7, 1.7, 1.7, 0.1), mgp = c(1.6, 0.6, 0))R> plot(exrates$date, exrates$CHF, type = 'l', main = 'Price of 1 EUR in CHF')R> dat <- logret(exrates$CHF, demean = TRUE)R> plot(exrates$date[-1], dat, type = 'l', main = 'Demeaned log returns')2000 2002 2004 2006 2008 2010 20121.11.31.51.7 Price of 1 EUR in CHFexrates$dateexrates$CHF2000 2002 2004 2006 2008 2010 2012−0.020.020.06Demeaned log returnsdatFigure 1: Levels and demeaned log returns of EUR in CHF.First, we fit a standard SV model to the data. Then, by specifying the argument priornu(a two-element vector containing the lower and upper bounds of the uniform prior for ν), wecan trigger the sampler allowing for heavy-tailed conditional innovations.R> res <- svsample(dat, priormu = c(-12, 1), priorphi = c(20, 1.1),+ priorsigma = 0.1)R> plot(res, showobs = FALSE)R> rest <- svsample(dat, priormu = c(-12, 1), priorphi = c(20, 1.1),+ priorsigma = 0.1, priornu = c(2, 100))R> plot(rest, showobs = FALSE)Gregor Kastner 3Results are displayed in Figure 2, where the top panel (rows 1 to 3) corresponds to the outputfrom the standard SV model and the bottom panel (rows 4 to 7) contains the output fromthe SV-t model. Row 4 depicts exp(ht/2) for t ∈ {1, . . . , n}; row 5 shows the time varyingstandard deviations given through√ν/(ν − 2) exp(ht/2) for t ∈ {1, . . . , n}; row 6 portraystrace plots and row 7 outlines the corresponding smoothed kernel density estimates for thefour parameters µ, φ, σ, and ν. It is worth noting that ν is estimated to lie between 6 and18 with high posterior probability, indicating evidence for the presence of heavy tails evenafter catering for stochastic volatility. The extra flexibility of the SV-t sampler seems to allowfor increased persistence φ and smaller variance of log-volatility σ2, resulting in smoothertime-varying volatility estimates.We investigate one-day-ahead out-of-sample predictive performance of an AR(1) model with(a) homoskedastic, (b) SV, (c) SV-t, (d) GARCH(1,1) errors, applied to the raw exchangerate data. Details about this procedure are provided in Chapter 5 of Kastner (2016a). Theresults, summarized in Figure 3, speak in favor of the SV-t model for this dataset.3. Necessary modifications in the sampling schemeThe Student’s t distribution appearing in Equation 1 can be conveniently expressed as a scalemixture of normal distributions,yt|ht, τt ∼ N (0, τt expht) ,τt|ν ∼ G−1(ν/2, ν/2) ,where N (µ, σ2η) denotes the normal distribution with mean µ and variance σ2η, and G−1(a, b)denotes the inverse gamma distribution with shape and scale parameters a and b, respectively.Treating τ = (τ1, . . . , τn)> as latent data and letting y˜t = yt/√τt for t ∈ {1, . . . , n}, we havey˜t|ht ∼ N (0, expht) ,and the AWOL sampler described in Kastner and Fru¨hwirth-Schnatter (2014) can directly beapplied to the transformed data. To obtain draws from the newly introduced variables τ andν, two additional steps are required.3.1. Sampling the auxiliary variablesIt is easy to see that the marginal posterior is given throughτt|yt, ht, ν ∼ G−1(ν + 12,ν + y2t exp(−ht)2),independently for each t ∈ {1, . . . , n}. Obtaining draws from this distribution is straightfor-ward.3.2. Sampling the degrees of freedom parameterThe full conditional posterior of the degrees of freedom parameter, ν|·, only depends on τ .Its density is given through the product of n univariate truncated inverse gamma densities4 Heavy-Tailed Innovations in the R Package stochvolEstimated volatilities in percent (5% / 50% / 95% posterior quantiles)0.00.51.01.52.02.50 500 1000 1500 2000 2500 30002000 4000 6000 8000 10000−13.5−12.5−11.5Trace of mu (thinning = 1)2000 4000 6000 8000 100000.9700.9800.990Trace of phi (thinning = 1)2000 4000 6000 8000 100000.140.180.220.26Trace of sigma (thinning = 1)−13.5 −13.0 −12.5 −12.0 −11.5 −11.00.00.51.01.5Density of mu0.970 0.975 0.980 0.985 0.990 0.995 1.000020406080100Density of phi0.15 0.20 0.25 