A Bayesian GED-Gamma stochastic volatility model for return data: a marginal likelihood approach
aa r X i v : . [ q -f i n . S T ] A ug A Bayesian GED-Gamma stochastic volatilitymodel for return data: a marginal likelihoodapproach
T. R. Santos ∗ Department of Statistics, Universidade Federal de Minas Gerais
Abstract
Several studies explore inferences based on stochastic volatility (SV)models, taking into account the stylized facts of return data. The com-mon problem is that the latent parameters of many volatility models arehigh-dimensional and analytically intractable, which means inferences re-quire approximations using, for example, the Markov Chain Monte Carloor Laplace methods. Some SV models are expressed as a linear Gaus-sian state-space model that leads to a marginal likelihood, reducing thedimensionality of the problem. Others are not linearized, and the latentparameters are integrated out. However, these present a quite restrictiveevolution equation. Thus, we propose a Bayesian GED-Gamma SV modelwith a direct marginal likelihood that is a product of the generalized Stu-dent’s t-distributions in which the latent states are related across timethrough a stationary Gaussian evolution equation. Then, an approxima-tion is made for the prior distribution of log-precision/volatility, withoutthe need for model linearization. This also allows for the computation ofthe marginal likelihood function, where the high-dimensional latent statesare integrated out and easily sampled in blocks using a smoothing proce-dure. In addition, extensions of our GED-Gamma model are easily madeto incorporate skew heavy-tailed distributions. We use the Bayesian es-timator for the inference of static parameters, and perform a simulationstudy on several properties of the estimator. Our results show that theproposed model can be reasonably estimated. Furthermore, we providecase studies of a Brazilian asset and the pound/dollar exchange rate toshow the performance of our approach in terms of fit and prediction.
Keywords: SV model, New sequential and smoothing procedures, Gen-eralized Student’s t-distribution, Non-Gaussian errors, Heavy tails, Skew-ness ∗ T. R. Santos Universidade Federal de Minas Gerais, Brazil. E-mail: [email protected]. The author received support from the FAPEMIG Foundation and CNPq-Brazil Introduction
There is evidence of non-Gaussianity, skewness, and heavy tails in the distri-bution of return data. Therefore, we need to choose more flexible models inorder to incorporate these stylized facts. Volatility is an important statisticalmeasure, representing the conditional variance of an underlying asset return,and plays a key role in finance [Tsay, 2010]. It is also an important componentof risk management, portfolio optimization, and options trading.Since volatility is a latent component, its estimation calls for specific tech-niques and suitable statistical inferences. Several models have been proposed forestimating the volatility of asset return data. For example, Engle [1982] intro-duced an autoregressive conditional heteroskedasticity (ARCH) model, wherevolatility is a function of past time series values. The generalized ARCH(GARCH) model was proposed by Bollereslev [1986], where volatility can de-pend on its own past. Taylor [1982] proposed a stochastic volatility (SV) modelwith an error term in its volatility equation, capturing some of the character-istics found in financial return time series in a better way than the GARCHmodel.The usual approach to using a SV model is to use linearization to con-vert it into a linear state-space model by a transformation in the return data.Harvey et al. [1994], who adopted this approach, presented the quasi-maximumlikelihood (QML) estimator from the classical perspective, considering the inno-vation distribution to be approximately Gaussian. Danielsson [1994] proposedthe simulated maximum likelihood (SML) method to estimate the SV model.Subsequently, Sandmann and Koopman [1998] discussed the Monte Carlo like-lihood estimation (MCL), and how a very efficient MCL estimator can be ob-tained, while keeping the linear state-space form under the classical inference.Their procedure first linearizes the SV model and then a better approximationof the observation equation error distribution is made using the MCL method.Further, Kim et al. [1998] used a normal mixture to approximate the observa-tion distribution once the SV model is linearized. They also proposed a differentestimation method based on Gibbs sampling under the Bayesian approach.Another interesting approach is the method of moments (MM). Several MMestimators have been introduced in the literature; for example, see Taylor [1986]and Melino and Turnbull [1990]. The latter used the generalized method of mo-ments (GMM) to estimate the SV model parameters. These estimators preventthe problems related to model linearization and full likelihood function evalu-ation. However, they have poor finite sample properties, and do not estimatethe underlying volatility directly [Broto and Ruiz, 2004].A Bayesian estimation approach to SV models using a Markov Chain MonteCarlo (MCMC) method and the full likelihood function was developed by Jacquier et al.[1994]. Their extensive simulation experiments showed that the MCMC methodperforms better than the QML and MM estimation techniques. Subsequently,Jacquier et al. [2004] introduced a new version of this model to accommodate fattails and correlated errors and Cappuccio et al. [2004] presented an interestingSkew-GED SV model. 