Energy Economics 33 (2011) 956–965

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Energy Economics

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Oil and stock market volatility: A multivariate stochastic volatility perspective

Minh Vo ⁎Metropolitan State University, Minnesota, USA

⁎ Tel.: +1 612 659 7305; fax: +1 612 659 7268.E-mail address: minh.vo@metrostate.edu.

0140-9883/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.eneco.2011.03.005

a b s t r a c t

a r t i c l e i n f o

Article history:Received 18 February 2010Received in revised form 7 February 2011Accepted 9 March 2011Available online 21 March 2011

JEL classification:C11C13C22G12G17

Keywords:Multivariate stochastic volatilityOil pricesStock marketMCMC

This paper models the volatility of stock and oil futures markets using the multivariate stochastic volatilitystructure in an attempt to extract information intertwined in both markets for risk prediction. It offers fourmajor findings. First, the stock and oil futures prices are inter-related. Their correlation follows a time-varyingdynamic process and tends to increase when the markets are more volatile. Second, conditioned on the pastinformation, the volatility in each market is very persistent, i.e., it varies in a predictable manner. Third, thereis inter-market dependence in volatility. Innovations that hit either market can affect the volatility in theother market. In other words, conditioned on the persistence and the past volatility in their respectivemarkets, the past volatility of the stock (oil futures) market also has predictive power over the future volatilityof the oil futures (stock) market. Finally, the model produces more accurate Value-at-Risk estimates thanother benchmarks commonly used in the financial industry.

ll rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Oil is one of the most important production factors in an economy.Not surprisingly, a growing theoretical and empirical literature has beendevoted to the study of oil and its impact on the economy. Rising oilprices lead to higher production costs which affect inflation, consumerconfidence and therefore economic growth. Several studies report aclear negative correlation between energy prices and aggregate outputor employment. For instance, Hamilton (1983) and Gisser and Goodwin(1986) demonstrate that rising oil prices are responsible for recessions.Rotemberg and Woodford (1996) estimate that a 10% increase in oilprices leads to an average GDP decline of 2.5% five or six quarters later.Jones et al. (2004) estimate that the oil price–GDP elasticity (the ratio ofpercentage change in GDP to percentage change in oil price) is around−0.06. However, Lee et al. (1996), Hamilton (1996), Huntington(1998), among others, report an asymmetric relationship between oilprices and the macroeconomy. Rising oil prices seem to decrease theaggregate economic activities more than falling oil prices stimulatethem. Furthermore, Bernanke (1983) and Pindyck (1991) show thatlarge oil price movements increase uncertainty about future prices andthus cause delays in business investments. Nevertheless, Hooker (1996)

indicates that the correlation between oil prices and economic activityappears to be much weaker in data since 1985, so the suggestion thatoil shocks contribute directly to the economic downturn remainscontroversial.

The connection between oil and stock prices appears to be quitenatural. Theoretically, the value of a firm is the present value ofexpected future cash flows. Rising oil prices affect the future cashflows of a firm, either negatively or positively depending on whetherthe firm is producing or consuming oil. In addition, oil prices alsoaffect interest rates in the economy via inflation and monetary policyof the central bank. Rising oil prices lead to high inflation whichincreases interest rates. Furthermore, the central bank often usescontractionary monetary policy to fight inflation. This further in-creases interest rates. As a result, the discount rate of the firm alsoincreases. Increasing discount rate leads to lower stock price, otherthings equal. Empirical findings, however, are mixed. Jones and Kaul(1996) find that there is an adverse impact of oil prices on stock pricesin the U.S., Canada, the U.K., and Japan in the period 1947–1991.Papapetrou (2001) shows that rising oil prices tend to reduce the realstock returns in Greece. At the industry level, Faff and Brailsford(1999), Sadorsky (2001) find that there is positive impact of oil priceson stock returns of oil and gas companies. However, El-Sharif et al.(2005) show that the impact of oil prices on non-oil and gas sectors isvery weak. Huang et al. (1996) find that oil shocks do not have anyimpact on the aggregate stock market.

Table 1Descriptive statistics for return series.

Statistics Oil futures S&P 500

Mean (%) .07 −.01Std. dev. (%) 2.62 1.40Skewness −.22 −.10Excess kurtosis 3.80 7.37Kolmogorov–Smirnov .047⁎ .079⁎

Jarque–Bera 1616.96⁎ 5988.52⁎

ARCH(1) 36.74⁎ 4.39⁎

ARCH(6) 53.95⁎ 64.21⁎

ARCH(12) 68.55⁎ 82.74⁎

Ljung–Box (6) 25.61⁎ 43.33⁎

Ljung–Box (12) 31.73⁎ 64.78⁎

Ljung–Box (6) on squared returns 543.16⁎ 1327.48⁎

Ljung–Box (12) on squared returns 889.38⁎ 2758.74⁎

⁎ Significant at the 5% level.

957M. Vo / Energy Economics 33 (2011) 956–965

The link between the oil and stock markets appears not only inreturn but also in volatility. Clark (1973), Tauchen and Pitts (1983),and Ross (1989) note that it is the volatility of an asset, not its return,that is related to the rate of information flow to a market. Thus,volatility is a good measure of information flow among markets.Exploring information flow may generate new insights. For instance,this study discovers a bidirectional dependence in volatility betweenstock and oil markets. That is, shocks to either market help predict notonly volatility in their own market but also that in the other market.

This paper attempts to extract information intertwined in stockand oil prices. It will not focus on the relationship between stock andoil returns, which many papers cited above have studied, but aimsmore at volatility models that can extract useful information andinformation flow with good forecasting power. Modeling andforecasting volatility are very important for at least two reasons.First, volatility is an important variable for pricing derivatives, whosetrading volume has quadrupled in recent years. Furthermore,volatility is an important input in risk management. For instance, itis used to construct optimal hedge ratios to hedge against risk and toestimate the value at risk, to name only two. Second, in makingefficient econometric inference about the mean of a variable, we needa correct specification of its volatility.

