Stochastic volatility models with leverage and heavy-tailed distributions: A Bayesian approach using scale mixtures

Computational Statistics and Data Analysis 55 (2011) 852–862

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Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda

Stochastic volatility models with leverage and heavy-tailed distributions: A Bayesian approach using scale mixtures Joanna J.J. Wang a,∗ , Jennifer S.K. Chan a , S.T. Boris Choy b a

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

b

Discipline of Operations Management and Econometrics, The University of Sydney, NSW 2006, Australia

article

abstract

info

Article history: Received 7 May 2009 Received in revised form 3 April 2010 Accepted 9 July 2010 Available online 18 July 2010

This paper studies a heavy-tailed stochastic volatility (SV) model with leverage effect, where a bivariate Student-t distribution is used to model the error innovations of the return and volatility equations. Choy et al. (2008) studied this model by expressing the bivariate Student-t distribution as a scale mixture of bivariate normal distributions. We propose an alternative formulation by first deriving a conditional Student-t distribution for the return and a marginal Student-t distribution for the log-volatility and then express these two Student-t distributions as a scale mixture of normal (SMN) distributions. Our approach separates the sources of outliers and allows for distinguishing between outliers generated by the return process or by the volatility process, and hence is an improvement over the approach of Choy et al. (2008). In addition, it allows an efficient model implementation using the WinBUGS software. A simulation study is conducted to assess the performance of the proposed approach and its comparison with the approach by Choy et al. (2008). In the empirical study, daily exchange rate returns of the Australian dollar to various currencies and daily stock market index returns of various international stock markets are analysed. Model comparison relies on the Deviance Information Criterion and convergence diagnostic is monitored by Geweke’s convergence test. © 2010 Elsevier B.V. All rights reserved.

Keywords: Stochastic volatility Scale mixture of normal Heavy tails Leverage Outlier diagnostics

1. Introduction The generalised autoregressive conditional heteroscedasticity (GARCH) models (Bollerslev, 1986) and the stochastic volatility (SV) models are commonly used in the literature to model time-dependent volatility. The former determines the conditional volatility by the past volatilities and past observations while the latter allows the conditional volatility to be modelled by an unobservable stochastic process. See Engle (1995) and Shephard (2005) for a comprehensive review of the GARCH and SV models, respectively. Comparatively speaking, the SV model offers a more flexible and realistic alternative to the GARCH type models, as it directly associates with the diffusion type process used in asset pricing (Melino and Turnbull, 1990). The basic SV model is given by 1/2

Return equation : yt |ht = Ht

t ,

t = 1 , . . . , n,

Volatility equation : ht +1 |ht , µ, φ, τ = µ + φ(ht − µ) + τ ηt , 2

(1) t = 1, . . . , n − 1,

(2)

where yt is the mean-corrected return of an asset at time t, ht and Ht = exp(ht ) are the log-volatility and volatility at time t, exp(µ/2) measures the model instantaneous volatility, |φ| < 1 is the persistence in the volatility and τ is the

∗

Corresponding author. Tel.: +61 (02) 9351 5805. E-mail address: [email protected] (J.J.J. Wang).

0167-9473/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2010.07.008

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853

scale parameter of the log-volatility. t and ηt are independent standard normal processes. The marginal distribution of log-volatility at t = 1 is given by h1 |µ, φ, τ 2 ∼ N



µ,

τ2 1 − φ2



.