2he MCMC procedure requires large, computer-intensive simulations [Broto and Ruiz,2004] and its computational implementation is not a simple task. Anotherproblem is the dimensionality of the parameter space, once the latent (log-volatility) and static parameters are simultaneously estimated using the fullposterior distribution, which is based on a full likelihood function, although itis does not necessarily require the linearization of the SV model. Two alterna-tives to the MCMC methods under the Bayesian perspective are the particlefilter [Pitt and Shephard, 1999, Lopes and Tsay, 2011, Malik and Pitt, 2011]and Laplace [Rue et al., 2009] approximations.Several studies have examined using SV models from a Bayesian perspec-tive, including those of Taylor [1994], Chib et al. [2002], Yu [2005], Omori et al.[2007], Raggi and Bordignon [2006], and Kastner and Fruhwirth-Schnatter [2014].Then, Ferrante and Vidoni [1998], Vidoni [1999], and Davis and Yam [2005]considered nonlinear and non-Gaussian state-space models. See also Watanabe[1999], Knight and Yu [2002], Feunou and Tedongap [2012], and Koopman and Bos[2012]. A detailed review of SV models can be found in Broto and Ruiz [2004].The family of non-Gaussian state-space models (NGSSM) was proposed byGamerman et al. [2013] and is an attractive alternative to both the SV andGARCH models. These models have a dynamic level associated with volatilityand a multiplicative Beta evolution equation. This evolution provides an ex-act marginal likelihood function and filtering and smoothing distributions. Inspite of the analytical tractably of this family of models, the evolution equationis a random walk in log-scale, and does not include drift (a quite restrictive).Pinho et al. [2016] presented several heavy-tailed distributions representing par-ticular cases of the NGSSM family. In this class, Shepard [1994] introduced localscale models, which were then generalized by Deschamps [2011].Several studies explore inferences based on stochastic volatility (SV) mod-els, taking into account the stylized facts of return data. The general problemwith these models is that the latent parameters are high-dimensional, whichmakes it difficult to integrate out or to use high-dimensional numerical inte-gration. Thus, inferences using these models require approximations using,for example, Markov Chain Monte Carlo or Laplace methods[Jacquier et al.,1994, 2004, Omori et al., 2007, Kastner and Fruhwirth-Schnatter, 2014]. TheGARCH model has been an attractive option among the users due to the dif-ficult in obtaining the marginal likelihood of the SV model (its computationalimplementation) according to Fridman and Harris [1998]. Some SV models areexpressed as a linear Gaussian state-space model, leading to an approximatedmarginal likelihood function and a marginal posterior distribution, which re-duces the dimensionality of the problem. However, the observation disturbanceis either Gaussian or requires approximations [Harvey et al., 1994, Danielsson,1994, Sandmann and Koopman, 1998, Kim et al., 1998]. Other models are notlinearized, and possess a marginal likelihood that is approximated using MonteCarlo integration/importance sampling [Davis and Yam, 2005].Thus, the main objective of this study is to develop a Bayesian GED-GammaSV model for return data with a new sequential analysis procedure and an ap-proximated marginal likelihood that is a product of the generalized Student’s3-distributions and is evaluated directly, where the inferential procedure is fastand easy to implement under the Bayesian approach. The latent states inour proposed GED-Gamma model are related across time through a station-ary Gaussian evolution equation, and an analytical approximation is made forthe prior distribution of the log-precision/volatility, without the need for modellinearization. This also allows us to approximate the marginal likelihood func-tion. Furthermore, the high-dimensional latent states are easily integrated outand sampled in blocks using a new approximated smoothing procedure that isintroduced, enabling inferences to be made for these states.The main advantages of the employed method are its mathematical andcomputational simplicity, and its ability to accommodate the stylized facts ofreturn data and a stationary Gaussian evolution equation. This circumventsthe problem of high-dimensional latent states, without the need for model lin-earization.Section 2 presents the GED-Gamma SV model. Then, Section 3 presentsa simulation, and Section 4 provides a case study of the proposed model usingreal return data. Finally, Section 5 concludes the paper, including an indicationof potential areas for future research.
