Parameter Estimation in LNSV Models:Griddy Gibbs versus Metropolis-Hastings

Abstract

This paper compares performances of two MCMC samplers, Griddy-Gibbs (GG) andMetropolis-Hastings (MH) samplers, to estimate parameters and latent variables inthe four log-normal stochastic volatility (LNSV) models: (1) LNSVn (standard), (2)LNSVt (with fat-tails), (3) LNSVL (with correlated errors), (4) LNSVLt (with fat-tailsand correlated errors). To illustrate the comparison of two samplers, we apply the mod-els and samplers to both simulated and real data sets. The stock indices we examineare TOPIX and three stocks of the TOPIX Core 30: Hitachi Ltd., Nissan Motor Co.Ltd. and Panasonic Corp.

Keywords: Log-normal stochastic volatility, Bayesian inference, single-move MarkovChain Monte Carlo, Griddy Gibbs, Metropolis-Hastings, TOPIX.

1 Introduction

Amongst modeling of financial time series, the stochastic volatility (SV) model is re-cognized as one of the most important class as it has the ability to capture the commonlyobserved change in variance of the observed stock index or exchange rate over time. Apopular and most widely used SV model is the log-normal (LN) SV model, which was firstintroduced by Taylor (1982). In his discrete time model, volatility process is modeled as afirst-order autoregression for the log-squared volatility. The LNSV model provides a morerealistic and adequate than the ARCH-type models (see, for example, Ghysels et al. (1996))and the GARCH-type models (see, for example, Kim et al. (1998), Yu (2002) and Carneroet al. (2001)).

Unfortunately, it is not possible to take random draws directly from the posterior forsome unknown parameters in SV models. An approach has become very attractive is theBayesian approach, which was first proposed by Shephard (1993) and Jacquier et al. (1994).Inference in this approach often requires advanced Bayesian computation, and here wefocus on Markov Chain Monte Carlo (MCMC) procedure to perform posterior samplingand statistical inference. MCMC permits to obtain the conditional posterior distributionsof the parameters by simulation rather than analytical methods. To summarize posterior

∗The email addresses of the authors are, respectively, bke76145@kwansei.ac.jp andmorimot@kwansei.ac.jp. The first author is a PhD candidate. The first author also wishes tothank Shuichi Nagata for some Matlab codes and helpful discussions.

1

2Didit B. Nugroho and Takayuki Morimoto1,2

Department of Mathematics, Satya Wacana Christian UniversityDepartment of Mathematical Sciences, Kwansei Gakuin University

June 25, 2012

2

1

distributions, it is necessary to have algorithms that are well-suited to updating randomsamples (e.g. Gibbs sampling, Metropolis-Hastings and Griddy-Gibbs sampler).

Comparison of Metropolis-Hastings (MH) algorithm and Griddy-Gibbs (GG) sampler hasalready been discussed in Bauwens and Lubrano (1998) for an asymmetric StudentGARCHmodel. They show that GG sampler is feasible and competitive compared with MH sampler.

In this paper we compare the performance between the use of the GG sampler and theMH sampler to estimate four LNSV models fitted to both simulated and real data sets. Thereal data we analysis includes the high frequency data sets for computing realized volatility(RV) and daily returns of TOPIX and three stocks of the TOPIX Core 30, which are HitachiLtd., Nissan Motor Co. Ltd. and Panasonic Corp., from January 5, 2004 until December30, 2011. RV is used as a proxy to evaluate the daily estimated volatility performances. Itwas showed by Anderson et al. (2003) that the daily RV obtained from the intra-day highfrequency data set is an consistent and robust estimator of the ”true” volatility.

The rest of this paper is organized as follows. The model specification is discussed inSection 2. Section 3 presents the description of the Bayesian MCMC method, conditionalposteriors and sampling algorithms. In Section 4, we apply the model and the samplersto the daily returns on three stocks to obtain volatilities and values of model parameters.Finally, Section 5 gives some concluding remarks.

2 MCMC Methods

The Gibbs sampling algorithm (Geman and Geman. (1984)) uses a sequence of drawsfrom conditional distributions to characterize the joint target distribution. Suppose that ourparameter vector θ has two components, making our target distribution f(θ1, θ2|y). To use

the Gibbs sampler, one begins by choosing starting values θ(0)2 (starting values are usually

chosen near the posterior mode or the maximum likelihood estimates). One then repeats,for i = 1, ..., N iterations (making sure to store the sequence of draws at each iteration):

Step 1: Sample θ(i+1)1 from p

(θ1|θ(i)2 , y

).

Step 2: Sample θ(i+1)2 from p

(θ2|θ(i+1)

1 , y)

.

Metropolis-Hastings algorithm is first introduced by Metropolis et al. (1953) and general-ized by Hastings (1970). The algorithm has many applications beyond Bayesian statistics;it is commonly used for all sorts of numerical integration and optimization. (It is alsothe case that the Gibbs sampling algorithm is a special case of the Metropolis-Hastingsalgorithm.) To simulate from our target distribution p(θ|y), we again start with sensiblestarting values: θ(0). For each iteration of the simulation i = 1, ..., N , we draw a proposal θ∗

from a known proposal distribution p(θ∗|θ(i)

). Unlike the Gibbs sampling algorithm, when

each move is automatically accepted, one accepts the proposal distribution probabilistically,sometimes moving to a value with a higher density value, sometimes moving to one with alower density value.

Unlike the MH sampler, in the GG sampler we do not need to find an efficient proposaldistribution.

When computing Monte Carlo estimates of quantities of interest, like the posterior mean,one discards the first set of samples to ensure the chain has reached steady state. The periodfor that samples is known as the burn-in period.

2

3 Accuracy

Without being able to plot the target distribution, which in most applications is of highdimension, it would be difficult, if not impossible, to assess whether the chain is samplingfrom the target distribution. An important part of any Bayesian analysis is assessing theconvergence of the simulation results. Indeed, no Monte Carlo estimates of posterior densitysummaries can be trusted unless the chain has reached its steady state.

4 LNSV Model with Fat-tails and Correlated Errors

We consider the lognormal stochastic volatility (LNSV) model given by

Rt = e12htωt, ωt = λ

− 12

t εt, εtiid∼ N (0, 1), t = 1, ..., T, (1)

ht+1 = α+ φ(ht − α) + τηt+1, ηt+1iid∼ N (0, 1), t = 1, ..., T − 1, (2)

λt ∼ p (λt|ν)

corr (εt, ηt+1) = ρ

where ht = lnσ2t , for the unobservable volatility σt of asset on day t, Rt is the assetreturn on day t from which the mean and autocorrelations are removed, 0 ≤ φ < 1,h1 ∼ N

(α, τ2/

(1− φ2

)), and N (·, ·) represents the normal distribution. In this model,

ν captures the fat-tailness and ρ captures the leverage effect, which refers to the asymmet-ric behavior that return movements are negatively correlated with volatility. As pointedout in Yu (2005), specification of corr (εt, ηt+1) = ρ implies that the above model is a mar-tingale difference sequence and clear to interpret the leverage effect instead of assumingcorr (εt, ηt) = ρ in Jacquier et al. (2004).