For SV modelling, statistical inference using the classical maximum likelihood method is difficult due to the complicated likelihood function which involves high dimensional integration of the unobserved volatilities. Amongst other methods, Melino and Turnbull (1990) used the generalised method of moments, Harvey et al. (1994) proposed the quasi maximum likelihood method, Danielsson (1994) adopted a simulated maximum likelihood method and Asai (2008) used the Monte Carlo likelihood method. In the past fifteen years, Bayesian approach using the simulation-based Markov Chain Monte Carlo (MCMC) algorithms has become popular in SV modelling. For instance, Jacquier et al. (1994) adopted the Gibbs sampling algorithm and Shephard and Pitt (1997) used the Metropolis–Hastings algorithm. To facilitate a more efficient MCMC algorithm, Kim et al. (1998) transformed the return to the logarithmic of the squared return such that the new return followed a log-χ 2 distribution with one degree of freedom. This log-χ12 distribution is then approximated by a mixture of seven normal distributions. The approximation is further improved by Omori et al. (2007). To reduce the dependence of the simulated posterior samples, Kim et al. (1998) used Gaussian simulation smoother for simulating the unobserved log-volatility while Shephard and Pitt (1997), Omori and Watanabe (2008) and Abanto-Valle et al. (2010) proposed multiplemove samplers to simulate the model parameters in blocks. In this paper, we adopt the Bayesian approach with simulationbased MCMC algorithms and the implementation of the SV model relies on the user-friendly Bayesian software WinBUGS (Windows version of Bayesian Analysis Using Gibbs Sampler) (Spiegelhalter et al., 2004). It is a well-known phenomenon that financial returns typically exhibit heavy tails (Gallant et al., 1997). However, this leptokurtosis cannot be fully explained by the changing volatility in SV models. A common modification is to replace the normal error distribution of the return and/or the volatility by heavy-tailed distributions. Liesenfeld and Jung (2000), Chib et al. (2002) and Asai (2008) studied SV models with Student-t error distributions for the returns. Meyer and Yu (2000) and Abanto-Valle et al. (2010) studied the Student-t SV model by expressing the Student-t density function into a scale mixture of normal (SMN) representation while Choy et al. (2008) used a scale mixture of uniform (SMU) form for the Student-t density function and showed that most of the full conditional distributions required for the Gibbs sampling scheme are of standard forms. Choy and Chan (2000) studied the SV model with an exponential power error distribution for both the return and the volatility via a SMU form. The advantage of using the SMN and SMU forms for the error distributions is that the SV models can be straightforwardly implemented using MCMC algorithms without substantially increasing the computing time. Furthermore, the mixing parameters arose from the SMN and SMU representations facilitate the diagnostics of potential outliers in the observed return and the unobserved volatility series. In addition to adopting a more flexible error distribution, the basic SV model specification can also be extended to enhance the flexibility of the model to capture important empirical regularities observed in real financial time series data. For example, Chib et al. (2002) extended the basic SV model to include covariates, Harvey et al. (1994) introduced two autoregressive AR(1) processes to the return, Berg et al. (2004) studied the SV models with jumps and Chen et al. (2008) considered the threshold SV models. In the situation where the volatility responses asymmetrically towards the positive and negative returns, a SV model with leverage (SVL model) is used. See Black (1976) and Jacquier et al. (2004). The leverage effect is measured by the correlation coefficient of the two error terms, t and ηt . Meyer and Yu (2000) and Omori et al. (2007) studied the SVL model by modelling the pair (t , ηt ) with a bivariate normal distribution while Choy et al. (2008) assumed a bivariate Student-t distribution. In contrast, Asai (2008) modelled t and ηt using the Student-t and normal distributions, respectively. This paper considers the SVL model with a bivariate Student-t error distribution. Choy et al. (2008) assessed this model using the SMN form for the bivariate Student-t distribution. This approach, hereafter referred to as Choy’s approach, has only one mixing parameter arose from the SMN representation of the Student-t density for each pair of (yt , ht +1 ) for outlier diagnostics. However, when an outlier is identified, Choy’s approach is unable to identify whether it is outlying in the return, log-volatility or both. We propose a new formulation for the SVL model where an alternative conditional Student-t distribution for the return is derived. Therefore, we have two different Student-t distributions in the return and volatility equations. Expressing these Student-t distributions as SMN distributions, our approach, hereafter referred to as Wang’s approach, allows two different mixing parameters for accurately identifying outliers in the return and volatility series and hence provides an improvement over Choy’s approach. Despite the growing number of advanced sampling schemes developed with various degree of sophistication and complexity, for example, the integrated sampler and the mixture sampler, the idea to trade off the easy-to-use techniques with more efficient but complicated techniques may be unattractive to general practitioners. Therefore, we adopt the WinBUGS software to implement the SVL model although WinBUGS uses a single-move sampler. Compared with the multiple-move sampler, the single-move sampler produces higher correlated posterior samples. However, such dependency can be compensated by running a longer Markov chain. On the other hand, the gain in efficiency in using integrated/mixture sampler is ‘largely outweighed by the ease of implementation’ in WinBUGS (Meyer and Yu, 2000). The remainder of the paper is organised as follows. Section 2 presents two different formulations of the Student-t SVL model. Choy’s approach is outlined and Wang’s approach is proposed. In Section 3, we compare Choy’s and Wang’s approaches using a simulation study. In Section 4, we analyse real exchange rate data and stock market index data. Deviance