There is growing literature investigating the stochastic volatility ofoil prices. Agnolucci (2009) shows that GARCH-type models forecastoil volatility better than the implied volatility obtained from invertingthe Black–Scholes equation. Wei et al. (2010) find that the non-linearGARCH-typemodels which account for long-memory and asymmetricvolatility perform better than the linear counterparts in forecastingover long horizon. Sadorsky (2006) reports that the GARCH(1,1)model outperforms more complex models, such as state space, vectorautoregression, bivariate GARCH(1,1). Fong and See (2002) and Vo(2009) show that the forecasting power of GARCH and stochasticvolatility models will improve if the structural breaks of the timeseries are taken into account. This paper differs from those studies inthat it attempts to model the volatility interaction between the stockand oil markets to extract the information in one market that has notyet been incorporated into the other market in order to improve theforecasting power of volatility models. I find that doing so willimprove the forecasting power of the model.

Based on a preliminary analysis of the data, I posit a bivariatemodel of vector of autoregression VAR(1) with stochastic volatility(SV) for the joint processes governing the returns of the Standard andPoor's 500 (S&P 500) stock index and the oil futures. I consider twovariants of the model: constant correlation and time-varyingcorrelation. For a comprehensive discussion of multivariate stochasticvolatility model, the reader is referred to Asai et al. (2006). EstimatingSV-type models is a challenge. Besides the inherent issues ofmultivariate models, such as high dimensions of parameter spaceand the requirement for positive semi-definiteness of the covariancematrix, these models do not have closed-form likelihood function dueto their latent structure of the variance. Thus, themaximum likelihoodmethod cannot be used directly. Several estimation methods havebeen proposed in the literature, including GMM by Melino andTurnbull (1990), and Sorensen (2000), the quasi-maximum likelihoodof Harvey et al. (1994), the indirect inference of Gourieroux et al.(1993), the efficient method of moments by Gallant et al. (1997), thesimulated maximum likelihood by Danielsson (1994), Durbin andKoopman (1997), and Sandmann and Koopman (1998), and theBayesian Markov Chain Monte Carlo (MCMC) method by Jacquier etal. (1994), and Kim et al. (1998). However, Andersen et al. (1999)show that MCMC is one of the most efficient methods. Therefore, thispaper uses the Bayesian MCMC method to estimate the models. Tocompare models in terms of goodness-of-fit, I use the devianceinformation criterion (DIC) of Spiegelhalter et al. (2002) which is ageneralization of the popular Akaike information criterion (AIC) forcomplex hierarchical models.

The empirical results of the paper offer four major findings. First,there is evidence that the correlation between the stock and oilmarkets is not constant but time-varying. It tends to increase with thevolatility in the market. Second, the daily volatility in each market isvery persistent. That is, it varies over time in a predictable manner,conditioned on the past volatilities. Third, there is a bidirectionaldependence in volatility between the two markets. Innovations thathit either market can affect the volatility in the other market. In otherwords, conditioned on the persistence and the past volatility in theirown markets, the past volatility of the stock (oil) market also haspredictive power over the future volatility of the oil (stock) market.Finally, in validating the informativeness of themodel by conducting aValue-at-Risk (VaR) analysis on the stockmarket, I find that themodelperforms much better than benchmark methods commonly used inthe financial industry in producing the value at risk estimates.

The remainder of the paper is organized as follows. Section 2describes the data set and the preliminary analysis. Section 3discusses the models used in the paper. Section 4 discusses theestimation methodology and presents the empirical results. The VaRanalysis is conducted in Section 5. Section 6 concludes the paper.

2. Data and preliminary analysis

The data set used in this study consists of a daily oil price time seriesand a daily broad market index, S&P 500, time series. Both series spanfrom January 06, 1999 to July 26, 2009. The dataset is divided into twoparts. The first one, from January 06, 1999 to December 31, 2008, is formodel estimation. The remainder is for forecasting assessment. The oiltime series is a daily oil futures price series of the West TexasIntermediate (WTI) crude oil futures contract traded on the New YorkMercantile Exchange. The contract is denominated in 1000 U.S. barrels(42,000 gal) of light sweet oil. It is obtained from the historicaldatabase of the U.S. Department of Energy. The S&P 500 stock indexseries is obtained from the Yahoo Finance database. For both series,daily returns, in percentage, are defined as rt=100[ln(pt)− ln(pt−1)].

Table 1 presents some descriptive statistics for both returns timeseries. Both return series have small means. For each one, the standarddeviation is much greater than the mean in absolute value, indicatingthat the mean is not significantly different from zero. Both series areslightly skewed to the left. Their excess kurtosis is significantlypositive, indicating that they have heavy tails relative to the normaldistribution. Both Kolmogorov–Smirnov and Jarque–Bera tests rejectthe null hypothesis that the return distributions are normal. Ljung–Box portmanteau tests on return and squared return series at 6 and 12lags indicate a high serial correlation in the first and secondmoments.Furthermore, the ARCH tests at 1, 6 and 12 lags reject the nullhypothesis of homoscedasticity in the data.

Fig. 1 graphs the S&P 500 stock index, its return and the returndistribution. We observe that the return displays volatility clustering.

0 500 1000 1500 2000 2500

800

1000

1400

1200

1600

0 500 1000 1500 2000 2500

-10

-50

510

-3 -2 -1 0 1 2 3

01-5-

05

01

Theoretical Quantiles

-10 -5 0 5 10

0.4

0.3

0.2

0.0

0.1

S&

P50

0 In

dex

S&

P50

0 R

etur

n

Den

sity

Sam

ple

Qua

ntile

s

Fig. 1. S&P 500 index, its return and the return distribution.

958 M. Vo / Energy Economics 33 (2011) 956–965

That is to say, large changes tend to follow large changes, and smallchanges tend to follow small changes. However, the sign of the changefrom one period to the next is unpredictable. The QQ-plot against thenormal distribution shows that the distribution has heavier tail thannormal. The same can be seen in Fig. 2 for oil futures.