The above model has a parameter vector (ν, α, φ, τ2, ρ) and two latent stochastic pro-cesses denoted, respectively, by h = (h1, h2, ..., hT ) and λ = (λ1, λ2, ..., λT ), which are tobe estimated simultaneously using simulation methods which will be discussed in the nextsection. Here, the value of φ measures the autocorrelation present in the logged squaredvolatility. Thus φ can be interpreted as the persistence in the volatility, the constant scal-ing factor exp

(12α)

as the modal instantaneous volatility, and τ is the volatility of thelog-volatility (cf. Kim et al. (1998)). It is common to assume that 0 ≤ φ < 1 because thevolatility is positively autocorrelated in most financial time series. Next, λt is a scale factor,p (λt|ν) is the mixing density. We assume that λt is distributed i.i.d gamma, or that νλtfollows a χ2 distribution with 3 ≤ ν ≤ 40 degrees of freedom, as discussed by Watanabeand Asai (2001) and Jacquier et al. (2004). This assumption implies that the marginal

distribution of λ− 1

2t εt is Student-t with ν degrees of freedom.

As in the previous literature (see, for example Zhang and King (2003, 2008), Jacquier etal. (2004) and Yu (2005)), it is convenient to write

ηt+1 = ρεt +√

1− ρ2wt+1, (3)

for t = 1, 2, ..., T − 1, where wt+1 is iid N (0, 1) and corr (εt, wt+1) = 0. Transformation (3)shows that var (ηt+1) = 1 and corr (ηt+1, εt) = ρ which satisfies the above LNSV modelspecification. Substituting (3) into (2), we obtain

ht+1 = α+ φ(ht − α) + ρτe−12htλ

12t Rt +

√1− ρ2τwt+1.

It is evident that a unit increase in stock return at time t results in a ρτσ−1t λ12t unit change

in the logarithm of the variance at time t + 1. If the correlation coefficient ρ is negative,

3

a unit fall in stock return at this period will result in exp

{ρτσ−1t λ

12t

}unit increase in the

variance at time t+ 1.1

Following Jacquier et al. (2004), and has been applied more recently by Yu (2005) andZhang and King (2003, 2008), we re-parameterized ρ and τ to ϕ = ρτ and ψ2 =

(1− ρ2

)τ2,

respectively, and obtain

Rt = e12htλ− 1

2t εt, εt

iid∼ N (0, 1), t = 1, ..., T, (4)

ht+1 = α+ φ(ht − α) + ϕe−12htλ

12t Rt + ψwt+1, wt+1

iid∼ N (0, 1), t = 1, ..., T − 1,(5)

where

h1 ∼ N(α,ψ2/

(1− φ2

))and ht|ht−1 ∼ N

(α+ φ(ht−1 − α) + ϕe−

12ht−1λ

12t−1Rt−1, ψ

2

)(6)

for t = 2, ..., T . We now have a new parameter vector denoted by θ = (ν, α, φ, ϕ, ψ2).Because εt and wt+1 are uncorrelated, we can now easily obtain the joint posterior of h, λand θ conditional on observations R, where R = (R1, R2, ..., RT ).

5 Prior Specification

Following standard practice, we assume that our model is completed by priors for theunknown structural parameters as follows:

λt ∼ G(12ν,12ν), ν ∼ G(aν , bν), (7)

φ ∼ B(A,B), ψ2 ∼ IG(a, b), α|ψ2 ∼ N(α0, ψ

2/p0), ϕ|ψ2 ∼ N

(ϕ0, ψ

2/q0), (8)

where B(·, ·), IG(·, ·) and G(·, ·) represent the beta, inverse gamma and gamma distributions,respectively. For parameter ν, the following alternative priors were used in the previousliterature: exponential by Geweke (1993), Watanabe and Asai (2001), uniform discrete onthe interval [3, 40] by Jacquier et al. (2004), Gaussian by Zhang and King (2008) and gammaon the interval (2, 40] by Abanto-Valle et al. (2010). These priors are chosen to prevent νfrom getting too large during MCMC iterations, because the update of ν has a negligibleeffect when ν is already large.

6 Bayesian Approach

In order to implement Gibbs sampling, all of the fully conditional posterior distributions(one for each component of θ) need to be derived from the joint posterior distribution. Theconditional posterior distribution is derived from the joint posterior distribution by pickingout the parts that involve the unknown parameter in question.

Denote θ−ν be the parameter vector after removing ν. By using Bayes’ theorem, thejoint posterior distribution of parameters and latent unobservable variables conditional onthe returns is

p(λ,h,θ|R) = p(R|h,λ)× p(λ|ν)× p(h|θ−ν)× p(θ), (9)

1See Selcuk (2008) for detail.

4

where

p(R|h,λ) =

T∏t=1

p(Rt|ht, λt) ∝T∏t=1

λ12t exp

{−ht +R2

tλte−ht

2

},

p(λ|ν) =T∏t=1

p (λt|ν) ∝T∏t=1

(λt)12ν−1 exp

{−νλt

2

},

p(h|θ−ν) = p(h1|θ−ν)×T∏t=2

p(ht|ht−1,θ−ν)

∝(ψ2)− 1

2T (

1− φ2) 1

2 exp

{−(1− φ2

)(h1 − α)2

2ψ2

}

×T∏t=2

exp

−[ht − α− φ (ht−1 − α)− ϕe−

12ht−1λ

12t−1Rt−1

]22ψ2

,

p(θ) = p(ν)× p(α)× p(φ)× p(ϕ)× p(ψ2)

∝ νaν−1e−bνν ×(ψ2

p0

)− 12

exp

{−p0 (α− α0)

2

2ψ2

}× φA−1 (1− φ)B−1

×(ψ2

q0

)− 12

exp

{−q0 (ϕ− ϕ0)

2

2ψ2

}×(ψ2)−a−1

exp

{− b

ψ2

},

which are obtained from the likelihood function of R, the conditional and unconditionalfunctions of h and the prior distributions in (7) and (8). A Gibbs sampler approach thenestimates the parameters and latent variables by drawing random samples from the followingconditional posterior distributions:

p(θ|h,λ,R), p(λ|θ,h,R), p(h|θ,λ,R).

7 Conditional Posterior Distributions

7.1 Sampling λ and ν

As suggested by Jacquier et al. (2004), we draw λ and ν as a block, p(λ|ν, ·)p(ν|·), ratherthan with a Gibbs cycle [p(λ|ν, ·)p(ν|λ, ·)], to avoid the problems that ν can get absorbedinto the lower bound. In this case, the conditional posterior for λ is obtained from the jointposterior (9) and the conditional posterior for ν is obtained from the product of the priordistribution of ν and the distribution of ωt.

