multivariate garch modeling of sector volatility transmission

The Quarterly Review of Economics and Finance47 (2007) 470–480

Multivariate GARCH modeling of sectorvolatility transmission

Syed Aun Hassan a,1, Farooq Malik b,∗a Department of Business Administration and Economics, Morningside College,

1501 Morningside Avenue, Sioux City, IA 51106, United Statesb Department of Economics and Finance, College of Business, University of Southern Mississippi,

730 East Beach Boulevard, Long Beach, MS 39560, United States

Received 22 June 2005; received in revised form 22 May 2006; accepted 26 May 2006Available online 19 April 2007

Abstract

This paper employs a multivariate GARCH model to simultaneously estimate the mean and conditionalvariance using daily returns among different US sector indexes from January 1, 1992 to June 6, 2005.Since different financial assets are traded based on these sector indexes, it is important for financial marketparticipants to understand the volatility transmission mechanism over time and across sectors in order tomake optimal portfolio allocation decisions. We find significant transmission of shocks and volatility amongdifferent sectors. These findings support the idea of cross-market hedging and sharing of common informationby investors in these sectors.© 2007 Board of Trustees of the University of Illinois. All rights reserved.

JEL classification: G1

Keywords: Volatility transmission; MGARCH; Sector indexes

1. Introduction

Globalization has resulted in more integration of international financial markets and financialmarket participants are interested in knowing how shocks and volatility are transmitted acrossmarkets over time. Some important papers that have studied this volatility transmission mech-anism across different markets include those by Hamao, Masulis, and Ng (1990), King and

∗ Corresponding author. Tel.: +1 228 865 4505; fax: +1 228 865 4588.E-mail addresses: [email protected] (S.A. Hassan), [email protected] (F. Malik).

1 Tel.: +1 712 274 5287.

1062-9769/$ – see front matter © 2007 Board of Trustees of the University of Illinois. All rights reserved.doi:10.1016/j.qref.2006.05.006

mailto:[email protected]

mailto:[email protected]

dx.doi.org/10.1016/j.qref.2006.05.006

S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 471

Wadhwani (1990), Engle and Susmel (1993), King, Sentana, and Wadhwani (1994), Lin, Engle,and Ito (1994), and Karolyi (1995). However, most of these studies have focused on some specificfinancial market(s) and no serious work has been undertaken to study the volatility transmis-sion mechanism among sector returns. There are generally two main lines of research in thiscontext; the first one is the cointegration analysis which was originally used by Kasa (1992) toinvestigate the transmission of shocks among stock prices and stock returns. This approach isnormally adopted to study the co-movements between different international financial marketsover a long period of time. The second line of research is to study the time path of volatilityin stock prices and stock returns. Researchers have mostly used the autoregressive conditionalheteroscedasticity (ARCH) to model time variant conditional variances. In recent years researchhas focused more on the persistence and transmission of volatility from one market to othermarkets.

This paper combines elements of the two lines of research by examining the volatility andshock transmission mechanism among six US sector indexes, i.e., financial, industrial, consumer,health, energy, and technology sectors. Specifically, we employ a trivariate GARCH model tosimultaneously estimate the mean and conditional variance using daily returns from January1, 1992 to June 6, 2005. We find significant volatility transmission among the sectors underinvestigation. Our results are important for building accurate asset pricing models, forecastingvolatility in sector returns, and will further our understanding of the equity markets. Additionally,since different financial assets are traded based on these sector indexes, it is important for financialmarket participants to understand the volatility transmission mechanism over time and acrosssectors in order to make optimal portfolio allocation decisions. We illustrate the usefulness of ourresults by calculating dynamic hedge ratios and risk minimizing portfolio weights for two sectors.Thus we fill another void in the literature since most empirical studies do not explicitly discussthe implications of their findings for market participants.

2. Literature review

There is a body of literature on how different markets and sectors interact over time. Ewing(2002) using generalized forecast error variance decomposition technique within a vector auto-regression (VAR) framework analyzed the interrelationship among five major sectors, i.e., capitalgoods, financials, industrials, transportation, and utilities. Using monthly data from S&P stockindexes from January 1988 to July 1997, he finds that unanticipated ‘news’ or shocks in onesector have significant impact on other sector returns. Ewing, Forbes, and Payne (2003) studiedthe effects of macroeconomic shocks on five major S&P sector-specific stock market indexesfor the post-1987 crash period. Using generalized impulse response analysis, they showed thatindividual asset prices are influenced more by unanticipated macroeconomic events as comparedwith some predictable events.

Fornari, Monticelli, Pericoli, and Tivegna (2002) used a trivariate GARCH model to analyzethe impact of political and economic ‘news’ on conditional volatility of several Italian financialvariables. They found a significant regime shift and seasonal daily pattern in the unconditionalvariance of the variables under study. Bernanke and Kuttner (2005) documented how monetarypolicy affects equity prices and explore the economic sources of the total impact. They find thatan unanticipated 25 basis point cut in the federal funds target rate will increase overall stockindexes by 1%. They also studied the impact of monetary policy on different sectors and foundthat monetary policy has less of an impact on individual sectors as compared with the broadindexes.