In short, the preliminary analysis shows that the data exhibitheavy tails, autocorrelation and heteroscedasticity. This suggests theimportance of using the stochastic volatility framework to modelconditional volatility of return in all series.

3. The model

Based on the preliminary analysis in the previous section, I positthe bivariate VAR(1)−SV model for the joint processes governing the

0 500 1000 1500 2000 2500

140

120

100

8060

4020

0 500 1000 1500 2000 2500

-15

-10

-50

510

15

Oil

Pric

eO

il R

etur

n

Fig. 2. Oil futures prices, their retur

S&P 500 index and the oil futures returns. Two variants of the model,the constant correlation and the dynamic correlation, are used to fitthe data. This section discusses the model and the estimationmethodology.

Let rt = rst r otð Þ′ be the vector of returns of the S&P 500 index andoil futures at time t (rate of change in log-prices between t-1 and t). rt ismodeled as a vector process, consisting of a deterministic partmt and astochastic part et. Specifically, rt follows

rt = mt + et : ð1Þ

The deterministic part mt is specified as a vector autoregressiveprocess of order one VAR(1):

mt = C + Brt−1;

-3 -2 -1 0 1 2 3

-15 -10 -5 0 5 10 15

-15

-10

-50

510

150.

150.

100.

050.

00

Theoretical Quantiles

Den

sity

Sam

ple

Qua

ntile

s

ns and the return distribution.

959M. Vo / Energy Economics 33 (2011) 956–965

where C = cs co½ �′ and B is a (2×2) matrix with Bf gij = bij for i,j=s,o. The subscripts s and o stand for stock and oil, respectively.

The stochastic part et = est eot½ �′ involves the multivariatestochastic volatility (MSV) element of the model and can be specifiedas follows:

et = ∑t ·�t ; ð2Þ

where �t is bivariate and normally distributed withmean E ztð Þ = 02×1

and variance

var �tð Þ = Zt =1 ρeρe 1

� �:

The correlation coefficient between ets and et

o, ρe, can be eitherconstant or time-varying, depending on the model. ∑t is a (2×2)diagonal matrix of standard deviation with the following form:

∑t =exp

hst2

� �0

0 exphot2

� �26664

37775;

where hts, ht

o are conditional log-variances at time t of rts, rt

o,respectively.

In the following subsections, I will discuss two MSV models, theconstant correlation (CC-MSV) and the dynamic correlation (DC-MSV) models, to model hts and ht

o.

3.1. The constant correlation MSV model

The VAR(1) model with constant correlation stochastic volatilitycan be specified as follows:

rt = C + Brt−1 + ∑t ·�t ;ht + 1 = μ + Φ ht−μð Þ + ηt ;

ð3Þ

where ht = hst hot� �

′ is the vector of log-variances of rts and rt

o,μ = μs μo½ �′ is the unconditional mean vector of ht , Φ={ϕij}, for i,j=s,o, represents the persistence and interaction between markets.The error term ηt = ηs

t ηot½ �′ in the log variance equation is a

bivariate normal random variable, ηt∼N(02×1,diag(σs2,σo

2)).According to Eq. (3), conditioned on the conditional mean,

C + Brt−1, the return series of S&P 500 and oil futures are jointlynormally distributed with variance–covariance matrix ∑tZt∑′

t . Thecorrelation ρe between the two series is assumed to be constant. Thesecond equation in (3) models the time-varying cross-dependentvolatilities of the two series. In this equation, the diagonal parametersof the matrix Φ (ϕss and ϕoo) capture the persistence of the volatilityin each market while the off-diagonal parameters, ϕso and ϕos,measure the dependence of the conditional volatility of the stockmarket on the oil market and that of the oil market on the stockmarket, respectively. If ϕso (ϕos) is different from zero, oil (stock)volatility may Granger-cause stock (oil) volatility.

3.2. The dynamic correlation MSV model

When the correlation between the two returns is allowed tovary over time, we get the dynamic correlation MSV model (DC-

MSV). The DC-MSV model with VAR(1) in the mean has the followingform:

rt = C + Brt−1 + ∑t�t ;

ht + 1 = μ + Φ ht−μð Þ + ηt ;

�t j∑t∼N 02×1;1 ρtρt 1

" # !

ρt =exp qtð Þ−1exp qtð Þ + 1

qt = ω + β qt−1−ωð Þ + σqzt ; zt ∼iid N 0;1ð Þ:

ð4Þ

This model captures the dynamic correlation between the twomarkets. To constrain ρt to the interval [−1,1], I use the Fishertransformation as suggested by Christodoulakis and Satchell (2002),in which ρt is a function of qt which follows an AR(1) stochasticprocess.

4. Estimation methodology

4.1. The algorithm

Estimating SV-type models is a challenge since they do not haveclosed-form likelihood functions due to the latent structure of thevariance. Several estimation methods have been proposed inliterature, such as the quasi maximum likelihood (QML) of Harveyet al. (1994), the generalized method of moments (GMM) byAndersen and Sorensen (1996), Melino and Turnbull (1990), andSorensen (2000), the efficient method of moments by Gallant et al.(1997), the simulated maximum likelihood by Danielsson (1994),Durbin and Koopman (1997), and Sandmann and Koopman (1998),and the Bayesian MCMCmethods by Jacquier et al. (1994), and Kim etal. (1998). However, Jacquier et al. (1994) find that theMCMCmethodis superior to both the QML and the GMM. Furthermore, while mostclassical methods rely on asymptotic arguments to make inference,MCMC uses the exact posterior distributions of parameters. This willyield more accurate results. In this paper, I use the BayesianMCMC formodel estimation. For a comprehensive discussion of Bayesianinference and the MCMC method, the reader is referred to Geweke(2005), Gammerman (1997), and Johannes and Polson (2006).