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Information Criterion (DIC) is used as a means for model comparison and Geweke’s convergence test is used for testing the convergence of the constructed Markov chains. Finally, a concluding remark is given in Section 5. 2. Bayesian SV model with leverage 2.1. Student-t distribution as SMN Andrews and Mallows (1974) presents the class of SMN distributions. Let X be a vector of continuous random variables with location vector µ and scale matrix 6. If the probability density function (pdf) of X can be expressed into the following mixture representation f (x|µ, 6) =

∞

Z

N (x|µ, κ(λ)6)π (λ)dλ, 0

where N (x|·, ·) is a multivariate normal pdf, κ(λ) is a positive function of λ and π (·) is a pdf defined on R+ , then we say that the pdf of X has a SMN representation. We refer λ to as the mixing parameter and π (·) as the mixing density of this SMN representation. For a multivariate Student-t distribution with location vector µ, scale matrix 6 and degrees of freedom ν , κ(λ) = λ and π(λ) is the pdf of the inverse gamma IG( ν2 , ν2 ) distribution where IG(a, b) has a pdf given by IG(λ|a, b) =

ba

Γ (a)

λ−(a+1) e−b/λ ,

λ, a, b > 0.

That is, the pdf of the Student-t distribution can be rewritten as t (x|µ, 6, ν) =

 ν ν

∞

Z 0

N (x|µ, λ6) IG λ| , dλ, 2 2

(3)

or we can express the Student-t distribution hierarchically as: X |µ, 6, ν, λ ∼ N (µ, λ6)

λ|ν ∼ IG

ν ν  , . 2 2

2.2. Bayesian SV model with leverage Black (1976) revealed that equity returns react asymmetrically to good and bad news. A drop in return due to bad news increases the future volatility but a rise in return of the same size due to good news will result in a smaller future volatility. This negative correlation between the current return and future volatility is known as the leverage effect. Harvey and Shephard (1996) proposed the univariate SVL model in discrete time. The model is given in (1) and (2) where

     t i.i.d 0 1 ∼N , ηt 0 ρ

ρ



1

,

(4)

ρ is the correlation coefficient between t and ηt and it is used to measure the leverage effect. To facilitate an efficient posterior inference using WinBUGS, Meyer and Yu (2000) showed that the SVL model in (1), (2) and (4) can be alternatively specified by

ρ

1/2

 (ht +1 − µ − φ(ht − µ)), Ht (1 − ρ 2 ) τ
ht +1 |ht , µ, φ, τ 2 ∼ N µ + φ(ht − µ), τ 2 . yt |ht , ht +1 , µ, φ, τ 2 , ρ ∼ N

Ht

To adequately account for the leptokurtic feature of the return and volatility, a bivariate Student-t distribution with ν degrees of freedom can replace the bivariate normal distribution and (4) becomes

     t i.i.d 0 1 ∼ tν , ηt 0 ρ

ρ



1

.

Choy et al. (2008) studied the robustness property of this Student-t SVL model by first expressing the bivariate Student-t error distribution into the following SMN representation

     t i.i.d 0 1 ∼N , λt ηt 0 ρ ν ν  i.i.d λt ∼ IG , 2 2

ρ 1



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and then adopted the formulation of Meyer and Yu (2000). Under Choy’s approach, the Student-t SVL model is reformulated as

ρ

1/2

 [ht +1 − µ − φ(ht − µ)], λt Ht (1 − ρ 2 ) τ
ht +1 |ht , µ, φ, τ 2 , λt ∼ N µ + φ(ht − µ), λt τ 2   λ0 τ 2 h1 |µ, φ, τ 2 , λ0 ∼ N µ, 1 − φ2 ν ν  λt |ν ∼ IG , . yt |ht , ht +1 , µ, φ, τ 2 , ρ, λt ∼ N

Ht

2 2

This approach not only simplifies the Gibbs sampling algorithm but also provides a means for outlier diagnostics. The SVL model can be implemented using the WinBUGS package in a similar way as in Meyer and Yu (2000). The mixing parameter λt serves as a proxy for identifying a potential outlier. An outlying pair (yt , ht +1 ) is associated with a λt value. However, once an outlier is identified, it is unclear whether it is an outlier because of the return, volatility or both. To determine which of these is the case, this paper proposes an alternative formulation for the SVL model where a different conditional Student-t distribution is derived for the return. The Student-t SVL is now given by yt |ht +1 , ht , µ, φ, τ 2 , ρ, ν ∼ tν+1