472 S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480

Additionally, Engle, Ito, and Lin (1990) argued that volatility in one foreign exchange market istransmitted to other foreign exchange markets like a “meteor shower”, while Ross (1989) showedthat volatility in asset returns depends upon the rate of information flow. Since the rate of infor-mation flow and the time used in processing that information varies with each individual market(sector), one should expect different volatility patterns across markets (sectors). The increasingintegration of major financial markets has generated strong interest in understanding the volatil-ity spillover effects from one market to another. These volatility spillovers are usually attributedto cross-market hedging and change in common information, which may simultaneously alterexpectations across markets.1 Another line of research to explain the mean and volatility spillovereffects is through financial contagion. Financial contagion is defined as a shock to one country’sasset market that causes changes in asset prices in another country’s financial market. Kodres andPritsker (2002) developed a multiple asset rational expectations model to explain financial marketcontagion. Through the channel of cross-market balancing, investors transmit shocks among mar-kets by adjusting their portfolio’s exposure to macroeconomic risks. They show that the extent ofthe financial contagion depends upon market sensitivities of shared macroeconomic risk factorsand the amount of information asymmetry among markets.

The ARCH model originally developed by Engle (1982), and later generalized by Bollerslev(1986), is one of the most popular methods used for modeling volatility of high-frequencyfinancial time series data.2 Multivariate generalized autoregressive conditional heteroscedasticity(MGARCH) models have been commonly used to estimate the volatility spillover effects amongdifferent markets. Among others, Kearney and Patton (2000) used a multivariate GARCH modelto document significant volatility transmission among different exchange rates in the EuropeanMonetary System. Poon and Granger (2003) surveyed the financial market volatility literatureand showed how it affects asset pricing, risk management, and monetary policy. They argue thatvolatility in financial markets is predictable.

In this paper, we use multivariate GARCH models to simultaneously estimate the mean andconditional variance of daily sector index returns, thus avoiding the generated regressor problemassociated with the two-step estimation process found in many earlier studies (Pagan, 1984). Inaddition, we employ the BEKK parameterization of the multivariate GARCH model which doesnot impose the restriction of constant correlation among variables over time. Specifically, we use atrivariate GARCH model which allows us to study the volatility transmission among three differentsectors simultaneously.3 Ewing and Malik (2005) have also used the BEKK parameterization ofthe multivariate GARCH model to study the volatility transmission mechanism between largeand small capitalization stocks.

3. Methodology

The following mean equation was estimated for each return series given as:

Ri,t = μi + αRi,t−1 + εit (1)

1 Fleming et al. (1998) developed a model that demonstrates how cross-market hedging and sharing of commoninformation could lead to transmission of volatility across markets over time.

2 See Engle (2002) for a detailed recent survey.3 A model including more than three variables would be superior since it would capture all the interaction in second

moments among variables simultaneously. Consequently, we tried a four-variable GARCH model but the model did notconverge. This is hardly surprising as multivariate GARCH models are notorious for convergence problems.


where Ri,t is the return on index i between time t − 1 and t, μi is a long-term drift coefficient,and εit is the error term for the return on index i at time t. Eq. (1) was then tested using the testdescribed in Engle (1982) for the existence of ARCH. All estimated series exhibited evidenceof ARCH effects.4 Since we are interested in the possibility of volatility transmission amongdifferent sectors, as well as persistence of volatility within each sector, we employ a variant ofthe multivariate GARCH model.

The two popular parameterizations for the multivariate GARCH model used in the literatureare the VECH and BEKK parameterizations.5 The traditional VECH parameterization techniquewas introduced by Bollerslev, Engle, and Wooldridge (1988) given as:

vech(Ht) = A0 +q∑

j=1

Bjvech(Ht−j) +p∑

j=1

Ajvech(εt−jε′t−j) (2)

where εt = H1/2t ηt , ηt ∼ iid N(0,I). The notation vech (Xt) in the above equation represents a

vector formed by stacking the columns of matrix Xt, and the term Ht describes the conditionalvariance matrix. In our trivariate case, we have a total number of 78 estimated elements for ourvariance equation. In order to ensure a positive semi-definite covariance matrix, all elements muststay positive during estimation.