5

The conditional posterior for λ is

p(λ|θ,h,R) ∝T∏t=1

(λt)ν+12−1 exp

−R2t e−ht + ν

2λt −

(ht −Rtλ

12t

)2

2ψ2It<T

∝

T∏t=1

(λt)ν+12−1 exp

{−R2t e−ht(1 + ϕ2ψ−2It<T

)+ ν

2λt

}

× exp

{ψ−2htRtλ

12t It<T

}∝

T∏t=1

fG (λt|Aλt , Bλt)× g1 (λt)

where

g1 (λt) = exp

{ψ−2htRtλ

12t It<T

}, ht = ht+1 − α− φ (ht − α) , Rt = ϕe−

12htRt,

I is an indicator function, and fG (λt|Aλt , Bλt) denote the Gamma density function withshape and inverse scale parameters being defined, respectively, by

Aλt =ν + 1

2and Bλt =

R2t e−ht(1 + ϕ2ψ−2It<T

)+ ν

2.

The above conditional posterior is only standard for t = T , and hence we sample directlyλ1 from Gamma distribution and sample λt, for t = 1, ..., T − 1, by using Griddy Gibbsand Metropolis-Hastings. Because the fG distribution does not depend on λt, using theIndependence Chain (IC) MH sampling method introduced by Tierney (1994), we samplea proposal λ∗t ∼ G (Aλt , Bλt) and accept it with probability min {1, g1 (λ∗t ) /g1 (λt)}.

The probability density function (PDF) of the standardized student-t distribution withdegrees of freedom ν > 2 is given by

f (ωt) =Γ(12(ν + 1)

)Γ(12ν)√

(ν − 2)π

(1 +

ω2t

ν − 2

)− ν+12

So the conditional posterior for ν is

p(ν|h,R) = p(ν)f (ωt) ∝ νaν−1e−bννT∏t=1

Γ(12(ν + 1)

)Γ(12ν) (ν − 2)

ν2

(ν − 2 +R2

t e−ht)− ν+1

2(10)

A problem arises here in that the above posterior is not standard, and hence we sample νusing Griddy Gibbs sampler and Metropolis-Hastings algorithm.

To sample ν by Metropolis-Hastings algorithm, we use the Independence Chain Metropolis-Hastings (IC-MH) sampling method, in which the proposal for ν is ν∗ ∼ N[3,40]

(µν , s

2ν

)and

the acceptance probability is min {1, p (ν∗) /p (ν)}. The mean µν and variance s2ν are deter-mined by using the method which is based on the behavior of the density about its mode(see Section 5.5 in Albert (2009)). Let the logarithm of the conditional posterior of ν in(10),

ln p(ν|h,R) ∝ (aν − 1) ln(ν)− bνν + T ln Γ

(ν + 1

2

)− T ln Γ

(ν2

)+Tν

2ln(ν − 2)

− ν + 1

2

T∑t=1

ln(ν − 2 +R2

t e−ht). (11)

6

In this following, we write the above log posterior as `(ν) since after the data are observed, νis only random quantity. We find the the posterior mode ν of `(ν), meaning that `′(ν) = 0,by numerical optimization, and then we take µν = ν and s2ν = −1/`′′(ν). A very simpleand robust method for finding the mode ν is provided by bisection method,

νm =(νl + νu)

2,

where 3 ≤ νl < νu ≤ 40 and `′(νl)`′(νu) < 0. To find the variance s2ν , the problem is that

`(ν) is no longer concave up, that is `′′(ν) can be positive.To see this, we use the result of Abramowitz and Stegun (1972) (see Eq. 6.1.38), in

which the ln Γ(ν/2) is approximated by

ln Γ(x) =ln(2π)

2+

2x− 1

2ln(x)− x+

k

12x, 0 < k < 1.

Hence, we have

`′′(ν) = −

[aν − 1

ν2+T

6· 3ν4 + 12ν3 + (12 + 6k)ν2 + (3 + 6k)ν + 2k

(ν + 1)3ν3+

T∑t=1

1

ν − 2 +R2t e−ht

+2T

(ν − 2)2

]+

[T

2· ν

(ν − 2)2+ν + 1

2

T∑t=1

1(ν − 2 +R2

t e−ht)2], 0 < k < 1.

Thus, with aν > 1, the first term is always negative, and the second term is always positive.Therefore, following Watanabe and Asai (2001) we set s2ν = −1/Dν(ν) where

Dν(ν) = min{`′′(ν),−0.0001

}.

We should note that `′′(ν) is always negative in our empirical analysis.Given the conditional posterior for w, we propose the following Griddy Gibbs sampler

procedure:

1. Denote the equally spaced grid of values for w, say, w1 ≤ w2 ≤ · · · ≤ wm.

2. Compute the value of w’s posterior at each of the grid points and denote the resultantvector by p(w) = (p(w1), . . . , p(wm)).

3. Normalize p(w) and denote the resultant vector by p∗(w) = (p∗(w1), . . . , p∗(wm)) .

4. Compute the empirical cumulative distribution function of w, Φ(w).

5. Generate u ∼ U (0,Φ(wm)) and invert Φ(w) by linear interpolation to obtain a randomsample of w.

7.2 Sampling α

For sampling α based on the joint posterior (9), we found that the logarithm of theconditional posterior for α is

ln p (α|h,λ,θ−ν ,R) ∝ −(1− φ2

)(h1 − α)2

2ψ2−

T∑t=2

[zt − (1− φ)α]2

2ψ2− p0 (α− α0)

2

2ψ2

∝ − 1

2s2αα2 +

µαs2αα,

7

where zt = ht − φht−1 − ϕe−12ht−1λ

12t−1Rt−1, for t = 2, ..., T,, and s2α and µα are given by

s2α =ψ2

(1− φ2) + (T − 1)(1− φ)2 + p0, and

µα =s2αψ2

((1− φ2

)h1 + (1− φ)

T∑t=2

zt + p0α0

).

Hence α can be sampled directly from N(µα, s

2α

), given R, h, λ and θ−ν,−α.

7.3 Sampling ϕ

Based on the joint posterior (9), we found that the logarithm of the conditional posteriorfor ϕ is

ln p (ϕ|h,λ,θ−ν ,R) ∝ −

∑Tt=2

[ht − α− φ (ht−1 − α)− ϕe−

12ht−1λ

12t−1Rt−1

]22ψ2

− q0 (ϕ− ϕ0)2

2ψ2

∝ − 1

2s2ϕϕ2 +

µϕs2ϕϕ,

where

s2ϕ =ψ2∑T

t=2 e−ht−1λt−1R2

t−1 + q0

µψ =s2ϕψ2

(T∑t=2

[ht − α− φ (ht−1 − α)] e−12ht−1λ

12t−1Rt−1 + ϕ0q0

).

Hence ϕ can be sampled directly from N(µϕ, s

2ϕ

), given R, h, λ and θ−ν,−ϕ.

