J Econ Finan (2013) 37:216–239DOI 10.1007/s12197-011-9178-7

Stochastic volatility model under a discretemixture-of-normal specification

Dinghai Xu · John Knight

Published online: 15 April 2011© Springer Science+Business Media, LLC 2011

Abstract This paper investigates the properties of a linearized stochasticvolatility (SV) model originally from Harvey et al. (Rev Econ Stud 61:247–264,1994) under an extended flexible specification (discrete mixtures of normal).General closed form expressions for the moment conditions are derived. Weshow that our proposed model captures various tail behavior in a more flexibleway than the Gaussian SV model, and it can accommodate certain correlationstructure between the two innovations. Rather than using likelihood-basedestimation methods via MCMC, we use an alternative procedure based on thecharacteristic function (CF). We derive analytical expressions for the joint CFand present our estimator as the minimizer of the weighted integrated mean-squared distance between the joint CF and its empirical counterpart (ECF).We complete the paper with an empirical application of our model to threestock indices, including S&P 500, Dow Jones 30 Industrial Average index andNasdaq Composite index. The proposed model captures the dynamics of theabsolute returns well and presents some consistent and supportive evidencefor the Taylor effect and Machina effect.

D. Xu (B)Department of Economics, University of Waterloo,Waterloo, ON, N2L 3G1, Canadae-mail: dhxu@uwaterloo.ca

J. KnightDepartment of Economics, University of Western Ontario,London, ON, N6A 5C2, Canadae-mail: jknight@uwo.ca

J Econ Finan (2013) 37:216–239 217

Keywords Stochastic Volatility Model · Mixtures of Normal ·Characteristic Function · Integrated Squared Error · Absolute Return

JEL Classif ication C01 · C13 · C14

1 Introduction

Beginning with seminal works by Mandelbrot (1963) and Fama (1965), sub-stantial empirical evidence indicates that most time series of returns onfinancial assets are not normally distributed, but instead exhibit significantleptokurtosis (fat tails relative to the normal distribution) and, in many cases,skewness. More recently, empirical research has indicated that financial timeseries also display some properties or stylized facts such as time-varying volatil-ity and volatility clustering. These findings have sparked considerable interestin searching for alternative non-normal model specifications to capture theseempirical characteristics. A benchmark model was developed by Engle (1982),called the Autoregressive Conditional Heteroscedasticity (ARCH) model. Ina standard ARCH model, the conditional variance is a linear function of pastsquared errors. Bollerslev (1986) proposed a Generalized ARCH (GARCH)specification allowing the conditional variance to be a linear function of boththe past squared errors and past conditional variances. Alternatively, Sto-chastic Volatility (SV) models by Taylor (1986) provide another specificationof dynamic properties. In the SV framework, the conditional variances arespecified to follow some latent stochastic process themselves. Therefore, twoinnovations give the time-varying characteristics in the SV specifications, whileone error process is specified in the GARCH families. There is a vast existingliterature discussing statistical properties of both model specifications, such asKim et al. (1998), Bai et al. (2003), Carnero et al. (2004), etc. In this paper, wewill not attempt to conduct a general survey,1 instead, we will provide relevantreferences relating to our work.

The SV model has an intuitive appeal and realistic modeling specification,but empirical applications have been limited due to the intractability of itslikelihood function. More specifically, since the volatility is modeled as a latentvariable, the objective likelihood function involves a series of integrals withthe dimension of the sample size. As is well-known, it is extremely difficult tosolve the integral in any analytical form and consequently, see e.g. Broto andRuiz (2004), alternative estimation methods have been devised and used forestimating the SV models.

Unlike the likelihood-based methods, those based on moments are rela-tively easy to implement because they avoid the high dimensional integration.

1Excellent survey papers include [GARCH]: Bollerslev et al. (1994); [SV]: Shephard (2004),Ghysels et al. (1996), Broto and Ruiz (2004) and etc.

218 J Econ Finan (2013) 37:216–239

Taylor (1986) and Melino and Turnbull (1990) proposed using the (Gener-alized) Method of Moments (GMM), matching a finite number of selected(weighted) theoretical moments with the corresponding sample moments.Andersen and Sorensen (1996) employed a modified weighting matrix toimprove the efficiency of the GMM procedure. Duffie and Singleton (1993)introduced Simulated Method of Moments (SMM) by selecting theoreticalmoments based on a simulated process. Jiang et al. (2005) provided generalanalytical expressions for both conditional and unconditional theoretical crossmoments, which allow us to examine the SV model’s statistical propertiesmore easily. However, these GMM type methods suffer from efficiency andconvergence problems, see Broto and Ruiz (2004). A Quasi Maximum Likeli-hood (QML) approach on a linearized transformed model was suggested byHarvey et al. (1994). Instead of concerning two error product processes inthe original SV specification, they transform the model to a linearized state-space structure. They then apply standard numerical optimization techniquesto the linearized model, treating the non-normal transformed error as ifit were normally distributed. Knight et al. (2002) considered an estimationusing the Empirical Characteristic Function (ECF) approach. There are alsoother alternative estimation procedures, such as Markov Chain Monte Carlo(MCMC) by Jacquier et al. (1994), Simulated Maximum Likelihood (SML)by Danielsson and Richard (1993), Efficient Method of Moments (EMM) byGallant and Tauchen (1996) etc. A recent survey paper, Broto and Ruiz (2004),collected various estimation methods for the SV models and compared themin several classifications.

In this paper, we retain the convenient linearized model originally fromHarvey et al. (1994). We extend the SV model under a discrete bivariatemixtures of normal (MN) specification and use a continuous empirical charac-teristic function (CECF) based method for the estimation. There are severalcharacteristics of our model which we now note. First, retaining the lineartransformation reduces some complexities from the original nonlinear productprocess. As Harvey et al. (1994) and Knight et al. (2002) mentioned, thetransformed data loses no information except the sign information on theoriginal series. Second, our proposed specification which models the log-transformed error with MN is more flexible for capturing various tail behaviorof error distributions. As is well known, any continuous distribution can be ap-proximated arbitrarily well by an appropriate finite MN. Third, our proposedmodel easily accommodates the possible correlations between the two randomprocesses. Consequently, the proposed model provides a better descriptionfor the empirical data than other competing alternatives. Furthermore, theevolution (dynamics) of the absolute returns can be well captured under ourmodel framework.