ρ τ

1/2

Ht

ht +1 − [µ + φ(ht − µ)] , M





ht +1 |ht , µ, φ, τ 2 , ν ∼ tν µ + φ(ht − µ), τ 2 h1 |µ, φ, τ 2 , ν ∼ tν

 µ,

τ2 1 − φ2








where

 M =

 h 2 i ν 1  (1 − ρ 2 )Ht 1 + 2 ht +1 − [µ + φ(ht − µ)] . ν+1 ντ

Now, Wang’s approach expresses the Student-t distributions as SMN distributions and the model becomes yt |ht +1 , ht , µ, φ, τ 2 , ρ, ν, λyt ∼ N

ρ τ

1/2

Ht

ht +1 − [µ + φ(ht − µ)] , λyt M



ht +1 |ht , µ, φ, τ 2 , ν, λht +1 ∼ N µ + φ(ht − µ), λht +1 τ 2








  λh1 τ 2 h1 |µ, φ, τ , ν, λh1 ∼ N µ, 1 − φ2   ν+1 ν+1 λyt ∼ IG , 2 2 ν ν  λht ∼ IG , 2

2 2

where λyt and λht , t = 1, . . . , n are the mixing parameters of the return and volatility equations respectively. Comparing with Choy’s approach, Wang’s approach has two mixing parameters, λyt and λht , to identify outliers in the return equation and volatility equation separately. To complete the Bayesian paradigm, we adopt the following prior distributions.

µ ∼ N (aµ , bµ ) τ 2 ∼ IG(aτ , bτ ) φ ∗ ∼ Be(aφ , bφ ) ρ ∼ U (−1, 1) ν ∼ Ga(aν , bν ) where φ ∗ =

φ+1 2

, Be(α, β) is the beta distribution with density

f (x|α, β) =

1 B(α, β)

xα−1 (1 − x)β−1

where B(·, ·) is the beta function, U (α, β) is the uniform distribution and Ga(α, β) is the gamma distribution with density f (x|α, β) =

β α α−1 −β x x e . Γ (α)

Now, the Student-t SVL models can be easily implemented using WinBUGS based on Wang’s approach.

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Table 1 Simulation results based on 100 replicates on n = 500 observations. Par.

Approach

ν

Wang’s Choy’s

True 5 5

µ

Wang’s Choy’s

φ

Estimate

Sd

95% CI

5.7937 5.7159

2.4708 3.2069

(3.7092, 13.1320) (3.3416, 15.2972)

−10 −10

−9.9758 −9.9999

0.1133 0.1600

Wang’s Choy’s

0.8 0.8

0.8107 0.8104

ρ

Wang’s Choy’s

0.8 0.8

τ

Wang’s Choy’s

0.2 0.2

PE (%)

MSE

PC (%)

15.87 14.32

2.3335 7.1341

98 92

(−10.1860, −9.7506) (−10.2836, −9.6827)

0.24 0.00

0.0111 0.0684

96 96

0.0747 0.0831

(0.6205, 0.9057) (0.6064, 0.9209)

1.35 1.30

0.0039 0.0035

99 98

0.7136 0.7146

0.1538 0.1810

(0.3421, 0.9044) (0.2977, 0.9279)

−10.80 −10.67

0.0169 0.0554

95 83

0.2085 0.1913

0.0735 0.0833

(0.1209, 0.3988) (0.0885, 0.4018)

4.24

−4.34

0.0045 0.0048

97 96

In the simulation and empirical studies presented in Section 3, a vague prior is assigned to µ, a uniform prior U (−1, 1) is assigned to ρ , a non-informative prior is assigned to τ 2 , a beta Be(20, 1.5) prior distribution is assigned to φ ∗ and a noninformative prior with restricted range (1, 40) is assigned to ν . In the Gibbs sampling scheme, a single Markov chain is run for 300,000 iterations. We discard the initial 50,000 iterations as the burn-in period to ensure convergence. Simulated values from the Gibbs sampler after the burn-in period are taken from every 20th iteration to avoid high autocorrelation and to mimic a random sample of size 12,500 from the joint posterior distribution for posterior inference. The convergence of the Markov chain is assessed using Geweke’s convergence test (Geweke, 2004) which can be performed using the BOA (Bayesian Output Analysis) package in the R software. See Smith (2005) for details. 3. A simulation study In this simulation study, we simulate 100 independent data sets from the Student-t SVL model with true parameter values