0.3005101520Density of sigmaEstimated conditional volatilities in percent (5% / 50% / 95% posterior quantiles)0.00.51.01.50 500 1000 1500 2000 2500 3000Estimated volatilities in percent (5% / 50% / 95% posterior quantiles)0.00.51.01.52.00 500 1000 1500 2000 2500 30002000 4000 6000 8000 10000−14−12Trace of mu (thinning = 1)2000 4000 6000 8000 100000.9800.9901.000 Trace of phi (thinning = 1)2000 4000 6000 8000 100000.120.160.20Trace of sigma (thinning = 1)2000 4000 6000 8000 10000681216Trace of nu (thinning = 1)−15 −14 −13 −12 −110.00.40.81.2Density of mu0.980 0.985 0.990 0.995 1.00004080120Density of phi0.10 0.14 0.18 0.2205101520Density of sigma5 10 15 200.000.100.20Density of nuFigure 2: Standard output of the plot method when applied to an svdraws object containingposterior draws from an SV model with Gaussian errors (top) and Student’s t errors (bottom).Gregor Kastner 5llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll−0.050.000.05Observed and predicted residualsl Observed residuals1% and 99% one−day−ahead predictive quantiles: SV1% and 99% one−day−ahead predictive quantiles: SVt1% and 99% one−day−ahead predictive quantiles: GARCH1% and 99% one−day−ahead predictive quantiles: homoskedastic2003−12−03 2004−12−17 2006−01−02 2007−01−18 2008−02−05 2009−02−23 2010−03−10 2011−03−23 2012−04−04Cumulative log predictive Bayes factors in favor of heteroskedasticity02004006008001000SVSVtGARCH2003−12−03 2004−12−17 2006−01−02 2007−01−18 2008−02−05 2009−02−23 2010−03−10 2011−03−23 2012−04−04Figure 3: Log predictive one-day-ahead Bayes factors in favor of SV, SV-t, and GARCH errorsover the homoskedastic model. The final log predictive Bayes factors aggregate to 1107.44(SV), 1115.36 (SV-t), and 1033.53 (GARCH), respectively, thus providing strong evidence forthe SV-t model.6 Heavy-Tailed Innovations in the R Package stochvolwhich may be written asp(ν|·) = p(ν|τ ) ∝(ν2)nν/2Γ(ν2)−n( n∏t=1τt)−ν/2exp{−ν2n∑t=11τt}(4)for ν ∈ (a, b) and zero elsewhere.For obtaining draws from this distribution, we use an independence Metropolis-Hastings up-date. We follow Chib and Greenberg (1994), who introduced the idea of specifying an inde-pendence proposal through numerical maximization of the log-density. For the problem athand, we consequently aim for optimizinglog p(ν|τ ) = nν2log(ν/2)− n log Γ(ν/2)− ν2n∑t=1(log τt +1τt)+ C, (5)with first and second derivatives given through∂ log p(ν|τ )∂ν=n2(1 + log(ν/2)− ψ(0)(ν/2))− 12n∑t=1(log τt +1τt), (6)∂2 log p(ν|τ )∂ν2=n2ν− n4ψ(1)(ν/2), (7)where ψ(m) denotes the polygamma function of order m. Using the above, it is easy tonumerically findνˆ = arg maxνlog p(ν|τ ),Bνˆ = −1/∂2 log p(ν|τ )∂ν2∣∣∣∣ν=νˆ,and a proposal candidate νprop may be drawn from a normal distribution with mean νˆ andvariance Bνˆ (the Laplace approximation). Letting φ(x|νˆ, Bνˆ) denote the corresponding den-sity function, the acceptance probability is equal to min{1, R} withR =p(νprop|τ )p(νold|τ ) ×φ(νold|νˆ, Bνˆ)φ(νprop|νˆ, Bνˆ) .4. ConclusionWe have shown how a simply data augmentation trick can be utilized to generalize the coresampler in stochvol in order to cater for potentially heavier-tailed innovation distributions.However, several caveats are called for:• Even though the uniform prior for ν has been used widely, more robust alternatives areprobably preferred, cf. Fru¨hwirth-Schnatter and Pyne (2010) and the references therein.