The paper is organized as follows. Section 2 details the model specificationand presents its statistical properties. Section 3 discusses an estimation methodbased the CECF. Section 4 considers empirical applications of the model andthe estimation procedure. We further apply the proposed model in examiningthe dynamics of the absolute returns and the associated Taylor effect and

J Econ Finan (2013) 37:216–239 219

Machina effect.2 Section 5 concludes the paper. All the proofs are collectedin the Appendix.

2 Model specification and statistical properties

There are two random processes in the SV specification: the return processand the volatility process. Following Taylor (1986), the standard discrete timeSV model is given as follows:

xt = exp

(ht

2

)et (1)

ht+1 = λ + αht + ηt (2)

where xt is defined as the logarithmic closing price differences in the usualway. et and ηt (t = 1, 2, ..., T) are assumed to be independently and identicallydistributed (i.i.d) normal random disturbances, i.e, et ∼ iid. N(0, 1) and ηt ∼iid. N(0, σ 2

η ) . The latent variable, ht, is the log volatility at time t, and isassumed to follow a stationary AR(1) process (|α| < 1). The estimation ofEqs. 1 and 2 via the standard Maximum Likelihood (ML) procedure involvesa T-dimensional integration evaluation.

To derive the general properties of the SV model, we follow Harvey et al.(1994), Mahieu and Schotman (1998) and Granger and Sin (2002) and firstlyrewrite Eq. 1 as:

x2t = exp (ht) e2

t (3)

As suggested by Harvey et al. (1994), Eq. 3 can be linearized by takinglogarithms on both sides, which results in,

yt = log(x2t ) = ht + εt (4)

where εt = log(e2t ). When et follows an i.i.d. standard normal distribution, it is

easy to see that εt follows a log-Chi-squared distribution with 1 dof.The dynamic characteristics of the transformed data yt are still captured

by the latent AR(1) process of ht, but in a linear specification. The newprocess, defined in Eqs. 2 and 4, contains all the parameters of interest.Some estimation techniques have been developed based on this linearizedspecification. One straightforward method is the QML. In the QML approach,

2The Taylor effect and Machina effect are first defined by Granger and Ding (1995a). We willprovide more details in the later empirical section.

220 J Econ Finan (2013) 37:216–239

a normal density is used to approximate the log χ21 distribution. As is well

known, the approximated normal distribution is characterized by a mean of−1.2704, and a variance of π2/2. Under a Gaussian state space, Kalman filtertechniques can be applied to the quasi-likelihood function. In essence, thequasi-likelihood function in the QML procedure is the first-order Edgeworthexpansion of the exact likelihood function. Approximation by truncating theseries expansion may yield inefficient estimates. Kim et al. (1998) constructa mixtures of seven normal distribution to efficiently estimate the log-Chi-squared distribution and conduct an MCMC procedure to estimate the SVparameters.

Instead of fixing the distribution for ε as a log-Chi-squared, Mahieu andSchotman (1998) use a flexible mixtures of three normals to accommodatethe shape of the transformed error and estimate the MN and AR parameterssimultaneously based on an EM algorithm. It is worth noting that in none of theaforementioned studies was correlation taken into account between the twoprocesses in the linearized specification. One obvious reason is from Harveyet al. (1994), in which they have proved that if e and η, in Eqs. 1 and 2, aregenerated from a joint symmetric distribution, then the covariance of ε and η

is always zero.However, in general, there is no reason to always assume the distribution

of e and η to be symmetric. For example, the empirical evidence showsthat the distribution of most financial asset returns is often left-skewed. Inother words, the marginal distribution of the innovation in the return processtends to be asymmetric. This indicates that e and η tend to be generatedfrom an asymmetric joint distribution. In addition, statistically, if correlationsbetween the two processes are ignored, it might affect the estimation results.Therefore, extending the framework based on Mahieu and Schotman (1998)to accommodate the dependence, we impose a discrete bivariate MN on thedisturbances, ε and η. That is,

(εt

ηt

)∼ pl N

((μl

0

) (σ 2

l ρlσlση

ρlσlση σ 2η

))(5)

where l = (1, 2, ... , L). L is the number of mixture components. pl is the mixing

proportion parameter, withL∑

l=1

pl = 1.

Note that the proposed SV model under assumption (5) is a generalizedversion of Harvey et al. (1994) and Mahieu and Schotman (1998). If we setρl = 0, (for l = 1, 2, ..., L), it collapses to the model in Mahieu and Schotman(1998). Furthermore, with zero correlation between the two processes, if weset the number of the mixture components to be 1, then the model collapsesto the one in Harvey et al. (1994). To further examine the statistical propertiesof the model, we derive analytical expressions for the general cross moments,given in Proposition 1.

J Econ Finan (2013) 37:216–239 221

Proposition 1 If a time series xt is specif ied under Eqs. 2 and 4, and ε andη satisfy the assumption (5), for m, n, k ≥ 0, the closed form cross momentsexpression for |xt| and |xt+k| is given as follows,3

E(|xt|m|xt+k|n) = exp

⎛⎝nλ

2

k∑j=1

α j−1

⎞⎠

× exp

(λ(m + nαk)

2(1 − α)+ σ 2

η (m + nαk)2

8(1 − α2)

)

× exp

⎛⎝n2σ 2

η

8

k∑j=1

α2(k− j)

⎞⎠

×L∑

l=1

pl exp

(mμl

2+ m2σ 2

l

8+ mnαk−1ρlσlση

4

)

×L∑

l=1

pl exp

(nμl

2+ n2σ 2

l

8

)(6)

Proof See Appendix. ��

With the above formulas, it is straightforward to show that our modelis more flexible than the standard SV model. Harvey (1998) derives themoments of powers of the absolute returns under the standard SV specification(for d > −1):

E(|xt|d

) =[

2d/2 (d/2 + 1/2)

(1/2)

]× exp

(dλ

2(1 − α)+ d2σ 2

η

8(1 − α2)

)(7)

For a simple comparison, we set ρl = 0, (l = 1, 2, ..., L), in the Proposition1. It yields:

E(|xt|d

) =[

L∑l=1

pl exp

(dμl

2+ d2σ 2

l

8

)]× exp

(dλ

2(1 − α)+ d2σ 2

η

8(1 − α2)

)(8)

The comparison between our model and the standard model regardingdifferent orders of moments boils down to the comparison between Eqs. 7and 8. For example, if we are interested in the tail behavior under both models,

3We use the convention thatb∑

j=a

f j = 0 for b < a, where f j is the functional form indexed by j.