µ = −10, τ = 0.2, φ = 0.8, ρ = 0.8 and ν = 5. The sample size for getting reasonable estimation results is investigated and a sample size of 500 observations is adopted. Choy’s and Wang’s approaches are used to provide posterior inference. To compare the performance of the two approaches, the following goodness-of-fit measures are used: the percentage error (PE), mean square error (MSE) and posterior coverage (PC), defined by

b θ −θ × 100% |θ | N 1 X (b θj − θ )2 MSE = PE =

N j=1

PC =

N 1 X

N j =1

I [θ ∈ (b θj,0.025 , b θj,0.975 )]

where b θ is the average of the posterior means b θj , j = 1, . . . , 100 for each parameter θ , (b θj,0.025 , b θj,0.975 ) is the 95% credible interval of θ in data set j and I (·) is the indicator function which is one if the condition is satisfied and is zero otherwise. Table 1 presents the posterior summaries, PE, MSE and PC of the five parameters under Choy’s and Wang’s approaches. The point estimates under the two approaches are very close to each other and are also close to their true values. Generally speaking, points estimates from Choy’s approach are marginally closer to the true values than those from Wang’s approach but they have larger posterior standard errors and hence wider credible intervals. Wang’s approach gives a smaller MSE and a better posterior coverage while Choy’s approach gives a smaller PE. 4. Empirical studies In this section, we analyse two sets of financial times series. The data are the mean-corrected returns of the daily exchange rate and the daily stock market closing index. We fit normal SVL model and Student-t SVL model using both Choy’s and Wang’s approaches. Comparison between the SVL models with different error distributions is based on the Deviance Information Criterion (DIC) defined as DIC = D(θ ) + pD where D(θ ) is the posterior mean deviance and pD is the effective number of parameters in the model. See Spiegehalter et al. (2002) for the development of the DIC in Bayesian model comparison using MCMC algorithms. To choose between different models, the model with the smallest DIC value is preferred. See Berg et al. (2004) for the use of DIC in SV modelling. Since the DIC value of a model can be automatically computed using WinBUGS, it is adopted as a user-friendly model

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Table 2 Summary statistics for mean-corrected exchange rate return data and stock market index return data.

Currency USD EUR GBP JPY HKD Market index AORD CAC DAX FTSE HSI N225 S&P 500

S.D.

Max

Min

Skewness

Kurtosis

0.00703 0.00578 0.00576 0.00951 0.00702

0.0258 0.0185 0.0159 0.0381 0.0254

−0.0400 −0.0398 −0.0273 −0.0643 −0.0386

−0.92 −1.12 −0.63 −1.69 −0.91

7.00 7.61 5.40 13.08 6.89

0.01267 0.01499 0.01428 0.01392 0.01908 0.01752 0.01462

0.0537 0.1061 0.1079 0.0939 0.1341 0.1326 0.1099

−0.0854 −0.0946 −0.0745 −0.0926 −0.1358 −0.1208 −0.0944

−0.65

9.24 14.30 15.68 13.72 13.44 13.74 16.65

0.10 0.27 −0.11 0.06 −0.54 −0.33

Table 3 p-values for Geweke’s convergence test for selected model parameters: AUS/JPY data S&P 500 data. Data set