Because of the stylized facts common to return data, we need to choose moreflexible models that allow for the use of non-Gaussian heavy-tailed skew distri-butions [Abad et al., 2014, Taylor, 1986]. The GED is a non-Gaussian distri-bution with the flexibility to capture heavy-tailed patterns, and is discussed indetail in Box and Tiao [1992] and used in Nelson [1991] and Deschamps [2011].Another possibility is the skew-GED distribution that was used and motivatedby Pinho et al. [2016] and Cappuccio et al. [2004]. However, we opt for a GEDdistribution that is a skew-GED distribution with the asymmetry parameter κ = 0, as in Deschamps [2011]. It is no difficult to extend the GED-Gamma SVmodel to other cases, as it will be shown in Subsection 2.3.The GED-Gamma SV model, which is a composing of the GED distributionwith precision distributed as a gamma distribution, for the return time series { y t } nt =1 is defined as follows: (A1) The observation equation is p ( y t | λ t , ϕ ) = r Γ(3 /r ) / /r ) / λ /rt exp ( − λ t ψ ( r ) | y t | r ) , (1)for y t ǫ ℜ , where ψ ( r ) = [Γ (3 /r ) / Γ (1 /r )] r/ , ϕ is a static parameter vector,the latent states λ t = h − t (precision), and h t is the volatility at time t . If r = 1,it is the Laplace model, and if r = 2, it is the normal model [Deschamps, 2011].We consider a correlation structure in the mean of the returns series, such that y t = ( R t − µ t ), where R t is the usual return series and µ t is the mean of thedata.The model is fully specified by the following remaining assumptions:4 (A2) The prior distribution is λ t | Y t − , ϕ ∼ Gamma( a t | t − , b t | t − ); • (A3) The evolution equation is ln( λ t ) = − α + φ ln( λ t − ) + η t , where η t ∼ i.i.d. N (cid:0) , σ η (cid:1) , α ∈ ℜ , φ ∈ [0 ,
1) and σ η > • The initial information is λ | Y , ϕ ∼ Gamma( a , b ), that is, ln( λ ) | Y ∼ Log-Gamma( f , q ) , where the mean is f = ln( a ) − γ ( b ) and the vari-ance q = γ ′ ( a ). Then, γ ( · ) and γ ′ ( · ) are the digamma and trigammafunctions, respectively.Note that Y t − = ( Y , y , . . . , y t − ) ′ is the information available up to time t −
1. Furthermore, the evolution equation (A3) in terms of the volatility h t canbe written as ln( h t ) = α + φ ln( h t − )+ η ⋆t , where η ⋆t ∼ N (cid:0) , σ η (cid:1) and E ( ε t η ⋆t ) = 0, { ε t } is the disturbance term of the the observation equation.Instead of approximations of the observation distribution, as in the QML,MCL, and MCMC [Kim et al., 1998] methods, our approach approximates thedistribution of the natural logarithm of the latent states, the log-precision, interms of the two first moments, using an analytical approximation approach.Once the distribution of the natural logarithm of the latent states is a normaldistribution or can be approximated by a normal, we can specify it in termsof its two first moments. Figure 1 shows a comparison of the log-gamma andnormal distributions for the states to illustrate and assess the quality of theapproximation in terms of two first moments. At the top, we have the shapeparameter ( a ) at 2 and the scale parameter ( b ) assuming the values 2 and 100and a reasonable approximation of the log-gamma distribution by the normaldistribution. When the shape parameter is large, the difference between the dis-tributions become indistinguishable, because of the central limit theorem. Thevalues of the parameters a and b were chosen based on the usual values of theshape and scale parameters of the updated distribution in our simulation ex-periments. This approach is similar to that adopted in the dynamic generalizedlinear model (DGLM) [West and Harrison, 1997].Hereafter, we present the proposed sequential analysis (inferential) procedureof this model, that is more similar to that of the Dynamic Linear Model (DLM)than the DGLM (see Figure 2). This consists of the one-step ahead predictiveand filtering (or online) distributions of the latent states λ = { λ t } t =1: n , andthe one-step ahead predictive distribution of the observations. If the model isdefined as proposed in this section, we can use an approximation of the statedistribution to obtain the following results: Proposition 1.
1. The one-step ahead predictive (prior) distribution of the latent states attime tλ t | Y t − , ϕ ˙ ∼ Gamma( a t | t − , b t | t − ), where a t | t − = ( φ a − t − + σ η ) − , (2) b t | t − = exp( α )( a t − /b t − ) − φ ( φ a − t − + σ η ) . (3)5. The update or online (posterior) distribution at time t λ t | Y t , ϕ ∼ Gamma( a t , b t ),where a t = a t | t − + 1 /r, (4) b t = b t | t − + ψ ( r ) | y t | r . (5)3. The one-step ahead predictive distribution of the observations at time t isgiven by p ( y t | Y t − , ϕ ) = Γ(1 /r + a t | t − ) r Γ(3 /r ) / /r ) / ( b t | t − ) a t | t − Γ( a t | t − )[ ψ ( r ) | y t | r + b t | t − ] /r + a t | t − , y t ∈ ℜ , (6)for t = 1 , . . . , n , where n is the number of observations of the time se-ries and Γ( · ) is the gamma function. This predictive distribution is thegeneralized Student’s t-distribution with 2 a t | t − degrees of freedom andif r = 2, then it is Student’s t-distribution [Triantafyllopoulos, 2008], aninteresting feature of the proposed model.The proof of this proposition is given in Appendix I. The important distributionof λ t | Y t − , ϕ in Part 1 of Proposition 1 preserves, in general, the mean of thedistribution of λ t − | Y t − , ϕ and increases the variance.The approximated marginal log-likelihood function, which is a product ofthe generalized Student’s t-distributions, is given byln L ( ϕ ; Y n ) = ln n Q t =1 p ( y t | Y t − , ϕ ) = n P t =1 ln Γ( a t | t − + 1 /r ) − ln Γ( a t | t − ) + a t | t − ln b t | t − + ln (cid:16) r Γ(3 /r ) / /r ) / (cid:17) − (1 /r + a t | t − ) ln[ ψ ( r ) | y t | r + b t | t − ] , (7)where ϕ is composed of α , φ , σ η , and r ; Y n = ( Y , y , . . . , y n ) ′ (when allinformation is available). Since the marginal posterior distribution of parameter vector ϕ is not analyt-ically tractable, a Bayesian inference for ϕ can be performed using a MCMC[Gamerman and Lopes, 2006] or an Adaptive Rejection Metropolis Sampling(ARMS) [Gilks et al., 1995] algorithm. The marginal posterior distribution of ϕ is given by p ( ϕ | Y n ) ∝ L ( ϕ ; Y n ) p ( ϕ ) , (8)where L ( ϕ ; Y n ) is the