222 J Econ Finan (2013) 37:216–239

then, we need to calculate both Eqs. 7 and 8 when d = 4. Realizing that thesecond component of both expressions are exactly the same, we only need tocompare the first component of each expression.

First part of Eq. 7 = 4 × (2.5)

(0.5)= 3

First part of Eq. 8 =L∑

l=1

pl exp(2μl + 2σ 2l )

The first part of Eq. 7 equals a fixed number, while the first part of Eq. 8is a function of the mixture parameters. It is not hard to see that our model ismore flexible for modeling various tail behavior than the conventional. Undercertain circumstance, the standard SV models can be viewed as one special caseof our model. For example, Omori et al. (2007) constructed a ten-componentMN to approximate the log χ2

1 distribution and applied MCMC algorithm toefficiently estimate the SV parameters. When those MN parameter values areplugged into Eq. 8, we find Eqs. 7 and 8 are almost identical.

Furthermore, having obtained the general closed form expressions for thecross moments of |xt| and |xt+k|, we can use the linearized transformationbetween xt and yt to back out the ACF of yt in a closed form. In essence,we use Eq. 6 as the joint moment generating function (MGF) of yt and yt+k

to derive the corresponding cumulant generating function (CGF). With theproperties of the CGF, the central moments are easily achieved by takingdifferent orders of the derivatives based on the CGF. Therefore, the ACF isgiven in the Proposition 2.

Proposition 2 If a time series yt is def ined in Eqs. 2 and 4, and ε and η satisfythe assumption (5), the ACF of yt is given as follows,

ACFk =

σ 2η

(1−α2)+ ση

α

L∑l=1

plρlσl

σ 2η

(1−α2)+

L∑l=1

pl(μ2

l + σ 2l

) −(

L∑l=1

plμl

)2 αk (9)

Proof See Appendix. ��

Notice that Eq. 9 can be also equivalently expressed as,

ACFk = σ 2h + 1

ασε,η

σ 2h + σ 2

ε

αk (10)

When ρl = 0 (for l = 1, 2, ..., L), Eq. 10 collapses to the formula derived inHarvey (1998) for the standard SV model. Equation 10 presents an explicit

J Econ Finan (2013) 37:216–239 223

relationship between the ACF and the model parameters, clearly, ignoringthe covariance component, σε,η, in Eq. 10 may yield some biases for otherparameter estimates with the ACF value fixed at its corresponding empiricalvalue. We expect that the directions of the biases would depend on the signsof the covariance between the two processes.

3 Estimation via the CECF procedure

In this section, we propose to use a procedure based on the CECF to estimatethe model. The main idea of the CECF method is to match the theoreticalCF with its empirical counterpart (ECF) continuously in Lr space with certainweighting measures. The CF is always uniformly bounded in the parameterspace and contains the same amount of information as the distribution functiondue to its one-to-one correspondence with it.

Another attractive characteristic of the CECF procedure in this model isthat there exists an analytical form of the theoretical CF. Xu and Knight (2011)showed that the CECF estimator is comparable to the MLE in the MN modelsand discussed some asymptotic efficiency issues associated with the Gaussiankernel. There are several reasons for selecting the exponential weighting form,w(r) = exp(−br′r), where b is non-negative. This weighting function puts moreweight on the interval around the origin, where the CF contains the mostinformation about the tail of the distribution. Furthermore, this weightingfunction retains certain properties of the Gaussian kernels, which has somecomputational advantages.

Due the aforementioned advantages, the CECF approach has been suc-cessfully used for estimating the standard SV model. Knight et al. (2002)was the first paper, in which they construct the CF approach in estimatingthe SV model parameters. However, in their paper, when the correlation isaccommodated in the standard SV specification, the CF is involved with avery complicated hypergeometric function. This would lead to some estimationdifficulties in practice. For this reason, the CECF approach is not empiricallyimplemented in the standard SV model with correlations in Knight et al.(2002). In this paper, the CF under the proposed model is derived in a relativelyeasy closed form expression, which leads to a straightforward implementationin practice.

Since our proposed SV specification implies that the volatility is autocorre-lated and hence time dependent, this dependence must be taken into accountwhen constructing the theoretical CF. Following Knight and Yu (2002), wefirst need to define the overlapping blocks of y1, y2, ..., yT as, z j = (y j, ...y j+q),j = 1, 2, ..T − q, where the block size is q + 1. Then the joint CF for the jthblock is defined as:

c(r, θ) = E(exp(ir′z j)) (11)

224 J Econ Finan (2013) 37:216–239

where the transformed variable r, r = (r1, r2, ..., rq+1), is a q + 1 dimensionalvector and θ is the unknown parameter vector from the parametric distribu-tional assumption. The empirical counterpart (ECF) of Eq. 11 is defined as:

cn(r) = 1

n

n∑j=1

exp(ir′z j) where n = T − q (12)

Feuerverger (1990) established that, under regularity conditions, cn(r)a.s→

c(r, θ).4 Based on this result, the CECF procedure is defined as minimizing acertain distance measure between Eqs. 11 and 12. Define the distance measurein L2 space as:

D(θ; y) =∫

...

∫|c(r, θ) − cn(r)|2w(r)dr1...drq+1 (13)

where w(r) is a general weighting function. In this paper, we take w(r) =exp(−r′ Br), where B = diag(b 1, b 2, ..., b q+1) and b js are non-negative realnumbers. Each moving block has q periods overlapping with its adjacent block.

Knight and Yu (2002) established the strong consistency and asymptoticnormality for the CECF estimator. The asymptotic covariance matrix of theCECF estimators can be expressed as,

1

n

[∫...