µ

φ

ρ

τ

ν

h100

λy100

λh100

AUD/JPY S&P 500

0.531 0.476

0.840 0.887

0.549 0.072

0.478 0.360

0.181 0.867

0.524 0.375

0.240 0.410

0.425 0.568

comparison criterion in this paper. Other model selection criteria can also be used, for example, Kim et al. (1998) calculated Bayes factors using the approach of Chib (1995). However, the calculation requires the marginal likelihood and it becomes computationally demanding when the number of unknown parameters, i.e. the latent volatilities in SV and SVL models, is large. See Han and Carlin (2001) for computing Bayes factors using MCMC algorithms. 4.1. Exchange rate data The data are the mean-corrected daily exchange rate return of the Australian dollar (AUD) against five other currencies. These currencies are the American Dollar (USD), Euro (EUR), Great British Pound (GBP), Japanese Yen (JPY) and Hong Kong Dollar (HKD). The exchange rate data cover the period from January 2005 to December 2007 and they were analysed using Choy’s approach in Choy et al. (2008). Table 2 presents the summary statistics of the data. It shows that the daily return of the AUD to the five currencies have excess kurtosis. We fit a SVL model with Student-t distribution to capture the high kurtosis of the data. The convergence of the Markov chain in the MCMC algorithm is assessed using Geweke’s convergence test. The p-values of Geweke’s convergence test for all parameters are all greater than 5%, indicating the convergence of all Markov chains in the MCMC algorithm. For the AUD to JPY exchange rate data, the p-values for different parameters are reported in Table 3. Table 4 exhibits the posterior mean and posterior standard error of the model parameters in the Student-t SVL model using Wang’s approach. These estimates are close to the values obtained by Choy’s approach (Choy et al., 2008) although the latter uses a slightly different prior distribution for the degrees of freedom and a smaller number of posterior sample. The persistence parameter φ of the log-volatility is quite high for all currencies. The correlation coefficient ρ ranges from −0.24 for AUD/GBP to 0.16 for AUD/USD and is insignificant at the 5% level. This result agrees with the findings in literature that the exchange rate data do not exhibit a leverage effect. The posterior mean of the degrees of freedom ν ranges from 6.5 for AUD/JPY to 18.0 for AUD/GBP, indicating that the bivariate Student-t distribution is more appropriate than the bivariate normal distribution in modelling the random errors (ut , vt ). The DIC values in Table 4 confirm that the Student-t SVL model is superior to the normal SVL model for all five currencies. In fact, the Student-t SVL model gives smaller point estimate and standard error for the standard deviation, τ , of the log-volatility than the normal SVL model. For the AUD/JPY data, the estimates of the log-volatility under the normal and Student-t SVL models are displayed in Fig. 1. Generally speaking, the volatility is higher under the normal SVL model. It is known that the Student-t distribution provides a robust inference by downweighing the distorting effects of outliers. By expressing the Student-t distribution as a SMN distribution, Choy and Smith (1997) first proposed to perform outlier diagnostics using the mixing parameter of the SMN representation. An outlier is associated with a large value of the mixing parameter which inflates the variance of the corresponding normal distribution to accommodate the outlier. Therefore, the extremeness of observations are closely associated with the magnitude of the mixing parameters. In practice, we identify outliers by comparing the posterior means of the mixing parameters of all observations. For the AUD/JPY data, Choy et al. (2008) found six pairs of possible outliers which are (yt , ht +1 ) for Day t, t = 173, 467, 104, 254, 294 and 394. Since Choy’s approach only involves a single mixing parameter, it is unclear whether these pairs are outlying in the return, volatility or both. Alternatively, Wang’s approach uses two mixing parameters, λyt and λht , to identify outlying returns and unobserved outlying volatilities separately. From Table 5, Wang’s approach identifies that

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Log-Volatilities for AUD/JPY daily exchange rate bivariate t

theta(t)

-12

-11

-10

-9

-8

bivariate normal

0

100

200

300 time t

400

500

Fig. 1. Posterior means of unobserved log-volatility under the normal and Student-t SVL models for the AUD/JPY exchange rate return data. Table 4 Parameter estimates (standard errors in parentheses) of the normal and Student-t SVL models for the exchange rate return data. Currency

USD

EUR

GBP

JPY

HKD

Model

DIC

µ

φ

Bivariate normal

−3719.4

−10.24

Bivariate t

−3727.1

(0.59) −10.34 (0.56)

0.9815 (0.0118) 0.9827 (0.0102)

Bivariate normal

−3919.9

−10.60

Bivariate t

−4088.1

(0.48) −10.77 (0.41)

Bivariate normal

−3905.2

−10.58

Bivariate t

−3957.2

(0.46) −10.71 (0.54)

Bivariate normal

−3582.6

−9.95

Bivariate t

−3643.3

(0.62) −10.27 (0.62)

Bivariate normal

−3719.8

−10.24

−3751.2

(0.52) −10.36 (0.61)

Bivariate t

ρ 0.1555 (0.1786) 0.0996 (0.1947)

0.9746 (0.0134) 0.9766 (0.0127)

−0.2189

0.9797 (0.0116) 0.9836 (0.0101)

−0.2413

0.9762 (0.0124) 0.9823 (0.0095)

−0.0961

0.9798 (0.0114) 0.9831 (0.0108)

(0.1720) −0.3657 (0.1943) (0.1870) −0.2906 (0.2195)

τ

ν

0.1489 (0.0349) 0.1268 (0.0270)

–

0.1654 (0.0348) 0.1359 (0.0314) 0.1450 (0.0305) 0.1211 (0.0280)

(0.1503) −0.2561 (0.1873)

0.2316 (0.0460) 0.1459 (0.0340)

0.1124 (0.1685) 0.1061 (0.1968)

0.1592 (0.0290) 0.1272 (0.0298)