The basic idea of Bayesian estimation lies on the Bayes' theoremwhich states that the posterior joint distribution function of theparameters is proportional to the product of their prior distributionfunction and the likelihood function of the data. In particular, letΩ = μ Φ …½ �′ be the vector of parameters to be estimated,Ht = h1 h2 … ht½ �′ be the vector of latent log volatility, andRt = r1 r2 … rt½ �′ be the vector of observed data. Bayesianinference is then based on the joint posterior distribution ofunobservables Ω Htð Þ given the data Rt . Let f(.) be the probabilitydensity function. Using Bayes' theorem, we get:

f Ω;Ht jRtð Þ∝f Rt jΩ;Htð Þf Ht jΩð Þf Ωð Þ:

In a sense, the posterior distribution function is a balance betweenthe prior belief, f Ht jΩð Þf Ωð Þ, and the likelihood of the data,L Ω;Ht jRtð Þ≡f Rt jΩ;Htð Þ. Inference is then conducted by evaluatingits moments. For simple and low-dimensional problems, the posteriordistribution function may have tractable forms and so evaluatingits moments is a simple task. However, in complex and higher-dimensional problems, such as the MSV models in this paper, theposterior distribution functions often do not have familiar functionalforms and we have to rely on simulation to make inference.

The Markov Chain Monte Carlo (MCMC) is a computationalmethod to generate random samples from a given distribution. Itwas first introduced by Metropolis et al. (1953) and subsequently

Table 2Estimates from VAR models.

Eq. (1) Eq. (2)

S&P 500 return Oil futures return

Parameters Coef. t-value Coef. t-value

Constant (cs/co) −.0128 −.48 .0565 1.13Lag S&P 500 returns (bss/bso) −.0771 −3.86⁎ .1854 4.98⁎

Lag oil returns (bos/boo) −.0234 −2.17⁎ −.0326 −1.62

⁎ Significant at the 5% level.

960 M. Vo / Energy Economics 33 (2011) 956–965

generalized by Hastings (1970). It is based on the construction of aMarkov Chain in the parameter space. Under some mild regularityconditions (see Tierney, 1994) the chain asymptotically converges tothe joint posterior distribution. Thus, the realized value of the chaincan be used to make inference about the joint posterior distribution.

An MCMC algorithm called Gibbs sampler, introduced by Gemanand Geman (1984), is based on the Clifford–Hammersley theoremwhich states that a joint distribution can be characterized by itscomplete conditional distributions. It is the natural choice in MCMCsampling when the conditionals from which samples are drawn canbe specified. This paper uses Gibbs sampler to generate MarkovChains. This is done through BUGS software.

4.2. Estimation

Let RESti for i=s,o, be the residual in the mean equation. Model (3)can be rewritten as follows:

RESst ≡ rst− cs + bssrst−1 + bsor

ot−1

� ; ð5Þ

RESot ≡ rot − co + bosrst−1 + boor

ot−1

� ; ð6Þ

RESst = exphst2

� �est ; ð7Þ

RESot = exphot2

� �eot ; ð8Þ

ρe = cov est ; eot

� ; ð9Þ

hst + 1 = μs + ϕss hst−μs�

+ ϕso hot−μo�

+ ηst ; ηs

t ∼iid N 0;σ2

s

�; ð10Þ

hot + 1 = μo + ϕos hst−μs�

+ ϕoo hot−μo�

+ ηot ; ηo

t ∼iid N 0;σ2

o

�;

ð11Þ

To estimate this system, I follow two steps as suggested inKalimipalli and Susmel (2004), and Ball and Torous (1999). First, I usethe maximum likelihood method to estimate the parameters C and Bin the mean Eqs. (5)–(6), then extract the residuals RESwhich will beused to estimate volatility Eqs. (7)–(11). In the second step, I employthe Bayesian MCMC method with Gibbs sampling algorithm toestimate parameters in Eqs. (7)–(11). To ensure that ht is stationary,I constrain the persistent coefficients ϕss and ϕoo to the interval(−1,1) by setting ϕii=2ϕii

⁎−1 for i=s,o, where ϕii⁎ has beta prior.

In the same manner, the DC-MSV model (4) can be expanded asfollows:

RESst ≡ rst− cs + bssrst−1 + bsor

ot−1

� RESot ≡ rot − co + bosr

st−1 + boor

ot−1

� RESst = exp

hst2

� �est

RESot = exphot2

� �eot

ρt =exp qtð Þ−1exp qtð Þ + 1

qt = ω + β qt−1−ωð Þ + σqzt ; zt ∼iid N 0;1ð Þ;

hst + 1 = μs + ϕss hst−μ s

� + ϕso hot−μo

� + ηs

t ; ηst ∼iid N 0;σ 2

s

�;

hot + 1 = μo + ϕos hst−μ s

� + ϕoo hot−μo

� + ηo

t ; ηot ∼iid N 0;σ 2

o

�:

ð12Þ

The estimation strategy used in the constant correlation MSVmodel is applied to this model. The following are chosen as priordistributions of parameters in both models:

• ϕss=2ϕss⁎−1, where ϕss

⁎∼beta(20,1.5),• ϕoo=2ϕoo

⁎ −1, where ϕoo⁎ ∼beta(20,1.5),

• μ s∼N(0,100),• μo∼N(0,100),• σs

2∼ Inverse−gamma(.001,.001),• σo

2∼ Inverse−gamma(.001,.001),• ρe∼Uniform(−1,1),• ϕso∼N(0,10),• ϕos∼N(0,10),• ω∼N(.7,10),• β=2β⁎−1Z, where β⁎∼beta(20,1.5),• σq

2∼ Inverse−gamma(.001,.001).