7.4 Sampling ψ2

For sampling ψ2 based on the joint posterior (9), we found that the conditional posteriorfor ψ2 is

p(ψ2|h,λ,θ−ν ,R

)∝

(ψ2)− 1

2T

exp

{−(1− φ2

)(h1 − α)2

2ψ2

}×

T∏t=2

exp

{− [zt − (1− φ)α]2

2ψ2

}

×(ψ2)− 1

2 exp

{−p0 (α− α0)

2

2ψ2

}×(ψ2)− 1

2 exp

{−q0 (ϕ− ϕ0)

2

2ψ2

}

×(ψ2)−a−1

exp

{− b

ψ2

}=

(ψ2)−a∗−1

exp

{− b∗

ψ2

},

where

a∗ =T + 2a+ 2

2

8

and

b∗ =1

2

{(1− φ2

)(h1 − α)2 +

T∑t=2

[zt − (1− φ)α]2 + p0 (α− α0)2 + q0 (ϕ− ϕ0)

2 + 2b

}.

Hence we can sample directly ψ2 from IG (a∗, b∗), given R, h, λ and θ−ν,−ψ. Once ϕ andψ2 are sampled, respectively, from their conditional posteriors, we can calculate τ2 and ρ

through τ2 = ϕ2 + ψ2 and ρ =ϕ

τ.

7.5 Sampling φ

Based on the joint posterior (9), the log of the conditional posterior for φ is given by

ln p (φ|h,λ,θ−ν ,R) ∝ ln p(φ) +ln(1− φ2

)2

−(1− φ2

)(h1 − α)2 +

∑Tt=2 [yt − φ(ht−1 − α)]2

2ψ2

∝ ln p(φ) +ln(1− φ2

)2

− 1

2s2φφ2 +

µφs2φφ,

where yt = ht − α − ϕe−12ht−1λ

12t−1Rt−1, for t = 2, . . . , T , and s2φ and µφ are respectively

given by

s2φ =ψ2

T∑t=3

(ht−1 − α)2and µφ =

s2φψ2

T∑t=2

yt(ht−1 − α).

Hence the conditional posterior for φ can be expressed as

p(φ|h, α, ϕ, ψ2

)∝ fN

(φ|µφ, s2φ

)× g2(φ),

where g2(φ) = p(φ) ×(1− φ2

) 12 and fN (·|µ, s2) denotes the normal density function with

mean µ and variance s2. The above conditional posterior is not standard, and hence wedraw φ by Griddy Gibbs sampler and Metropolis-Hastings algorithm.

We note that a fN

(φ|µφ, s2φ

)distribution does not depend on φ. Therefore, using the

IC-MH sampling method, we sample a proposal φ∗ ∼ N[0,1)

(µφ, s

2φ

)and accept it with

probability min {1, g2 (φ∗) /g2(φ)}.

7.6 Sampling h

Denote h−t be the latent stochastic vector after removing ht. Based on the joint posterior(9), the log of the conditional posterior for h1, hk, hT , for k = 2, ..., T − 1 are respectivelygiven by

ln p (h1|h−1, λ1,θ, R1) ∝ −h1 +R2

1e−h1λ1

2−(1− φ2

)(h1 − α)2 + w2

1

2ψ2,

ln p (hk|h−k, λk−1, λk,θ, Rk−1, Rk) ∝ −hk +R2

ke−hkλk

2−w2k−1 + w2

k

2ψ2,

ln p (hT |h−T , λT−1, λT ,θ, RT−1, RT ) ∝ −hT +R2

T e−hT λT

2−w2T−1

2ψ2,

(12)

9

where wt = ht − Rtλ12t , for t = 1, ..., T − 1. In order to sample h, we use the Griddy

Gibbs sampler and the random walk Metropolis-Hastings algorithm, in which the proposaldensity is the standar normal and the acceptance probability is computed through (12). Inthis case, the proposal value h∗t at (i+ 1)th iteration is generated according to the process

h∗t = h(i)t + ε

(i+1)t , where ε

(i+1)t ∼ N

(0,(ψ2)(i+1)

).

7.7 Single Move MCMC

Since the conditional posteriors of ν, φ and h do not have closed form, we first samplethe latent variable λ and parameters α, ϕ and ψ2. The Gibbs sampling scheme is describedas follows. Choose arbitrary starting values θ(0), λ(0) and h(0), and let i = 0.

1. Sample λ(i+1) from p(λ|ν(i),h(i),R

).

2. Sample ν(i+1) from p(ν|h(i),R

).

3. Sample α(i+1) from p(α|λ(i+1),h(i), φ(i), ϕ(i),

(ψ2)(i)

,R)

.

4. Sample ϕ(i+1) from p(ϕ|λ(i+1), α(i+1),h(i), φ(i),

(ψ2)(i)

,R)

.

5. Sample(ψ2)(i+1)

from p(ψ2|λ(i+1), α(i+1), ϕ(i+1),h(i), φ(i),R

).

6. Sample φ(i+1) from p(φ|λ(i+1), α(i+1), ϕ(i+1),

(ψ2)(i+1)

,h(i),R)

.

7. Sample h(i+1) from p(h|λ(i+1),θ

(i+1)−ν ,R

).

8. Set i = i+ 1 and go to step 1 until convergence is achieved.

8 Comparison of MCMC Methods

8.1 Simulated Data

8.2 Stock Market Data

8.2.1 Data Description

In this section we apply the methods and LNSV models used previously to the seriesof daily closing prices {St} of the TOPIX and three stocks of the TOPIX Core 30, whichare Hitachi Ltd. (HIT), Nissan Motor Co. Ltd. (NIS) and Panasonic Corp. (PAN), fromJanuary 5, 2004 until December 30, 2011, for a total of 1960 observations. The series ofreturn are daily percentage mean-corrected returns, Rt, given by the transformation

Rt = 100×

(ln

StSt−1

− 1

T

T∑i=1

lnSiSi−1

), t = 1, ..., T.

10

Figure 1: Time series plots for percentage daily mean-corrected returns of TOPIX, HIT (un-smooth), NIS and PAN from January 2004 until December 2011.

Table 1 reports summary statistics for the returns data, and Figure 1 displays the seriesof returns data. As usual, the kurtosis of the returns is significantly above three, indi-cating leptokurtic return distributions. The Ljung-Box (LB) test statistics indicate thatthe returns for TOPIX, NIS and PAN stock price indices are serially uncorrelated, whilethe returns for HIT index is serially correlated at the 5% level. One simple way to adjustthese autocorrelated returns is to unsmooth the return series such that the adjusted returnsdisplay no serial correlation. The procedure is given as follows (see De Souza and Gokcan(2004)):

R∗t =Rt − ρkRt−k

1− ρkwhere ρk is the kth-order autocorrelation of the autocorrelated return series Rt, and Rt−kis the k-period lagged return.

Table 1: Descriptive statistics of the daily returns for TOPIX and TOPIX Core 30.