16.66 (5.67) – 13.24 (5.73) – 17.98 (5.59) – 6.47 (2.50) – 16.50 (6.00)

y173 , y476 , y104 , y254 , y294 and h395 are outlying, indicating that, the first five outliers found by Choy’s approach are outlying in return only and the sixth outlier is outlying in volatility only. Fig. 2 displays the posterior means of λyt and λht . This graph shows that the first five outliers have the most outlying returns while the sixth outlier has the largest log-volatility. 4.2. Stock market index data In this empirical study, we analyse the mean-corrected daily return of seven international stock market indices, namely the Standard & Poors 500 (S&P 500), Financial Times Stock Exchange 100 (FTSE), Deutscher Aktien (DAX), Cotation Assiste en Continu 40 (CAC), All Ordinaries (AORD), Nikkei 225 (N225) and Hang Seng Index (HSI), covering the period from January 2005 to December 2008. Summary statistics in Table 2 show that the data have excess kurtosis. For each stock market index, the data are fitted by a Student-t SVL model using both Choy’s and Wang’s approaches. Table 6 reports the DIC of the model and the posterior means and standard errors of all model parameters for each index. First of all, the DIC values evaluated from the two approaches are quite close, in most cases within a percentage difference of 2.5% except for the S&P 500 index which is 3.1% and the Nikkei 225 index which is 5.8%. The percentage difference is less than 5% for µ and τ and is less than 1% for φ . For ρ , the percentage difference is less than 4% except for the AORD and

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(a) Mixing parameter for return

859

1.4 1.3

lambda_h(t)

1.2 1.0

1.1

1.5 1.0

lambda_y(t)

2.0

1.5

1.6

(b) Mixing parameter for log-volatility

0

100

200

300 time t

400

500

0

100

200

300 time t

400

500

Fig. 2. Outlier diagnostics using (a) mixing parameter, λyt , for return and (b) λyt and (c) λht in Wang’s approach for the AUD/JPY exchange rate return data. Large values are associated with possible outliers. Table 5 Bayes estimates of the mixing parameters for possible outliers: AUD/JPY data and S&P 500 data. Day t

Wang’s

Choy’s

λyt

λht +1

λt

AUD/JPY Day 173 Day 467 Day 104 Day 254 Day 294 Day 394

2.387 2.213 2.151 2.098 1.940 1.371

1.189 1.353 1.178 1.239 1.200 1.583

4.055 4.574 3.693 4.181 3.397 2.105

S&P 500 Day 542 Day 943

1.299 1.048

2.610 1.751

3.057 2.263

HSI indices which are 6.7% and 13.3%, respectively. For the degrees of freedom, ν , Wang’s approach gives a higher posterior mean with a smaller standard error than Choy’s approach but the difference is less than 3.5 degrees of freedom. Asai (2008) also analysed the daily S&P 500 index return using a SVL model with a Student-t innovation for the return and a normal innovation for the volatility. He found a smaller degrees of freedom for the return compared with the degrees of freedom of the bivariate Student-t distribution in the SVL model. However, no direct comparison can be made because Asai (2008) analysed a longer time series which had no overlap with the time series used in this paper. However, the two studies give a quite consistent estimate of the persistence in volatility, φ . Since the ρ values of the seven indices are significantly negative, this finding is in agreement with the fact that the leverage effect is more profound in stock market index return data (Harvey and Shephard, 1996). For the S&P 500 index, Table 3 presents the result of Geweke’s convergence test. Since all tests are insignificant, the constructed Markov chain converges. The estimates of the unobserved log-volatilities, ht , under Choy’s and Wang’s approaches are given in Fig. 3. One can see that the estimates from the two approaches are comparable and the market was very volatile in September and October 2008 because of the US financial tsunami. On February 27, 2007 (Day 542), the S&P 500 index tumbled 3.5%, its worst one-day percentage loss since March 2003, due to the worries of the global economic growth. On September 29, 2008 (Day 943), the US Congress unexpectedly rejected the President’s financial rescue plan and the S&P 500 index fell 107 points or 8.8%. Fig. 4 plots the posterior means of the mixing parameters λt of Choy’s approach and λyt and λht of Wang’s approach. These two observations are successfully identified to be the two most significant outliers by Choy’s approach. However, Wang’s approach reveals that the data on February 29, 2007 is outlying in both return and volatility while the data on September 29, 2008 is outlying in volatility only (see Table 5). 5. Conclusion This paper proposes an alternative formulation for the Student-t SVL model of Choy et al. (2008). Both Choy’s and Wang’s approaches express the Student-t distribution as a SMN distribution at different stages and the mixing parameters arose from the SMN representation play the role of identifying possible outliers. Both Choy’s and Wang’s approaches, being the extension of the formulation of the normal SVL model proposed by Meyer and Yu (2000), allow Bayesian analysis of the Student-t SVL models to be easily implemented using the WinBUGS software. A simulation study confirms that both Choy’s and Wang’s approaches give very close and sensible results but Wang’s approach achieves a smaller standard error