4.3. The results

4.3.1. Mean estimatesAs indicated above, the estimation process involves two steps.

First, the mean part (VAR(1)) of both models (3) and (4) is estimatedand the residuals are extracted. Then, the volatility part is estimatedfrom the residuals using the Bayesian MCMC method. The mean partis estimated by the maximum likelihoodmethod. Table 2 presents theresults. Both the coefficient estimates and the associated t-values arereported. We observe that in Eq. (1) of the mean part, the return onthe S&P 500 stock index is negatively correlated with its own lag andthe lag of oil futures return. However, the correlations are quite weak.On average, when oil futures return increases by 1%, the broad stockmarket index decreases approximately by .02% the next day, otherthings being equal. On the other hand, Eq. (2) tells us that oil futuresreturn is not significantly affected by its own lag but is positivelycorrelated with stock return (bso=.1854 and significant at 5%). Thus,on average, when the stock market increases by 1%, oil futures priceincreases roughly by .19% the next day, other things being equal. Onepossible explanation for this phenomenon is that higher oil prices leadto higher production costs (since oil is a factor of production) which,in turn, reduce business profits and stock prices. However, higherstock prices reflect an optimistic view that businesses are doing wellwhich implies the demand for oil is going up. This causes oil futuresprices to rise.

4.4. Volatility estimates

Table 3 presents the estimation result for the volatility part of bothCC-MSV and DC-MSV models. For the CC-MSV model, the parametersϕss and ϕoo are close to one (.9852 and .9295 respectively), indicatingthat a shock to volatility in eachmarket is quite persistent. In addition,the fact that parameters ϕso and ϕos are positive indicates that avolatility shock to one market is also transmitted into the othermarket. In particular, on average, a shock that increases the volatilityin the oil market by 1% will increase the volatility in the stock marketby about .014% a day later, other things being equal. Similarly, a shockthat increases the volatility in the stock market by 1% will be

Table 3Posterior estimates of MSV models for oil and the S&P 500 index returns.

Para. Stat. CC-MSV DC-MSV Para. Stat. CC-MSV DC-MSV

μs Mean .4909 .3053 μo Mean 1.7726 1.6728SD .3190 .2071 SD .1422 .100995%CI (−.1,1.23) (−.08,.73) 95%CI (1.53,2.08) (1.48,1.88)

ϕss Mean .9852 .9851 ϕoo Mean .9295 .9205SD .0057 .0069 SD .0215 .019595%CI (.97,.99) (.97,.99) 95%CI (.89,.97) (.88,.95)

ϕso Mean .0136 .0081 ϕos Mean .0268 .0281SD .0085 .0089 SD .0096 .009295%CI (0,.03) (−.01,.03) 95%CI (.01,.05) (.01,.05)

σs Mean .1378 .1401 σo Mean .1873 .2075SD .0160 .0213 SD .0244 .024095%CI (.11,.18) (.11,.18) 95%CI (.14,.23) (.16,.25)

ρe Mean −0.0182 ω Mean .0707SD .0215 SD .045195%CI (−.06,.02) 95%CI (−.02,.16)

β Mean .9156 σq Mean .1356SD .0248 SD .016195%CI (.87,.95) 95%CI (.11,.17)

DIC 18273 18187

961M. Vo / Energy Economics 33 (2011) 956–965

transmitted into the oil market and raises the volatility in this marketby about .027% a day later.

For the correlation between the stock and oil futures markets,there are two opposite effects. On one hand, higher oil prices increaseproduction costs and thus decrease business profits (in aggregate, theU.S. is net consumer of oil). This has negative impact on the stockmarket and so the correlation should be negative. On the other hand,higher stock market reflects an optimistic view of the business which

-0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Density of μs

1.2 1.4 1.6 1

0.0

0.5

1.0

1.5

2.0

2.5

Dens

0.85 0.90 0.95 1.00

05

1015

Density of φoo

-0.02 0.00

010

2030

400

1020

3040

Dens

-0.10 -0.05 0.00 0.05

05

1015

Density of ρe

Den

sity

Den

sity

Den

sity

Den

sity

Den

sity

Den

sity

0.10 0.12 0.14

Dens

ρe

φoo

μs

Fig. 3. The estimated densities of pa

translates into high future demand for oil. Since oil supply cannotincrease overnight, oil prices should go up. This establishes thepositive correlation between the two markets. The CC-MSV modelindicates that the first effect likely dominates the second, so weobserve a negative correlation between the twomarkets. Despite that,the correlation is quite weak (ρe=−.0182). Fig. 3 graphs theposterior densities of the parameters estimated in the CC-MSVmodel.

Fig. 4 shows the volatility estimates in both markets using the CC-MSV model. They reflect very well the variation of the absolute valueof the residuals in the mean equations. Volatility in both markets hasincreased significantly since September 2008 when the financial crisisentered the critical stage.

To investigate the possibility of time-varying correlation, I estimatethe DC-MSV model. For comparison the results are reported in Table 3,next to those of the CC-MSV model. We see that when time-varyingcorrelation is allowed, the coefficient ϕso, which represents theinformation feedback from the oil market to the stock market, getssmaller. However, the coefficient ϕos, which represents feedback fromthe stock market to the oil market, is almost unchanged. Thus, time-varying correlation coefficient captures some information feedbackfrom the oil to the stock market but not the reverse. Fig. 5 shows theposterior probability densities of the parameters in the DC-MSVmodel.

Fig. 6 shows the estimated volatilities of both markets along withthe time-varying correlation coefficient. For comparison, I show thecorrelation coefficient under both stock and oil volatility graphs. As inthe constant correlation case, the estimated volatilities reflect well thevariation in each market. We also observe that the correlationcoefficient varies in the similar fashion as volatility. It tends to go upwhen the market is more volatile.

.8 2.0 2.2 2.4

ity of μo

0.96 0.97 0.98 0.99 1.00

020

4060

Density of φss

0.02 0.04

010

2030

40

ity of φso

0.00 0.02 0.04 0.06

Density of φos

Den

sity

Den

sity

Den

sity

0.16 0.18

ity of σs

0.15 0.20 0.25

05

1015

Density of σo

σs σo

φso φos

φssμo

rameters in the CC-MSV model.