∫ (∂ Rec(r, θ)

∂θ

∂ Rec(r, θ)

∂θ ′ + ∂ Imc(r, θ)

∂θ

∂ Imc(r, θ)

∂θ ′

)w(r)dr1...drq+1

]−1

× ×[∫

...

∫ (∂ Rec(r, θ)

∂θ

∂ Rec(r, θ)

∂θ ′ + ∂ Imc(r, θ)

∂θ

∂ Imc(r, θ)

∂θ ′

)

× w(r)dr1...drq+1

]−1

(14)

where Re c(r, θ) and Im c(r, θ) stand for the real part and the imaginary partof the CF respectively. The expression of can be found in the Appendix.

The implementation of the CECF procedure essentially requires mini-mizing Eq. 13 with respect to the unknown parameter vector θ , i.e, θ̂ =argmin[D(θ; y)]. As mentioned, we are able to derive the closed form expres-sion for the theoretical CF.

4 a.s→ stands for convergence almost surely.

J Econ Finan (2013) 37:216–239 225

Table 1 Summary statistics for sample data

Length Mean STD Skewness Kurtosis Minimum Maximum

S&P 500 1,507 −0.0102 1.1924 0.1181 5.2160 −5.9391 5.5846Dow-Jones 1,507 −0.0038 1.1599 −0.0265 6.4456 −7.5334 6.1585Nasdaq 1,511 −0.0415 2.0481 0.2103 6.2553 −10.1684 13.2546

Proposition 3 If a transformed time series yt is def ined under Eqs. 2 and 4 andunder assumption (5), the closed form expression of the joint CF for yt...yt+q

is given as follows:

c (r1, ...rk, θ)

= exp

⎛⎜⎝ iλ

(1 − α)

k∑j=1

r j −σ 2

η

2(1 − α2)

⎛⎝ k∑

j=1

αk− jr j

⎞⎠

2⎞⎟⎠

×k∏

j=1

⎡⎣ L∑

l=1

pl exp

(iμlr j − 1

2σ 2

l r2j

)

× exp

⎛⎝−σ 2

η

2

( j∑m=2

α j−mrm−1

)2

− ρlσlσηr j

j∑m=2

α j−mrm−1

⎞⎠

⎤⎦ (15)

Proof See Appendix. ��

4 Empirical application

In this section, we report on the application of our model and the empiricalestimates associated with the stock index data.5 The sample data consists ofthree stock indices, including S&P 500 index, Dow Jones 30 Industrial Averageindex and Nasdaq composite index. The sample period covers from 2 January2000 to 31 December 2005.

Table 1 presents the summary statistics of the empirical data. Excess kur-tosis values imply that the steady state distributions of the returns are non-normal. Furthermore, we find that, in general, the autocorrelations of thesquared returns are bigger than those of the raw return data, which is consistentwith the volatility clustering characteristics in the empirical literature.

5In this section, we empirically estimate the parameters based on Eq. 15 with k = 2 and L = 2. Wehave also tried different block sizes and different numbers of the mixture components. The resultsare not significantly changed. Since we have general formulation of the CF (see Proposition 3) forany mixture component L, the models can be easily extended.

226 J Econ Finan (2013) 37:216–239

Table 2 Empirical estimates

Parameter λ α ση σ 2ε ρ∗

SV-NS&P 500 0.0392 0.8255 0.1027 – –

(0.0351) (0.0864) (0.0465) – –Dow-Jones 0.0197 0.8720 0.0834 – –

(0.0214) (0.0946) (0.0937) – –Nasdaq 0.0461 0.9621 0.2105 – –

(0.0412) (0.0574) (0.0672) – –SV-MN

S&P 500 −0.0626 0.9237 0.2266 4.4407 –(0.0325) (0.0417) (0.0429) (0.3079) –

Dow-Jones −0.0787 0.9309 0.2161 4.2997 –(0.0660) (0.0596) (0.0634) (0.2976) –

Nasdaq 0.0382 0.9369 0.3267 4.5291 –(0.0208) (0.0491) (0.0853) (0.4284) –

SV-MN-CS&P 500 −0.0523 0.9219 0.2620 4.3318 −0.2041

(0.0277) (0.1040) (0.1267) (0.2923) (0.0279)Dow-Jones −0.0880 0.9081 0.2221 4.3703 0.1376

(0.1017) (0.0226) (0.0925) (0.3150) (0.0736)Nasdaq 0.0365 0.9424 0.3035 4.5852 0.0354

(0.0755) (0.1059) (0.1481) (0.3862) (0.0245)

Three linearized specifications of the SV models are applied to fit thesample data respectively. The first specification is based on the conventionalmodel from Harvey et al. (1994), in which we utilize a N(−1.2704, π2/2) forapproximating the log-χ2

1 distribution, denoted as SV-N. The second and thirdspecifications are the SV models under a flexible MN with zero correlation(SV-MN) and lagged inter-temporal dependence structure (SV-MN-C).

The empirical results are reported in Table 2. Overall, the empirical results6

are consistent with the SV literature findings. The high values for the persistentparameter, α, indicate the volatility clustering characteristics in these threestock indices, which is consistent with the stylized fact. We also observe thatthe persistent parameter estimates from the SV-MN and SV-MN-C are, ingeneral, greater than those from the SV-N (except for Nasdaq case). Anotherinteresting aspect from Table 2 is that a significant correlation between thereturn process and the transformed volatility process is detected for the S&P500 series. This empirical result indicates that in practice, the correlation factorshould be taken into account when we estimate the SV parameters.7

6To save space, we do not report all the estimates. In this paper, we provide the estimates of mostinterest. σ 2

ε is defined as the variance of the transformed error, σ 2ε = ∑L

l=1 pl[σ 2l + (μl − μ∗)2],

where μ∗ = ∑Ll=1 plμl . ρ∗ is the correlation coefficient between ε and η.

7In a companion paper of ours, a larger empirical data set is used to analyze the significance anddirection of the correlation coefficients. In this paper, the empirical application is constructed onlyfor illustration of the model specification and the estimation methodology.