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Table 6 Parameter estimates (standard errors in parentheses) of the Student-t SVL models for the stock market index data under Wang’s and Choy’s approaches. Market index

AORD

Approach

DIC

µ

φ

ρ

τ

ν

Wang’s

−7169.5

−9.253

−0.8315*

Choy’s

−7097.0

−9.421

0.9663 (0.0076) 0.9627 (0.0085)

0.2018 (0.0298) 0.2119 (0.0299)

21.94 (4.70) 18.38 (6.06)

Wang’s

−7421.3

−9.090

−0.9042*

Choy’s

−7607.9

(0.127) −9.211 (0.123)

0.9689 (0.0068) 0.9675 (0.0062)

0.2123 (0.0284) 0.2145 (0.0243)

19.37 (5.33) 16.85 (5.62)

Wang’s

−7038.6

−9.193

Choy’s

−6942.5

0.9665 (0.0087) 0.9635 (0.0076)

−0.7988*

(0.151) −9.293 (0.146)

0.1903 (0.0310) 0.1975 (0.0240)

15.06 (5.45) 13.91 (5.68)

Wang’s

−7242.6

−9.309

Choy’s

−7220.6

0.9808 (0.0046) 0.9802 (0.0046)

−0.8215*

(0.215) −9.517 (0.190)

0.1833 (0.0212) 0.1848 (0.0208)

20.24 (5.20) 17.02 (5.75)

Wang’s

−5935.5

−8.648

Choy’s

−5893.6

0.9874 (0.0056) 0.9879 (0.0058)

−0.4531*

(0.578) −9.078 (0.379)

0.1549 (0.0272) 0.1486 (0.0271)

16.63 (5.75) 15.09 (5.67)

Wang’s

−6394.4

−8.770

Choy’s

−6044.9

0.9744 (0.0077) 0.9721 (0.0083)

−0.6701*

(0.226) −8.947 (0.206)

0.1948 (0.0291) 0.1980 (0.0292)

19.80 (5.41) 18.77 (5.58)

Wang’s

−7331.3

−9.316

−7561.3

0.9822 (0.0048) 0.9819 (0.0049)

−0.8467*

(0.231) −9.527 (0.191)

0.1682 (0.0223) 0.1637 (0.0234)

11.84 (4.08) 10.16 (3.92)

(0.130) (0.149)

CAC

DAX

FTSE

HSI

N225

S&P 500

*

Choy’s

(0.0647)

−0.7795* (0.0613) (0.0448) −0.8905* (0.0454) (0.0752) −0.7941* (0.0661) (0.0623) −0.8210* (0.0748) (0.1087) −0.3998* (0.1179) (0.0666) −0.6502* (0.0759) (0.0654) −0.8751* (0.0572)

Indicates that ρ is significant at the 5% level.

-6

Log-Volatilities for S&P 500

-9 -11

-10

theta(t)

-8

-7

Wang’s approach Choy’s approach

0

200

400

600

800

1000

time t Fig. 3. Comparison of Choy’s and Wang’s approaches in the estimation of the unobserved log-volatilities under a Student-t SVL model for the S&P 500 Index Return data.

in parameter estimation, a smaller MSE and a higher posterior coverage than Choy’s approach. In the empirical study of exchange rate return data, we confirm that the exchange rate returns of AUD to five other currencies do not have leverage effects and a Student-t distribution is more appropriate than a normal distribution in modelling the random error terms in the return and volatility equations. In the empirical study of stock market return data, all seven stock market indices show a very high degree of persistence in volatility and a significant leverage effect. In identifying possible outliers in the return and volatility equations, Choy’s approach can only pick up outlying pairs of return and log-volatility but Wang’s approach

J.J.J. Wang et al. / Computational Statistics and Data Analysis 55 (2011) 852–862

(a) Mixing parameters for S&P 500

lambda_y(t) 200

400

600 time

800

1000

0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30

2.0 1.5

lambda(t)

2.5

3.0

(b) Mixing parameter for return

1.0 0

861

0

200

400

600 time t

800

1000

2.0 1.0

1.5

lambda_h(t)

2.5

(c) Mixing parameter for log-volatility

0

200

400

600 time t

800

1000

Fig. 4. Outlier diagnostics using (a) λt in Choy’s approach and (b) λyt and (c) λht in Wang’s approach for the S&P 500 index return data. Large values are associated with possible outliers.

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