2000 2002 2004 2006 20080

24

68

10

Time

Abs

.Val

ue o

f Sto

ck R

esid

ual

Abs

.Val

ue o

f Oil

Res

idua

l

2000 2002 2004 2006 2008

12

34

5

Sto

ck V

olat

ility

Oil

Vol

atili

ty

2000 2002 2004 2006 2008

05

1015

Time

Time Time

2000 2002 2004 2006 2008

23

45

67

Fig. 4. Estimated volatilities of the S&P 500 index and oil futures in the constant correlation case.

962 M. Vo / Energy Economics 33 (2011) 956–965

4.5. Model checking

This section checks the goodness-of-fit of each model andcompares them. To check the goodness-of-fit, I test whether thevolatility in each market indicated by each model is sufficient toexplain the stylized facts, such as heavy tail, volatility clustering, etc.In particular, I test if the residuals in the mean equations, having been

-0.5 0.0 0.5 1.0

0.00.1

Density of μs

μs

Den

sity

Den

sity

Den

sity

Den

sity

Den

sity

Den

sity

Den

sity

Den

sity

1.4 1.6

02

4

Dens

0.85 0.90 0.95

001

02

Density of φoo

φoo

-0.02 0.00

00

20

4

Dens

0.10 0.12 0.14 0.16 0.18 0.20

001

02

Density of σs

σs

0.15 0.

05

51

Dens

0.15 0.20 0.25

05

51

Density of β

β0.10 0.12 0

001

52

Dens

Fig. 5. Estimated densities of para

standardized by the corresponding volatility in the variance equa-tions, are standard normal. For comparison purposes, I also report thetest results for non-standardized residuals. Table 4 presents the testresults.

We observe that the mean of standardized residual in each marketis close to zero while its standard deviation is close to one.Furthermore, the kurtosis for both markets in both models are close

Den

sity

Den

sity

Den

sity

1.8 2.0

ity of μo

μo

0.96 0.97 0.98 0.99 1.00

002

05

Density of φss

φss

0.02 0.04

ity of φso

φso

0.00 0.02 0.04 0.06

002

04

Density of φos

φos

20 0.25

ity of σo

σo

-0.1 0.0 0.1 0.2

04

8

Density of ω

ω

.14 0.16 0.18

ity of σq

σq

meters in the DC-MSV model.

2000 2002 2004 2006 20080

24

68

10

Time

s.b

As.

eR

kcot

S

2000 2002 2004 2006 2008

12

34

Time

Vkc

otS

l.o

2000 2002 2004 2006 2008

-0.2

0.0

0.2

Time

r.ro

Cci

ma

nyD

r. ro

Cci

ma

nyD

2000 2002 2004 2006 2008

05

1015

Time

s.b

As.

eR

liO

2000 2002 2004 2006 2008

12

34

56

Time

Vli

Ol.

o2000 2002 2004 2006 2008

-0.2

0.0

0.2

Time

Fig. 6. Estimated volatilities of the S&P 500 index and oil futures in the dynamic correlation case.

963M. Vo / Energy Economics 33 (2011) 956–965

to 3, indicating that the thick tail has been removed. Both the Jarque–Bera and the Kolmogorov–Smirnov tests are used to test for normalityof the residual and standardized residual. For the residual, both rejectthe null hypothesis of normality. In the case of standardized residual,the Jarque–Bera test indicates that the null hypothesis of normalitycannot be rejected at 5% level of significance for S&P 500 in bothmodels (the p-values are .4351 and .3997). However, for oil futures,the null is rejected at 5% level of significance for both models. TheKolmogorov–Smirnov test, on the other hand, cannot reject the nullhypothesis of normality of oil futures at 5% but reject the normality forS&P 500. Thus, the tests are inconclusive about the normality of thestandardized residuals. However, it is clear that the non-normality isless pronounced in the standardized residuals. Thus, both CC-MSV andDC-MSV models do a fine job in modeling heteroscedasticity in thedata.

To decide which model, CC-MSV or DC-MSV, is more appropriategiven the data, I use the deviance information criterion (DIC)proposed by Spiegelhalter et al. (2002) which is a Bayesian versionof the Akaike Information Criterion (AIC) (Akaike, 1973) and closelyrelated to the Bayesian (or Schwarz) Information Criterion (BIC)(Schwarz, 1978). Like the AIC and the BIC, it trades off the model

Table 4Model checking.

Residual Standardized residual

Statistics CC-MSV DC-MSV

S&P 500 Oil fut. S&P 500 Oil fut. S&P 500 Oil fut.

Mean 0 0 .0270 .0221 .0275 .0221Std 1.3387 2.4974 .9489 .9829 .9593 .9893Skewness −.2613 −.3173 −.0631 −.1543 −.0641 −.1472Kurtosis 11.3956 6.6126 2.9893 3.0259 2.9630 2.9551Jarque–Bera 7374.943 1403.229 1.6643 9.9977 1.8342 9.2063(p-value) (.0000) (.0000) (.4351) (.0067) (.3997) (.0100)Kolmogorov–Smirnov

.0744 .0431 .0218 .0166 .0219 .0178

(p-value) (.0000) (.0000) (.0080) (.0964) (.007) (.0650)DIC 18273 18187

adequacy, measured by the log-likelihood, against the modelcomplexity, measured by the number of free parameters. The smallerthe DIC, the better the model is. For a discussion of the use of the DICto compare SV models, the reader is referred to Berg et al. (2004). TheDIC measures of both models are reported in Table 4. The DC-MSVmodel has slightly lower DIC than the CC-MSV model; thus, it is themore suitable model given the data. In the next section, we willexamine their forecasting power.