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Fig. 1 Densities of yt .Solid line: data; dashed line:GARCH*-t; dotted-dashedline: SV-N; dotted line:SV-MN-C

0 5 10 15

0

0.05

0.1

0.15

0.2

0.25

0.3Data

S&P500

0 5 10 15

0

0.05

0.1

0.15

0.2

0.25

0.3Data

Dow-Jones

0 5 10 15

0

0.05

0.1

0.15

0.2Data

Nasdaq

Due to the flexibility of our modeling structure, instead of restricting thetransformed error to be a logχ2

1 distribution, we use the empirical estimatesfrom MN to construct the implied error distribution. To illustrate the overallgoodness of fit, we plot the empirical densities of yt from the data againstthe implied densities generated from the empirical estimates, see Fig. 1.As expected, the SV-N has some drawbacks in capturing the shape of theempirical density from the data, especially in the tail section, while, in general,

228 J Econ Finan (2013) 37:216–239

Table 3 Empirical momentscomparison of yt

Mean Variance Skewness Kurtosis

S&P 500Data −1.2005 4.6884 −0.6240 2.9388GARCH(1,1)-N −1.3044 5.4090 −1.3918 3.5466GARCH(1,1)-t −1.5035 5.1281 −1.3702 3.3720GARCH*-t −1.2999 5.6825 −1.2987 3.7807SV-N −1.0019 4.9565 −0.0205 2.6320SV-MN −1.1990 4.7620 −0.6346 3.1759SV-MN-C −1.2429 4.7454 −0.6418 3.1329

Dow-JonesData −1.2515 4.5416 −0.6210 2.9914GARCH(1,1)-N −0.8704 6.3311 −0.8262 2.4819GARCH(1,1)-t −1.3225 5.2065 −1.4590 4.0309GARCH*-t −1.2200 5.3225 −0.8732 3.5443SV-N −1.0686 4.9293 −0.0062 2.8800SV-MN −1.2474 4.7901 −0.6375 3.1312SV-MN-C −1.2637 4.6585 −0.6395 3.1785

NasdaqData −0.2676 5.3859 −0.7140 3.2010GARCH(1,1)-N −0.6839 5.4972 −1.2414 2.7233GARCH(1,1)-t −1.0317 5.6610 −1.0853 3.3404GARCH*-t −0.5903 5.1447 −1.0635 3.1657SV-N 0.0100 5.6157 −0.0040 3.0126SV-MN −0.1879 5.3685 −0.6458 3.3505SV-MN-C −0.3008 5.5320 −0.6212 3.2953

the proposed SV model under the discrete MN specification provides a good fitfor the sample data. However, graphically, no significant difference is observedacross the implied densities with the SV-MN and SV-MN-C.

Furthermore, we also conduct the comparisons of the proposed modelagainst some popular competing alternative specifications in terms ofmoments. Specifically, we estimate the three stock indices data us-ing GARCH(1,1) with Gaussian error (denoted as GARCH(1,1)-N),GARCH(1,1) with student-t error (denoted as GARCH(1,1)-t) and optimal-order GARCH with student-t error (GARCH*-t) models.8 The first fourempirical-moment of yt implied from various models are provided in Table 3.In general, the GARCH-N and SV-N underestimates the kurtosis in theempirical data. The GARCH models tend to overestimate the variance. It alsocan be seen that the first four moments from the proposed specifications (boththe SV-MN and SV-MN-C) are fairly close to the corresponding momentsfrom the data. The SV-MN-C performs slightly better. This is consistent withour expectation since the SV-MN-C assumes a more generalized structure thanthe SV-MN and accommodates the possible correlations between the returnand the underlying volatility processes.

8We select the optimal orders of GARCH models based on the Bayesian Information Criteria(BIC).

J Econ Finan (2013) 37:216–239 229

Table 4 ACF comparisons

GARCH(1,1)-N GARCH(1,1)-t GARCH*-t SV-N SV-MN SV-MN-C

S&P 500 0.1926 0.2048 0.1892 0.1466 0.0877 0.0561Dow-Jones 0.1793 0.1085 0.0925 0.1211 0.1035 0.0819Nasdaq 0.3764 0.2699 0.1974 0.1899 0.1345 0.1209

The table presents the relative distance measure, which is defined as the summation of the absolutedistance between the model ACF and the empirical ACF values over 50 lags

As a further step, we investigate the implications on the dynamics ofthe absolute returns based on the proposed model. In recent years, therehas been an increasing interest in examining and modeling the behaviorof the absolute returns, see for example Granger and Ding (1995a, b),Granger and Hyung (2004), Forsberg and Ghysels (2007) and etc. In par-ticular, Forsberg and Ghysels (2007) provide both theoretical foundationsand empirical findings that absolute returns show more persistence than thesquare returns. The predictive power of the absolute returns outperformsother quadratic variations of the returns. Given the flexible structure in theproposed model, we expect that our model would provide a better description(model framework) of the absolute returns. The ACF is commonly used asa persistency measure in the literature. To conduct the comparison, we backout the ACFs of the absolute returns across different model specifications(including GARCH(1,1)-N, GARCH(1,1)-t, GARCH*-t, SV-N, and SV-MN)using the empirical estimates for the three stock indices respectively. We usethe empirical ACF as the benchmark for constructing the relative distance9

between the benchmark ACF and the ACF from each model over 50 lags.Table 4 presents the comparison results. As expected, the SV-MN-C producesthe smallest magnitude against the competing alternatives for all three indices.To visualize the ACF dynamic pattern, we plot the ACF from the benchmark(empirical data), GARCH*-t, SV-N and SV-MN-C in Fig. 2.10 The comparisonfrom the Fig. 2 reenforces the results from Table 4. The proposed modelperforms the best in capturing the evolution of the absolute return ACF.

Another interesting aspect in this context is to examine the so-called Tayloreffect (TE) and Machina effect (ME), see Granger and Ding (1995a), Rydenet al. (1998), Mora-Galan et al. (2004), Veiga (2009), etc. The TE is commonlyknown as that the autocorrelations of the absolute financial returns are largerthan those of the squared returns. If we define ϕk(a, b) = corr(|xt|a, |xt+k|b ),

9The relative distance measure is defined as the summation of the absolute distance between themodel ACF and the empirical ACF values over each lag (k).10To simplify the plots, we do not plot the ACFs from all the competing models. The ones reportedin Fig. 2 are the representatives. To reduce the numbers of figures, we only provide Nasdaq as anempirical illustration.