likelihood function defined in (7), and p ( ϕ ) is the priordistribution of ϕ . In this work, independent proper uniform priors are adoptedfor ϕ , as in Gamerman et al. [2013] and Cappuccio et al. [2004]. The idea is tointroduce vague uniform priors with a large variance, if we have no knowledge6bout the value of the parameters. However, other priors for the componentsof ϕ could be α ∼ N ( µ α , σ α ), φ +12 ∼ B ( a φ , b φ ), σ ∼ InvGamma( a σ , b σ ), and r ∼ Gamma( a r , b r ) [Kastner and Fruhwirth-Schnatter, 2014, see].Once a sample ϕ (1) , . . . , ϕ ( M ) is provided by the ARMS or MCMC algo-rithm, the approximated posterior mean, median and percentiles can be calcu-lated. The posterior mode can be obtained by maximizing function (8). Thistask is typically performed numerically using a maximization algorithm, such asthe Broyden -- Fletcher -- Goldfarb -- Shanno (BFGS) and sequential quadraticprogramming (SQP) algorithms [Avriel, 2003]. In general, ϕ may be re-parameterizedin order to utilize these algorithms.An inference for the latent variables can be made using the output fromthe MCMC and ARMS algorithms. Once a sample ϕ (1) , ..., ϕ ( M ) is available,the predictive, filtering or smoothed distributions of the latent states can becalculated in the following way. Note that p ( λ t + h | Y t ) = Z p ( λ t + h | Y t , ϕ ) p ( ϕ | Y t ) d ϕ . (9)Thus, the h -step-ahead predictive or filtering distributions can be approximatedby 1 M M X j =1 p ( λ t + h | Y t , ϕ ( j ) ) , from which summaries such as means, variances, and credibility intervals canbe obtained. Since p ( λ t + h | Y t ) is not available analytically, a draw λ ( s ) t + h from p ( λ t + h | Y t ) can be obtained from (9) by sampling ϕ ( s ) from p ( ϕ | Y n ), and thensampling λ ( s ) t + h from p ( λ t + h | Y t , ϕ ( s ) ). In addition, smoothing procedures maybe built, following Gamerman [1991]. See also Migon et al. [2005]. In order to infer the latent states λ = ( λ , . . . , λ n ) ′ , we can utilize an approx-imated smoothed distribution for ln( λ ), and apply the inverse transformation.If the model is defined as proposed here, we can use the results of the sequentialanalysis to obtain the following smoothed distribution. The joint distributionof ln( λ ) | Y n , ϕ has density p (ln( λ ) | ϕ , Y n ) = p ( ϕ | Y n ) p (ln( λ n ) | ϕ , Y n ) n − Y t =1 p (ln( λ t ) | ln( λ t +1 ) , ϕ , Y t ) . (10) Proposition 2.
The distribution p (ln( λ t ) | ln( λ t +1 ) , Y t , ϕ ) . = N ( µ ⋆t , σ ⋆t ) , (11)where σ ⋆t = (cid:16) φ σ η + q t (cid:17) − , µ ⋆t = σ ⋆t h φ (ln( λ t +1 )+ α ) σ η + f t q t i , f t = ln( a t ) − γ ( b t )and q t = γ ′ ( a t ), which depend on the shape and scale parameters of the filteringdistribution of λ t . The proof of Proposition 2 is given in Appendix II.7he inference for the latent variables or states can be made using the out-put from the MCMC and ARMS algorithms. Once a sample ϕ (1) , ..., ϕ ( M ) isavailable, posterior samples ln( λ ) (1) , ..., ln( λ ) ( M ) from the latent variables areobtained according to the following procedure. Smoothing procedure:
1. set j = 1;2. sample the static parameter ϕ ( j ) from the MCMC or ARMS algorithm;3. sample the set ln( λ ) ( j ) of latent variables from p (ln( λ ) | ϕ ( j ) , Y n ) in (10);4. set j → j + 1 and return to 2, if j ≤ M ; otherwise, stop. The model, for the observations, in Equation (1) can be generalized using a scalemixture for the observation disturbance to obtain other (skew) heavy-tailed dis-tributions directly, such as the (skew) Student’s t-distribution [Nakajima and Omori,2009, Gamerman et al., 2013, see]. If ε ⋆t = γ − / t ε t is the observation distur-bance of the model, where γ t ∼ Gamma( ν/ , ν/
2) and ε t ∼ GED( r = 2 , µ =0 , σ = 1), ε ⋆t will have a Student’s t-distribution, with ν degrees of freedom[Gamerman et al., 2013]. Furthermore, other probability distributions may beconsidered for γ t , leading to other (skew) heavy-tailed distributions for ε ⋆t . How-ever, the (skew) GED specification in Equation (1) leads to the one-step aheadpredictive (skew) generalized Student’s t-distribution for the observations (seeEquation (6)) and the marginal likelihood that is a product of the (skew) gen-eralized Student’s t-distributions, which was also used by Wang et al. [2013] formodelling volatility data. We assess the performance of the proposed model using a Monte Carlo simu-lation, following the design of Sandmann and Koopman [1998], Jacquier et al.[1994].The values of ϕ = ( α, φ, σ η , r ) T are chosen in the following manner. First,we set the autoregressive parameter φ to 0.90, 0.95, and 0.98. Next, we take thevalues of σ η for each value of φ , so as to ensure that the coefficient of variation(CV) exp( σ η − φ ) − α , such that the expected variance is equal to 0.0009. Finally, we setparameter r to 2 and 1 and, thus, assume a Gaussian distribution (GED( r = 2))and Laplace distribution (GED( r = 1), the heavy-tailed case), respectively, forthe observation disturbance.For each parameter setting, we generate 500 time series of length n = 500with normal and Laplace errors. We then estimate the proposed model and8alculate the mean and mean squared error (MSE) of the posterior mode es-timates. We adopt proper uniform priors for ϕ . The prior distributions are φ ∼ Unif(0 , α ∼ Unif( − , ), σ η ∼ Unif(0 , ), r ∼ Unif(0 , ),and λ | Y ∼ Gamma(0 . , . α has the largest bias, in general.The bias for α and σ η , with CV = 10, is larger than with CV = 1 and 0 .