StatisticsTOPIX HIT NIS PAN

Raw Data Raw data Unsmooth Raw data Raw data

Sample size 1961 1961 1959 1961 1961Mean 0.000 0.000 −0.001 0.000 0.000

Standard deviation 1.479 2.195 2.065 2.360 2.113Kurtosis 11.245 11.390 11.178 8.669 8.353LB(8) 9.026 18.227 10.295 15.136 12.682

p-value LB(8) 34.00% 1.96% 24.49% 5.65% 12.33%Autocorrelation no yes no no no

NOTE: The lag length s = 8 for the LB(s) statistic is selected based on the choiceof s ≈ ln(1961) (see Tsay (2010)).

11

10

HIT

-15 -20....._ _ __. __ _._ _ ___. __ _._ _ __. __ ...._ _ _._ __ ..__,

2004 2005 2006 2007 2008 2009 2010 2011 2012 2004 2005 2006 2007 2008 2009 2010 2011 2012

NIS

15

10

-10

PAN

10

-15 -15....._--~--'-----'--~----'---~---'---.L....I 2004 2005 2006 2007 2008 2009 2010 2011 2012 2004 2005 2006 2007 2008 2009 2010 2011 2012

-10

8.2.2 Estimation Results

In our analysis, we compare two proposed MCMC methods for the following four candi-date LNSV models:

1. Model SVn: the LNSV model without both fat-tails and correlated errors. The errorin return is assumed to follow normal distribution (λt = 1 for all t), and there is nocorrelation between errors in return and volatility (ρ = 0).

2. Model SVt: the LNSV model with fat-tails. The error in return is assumed to Student-t distribution with unknown degrees of freedom, and there is no correlation betweenerrors in return and volatility.

3. Model SVL: the LNSV model with correlated errors. The error in return is assumedto follow normal distribution, and there is a correlation between errors in return andvolatility (ρ 6= 0).

4. Model SVLt: the LNSV model with fat-tails and correlated errors. The error in returnis assumed to Student-t distribution with unknown degrees of freedom, and there is acorrelation between errors in return and volatility.

The hyperparameters in (7) and (8) required in the joint prior distribution were set toaν = 12, bν = 0.8, A = 30, B = 1.5, a = 1, b = 0.005, α0 = 0, p0 = 0.2, ϕ0 = 0 and q0 = 0.5.For initial parameter values and log-volatilities, respectively, we set θ(0) = (20, 1, 0.9, 0.1, 0.1)

and h(0)t = 2 lnRV . In particular, Griddy Gibbs samples of ν, φ and ht are, respectively,

drawn using 100, 200 and 400 equally spaced points. The possible range of ht for the ith

Gibbs iteration is [h(i−1)t − 1, h

(i−1)t + 1]. In all cases, we simulated the ht’s in a single-

move fashion. All the calculations were performed running stand alone code developed bythe authors using MATLAB version 7.8.0(R2009a) on a AMD Phenom(tm) II X6 1035TProcessor 2.60GHz.

For all the models and methods, we ran the MCMC simulation for 15000 iteration butdiscarded the first 5000 draws. With the resulting N = 10000 values, we calculated theposterior means, standard deviations (SD), the 95% credible interval, the Monte Carlostandard error (MCSE) and the simulation inefficiency factors (SIF). Table 2, 4, 6 and 8summarize these results. According to both MCSE and SIF, the two proposed samplingalgorithms have been mixing quite good. The 95% credible intervals are calculated usinga highest posterior density (HPD) proposed by Chen and Shao (1999). The MCSE isestimated by σf/

√N (see Roberts (1996)), where σ2f id defined as the variance of the

posterior mean from correlated draws. Here, the number of batches was 50 and there were200 draws in each batch. The SIF can be interpreted as the number of successive iterationsneeded to obtain near independent samples and is calculated by σ2f/σ

2f , where σf is the

standard deviation of the posterior mean.From the tables of posterior analysis we obtain the following results. First, the poste-

rior means for the modal of the stationary distribution of volatility, exp(12 α), in GG

sampler are smaller than those in MH sampler. The only exception is for Hitachi inSVLt model and Nissan in SV models with correlated errors. The highest posterior meanis for SVn model by MH sampler. Second, the posterior means of the volatility per-sistence, φ, are close to one, which indicates a well-known high persistence of volatilityfor all stocks. In all cases, the posterior means by GG sampler are lower than those byMH sampler, where the lowest posterior mean is for the SVL model, and the highest one isfor SVLt model except for TOPIX. It means that GG sampler exhibits a shorter half-life

12

of shock to volatility, HL = ln(0.5)/ ln(φ) (see Hall (2005)). Third, the posterior means forthe conditional variance of volatility, τ2, are smaller by the MH sampler thanthose by the GG sampler for all cases, indicating that MH sampler exhibits a lower vari-ability of volatility. The lowest posterior mean is for SVLt model in MH sampler exceptfor TOPIX, and the highest one is for SVL model in GG sampler. Fourth, the posteriormeans of the degrees of freedom, ν, in SV models with fat-tails are between 13 (for theHitachi stock in the SVLt model by MH sampler) and 24 (for the TOPIX in the SVt by GGsampler), indicating that the normal conditional distribution would be strongly rejectedby the data. In all cases, GG sampler exhibits the higher posterior means of thedegrees of freedom of the Student-t distribution. Fifth, the posterior means of theleverage effect parameter, ρ, in all cases of SV models with correlated errors are inthe negative interval, where posterior means in GG sampler suggest that some thoughweaker evidence of negative correlation between returns and volatility process.

To evaluate the daily estimated volatility performances, the standard realized volatility(RV) is used as a proxy. It was showed by Anderson et al. (2003) that the daily RVobtained from the intra-day high frequency data set is an consistent and robust estimatorof the ”true” volatility as long as there are no jumps. The percentage RV is defined by

RVt = 100×√∑Nt

k=2[p (t, k)− p (t, k − 1)]2,

where Nt is the number of observations in day t and p (t, k) denotes the log-price at thek’th observation in day t. To measure the closeness between the estimated volatilities andRV, six loss functions are used, namely, root mean-squared error (RMSE) for volatilityand variance, mean absolute error (MAE) for volatility and variance, logarithmic loss (LL)and Gaussian quasi-maximum likelihood function (GMLE), as discussed by Bollerslev et al.(1994) and Lopez (2001).