230 J Econ Finan (2013) 37:216–239

Fig. 2 ACF for absolutereturns. Solid line: data;dashed line: SV-N;dotted-dashed line: SV-MN;dotted line: SV-MN-C

0 10 20 30 40 500.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Data

Nasdaq

the hypothesis of TE can be expressed as ϕk(1, 1) > ϕk(2, 2). The ME hy-pothesis can be written as ϕk(a, 1) > ϕk(a, b), for b = 1. Both the TE andME have been established as important empirical persistence properties ofthe financial absolute returns in the literature, see for example, Forsberg andGhysels (2007). The absolute return would be a more appropriate risk measurecandidate alternative to the usual squared return risk proxy. Given that ourproposed model is well suited to examine the absolute returns, we constructTables 5 and 6 for examination of the TE and ME implied from our model

Table 5 Examination of TEand ME at k = 1

The table presents the ϕ

values across different a andb combinations. The boldnumber highlights the largestvalue in the whole correlationmatrix for each stock index,respectively

a = 0.5 a = 1.0 a = 1.5 a = 2.0 a = 2.5 a = 3.0

S&P 500b = 0.5 0.1439 0.1463 0.1385 0.1247 0.1093 0.0957b = 1.0 0.1428 0.1469 0.1406 0.1276 0.1124 0.0986b = 1.5 0.1349 0.1402 0.1356 0.1242 0.1103 0.0973b = 2.0 0.1225 0.1283 0.1253 0.1159 0.1039 0.0923b = 2.5 0.1088 0.1143 0.1125 0.1051 0.0952 0.0855b = 3.0 0.0964 0.1012 0.1002 0.0944 0.0865 0.0786

Dow-Jonesb = 0.5 0.1363 0.1382 0.1288 0.1124 0.0955 0.0821b = 1.0 0.1388 0.1408 0.1313 0.1146 0.0974 0.0838b = 1.5 0.1286 0.1303 0.1217 0.1068 0.0917 0.0798b = 2.0 0.1110 0.1119 0.1050 0.0933 0.0816 0.0724b = 2.5 0.0935 0.0936 0.0883 0.0798 0.0714 0.0650b = 3.0 0.0802 0.0796 0.0755 0.0695 0.0637 0.0594

Nasdaqb = 0.5 0.1971 0.1975 0.1733 0.1334 0.0929 0.0620b = 1.0 0.1961 0.2006 0.1792 0.1402 0.0992 0.0672b = 1.5 0.1717 0.1792 0.1625 0.1288 0.0922 0.0631b = 2.0 0.1330 0.1412 0.1296 0.1036 0.0745 0.0512b = 2.5 0.0941 0.1012 0.0937 0.0751 0.0539 0.0368b = 3.0 0.0644 0.0700 0.0651 0.0520 0.0368 0.0247

J Econ Finan (2013) 37:216–239 231

Table 6 Examinationof TE and ME at k = 2

The table presents the ϕ

values across different aand b combinations.The bold number highlightsthe largest value in the wholecorrelation matrix for eachstock index, respectively

a = 0.5 a = 1.0 a = 1.5 a = 2.0 a = 2.5 a = 3.0

S&P 500b = 0.5 0.1989 0.2029 0.2001 0.1895 0.1708 0.1534b = 1.0 0.2046 0.2092 0.2023 0.1873 0.1691 0.1520b = 1.5 0.1925 0.1967 0.1906 0.1773 0.1611 0.1457b = 2.0 0.1755 0.1785 0.1732 0.1622 0.1488 0.1362b = 2.5 0.1573 0.1587 0.1542 0.1455 0.1353 0.1259b = 3.0 0.1413 0.1412 0.1372 0.1308 0.1235 0.1170

Dow-Jonesb = 0.5 0.2028 0.2069 0.2008 0.1861 0.1693 0.1563b = 1.0 0.2037 0.2081 0.2019 0.1865 0.1690 0.1556b = 1.5 0.1949 0.1990 0.1932 0.1793 0.1636 0.1516b = 2.0 0.1795 0.1822 0.1772 0.1662 0.1542 0.1453b = 2.5 0.1638 0.1648 0.1606 0.1528 0.1448 0.1391b = 3.0 0.1525 0.1523 0.1487 0.1432 0.1382 0.1348

Nasdaqb = 0.5 0.1902 0.1893 0.1595 0.1155 0.0775 0.0519b = 1.0 0.1844 0.1928 0.1719 0.1326 0.0950 0.0676b = 1.5 0.1512 0.1679 0.1602 0.1326 0.1012 0.0758b = 2.0 0.1065 0.1269 0.1302 0.1151 0.0925 0.0719b = 2.5 0.0693 0.0892 0.0981 0.0916 0.0764 0.0606b = 3.0 0.0450 0.0625 0.0732 0.0713 0.0608 0.0486

with lag orders 1 and 2, respectively.11 Specifically, Tables 5 and 6 present the ϕ

values across different combinations of a and b . For all the three stock indices,we find that the position (a = 1, b = 1) gives the largest values in the wholematrix, which supports the TE. In general, a = 1 column across the tablespresent the largest values comparing to other columns. This confirms that theME hypothesis is satisfied in these tables. Although we can not make a generalstatement that our proposed model can always generate the TE and ME, atleast for the three stock indices examined in this paper, we do detect someconsistent findings with the stylized TE and ME in the literature. This will beuseful for further research on modeling and exploring the observed absolutereturn as an alternative measure of risk on the financial market.

5 Conclusion

In this paper, we investigate the properties of a stochastic volatility modelunder a discrete MN specification. General closed form expressions for mo-ment conditions are derived. We can easily accommodate certain correlationspecification between the two innovations. Due to difficulties involved in thelikelihood-based estimation methods, we propose an alternative procedurebased on the CECF. We also derive an analytical expression for the jointCF under the lagged inter-temporal dependence structure. In the empirical

11To reduce the number of tables, we only report the results from the first two lags. But, in practice,we did examine the lags up to 10. Those results are available upon request.