1. For CV = 10, the bias of our method for α is slightly larger than those of the MCMC,QML [Sandmann and Koopman, 1998, see], IS, and AIS [Davis and Yam, 2005,see] methods. However, for CV = 0 .
1, the estimates are not as biased as they arein Davis and Yam [2005] and Sandmann and Koopman [1998]. For the heavy-tailed case (Laplace errors of the observation equation), clearly, the bias andMSE of the normal-Gamma model are larger than those in the GED-Gammamodel. This indicates a need for more flexible heavy-tailed models, such as theproposed GED-Gamma model in this work, and that ignoring flexible tails maylead you a poor scenario in terms of estimation.
This case study uses the daily return data a Petrobr´as (a Brazilian company)asset and the pound/dollar exchange rate. The first is for the period 02/01/2001to 06/02/2015 (3546 observations), and the second is for the period 10/01/1981to 06/28/1985 (946 observations). The data can be found at the Yahoo financewebsite, and the second data set is also available in Durbin and Koopman [2001].Here, the return series at time t is defined as y t = R t = 100 ln (cid:16) P t P t − (cid:17) , centeredaround the sample mean, where P t is the daily closing spot price. For thesecond data set, P t represents the daily closing exchange rate. Data irregularitydue to holidays and weekends is ignored. We perform our case study usingOx [Doornik, 2009] installed on a Pentium dual-core computer, with a 2.3 GHzprocessor and 4GB of RAM. The codes are available upon authors request.Figure 3 presents the time series plots of the Petrobr´as and pound/dollarreturns. The Pound/Dollar return data set was analyzed by Harvey et al. [1994]and then reanalyzed by Davis and Yam [2005]. A distinctive feature of financialtime series is that they usually present nonconstant variance or volatility (seeFigure 3). Descriptive statistics are shown in Table 2. The Petrobr´as return se-9able 1: Comparison of static parameter estimates of the proposed GED-Gamma model, with different CV values and normal and Laplace errors, basedon 500 replications. For each parameter, the posterior mode estimate and meansquare error are presented. GED(r=2) (Normal) ErrorsCV=10 σ η φ µ r σ η φ µ r σ η φ µ r True
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GED(r=1) (Laplace) ErrorsCV=10 σ η φ µ r σ η φ µ r σ η φ µ r True
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Petrobr´as Pound/DollarNo. of obs
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P-value (Normality test) ries presents an excess of kurtosis compared to that of the pound/dollar returns.Both series have a slight positive skewness.The proposed GED-Gamma model is fitted to the return data of these assetsusing the Bayesian approach and the Metropolis–Hastings (MCMC) and BFGSalgorithms, implemented in Ox [Doornik, 2009]. For our model, we adopt inde-pendent proper uniform priors for ϕ . The prior distributions for the parametersare φ ∼ Unif(0 , α ∼ Unif( − , ), σ η ∼ Unif(0 , ), r ∼ Unif(0 , ),and λ | Y ∼ Gamma(0 . , . BF = 0 . AssetsMethods Criterion Pound/Dollar Petro3GED
LogLik † -925.74 -8390.52MLogLik ⋆⋆ -928.94 -8393.00 Normal ⋆ LogLik † -925.97 -8396.45MLogLik ⋆⋆ -928.03 -8398.00BF 2.48 0.0067 Note : Bayes factors against the proposed GED-Gamma model. † at the posterior mode; ⋆ the normal model is the proposed GED-Gamma model, with r = 2; ⋆⋆ marginallog-likelihood after integrating out the parameters. .1 Comparisons of the competing models In this section, we compare the GED-Gamma model to other models, includingthe Kim, Shephard, and Chib SV model (SV-KSC) [Kim et al., 1998], and theBayesian GARCH(1,1) model with Student’s t-disturbances [Ardia, 2008] (t-GARCH(1,1)). The former is implemented using Ox, by Pelagatti [2011], andlatter using the R package ’bayesGARCH’ [Ardia, 2015].For the SV-KSC approach, the simplest SV model is linearized, so log( y t )is used. An approximation for the log χ distribution is performed using aseven-component Gaussian mixture. Therefore, conditional on latent indicatorvariables w t ∈ { , . . . , } , t = 1 , . . . , n , the SV-KSC model is given bylog y t − µ w t = h t + log ε t , ε t ∼ N (0 , σ w t ) , (12) h t = α + φh t − + η t , η t ∼ N (0 , σ η ) , (13) t = 1 , . . . , n . We specify the independent noninformative priors for the modelparameters given by p ( α, φ ) ∝ p N ( α, φ ) and p ( σ η ) ∝ p IG ( σ η ) , where IG denotesthe inverse gamma distribution. The hyperparameters of the priors are thepackage default values.Another competing model is the GARCH(1,1) model with Student’s t-innovations[Ardia, 2008], given as: y t = (cid:18) ( ν − h t ν (cid:19) − / ε t , (14) t = 1 , . . . , n , where the conditional variance equation is h t = α + α y t − + βh t − ,α > α ≥ β ≥ ε t i.i.d. ∼ Student-t( ν ). The independent prior used in the t-GARCH(1,1)model is: p ( α ) ∝ p N ( α ; 0 , I [ α > , p ( α ) ∝ p N ( α ; 0 , I [ α > , p ( β ) ∝ p N ( β ; 0 , I [ β> and p ( ν ) = 0 .