RMSE1 =

√√√√ 1

T

T∑t=1

(σt −RVt)2, RMSE2 =

√√√√ 1

T

T∑t=1

(σ2t −RV 2

t

)2GMLE =

1

T

T∑t=1

[log(RV 2

t

)+ σ2tRV

−2t

], LL =

1

T

T∑t=1

log(σ2tRV

−2t

)MAE1 =

1

T

T∑t=1

|σt −RVt| , MAE2 =1

T

T∑t=1

∣∣σ2t −RV 2t

∣∣Table 3, 5, 7 and 9 present the value and ranking (indicated by subscripts) of all four

SV models and two samplers under the six statistical loss functions. An interesting resultis that for the TOPIX series the GG sampler provides the most accurate estimatesagainst the MH sampler in each model. In fact, of the eight cases the four best estimates areperformed by the GG sampler, where the best performance is for SVLt model. For Hitachi,excluding GMLE loss function, the best estimate is provided by MH sampler, wherefive loss functions indicate that the SVL model provides the most accurate estimation.Excluding GMLE loss function, MH sampler for SVLt model is the best sampler toestimate volatility of Nissan indicated by five loss functons. The results for Panasonicseries indicates no clear pattern.

The resulting plots of the corresponding absolute returns and posterior mean of volatil-ities are shown in Figure 2, together with the RV for TOPIX series. The posterior meanof volatilities for the other series are not plotted to save space. Those plots confirm that

13

the posterior means of volatilities from the MH sampler exhibit smoother movements thanthose from the GG sampler. The figure is also nicely illustrated that RV and the estimatedvolatility closely mimic the movements of |Rt| plotted in the top panel supporting the ideaof using mean-corrected returns as proxies for ht.

9 Conclusion

The results, based on daily observations from all four stocks using four LNSV models,reveal that volatility by MH sampler in all cases is more persistent and less variable thanthose by GG sampler. In the case of LNSV models with a fat-tails component, the degrees offreedom shows that the student-t distribution by MH sampler appears flatters with heaviertails. In case the LNSV models with correlated errors, the posterior mean of leverageeffect parameter involving the use of MH sampler show a stronger evidence of negativeasymmetric returns-volatility relation. After comparing the estimation performance of allcases, where standard daily RV is used as a proxy, it was found that the GG sampler issuperior according to all six loss functions for TOPIX in the LNSVLt model, while the MHsampler is superior according to five loss functions for Hitachi in the LNSVL model and forNissan in the LNSVLt model and three loss functions for Panasonic in the LNSVt model.Furthermore, GMLE (Gaussian quasi-maximum likelihood function) loss function is onlyminimized on all stocks in the same model and sampler: LNSVLt model by GG sampler.

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Table 2: Posterior analysis for daily TOPIX.

Parameter StatisticSVn SVt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.015 (0.086) 1.139 (0.124) 0.963 (0.086) 1.099 (0.124)95% HPD (0.850, 1.185) (0.897, 1.384) (0.796, 1.141) (0.862, 1.351)

MCSE (SIF) 0.0020 (5.60) 0.0022 (3.13) 0.0024 (8.03) 0.0023 (3.50)

φ

Mean (SD) 0.969 (0.007) 0.978 (0.006) 0.972 (0.006) 0.981 (0.005)95% HPD (0.963, 0.991) (0.966, 0.990) (0.958, 0.985) (0.969, 0.991)

MCSE 0.0004 (35.16) 0.0003 (36.85) 0.0004 (33.96) 0.0003 (38.07)

τ2Mean (SD) 0.043 (0.007) 0.033 (0.006) 0.039 (0.007) 0.026 (0.005)95% HPD (0.028, 0.060) (0.021, 0.046) (0.025, 0.053) (0.017, 0.038)

MCSE (SIF) 0.0007 (77.78) 0.0006 (93.15) 0.0006 (85.87) 0.0006 (124.35)

νMean (SD) - - 23.591 (4.335) 20.885 (2.732)95% HPD - - (15.544, 32.141) (15.525, 26.160)

MCSE (SIF) - - 0.1498 (11.94) 0.1044 (14.62)

Parameter StatisticSVL SVLt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.038 (0.053) 1.160 (0.063) 0.979 (0.052) 1.125 (0.065)95% HPD (0.938, 1.147) (1.040, 1.291) (0.880, 1.085) (0.995, 1.253)

MCSE (SIF) 0.0021 (15.69) 0.0022 (12.30) 0.0023 (19.82) 0.0035 (29.63)

φ

Mean (SD) 0.951 (0.008) 0.9668 (0.006) 0.955 (0.008) 0.970 (0.006)95% HPD (0.934, 0.965) (0.954, 0.978) (0.938, 0.971) (0.956, 0.982)

MCSE (SIF) 0.0005 (45.52) 0.0005 (79.00) 0.0007 (77.75) 0.0007 (118.88)

τ2Mean (SD) 0.058 (0.009) 0.042 (0.007) 0.051 (0.010) 0.036 (0.007)95% HPD (0.040, 0.078) (0.025, 0.056) (0.033, 0.073) (0.023, 0.050)

MCSE (SIF) 0.0008 (80.67) 0.0009 (157.97) 0.0011 (118.37) 0.0010 (167.61)

νMean (SD) - - 23.562 (4.222) 21.269 (2.798)95% HPD - - (16.005, 32.156) (15.636, 26.632)

MCSE (SIF) - - 0.1387 (10.79) 0.1574 (31.64)

ρMean (SD) −0.539 (0.050) −0.606 (0.054) −0.552 (0.054) −0.639 (0.061)95% HPD (−0.636,−0.442) (−0.705,−0.493) (−0.654,−0.440) (−0.749,−0.521)

MCSE (SIF) 0.0036 (50.45) 0.0063 (135.78) 0.0045 (70.919) 0.0074 (148.755)

Table 3: Estimation performance of the estimated volatility for TOPIX under six loss func-tions.

LossFunction

SVn SVt SVL SVLtGG MH GG MH GG MH GG MH

RMSE1 0.5284 0.6618 0.4632 0.6056 0.5113 0.6497 0.4371 0.5955RMSE2 1.9884 2.4948 1.6512 2.1686 1.8353 2.3517 1.5311 2.0495GMLE 2.0154 2.7758 1.7032 2.5256 1.9373 2.7217 1.5751 2.4425LL 1.1344 1.6308 0.9332 1.4726 1.0873 1.6067 0.8521 1.4245MAE1 0.4254 0.5578 0.3682 0.5106 0.4183 0.5567 0.3481 0.5065MAE2 0.9694 1.3388 0.8072 1.1796 0.9443 1.3247 0.7561 1.1685Time (min) 47.9 7.8 51.6 9.6 52.9 9.2 99.0 11.1

16

Table 4: Posterior analysis for daily Hitachi.