• Sampling the degrees of freedom parameter ν conditionally on τ can be very inefficientif ν becomes large (and thus the plain vanilla SV model suffices). We recommend toresort to the original SV sampler in this case.Gregor Kastner 7• Leaving aside the additional computational burden, it is trivial to incorporate this ex-tension into samplers employing stochvol as part of a larger MCMC scheme (e.g. Huber2014; Kastner, Fru¨hwirth-Schnatter, and Lopes 2014; Dovern, Feldkircher, and Huber2015). Nevertheless, at the current stage of development, this should be conducted withcaution by carefully investigating the convergence of the posterior draws.AcknowledgmentsThe author would like to thank the attendants of the Spring 2015 Brown Bag Seminar of theInstitute for Statistics and Mathematics, WU Vienna University of Economics and Business,in particular Sylvia Fru¨hwirth-Schnatter, Mark Jensen, and Kurt Hornik, for stimulatingcomments and suggestions.ReferencesChib S, Greenberg E (1994). “Bayes Inference in Regression Models with ARMA(p, q) Errors.”Journal of Econometrics, 64, 183–206. doi:10.1016/0304-4076(94)90063-9.Chib S, Nardari F, Shephard N (2002). “Markov Chain Monte Carlo Methods for StochasticVolatility Models.” Journal of Econometrics, 108, 281–316. doi:10.1016/S0304-4076(01)00137-3.Delatola EI, Griffin JE (2011). “Bayesian Nonparametric Modelling of the Return Distributionwith Stochastic Volatility.” 6, 901–926. doi:10.1214/11-BA632.Dovern J, Feldkircher M, Huber F (2015). “Does Joint Modeling of the World EconomyPay Off? Evaluating GVAR Forecasts from a Multivariate Perspective.” Discussion PaperSeries 590, University of Heidelberg, Department of Economics. URL http://www.ub.uni-heidelberg.de/archiv/18586.Fru¨hwirth-Schnatter S, Pyne S (2010). “Bayesian inference for finite mixtures of univariateand multivariate skew-normal and skew-t distributions.” Biostatistics, 11(2), 317–336.doi:10.1093/biostatistics/kxp062.Harvey AC, Ruiz E, Shephard N (1994). “Multivariate Stochastic Variance Models.” TheReview of Economic Studies, 61(2), 247–264. doi:10.2307/2297980.Huber F (2014). “Density Forecasting using Bayesian Global Vector Autoregressions withCommon Stochastic Volatility.” Department of Economics Working Paper Series 179, WUVienna University of Economics and Business. URL http://epub.wu.ac.at/id/eprint/4280.Jensen MJ, Maheu JM (2010). “Bayesian semiparametric stochastic volatility modeling.”Journal of Econometrics, 157(2), 306–316. doi:10.1016/j.jeconom.2010.01.014.Kastner G (2016a). “Dealing with Stochastic Volatility in Time Series Using the R Packagestochvol.” Journal of Statistical Software, 69(5), 1–30. doi:10.18637/jss.v069.i05.8 Heavy-Tailed Innovations in the R Package stochvolKastner G (2016b). stochvol: Efficient Bayesian Inference for Stochastic Volatility (SV)Models. R package version 1.2.3, URL http://CRAN.R-project.org/package=stochvol.Kastner G, Fru¨hwirth-Schnatter S (2014). “Ancillarity-Sufficiency Interweaving Strategy(ASIS) for Boosting MCMC Estimation of Stochastic Volatility Models.” ComputationalStatistics & Data Analysis, 76, 408–423. doi:10.1016/j.csda.2013.01.002.Kastner G, Fru¨hwirth-Schnatter S, Lopes HF (2014). “Analysis of Exchange Rates via Mul-tivariate Bayesian Factor Stochastic Volatility Models.” In E Lanzarone, F Ieva (eds.), TheContribution of Young Researchers to Bayesian Statistics – Proceedings of BAYSM2013,volume 63 of Springer Proceedings in Mathematics & Statistics, pp. 181–185. Springer-Verlag. doi:10.1007/978-3-319-02084-6_35.Nakajima J, Omori Y (2012). “Stochastic Volatility Model with Leverage and AsymmetricallyHeavy-Tailed Error Using GH Skew Student’s t-Distribution.” 56(11), 3690–3704. doi:10.1016/j.csda.2010.07.012.R Core Team (2016). R: A Language and Environment for Statistical Computing. R Founda-tion for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.Silva RS, Lopes HF, Migon HS (2006). “The Extended Generalized Inverse Gaussian Distri-bution for Log-Linear and Stochastic Volatility Models.” Brazilian Journal of Probabilityand Statistics, 20(1), 67–91.Affiliation:Gregor KastnerInstitute for Statistics and MathematicsDepartment of Finance, Accounting and StatisticsWU Vienna University of Economics and BusinessWelthandelsplatz 1, Building D4, Level 41020 Vienna, AustriaTelephone: +43/1/31336-5593Fax: +43/1/31336-90-5593E-mail: gregor.kastner@wu.ac.atURL: http://statmath.wu.ac.at/~kastner/

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