01 exp( − . ν − I [ ν> .For the t-GARCH(1,1) and the SV-KSC models, we utilized one chain,12,000 iterations of the MCMC algorithm, and a burn-in of 10,000 iterationsfor the Bayesian methods. The convergence of the chain was checked usingmethods, such as graphs.Table 4 presents the static parameter estimates of the proposed approach,the SV-KSC, and t-GARCH(1,1) methods. The estimates of our proposed modelwith normal innovations are very close to those of the SV-KSC method, withnormal innovations, in most of the cases for the two assets. The interval esti-mates (Table 4) show that the needed of a more flexible model than the normal-Gamma, as the GED-Gamma model.For the pound/dollar returns, our parameter estimates (Table 4) are veryclose to those obtained by Davis and Yam [2005, p.397] and Durbin and Koopman[2001, p.236]. The QML estimates from Harvey et al. [1994] are similar to ours,but mainly in the normal-Gamma case. Furthermore, the parameter estimatesof our model are very similar to those of the SV-KSC (Table 4). The parameters α , α , β , and ν belong to the t-GARCH(1,1). The estimate of ν indicates aheavy-tailed pattern for the two return series.12able 4: Static parameter estimates of the models fitted to the Petrobr´as andpound/dollar returns. MethodsGED § Normal §§ SV-KSC t-GARCH(1,1)P. Mode P. Mode P. Mean P. Mean(P. Mean) (P. Mean)Estimates Petrobr´as ˆ σ η CI [0.0065;0.0226] [0.0140;0.0310] [0.0161;0.0375] -ˆ φ CI [0.9685;0.9929] [0.9670;0.9868] [0.9680;0.9889] -ˆ α CI [0.0083;0.0426] [0.0167;0.0490] [0.0198;0.0588] -ˆ r CI [1.5972;1.9041] - - -ˆ α - - - 0.205 CI - - - [0.13498;0.2900]ˆ α - - - 0.082 CI - - - [0.06468;0.1023]ˆ β - - - 0.894 CI - - - [0.87091;0.9154]ˆ ν - - - 7.289 CI - - - [5.7441;9.0390] Estimates Pound/Dollar ˆ σ η CI [0.0139; 0.0999] [0.0159;0.0880] [0.0155;0.0681] -ˆ φ CI [0.9107;0.9847] [0.9054;0.9878] [0.9376;0.9917] -ˆ α -0.028 (-0.063) -0.036 (-0.059) -0.031 - CI [-0.1755;-0.0193] [-0.1288;-0.0191] [-0.0669;-0.0057] -ˆ r CI [1.7097;2.4793] - - -ˆ α - - - 0.031 CI - - - [0.0148;0.0557]ˆ α - - - 0.138 CI - - - [0.08970;0.20394]ˆ β - - - 0.810 CI - - - [0.7323;0.8664]ˆ ν - - - 8.635 CI - - - [4.9359;13.7402] Note : § The proposed GED-Gamma model; §§ the proposed GED-Gamma model, with r = 2(normal case); CI: 95% percentile credibility interval. For the in-sample analysis of the GED-Gamma model, we use the smoothedmean of volatility, calculated using the smoothing procedure of Subsection 2.2.For the SV-KSC and t-GARCH(1,1) models, we use the posterior mean of thevolatility. The square of the log-return is used as a proxy for the true unobservedvolatility σ t [Bauwens et al., 2012]. Thus, the square root of the mean squarederror, SRMSE = s n P t =1 ( y t − ˆ σ t ) n , and mean absolute error, MAE = n P t =1 | y t − ˆ σ t | n , areused to compare the models. For the in-sample analysis of the pound/dollarreturns, the GED-Gamma model has the smallest SRMSE value (Table 5). Forthe Petrobr´as return series, the smallest MAE value. In most cases, the GED-Gamma model is the best or second best of the competing models, indicatingthat it performs well in terms of fit. 13able 5: The SRMSE and MAE of in-sample estimation of the volatility of theproposed GED-Gamma, SV-KSC and t-GARCH models fitted to the Petrobr´asand pound/dollar returns. Assets GED Model SV-KSC GARCH-tPetrobr´as SRMSE
MAE
Pound/Dollar SRMSE
MAE
Note:
The numbers in parentheses denote the ranking among the competing models.