Parameter StatisticSVn SVt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.499 (0.108) 1.638 (0.155) 1.375 (0.112) 1.549 (0.163)95% HPD (1.286, 1.713) (1.327, 1.942) (1.155, 1.599) (1.237, 1.876)

MCSE (SIF) 0.0020 (3.45) 0.0029 (3.61) 0.0039 (12.18) 0.0031 (3.71)

φ

Mean (SD) 0.956 (0.009) 0.972 (0.008) 0.966 (0.008) 0.979 (0.006)95% HPD (0.937, 0.974) (0.956, 0.987) (0.949, 0.983) (0.966, 0.990)

MCSE (SIF) 0.0006 (47.23) 0.0006 (73.73) 0.0007 (63.67) 0.0004 (54.35)

τ2Mean (SD) 0.068 (0.012) 0.041 (0.009) 0.047 (0.011) 0.030 (0.007)95% HPD (0.043, 0.092) (0.024, 0.061) (0.027, 0.070) (0.018, 0.044)

MCSE (SIF) 0.0012 (96.02) 0.0011 (137.79) 0.0013 (124.58) 0.0007 (120.64)

νMean (SD) - - 17.812 (3.580) 15.415 (2.181)95% HPD - - (11.520, 24.977) (11.269, 19.734)

MCSE (SIF) - - 0.2085 (33.91) 0.1234 (32.01)

Parameter StatisticSVL SVLt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.485 (0.098) 1.562 (0.130) 1.337 (0.101) 1.328 (0.153)95% HPD (1.294, 1.679) (1.298, 1.812) (1.140, 1.538) (1.014, 1.613)

MCSE (SIF) 0.0027 (7.88) 0.0053 (16.83) 0.0055 (29.83) 0.0108 (50.11)

φ

Mean (SD) 0.952 (0.010) 0.970 (0.009) 0.964 (0.009) 0.985 (0.005)95% HPD (0.931, 0.973) (0.952, 0.987) (0.944, 0.979) (0.974, 0.996)

MCSE (SIF) 0.0008 (57.08) 0.0009 (111.88) 0.0008 (77.54) 0.0006 (105.96)

τ2Mean (SD) 0.071 (0.015) 0.043 (0.011) 0.049 (0.012) 0.019 (0.004)95% HPD (0.042, 0.099) (0.026, 0.066) (0.026, 0.074) (0.011, 0.029)

MCSE (SIF) 0.0015 (106.53) 0.0014 (168.46) 0.0014 (124.15) 0.0006 (169.78)

νMean (SD) - - 17.323 (3.630) 13.657 (1.746)95% HPD - - (11.139, 24.705) (10.405, 17.183)

MCSE (SIF) - - 0.2344 (41.68) 0.1361 (60.82)

ρMean (SD) −0.215 (0.063) −0.257 (0.073) −0.239 (0.070) −0.330 (0.121)95% HPD (−0.339,−0.090) (−0.393,−0.101) (−0.376,−0.098) (−0.549,−0.093)

MCSE (SIF) 0.0036 (32.90) 0.0070 (91.30) 0.0048 (46.43) 0.0158 (168.28)

Table 5: Estimation performance of the estimated volatility for Hitachi under six loss func-tions.

LossFunction

SVn SVt SVL SVLtGG MH GG MH GG MH GG MH

RMSE1 1.0446 0.9452 1.1477 1.0073 1.0425 0.9381 1.1598 1.0134RMSE2 6.8334 6.6142 7.3257 6.9435 6.7933 6.5461 7.3578 7.0416GMLE 2.1514 2.2487 2.0602 2.1776 2.1493 2.2538 2.0461 2.1675LL 0.9885 0.7172 1.2737 0.8493 0.9906 0.7061 1.3208 0.8714MAE1 0.8265 0.7262 0.9347 0.7813 0.8276 0.7191 0.9538 0.7894MAE2 3.8096 3.5042 4.1727 3.6803 3.8095 3.4631 4.2318 3.7094Time (min) 47.7 7.5 42.4 9 43.7 9.7 99.1 11.3

17

Table 6: Posterior analysis for daily Nissan.

Parameter StatisticSVn SVt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.576 (0.162) 1.769 (0.273) 1.436 (0.167) 1.696 (0.342)95% HPD (1.263, 1.902) (1.261, 2.315) (1.106, 1.757) (1.107, 2.303)

MCSE (SIF) 0.0039 (5.82) 0.0046 (2.94) 0.0050 (9.03) 0.0042 (1.53)

φ

Mean (SD) 0.975 (0.006) 0.986 (0.004) 0.980 (0.005) 0.989 (0.003)95% HPD (0.963, 0.989) (0.976, 0.994) (0.969, 0.989) (0.982, 0.997)

MCSE (SIF) 0.0004 (48.64) 0.0003 (44.52) 0.0003 (33.87) 0.0002 (36.81)

τ2Mean (SD) 0.044 (0.010) 0.027 (0.005) 0.032 (0.006) 0.019 (0.004)95% HPD (0.026, 0.065) (0.017, 0.038) (0.021, 0.046) (0.012, 0.027)

MCSE (SIF) 0.0010 (113.44) 0.0006 (114.26) 0.0006 (99.14) 0.0004 (124.28)

νMean (SD) - - 18.232 (3.451) 16.897 (2.274)95% HPD - - (12.065, 25.131) (12.551, 21.424)

MCSE (SIF) - - 0.1818 (27.75) 0.1331 (34.27)

Parameter StatisticSVL SVLt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.473 (0.134) 1.401 (0.221) 1.291 (0.133) 1.100 (0.117)95% HPD (1.206, 1.729) (0.969, 1.800) (1.023, 1.547) (0.871, 1.325)

MCSE (SIF) 0.0053 (15.91) 0.0181 (67.25) 0.0082 (38.32) 0.0042 (12.93)

φ

Mean (SD) 0.973 (0.006) 0.986 (0.006) 0.979 (0.005) 0.995 (0.001)95% HPD (0.959, 0.984) (0.973, 0.997) (0.969, 0.989) (0.992, 0.999)

MCSE (SIF) 0.0004 (45.07) 0.0007 (122.43) 0.0004 (55.30) 0.0001 (37.43)

τ2Mean (SD) 0.046 (0.009) 0.028 (0.009) 0.032 (0.008) 0.010 (0.001)95% HPD (0.028, 0.066) (0.014, 0.046) (0.018, 0.047) (0.006, 0.014)

MCSE (SIF) 0.0010 (109.69) 0.0012 (185.52) 0.0009 (128.85) 0.0002 (152.11)

νMean (SD) - - 16.936 (3.699) 15.639 (1.853)95% HPD - - (10.649, 24.430) (12.123, 19.342)

MCSE (SIF) - - 0.3159 (72.93) 0.1124 (36.77)

ρMean (SD) −0.295 (0.066) −0.351 (0.085) −0.328 (0.072) −0.478 (0.111)95% HPD (−0.420,−0.162) (−0.523,−0.195) (−0.466,−0.187) (−0.674,−0.268)

MCSE (SIF) 0.0043 (42.81) 0.0099 (135.65) 0.0062 (74.28) 0.0146 (171.93)

Table 7: Estimation performance of the estimated volatility for Nissan under six loss functions.LossFunction

SVn SVt SVL SVLtGG MH GG MH GG MH GG MH

RMSE1 0.5434 0.5998 0.5475 0.5383 0.5342 0.5977 0.5576 0.4891

RMSE2 4.0426 4.7377 3.6433 4.0225 3.9814 4.9168 3.5242 3.4741

GMLE 2.0474 2.3038 1.8682 2.1856 2.0243 2.2817 1.8041 2.1535LL 0.2606 0.2334 0.3395 0.2132 0.2607 0.2273 0.4018 0.1951

MAE1 0.3644 0.3786 0.3957 0.3462 0.3593 0.3755 0.4198 0.3171

MAE2 1.6975 1.9407 1.6894 1.6813 1.6602 1.9558 1.7206 1.4901

Time (min) 47.4 7.6 49.0 10.0 53.0 9.2 99.1 11.1

18

Table 8: Posterior analysis for daily Panasonic.