232 J Econ Finan (2013) 37:216–239

study, we apply our models to a 6-year daily return data set of three stockmarket indices, which includes S&P 500, Dow Jones 30 Industrial Averageindex and Nasdaq Composite index. In general, the empirical estimates pro-vide reasonable fit to the data. Based on the empirical results, we also findsome empirical evidence of the significant correlation structure between thetransformed return process and the underlying volatility process. In addition,the proposed model does a better job describing the dynamics of the absolutereturns against the competing models. We do find consistent and supportiveresults implied from the proposed model for the Taylor effect and Machinaeffect.

Appendix

Proof of Proposition 1

For convenience, throughout the Appendix, we notationally set ηt−1 equivalentto vt. Obviously, vt ∼ N(0, σ 2

v ), with σv = ση. Therefore, ht+1 = λ + αht + ηt

can be equivalently expressed as ht = λ + αht−1 + vt. Since the latent variableht follows an AR(1) process, we can write:

ht+k = αkht + λ

k∑j=1

α j−1 +k∑

j=2

αk− jvt+ j + αk−1vt+1

Hence, we have,

E(|xt|m|xt+k|n

) = E

⎡⎣exp

⎛⎝m

2ht + m

2εt + nαk

2ht + nλ

2

k∑j=0

α j−1

+ n2

k∑j=2

αk− jvt+ j + n2αk−1vt+1 + n

2εt+k

⎞⎠

⎤⎦

= E exp

⎛⎝m + nαk

2ht + nλ

2

k∑j=0

α j−1 + n2

k∑j=2

αk− jvt+ j

+ n2αk−1vt+1 + m

2εt + n

2εt+k

⎞⎠

J Econ Finan (2013) 37:216–239 233

and we substitute ht = λ1−α

+∞∑j=0

α jvt− j into the above expression. Then,

it yields

E(|xt|m|xt+k|n

) = E

⎡⎣exp

λ(m + nαk)

2(1 − α)+ m + nαk

2

∞∑j=0

α jvt− j + nλ

2

k∑j=1

α j−1

+ n2

k∑j=2

αk− jvt+ j + n2αk−1vt+1 + m

2εt + n

2εt+k

⎤⎦

= exp

(λ(m + nαk)

2(1 − α)

)× exp

⎛⎝nλ

2

k∑j=1

α j−1

⎞⎠

×E exp

⎛⎝m + nαk

2

∞∑j=0

α jvt− j

⎞⎠ × E exp

⎛⎝n

2

k∑j=2

αk− jvt+ j

⎞⎠

×E exp

(m2

εt + nαk−1

2vt+1

)× E exp

(n2εt+k

)

Since the marginal distribution of v is N(0, σ 2v ), we have,

E exp

⎛⎝m + nαk

2

∞∑j=0

α jvt− j

⎞⎠ = exp

(σ 2

v (m + nαk)2

8(1 − α2)

)

E exp

⎛⎝n

2

k∑j=2

αk− jvt+ j

⎞⎠ = exp

⎛⎝n2σ 2

v

8

k∑j=2

α2(k− j)

⎞⎠

By definition of joint moment generating function of εt and vt+1 and underassumption (5), we have,

Mεt,vt+1(r1, r2) = E exp(r1εt + r2vt+1

)

=L∑

l=1

[pl exp

(r1μl + 1

2r2

1σ2l + 1

2r2

2σ2v + ρlr1r2σlσv

)]

234 J Econ Finan (2013) 37:216–239

It is straightforward to get:

E exp

(m2

εt + nαk−1

2vt+1

)=

L∑l=1

pl exp

(mμl

2+ m2σ 2

l

8

+ n2α2k−2σ 2v

8+ mnαk−1ρlσlσv

4

)

E exp(n

2εt+k

)=

L∑l=1

pl exp

(nμl

2+ n2σ 2

l

8

)

Combining all the above expressions, we have the general moment conditionsstated in Proposition 1. ��

Proof of Proposition 2

Let m = 2s and n = 2r in Eq. 6, the joint MGF of yt and yt+k is given as follows,

My(s, r) = E (exp(syt) exp(ryt+k))

= exp

(λ

(1 − α)(s + r)

)exp

(σ 2

v

2(1 − α2)

(s2 + r2 + 2srαk))

×L∑

l=1

pl exp

(sμl + s2

2σ 2

l + srαk−1ρlσlσv

)

×L∑

l=1

pl exp

(rμl + r2

2σ 2

l

)

Define g1(s, r) = s2 + r2 + 2srαk, g2(s, r) = sμl + s2

2 σ 2l + srαk−1ρlσlσv and

g3(r) = rμl + r2

2 σ 2l . Therefore,

g1(s, r)|s=r=0 = 0 g2(s, r)|s=r=0 = 0 g3(r)|r=0 = 0

∂g1

∂s|s=r=0 = 0

∂g2

∂s|s=r=0 = μl

∂g2

∂r|s=r=0 = 0

∂2g1

∂s2|s=r=0 = 2

∂2g2

∂s2|s=r=0 = σ 2

l∂2g1

∂s∂r|s=r=0 = 2αk

∂2g1

∂s∂r|s=r=0 = ρlσlσvα

k−1

J Econ Finan (2013) 37:216–239 235

Therefore,

var(yt) = ∂φ2(s, r)∂s2

|s=r=0 = σ 2v

(1 − α2)+

L∑l=1

pl(μ2l + σ 2

l ) −(

L∑l=1

plμl

)2

cov(yt, yt+k) = ∂φ2(s, r)∂s∂r

|s=r=0 =(

σ 2v

(1 − α2)+ σ 2

v

α

L∑l=1

plρlσl

)αk

Combining the above expressions following the ACF formula, we completethe proof. ��

Proof of Proposition 3

Without loss of generality, we also derive the joint CF under assumption (5)when k = 2.

By definition, c(r1, r2; θ) = E [exp(ir1 yt + ir2 yt−1)] = E [exp(ir1ht + ir1εt +ir2ht−1 + ir2εt−1)].