For the out-of-sample analysis, a direct comparison of volatility forecasts isadopted using the square of the log-return as a proxy for the true unobservedvolatility σ t +1 [Bauwens et al., 2012]. For the proposed GED-Gamma model,the one-step ahead forecast volatility ˆ σ t +1 is calculated using the distributionof Item 1 on page 5. Under the Bayesian approach, the SRMSE and MAEare computed using the one-step ahead forecast ˆ σ t +1 of the competing models,leaving the out last five observations, then the last four observations out, andso on, until the last observation is left out. Finally, the SRMSE and MAE arecomputed as s P k =1 ( y t + k − ˆ σ t + k ) and P k =1 | y t + k − ˆ σ t + k | , respectively, where the index k varies over the last five observations.Table 6 presents the SRMSE and MAE of one-step ahead forecasts of theproposed GED-Gamma, SV-KSC, and t-GARCH(1,1) models. For the out-of-sample analysis of the pound/dollar returns, the SRMSE and MAE values of theGED-Gamma and SV-KSC models are similar, while the MAE and MSE of theGED-Gamma model are smaller than those of the SV-KSC and t-GARCH(1,1)models for the Petrobr´as return series. In most cases, the GED-Gamma modelis the best or second best of the three competing models. This indicates thatthe proposed GED-Gamma model is also a good option in terms of prediction.Table 6: The SRMSE and MAE of the one-step ahead forecasts for the volatilityof the proposed GED-Gamma, SV-KSC and t-GARCH models fitted to thePetrobr´as and pound/dollar returns. Assets GED Model SV-KSC t-GARCH(1,1)Petrobr´as SRMSE
MAE
Pound/Dollar SRMSE
MAE
Note:
The numbers in parentheses denote the ranking among the competing models. Conclusion
In this study, we introduced a GED-Gamma SV model for return data with anapproximated expression for the marginal likelihood, which can be evaluateddirectly, under the Bayesian approach. Using the model, we propose new se-quential analysis and smoothing procedures and a marginal likelihood that isa product of the generalized Student’s t-distributions based on an analyticalapproximation for the distribution of the latent states. The main advantages ofthe proposed method are its mathematical and computational simplicity and itsability to accommodate the stylized facts of return data and a stationary Gaus-sian evolution equation, circumventing the problem of the high-dimensional la-tent states. There is no need to linearize the model; that is, the data scaleis not changed and is free from approximation of the observation distribution.Non-Gaussian, heavy-tailed skew distributions for the observations are naturallyaccommodated. Beyond of the approximated sequential analysis procedure, thesmoothing procedure is provided. Another interesting feature is the availabilityof the one-step ahead predictive distribution, which is the generalized Student-tdistribution.A limitation of the model is the use of approximations for the distributionof the latent states in terms of the two first moments, because it was devel-oped as a DGLM [West and Harrison, 1997, Souza et al., 2018]. The qualityof this approximation depends on the quality of the normal approximation tothe log-gamma prior distribution of the latent states. The DGLM has a dy-namic structure in the mean of the data, which here is volatility. Both methodspreserve the sequential analysis of the data.Our approach performed well in the parameter estimation of the GED-Gamma SV model using the posterior mode, mean and quantiles under theBayesian perspective. The empirical results are competitive compared to othermethods in the literature in terms of fit and prediction. Thus, we achievedour primary objective of introducing a Bayesian GED-Gamma SV model thatcan be implemented in a fast and easy way, and that is free of approximationsfor the observation equation. The results and the proposed procedures of thisstudy are also useful to the closely related time series model. For example,the dynamic linear models proposed by West and Harrison [1997], for normalobservations with time varying means and variances, allows a stationary evo-lution equation for the volatility. Our results can also be used in the modelof Nakajima and Omori [2009], without the need to linearize the model for thevolatility sampling.Future works could include a study of other distributions for the observationequation (especially skew distributions), the inclusion of exogenous explanatoryvariables on volatility. 15 ppendix I
This appendix presents the proof of proposition 1 of the inferential procedurein the text.
Propositon 1.