5. Application: Value at Risk (VaR)

Risk management has long been a concern of both financialregulators and executives. Triggered by the 1987 market crash, VaRemerged as one of the popular measures to assess risk by financialregulators and bank supervisory committees. The reader is referred toJorion (2001) for a detailed introduction to VaR. As part of the modelevaluation, this section assesses the forecasting power of the eachmodel by conducting a VaR analysis and then comparing the resultwith those from other popular approaches used in the financialindustry, such as the normal model, and the GARCH(1,1) model.

VaR measures the possible loss under normal market conditions,over a specific time interval at a given confidence level. Specifically, letpt be the portfolio value at time t. Value at risk at level α over intervalh, denoted by VaRα

h , is defined as the loss in portfolio value thatsatisfies

P pt−pt + h≥VaRhα

�= α;

where P(.) is the probability function. The time interval h may rangefrom a few hours for an active trading desk to as long as a year for apension fund. The level α also varies. For external regulatoryrequirements, for instance, the bank capital requirements, α istypically very small, say 1%. However, for internal risk managementwithin the firm, typical α is around 5%. Statistically, VaRαh correspondsto the empirical quantile at α for a sample of returns over h.Intuitively, suppose that a portfolio manager has a daily VaR of

Table 5Value at Risk analysis.

In-sample Out-of-sample

Failure rate (%)DC-MSV 4.32 6.0CC-MSV 4.32 5.0GARCH(1,1) 5.52 6.0Normal model 3.68 10.0

964 M. Vo / Energy Economics 33 (2011) 956–965

$10 million at 5%, then there is a probability of 5% that his daily loss isgreater than $10 million under normal market condition. In thissection, I choose α=5% and h=1 day to conduct a VaR analysis.

I use five-year data, starting in January 2004, to estimate eachmodel whereby one-step-ahead VaR5%

1 estimates are generated. Themodels are then re-estimated successively using a rolling-window byadding 5 observations each time. Implicitly, this implies a “stabilitywindow” of 5 trading days for our parameters. The procedure isiterated until we generate a sample of 100 out-of-sample VaRforecasts. Each model is evaluated by computing the failure ratewhich is the number of times the absolute value of negative returnsexceeds the forecast VaR. If the model is correctly specified, the failurerate should be close to the VaR level α which is 5% in this analysis.Table 5 reports the results. The normalmodel does not performwell inboth in-sample and out-of-sample compared to others. The GARCH(1,1), DC-MSV and CC-MSV models are doing well in in-sampleforecast. However, the CC-MSV performs the best in out-of-sampleforecast. Thus, although the DC-MSV is more sophisticated, it does notoutperform the CC-MSV.

6. Conclusions

This paper studies the possible connection between the stock andthe oil futures markets. Instead of analyzing the relationship of theirreturns which has been done extensively in the literature, I focus onthe volatility association of both markets in an attempt to extractinformation intertwined in the two in order to improve the volatilityforecast of the stock market. Based on the preliminary analysis, Imodel the joint processes governing the returns of the S&P 500 stockindex and the oil futures as a bivariate VAR(1) with stochasticvolatility. The correlation between the two processes is also allowedto vary over time. In addition, the volatility in each market is allowedto Granger-cause that in the other. Based on its documentedsuperiority in dealing with the SV-type models, the Bayesian MCMCmethod is used to estimate the models. The paper offers four mainfindings. First, the stock and oil futures prices are inter-related. Theircorrelation follows a time-varying dynamic process and tends toincrease when the markets are more volatile. Second, conditioned onthe past information, the volatility in each market is very persistent,i.e., it varies in a predictable manner. Third, there is inter-marketdependence in volatility. Innovations that hit either market can affectthe volatility in the other market. In other words, conditioned on thepersistence and the past volatility in their respective markets, the pastvolatility of the stock (oil futures) market also has predictive powerover the future volatility of the oil futures (stock) market. Finally, theCC-MSV model produces more accurate Value-at-Risk estimates thanother benchmarks commonly used in the financial industry, such asthe GARCH(1,1), the normal models.

Acknowledgement

I would like to thank an anonymous referee, Liang Ding, CyrusBina, Allen Bellas, and seminar participants at the Metropolitan StateUniversity Faculty Seminar for valuable comments and suggestions

which have helped me significantly improve the paper. All remainingerrors rest with me.

References

Agnolucci, P., 2009. Volatility in crude oil futures: a comparison of the predictive abilityof GARCH and implied volatility models. Energy Economics 316–321.

Akaike, H., 1973. Information Theory and an Extension of the Maximum LikelihoodPrinciple. In: Petrov, B.N., Csaki, F. (Eds.), Proceedings of the Second InternationalSymposium on Information Theory, Budapest. Akademiai Kiado, pp. 267–281.

Andersen, T., Sorensen, B., 1996. GMM estimation of a stochastic volatility model: aMonte Carlo study. Journal of Business and Economic Statistics 329–352.

Andersen, T., Chung, H., Sorensen, B., 1999. Efficient method ofmoments estimation of astochastic volatility model: a Monte Carlo study. Journal of Econometrics 61–87.

Asai, M., McAleer, M., Yu, J., 2006. Multivariate stochastic volatility: a review.Econometric Review 145–175.

Ball, C., Torous, W.N., 1999. The stochastic volatility of short-term interest rates: someinternational evidence. Journal of Finance 2339–2359.

Berg, A., Myer, R., Yu, J., 2004. Deviance information criterion for comparing stochasticvolatility models. Journal of Business and Economic Statistics 107–120.

Bernanke, B.S., 1983. Irreversibility, uncertainty, and cyclical investment. QuarterlyJournal of Economics 85–106.

Christodoulakis, G.A., Satchell, S.E., 2002. Correlated ARCH (CorrARCH): modelling thetime-varying conditional correlation between financial asset returns. EuropeanJournal of Operational Research 351–370.

Clark, P., 1973. A subordinate stochastic process model with finite variance forspeculative prices. Econometrica 135–155.

Danielsson, J., 1994. Stochastic volatility in asset prices: estimation with simulatedmaximum likelihood. Journal of Econometrics 375–400.