From the AR(1) expression in Eq. 2, we have, ht = λ + αht−1 + vt. Hence,

c(r1, r2; θ) = E[exp(ir1λ + i(r1α + r2)ht−1 + ir1εt + ir2εt−1 + ir1vt)

]= exp (ir1λ) × E exp (i(αr1 + r2)ht−1) × E exp (ir1εt)

× E exp (ir2εt−1 + ir1vt)

It is straightforward to solve the each expectation in the above expression.

E exp[i (αr1 + r2) ht−1

] = exp

[i (αr1 + r2)

λ

(1 − α)− σ 2

v

2(1 − α2)(r1α + r2)

2

]

E exp (ir1εt) =L∑

l=1

pl exp

(iμlr1 − 1

2σ 2

l r21

)

and, by using the joint moment generating function of εt−1 and vt, we get,

E exp(ir2εt−1 + ir1vt) =L∑

l=1

exp

(ir2μl − r2

1σ2v

2− r2

2σ2l

2− ρlr1r2σlσv

)

236 J Econ Finan (2013) 37:216–239

Collecting all the above expressions, it yields,

c(r1, r2, θ) = exp

⎛⎜⎝ iλ

(1 − α)

2∑j=1

r j − σ 2v

2(1 − α2)

⎛⎝ 2∑

j=1

α2− jr j

⎞⎠

2⎞⎟⎠

×2∏

j=1

⎡⎣ L∑

l=1

pl exp(

iμlr j − (1/2)σ 2l r2

j

)

× exp

⎛⎝−σ 2

v c2

8

( j∑m=2

α j−mrm−1

)2

− ρlσlσvr j

j∑m=2

α j−mrm−1

⎞⎠⎤⎦

Similar process (for k = 3, 4, ..., K) will yield the general closed form expres-sion stated in Proposition 3. ��

Derivation of the asymptotic covariance matrix of CECF estimator

Without loss of generality, we derive the asymptotic covariance structure whenk = 2. Referring to Knight and Yu (2002), we have the following generalexpression for :

=∫

..

∫ ⎡⎣∂ Rec(r, θ)

∂θ

∂ Rec(s, θ)

∂θ ′1

n2

n∑j=1

n∑k=1

Cov(cos(r′x j), cos(s′xk))

+ ∂ Rec(r, θ)

∂θ

∂ Imc(s, θ)

∂θ ′2

n2

n∑j=1

n∑k=1

Cov(cos(r′x j), sin(s′xk))

+ ∂ Imc(r, θ)

∂θ

∂ Imc(s, θ)

∂θ ′1

n2

n∑j=1

n∑k=1

Cov(sin(r′x j), sin(s′xk))

⎤⎦

× w(r′)w(s′)dr′ds′

The double summation covariance expressions are readily found in Knightand Satchell (1997) and Yu (1998). These are given by:

1

n2

n∑j=1

n∑k=1

Cov(cos(r′x j), cos(s′xk))

= 1

2n[Re c(r + s, θ) + Re c(r − s, θ)] − Re c(r)Re c(s)

+ 1

2n2

n−1∑k=1

[(n−k) (Re �k(r, s)+Re �k(r, −s)+Re �k(s, r)+Re �k(s, −r))

]

J Econ Finan (2013) 37:216–239 237

2

n2

n∑j=1

n∑k=1

Cov(cos(r′x j), sin(s′xk))

= 1

n[Im c(r + s, θ) − Im c(r − s, θ)] − 2Re c(r)Im c(s)

+ 1

n2

n−1∑k=1

[(n−k) (Im �k(r, s)− Im �k(r, −s)+ Im �k(s, r)+ Im �k(s, −r))

]

1

n2

n∑j=1

n∑k=1

Cov(sin(r′x j), sin(s′xk))

= 1

2n[Re c(r − s, θ) − Re c(r + s, θ)] − Im c(r)Im c(s)

+ 1

2n2

n−1∑k=1

[(n−k) (Re �k(r, −s)−Re �k(r, s)+Re �k(s, −r)−Re �k(s, r))

]

In order to calculate the , we also need to derive �k(r, s). That is,

�k(r, s) = E (exp (ir1 yt + ir2 yt−1 + is1 yt+k + is2 yt+k−1))

To work out the above expectation, we need to use the assumption (5). Tediousbut straightforward steps yield the following results.

�k(r, s)

= exp

(ir1λ+ iλ

(r1α + r2+s1α

k+1+s2αk)

(1 − α)− σ 2

v

(r1α + r2 + s1α

k+1 + s2αk)2

2(1 − α2)

)

×L∑

l=1

exp

(ir2μl − 1

2r2

2σ2l − σ 2

v

2(r1 + s1α

k + s2αk−1)

− ρlr2(r1 + s1αk + s2α

k−1)σlσv

)× exp

(−1

2(r1α + r2)

2σ 2v

)

×L∑

l=1

exp

(ir1μl − 1

2r2

1σ2l − σ 2

v

2(s1α

k−1+s2αk−2)−ρlr1(s1α

k−1+s2αk−1)σlσv

)

× exp

⎛⎝is1λ

k+1∑j=1

α j−1 + is2λ

k∑j=1

α j−1

⎞⎠

238 J Econ Finan (2013) 37:216–239

× exp

⎛⎝−σ 2

v

2(s1

k−1∑j=2

αk− j + s2

k−1∑j=2

αk− j−1)2

⎞⎠

×L∑

l=1

pl exp

(is2μl − 1

2s2

2σ2l − 1

2s2

1σ2v − ρls1s2σlσv

)

×L∑

l=1

pl exp

(is1μl − 1

2s2

1σ2l

)

Lastly, based on the above results for , the asymptotic covariance matrixcan be calculated as:

1

n

[∫ ∫ (∂ Rec(r, θ)

∂θ

∂ Rec(r, θ)

∂θ ′ + ∂ Imc(r, θ)

∂θ

∂ Imc(r, θ)

∂θ ′

)w(r)dr1dr2

]−1

×

[∫ ∫ (∂ Rec(r, θ)

∂θ

∂ Rec(r, θ)

∂θ ′ + ∂ Imc(r, θ)

∂θ

∂ Imc(r, θ)

∂θ ′

)w(r)dr1dr2

]−1

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