We first provide the proofs of Parts 1 to 3 relating to the basic sequen-tial inference of the proposed model. For ease of notation, we omit the staticparameter vector ϕ from the proofs. Proof of Part 1 :Assume from the hypothesis that λ t − | Y t − ∼ Gamma ( a t − , b t − ); thus, ac-cording to West and Harrison [1997, Chapter 14],ln( λ t − ) | Y t − ∼ Log-Gamma [ f t − = γ ( a t − ) − ln( b t − ) , q t − = γ ′ ( a t − )] , where γ ( b t − ) and γ ′ ( a t − ) are the digamma and trigamma functions, respec-tively. Next, we approximate the log-gamma distribution by the normal distri-bution in terms of the two first moments. Then,ln( λ t − ) | Y t − ˙ ∼ Normal ( f t − , q t − ) . Now, we combine the above approximated distribution of ln( λ t − ) with theevolution equation ln( λ t ) | ln( λ t − ) ∼ Normal (cid:0) − α + φ ln( λ t − ) , σ η (cid:1) to obtain p (ln( λ t ) | Y t − ). Using the properties of the multivariate normal distribution[Harvey, 1989, West and Harrison, 1997], we have p (ln( λ t ) | Y t − ) ˙= Z p (ln( λ t − ) | Y t − ) p (ln( λ t ) | ln( λ t − )) d ln( λ t − ) d = Normal (cid:0) f t | t − , q t | t − (cid:1) , where f t | t − = − α + φf t − and q t | t − = φ q t − + σ η .Since λ t | Y t − ∼ Gamma (cid:0) a t | t − , b t | t − (cid:1) and ln( λ t ) | Y t − ˙ ∼ Normal (cid:0) f t | t − , q t | t − (cid:1) , the pair ( a t | t − , b t | t − ) can be elicited in terms of the two first moments f t | t − = γ ( a t | t − ) − ln( b t | t − ) and q t | t − = γ ′ ( a t | t − ). With suitable approximationsfor the digamma and trigamma functions [Abramovitz and Stegun, 1964], wehave f t | t − ≈ ln( a t | t − ) − ln( b t | t − ) and q t | t − ≈ a t | t − , and then a t | t − = q − t | t − and b t | t − = exp( − f t | t − ) q − t | t − . Now, by replacing f t | t − and q t | t − bytheir respective expressions, we have a t | t − = ( φ a − t − + σ η ) − and b t | t − = exp( α )( a t − /b t − ) − φ ( φ a − t − + σ η ) .Therefore, λ t | Y t − ˙ ∼ Gamma( a t | t − , b t | t − ) , where a t | t − = ( φ a − t − + σ η ) − and b t | t − = exp( α )( a t − /b t − ) − φ ( φ a − t − + σ η ) , to complete the proofof Part 1. 16 Proof of Part 2 :To calculate the on-line or update distribution of λ t , we have p ( λ t | Y t ) ∝ p ( y t | λ t ) p ( λ t | Y t − ) ∝ λ ( a t | t − +1 /r ) − t exp[ − λ t ( b t | t − + ψ ( r ) | y t | r )].Thus, it follows that λ t | Y t ∼ Gamma ( a t , b t ), where a t = a t | t − + 1 /r and b t = b t | t − + ψ ( r ) | y t | r , completing the proof. (cid:3) Proof of Part 3 : p ( y t | Y t − ) = ∞ Z p ( y t | λ t ) p ( λ t | Y t − ) dλ t = (cid:16) r Γ(3 /r ) / /r ) / (cid:17) Γ( a t | t − )( b t | t − ) − a t | t − ∞ Z h λ /r + a t | t − − t exp (cid:0) − λ t ( ψ ( r ) | y t | r + b t | t − ) (cid:1)i dλ t = Γ (cid:0) /r + a t | t − (cid:1) (cid:16) r Γ(3 /r ) / /r ) / (cid:17) ( b t | t − ) a t | t − Γ( a t | t − ) (cid:0) ψ ( r ) | y t | r + b t | t − (cid:1) a t | t − +1 /r , y t ∈ ℜ , where a t | t − = ( φ a − t − + σ η ) − and b t | t − = exp( α )( a t − /b t − ) − φ ( φ a − t − + σ η ) are parameters of the prior distribution of λ t in Part 1 of the results. (cid:3) Appendix II
Propositon 2.
This appendix presents the proof of Proposition 2 of the smoothing proce-dure in the text. We omit the static parameter vector ϕ from the proof. Samplesare taken from the smoothed log-precision ln( λ ) distribution. Consequently, weobtain samples from the precision λ and h = λ − volatility distributions.17 (ln( λ t ) | ln( λ t +1 ) , Y t ) = p (ln( λ t +1 ) | ln( λ t ) , Y t ) × p (ln( λ t ) | Y t ) p (ln( λ t +1 ) | Y t ) p (ln( λ t ) | ln( λ t +1 ) , Y t ) . = N ( − α + φ ln( λ t ) , σ η ) × N ( f t , q t ) N ( f t +1 | t , q t +1 | t ) p (ln( λ t ) | ln( λ t +1 ) , Y t ) ∝ exp[ − φ σ η + q t ) − × (ln( λ t ) − λ t ) × ( φ σ η + 1 q t ) − × ( φ (ln( λ t +1 ) + α ) σ η + f t q t ))] . Therefore, p (ln( λ t ) | ln( λ t +1 ) , Y t ) is a normal distribution, with approximatemean µ ⋆t = σ ⋆t × (cid:16) φ (ln( λ t +1 )+ α ) σ η + f t q t (cid:17) and variance σ ⋆t = (cid:16) φ σ η + q t (cid:17) − . (cid:3) References
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Figure 1: The log-gamma and normal distributions of the states for some valuesof the shape (a) and scale (b) parameters.Figure 2: The sequential analysis procedure.23 R t ( % ) − Petro3 t R t ( % ) − Pound−Dollar
Figure 3: The Petrobr´as and pound/dollar return series.24 S t d . D e v Petro3