Durbin, J., Koopman, S., 1997. Monte Carlo maximum likelihood estimation for non-Gaussian state space models. Biometrika 669–684.

El-Sharif, I., Brown, D., Burton, B., Nixon, B., Russell, A., 2005. Evidence on the nature andextent of the relationship between oil prices and equity values in the U.K. EnergyEconomics 819–830.

Faff, R., Brailsford, T., 1999. Oil price risk and the Australian stock market. Journal ofEnergy Finance and Development 69–87.

Fong, W.M., See, K.H., 2002. A Markov switching model of the conditional volatility ofcrude oil futures prices. Energy Economics 71–95.

Gallant, A.R., Hsieh, D., Tauchen, G., 1997. Estimation of stochastic volatility modelswith diagnostics. Journal of Econometrics 159–192.

Gammerman, D., 1997. Markov Chain Monte Carlo: Stochastic Simulation for BayesianInference. Chapman & Hall.

Geman, S., Geman, D., 1984. Stochastic relaxation, Gibbs distributions, and the Bayesianrestoration images. IEEE Transactions on Pattern Analysis and Machine Intelligence721–741.

Geweke, J., 2005. Contemporary Bayesian Econometrics and Statistics. Wiley & Sons.Gisser, M., Goodwin, T.H., 1986. Crude oil and themacroeconomy: tests of some popular

notions. Journal of Money, Credit, and Banking 95–103.Gourieroux, C., Monfort, A., Renault, E., 1993. Indirect inference. Journal of Applied

Econometrics 85–118.Hamilton, J.D., 1983. Oil and the macroeconomy since World War II. Journal of Political

Economy 228–248.Hamilton, J.D., 1996. This is what happened to the oil price–macroeconomy

relationship. Journal of Monetary Economics 215–220.Harvey, A.C., Ruiz, E., Shephard, N., 1994. Multivariate stochastic variance models.

Review of Economic Studies 247–264.Hastings, W.K., 1970. Monte Carlo sampling methods using Markov Chains and their

applications. Biometrika 1, 97–109.Hooker, M.A., 1996. What happened to the oil price-macroeconomy relationship.

Journal of Monetary Economics 195–213.Huang, R.D., Masulis, R.W., Stoll, H.R., 1996. Energy shocks and financial markets.

Journal of Futures Markets 1–27.Huntington, H.G., 1998. Crude oil prices and U.S. economic performance: where does

the asymmetry reside? Energy Journal 4, 107–132.Jacquier, E., Polson, N.G., Rossi, P.E., 1994. Bayesian analysis of stochastic volatility

models. Journal of Business and Economic Statistics 371–389.Johannes, M., Polson, N., 2006. MCMC Methods for Continuous-time Financial

Econometrics. In: Aït-Sahalia, Y., Hansen, L. (Eds.), Handbook of FinancialEconometrics. Elsevier.

Jones, C., Kaul, G., 1996. Oil and the stock markets. Journal of Finance 463–491.Jones, D.W., Leiby, P.N., Paik, I.K., 2004. Oil price shocks and the macroeconomy: what

has been learned since 1996. Energy Journal 1–32.Jorion, P., 2001. Value-at-Risk: The New Benchmark for Controlling Market Risk.

McGraw-Hill, Chicago.Kalimipalli, M., Susmel, R., 2004. Regime-switching stochastic volatility and short-term

interest rates. Journal of Empirical Finance 309–329.Kim, S., Shephard, N., Chib, S., 1998. Stochastic volatility: likelihood inference and

comparison with ARCH models. Review of Economic Studies 363–393.Lee, K., Ni, S., Ratti, R.A., 1996. Oil shocks and the macroeconomy: the role of price

variability. Energy Journal 4, 39–56.Melino, A., Turnbull, S.M., 1990. Pricing foreign currency options with stochastic

volatility. Journal of Econometrics 239–265.Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E., 1953.

Equations of state calculations by fast computing machines. Journal of ChemicalPhysics 6, 1087–1092.

965M. Vo / Energy Economics 33 (2011) 956–965

Papapetrou, E., 2001. Oil price shocks, stockmarket, economic activity and employmentin Greece. Energy Economics 511–532.

Pindyck, R., 1991. Irreversibility, uncertainty, and investment. Journal of EconomicLiterature 110–148.

Ross, S., 1989. Information and volatility: the no-arbitrage martingale approach totiming and resolution irrelevancy. Journal of Finance 1–17.

Rotemberg, J.J., Woodford, M., 1996. Imperfect competition and the effects of energyprice increases on economic activity. Journal ofMoney, Credit and Banking 549–577.

Sadorsky, P., 2001. Risk factors in stock returns of Canadian oil and gas companies.Energy Economics 17–28.

Sadorsky, P., 2006. Modeling and forecasting petroleum futures volatility. EnergyEconomics 467–488.

Sandmann, G., Koopman, S.J., 1998. Estimation of stochastic volatility models via MonteCarlo maximum likelihood. Journal of Econometrics 271–301.

Schwarz, G., 1978. Estimating the dimension of a model. Annals of Statistics 461–464.Sorensen, M., 2000. Prediction-based estimating functions. Econometrics Journal

123–147.Spiegelhalter, D.J., Best, N.G., Carlin, B.P., der Linde, A.V., 2002. Bayesian measures of

model complexity and fit. Journal of the Royal Statistical Society 583–639.Tauchen, G., Pitts, M., 1983. The price variability–volume relationship on speculative

markets. Econometrica 485–505.Tierney, L., 1994. Markov Chains for exploring posterior distributions. Annals of

Statistics 4, 1701–1762.Vo, M.T., 2009. Regime-switching stochastic volatility: evidence from the crude oil

market. Energy Economics 779–788.Wei, Y., Wang, Y., Huang, D., 2010. Forecasting crude oil market volatility: further

evidence using GARCH-class models. Energy Economics 1477–1484.