Parameter StatisticSVn SVt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.552 (0.108) 1.694 (0.147) 1.414 (0.111) 1.596 (0.153)95% HPD (1.351, 1.774) (1.407, 1.982) (1.199, 1.638) (1.311, 1.924)

MCSE (SIF) 0.0025 (5.44) 0.0027 (3.36) 0.0049 (19.71) 0.0034 (4.92)

φ

Mean (SD) 0.957 (0.009) 0.970 (0.008) 0.964 (0.008) 0.977 (0.006)95% HPD (0.939, 0.973) (0.952, 0.986) (0.945, 0.980) (0.964, 0.990)

MCSE (SIF) 0.0005 (36.20) 0.0008 (85.82) 0.0006 (53.83) 0.0004 (49.71)

τ2Mean (SD) 0.061 (0.011) 0.043 (0.011) 0.046 (0.010) 0.031 (0.006)95% HPD (0.040, 0.082) (0.021, 0.066) (0.027, 0.068) (0.020, 0.044)

MCSE (SIF) 0.0009 (75.35) 0.0014 (145.48) 0.0010 (105.24) 0.0007 (119.50)

νMean (SD) - - 16.447 (3.316) 14.883 (2.191)95% HPD - - (10.776, 23.058) (10.741, 19.240)

MCSE (SIF) - - 0.1788 (29.09) 0.1404 (41.06)

Parameter StatisticSVL SVLt

GG MH GG MH

exp( 12 α)

Mean (SD) 1.516 (0.098) 1.575 (0.132) 1.349 (0.106) 1.424 (0.130)95% HPD (1.321, 1.706) (1.308, 1.832) (1.132, 1.559) (1.148, 1.669)

MCSE (SIF) 0.0034 (12.16) 0.0054 (16.91) 0.0080 (57.25) 0.0071 (29.82)

φ

Mean (SD) 0.951 (0.010) 0.970 (0.009) 0.963 (0.009) 0.978 (0.006)95% HPD (0.932, 0.972) (0.950, 0.985) (0.944, 0.979) (0.965, 0.991)

MCSE (SIF) 0.0008 (59.88) 0.0010 (118.94) 0.0008 (76.95) 0.0005 (84.26)

τ2Mean (SD) 0.071 (0.014) 0.046 (0.012) 0.050 (0.013) 0.030 (0.006)95% HPD (0.043, 0.097) (0.028, 0.071) (0.025, 0.078) (0.016, 0.041)

MCSE (SIF) 0.0014 (104.23) 0.0017 (180.21) 0.0015 (136.17) 0.0007 (160.94)

νMean (SD) - - 15.961 (3.624) 14.498 (2.077)95% HPD - - (9.093, 23.048) (10.759, 18.708)

MCSE (SIF) - - 0.3027 (69.74) 0.1741 (70.30)

ρMean (SD) −0.294 (0.069) −0.366 (0.074) −0.329 (0.074) −0.404 (0.079)95% HPD (−0.425,−0.151) (−0.514,−0.223) (−0.471,−0.186) (−0.558,−0.249)

MCSE (SIF) 0.0046 (46.07) 0.0083 (126.93) 0.0062 (71.11) 0.0089 (128.28)

Table 9: Estimation performance of the estimated volatility for Panasonic under six loss func-tions.

LossFunction

SVn SVt SVL SVLtGG MH GG MH GG MH GG MH

RMSE1 0.6383 0.6546 0.6495 0.6181 0.6424 0.6668 0.6577 0.6232RMSE2 3.8886 4.2587 3.6112 3.8214 3.8765 4.3348 3.5571 3.6243GMLE 2.0844 2.2738 1.8962 2.1296 2.0823 2.2647 1.8651 2.0825LL 0.4425 0.3982 0.5197 0.3901 0.4586 0.4154 0.5698 0.4313MAE1 0.4643 0.4684 0.4736 0.4451 0.4705 0.4777 0.4858 0.4562MAE2 1.9165 2.0457 1.8311 1.8584 1.9386 2.0968 1.8452 1.8593Time (min) 48.1 7.6 49.3 9.2 53.6 9.5 100.2 12.4

19

Figure 2: Time series plots for the absolute returns (top panel) and the posterior mean of

volatility, σt = exp( 12 ht), for TOPIX based on the use of GG and MH samplers for

sampling ht, φ and ν in the four LNSV models, together with realized volatility.

20

12

10

absolute returns

8

6

4

2

0 2004/1/6 2005/1/4 2006/1/4 6~~~~~~~~~~~~~~~~~--.-~~~~~~~~~~~~~~~~~

2007/1/4 2008/1/4 2009/1/5 2010/1/4 2011/1/4 2011/12/30

SV-normal 5

4

3

2

o .......... ~~~~..__~~~-'-~~~-'-~~~_._~~~---'~~~~.__~~~-'-~~~ .............. 2004/1/6 2005/1/4 2006/1/4 6~~~~~~~~~~~~~~~~~--.-~~~~~~~~~~~~~~~~~

2007/1/4 2008/1/4 2009/1/5 2010/1/4 2011/1/4 2011/12/30

RV SVtGG sv~H

SV with fat-tails 5

4

3

2

o .......... ~~~~..__~~~-'-~~~-'-~~~_._~~~---'~~~~.__~~~-'-~~~ .............. 2004/1/6 2005/1/4 2006/1/4 6~~~~~~~~~~~~~~~~~--.-~~~~~~~~~~~~~~~~~

2007/1/4 2008/1/4 2009/1/5 2010/1/4 2011/1/4 2011/12/30

5 RV SVLGG sv~H

SV with correlated errors

4

3

2

o .......... ~~~~..__~~~-'-~~~-'-~~~-'-~~~---'~~~~'--~~~-'-~~~ .............. 2004/1/6 2005/1/4 2006/1/4 6....--.~~~~..--~~~~~~~--.-~~~--.-~~~---.~~~~.--~~~~~~~........-.

2007/1/4 2008/1/4 2009/1/5 2010/1/4 2011/1/4 2011/12/30

SV with fat-tails and correlated errors RV SVUGG SVL~H

5

4

3

2

o .......... ~~~~..__~~~-'-~~~-'-~~~-'-~~~---'~~~~'--~~~-'-~~~ .............. 2004/1/6 2005/1/4 2006/1/4 2007/1/4 2008/1/4 2009/1/5 2010/1/4 2011/1/4 2011/12/30