A more practicable alternative is the BEKK6 model given by Engle and Kroner (1995). Thismodel is designed in such a way that the estimated covariance matrix will be positive semi-definite, which is a requirement needed to guarantee non-negative estimated variances. The BEKKparameterization is given as:

Ht+1 = C′C + A′εtε′tA + B′HtB (3)

The individual elements for C, A, and B matrices in Eq. (3) are given as follows:

A =

⎡⎢⎣

a11 a12 a13

a21 a22 a23

a31 a32 a33

⎤⎥⎦ B =

⎡⎢⎣

b11 b12 b13

b21 b22 b23

b31 b32 b33

⎤⎥⎦ C =

⎡⎢⎣

c11 0 0

c21 c22 0

c31 c32 c33

⎤⎥⎦ (4)

where C is a 3 × 3 lower triangular matrix with six parameters. A is a 3 × 3 square matrix ofparameters and shows how conditional variances are correlated with past squared errors. Theelements of matrix A measure the effects of shocks or ‘news’ on conditional variances. B is also a3 × 3 square matrix of parameters and shows how past conditional variances affect current levelsof conditional variances. The total number of estimated elements for the variance equations forour trivariate case is 24.

4 Specifically, the test statistic is distributed as �2 with degrees of freedom equal to the number of restrictions. We foundsignificant ARCH effects in each return series which suggests that past values of volatility can be used to predict currentvolatility.

5 Some researchers have used the constant correlation model, which by assuming constant correlations among variablesover time significantly reduces the number of estimated parameters. Bollerslev (1990) and Karolyi (1995) have used thismodel. However, Longin and Solnik (1995) argue that in equity markets the assumption of constant correlations amongvariables does not hold over time. Additionally, the assumption does not permit volatility spillovers across markets. Ourapproach in this paper does not make any of these restrictive assumptions.

6 The acronym BEKK is used in the literature as earlier unpublished work was undertaken by Baba, Engle, Kraft, andKroner (1990).


The conditional variance for each equation, ignoring the constant terms, can be expanded forthe trivariate GARCH(1,1) as:

h11,t+1 = a211ε

21,t + 2a11a12ε1,tε2,t + 2a11a31ε1,tε3,t + a2

21ε22,t + 2a21a31ε2,tε3,t

+ a231ε

23,t + b2

11h11,t + 2b11b12h12,t + 2b11b31h13,t + b221h22,t + 2b21b31h23,t

+ b231h33,t (5)

h22,t+1 = a212ε

21,t + 2a12a22ε1,tε2,t + 2a12a32ε1,tε3,t + a2

22ε22,t + 2a22a32ε2,tε3,t

+ a232ε

23,t + b2

12h11,t + 2b12b22h12,t + 2b12b32h13,t + b222h22,t + 2b22b32h23,t

+ b232h33,t (6)

h33,t+1 = a213ε

21,t + 2a13a23ε1,tε2,t + 2a13a33ε1,tε3,t + a2

23ε22,t + 2a23a33ε2,tε3,t + a2

33ε23,t

+ b213h11,t + 2b13b23h12,t + 2b13b33h13,t + b2

23h22,t + 2b23b33h23,t + b233h33,t

(7)

Eqs. (5), (6), and (7) show how shocks and volatility are transmitted across sectors and overtime.7 Since we have six sectors, we study the transmission mechanism by estimating two trivariateGARCH models where each model contains three sectors.

We maximized the following likelihood function assuming that errors are normally distributed:

L(θ) = −T ln(2π) − 1

2

T∑t=1

(ln |Ht| + ε′tH

−1t εt) (8)

where θ is the estimated parameter vector and T is the number of observations. Numerical max-imization techniques were utilized in order to maximize this non-linear log likelihood function.Initial conditions were obtained by performing several initial iterations using the simplex algo-rithm as recommended by Engle and Kroner (1995). The BFGS algorithm was then used to obtainthe final estimate of the variance-covariance matrix with corresponding standard errors.8

4. Data

We used daily close returns from January 1, 1992 to June 6, 2005 obtained from Dow Jones.9

We used financial, industrial, consumer (services), health, energy (oil and gas), and technologysectors in our analysis. The Dow Jones’ indexes are especially important to examine because

7 Since the coefficient terms in Eqs. (5), (6), and (7) are a non-linear function of the estimated elements from Eq. (3),we used a first-order Taylor expansion around the mean to calculate the standard errors for the coefficient terms. Kearneyand Patton (2000) provided a detailed discussion on how they used this method.

8 Quasi-maximum likelihood estimation was used and robust standard errors were calculated by the method given byBollerslev and Wooldridge (1992). All calculations were performed using RATS version 5.01 (Regression Analysis ofTime Series).

9 The use of daily data was motivated to be consistent with almost all earlier studies. Daily (rather than weekly ormonthly) data give precise estimates as it yields more degrees of freedom per estimated parameter of the covariancematrix. Additionally, one can generate forecast at longer horizons (weekly or monthly) from daily data but the converseis not true.


Table 1Descriptive statistics

Consumer Energy Financial Health Industrial Technology

Mean 0.0003 0.0003 0.0004 0.0003 0.0002 0.0004Median 0.0002 0.0000 0.0001 0.0001 0.0001 0.0007Maximum 0.0736 0.0792 0.0783 0.0760 0.0668 0.1624Minimum −0.0926 −0.0702 −0.0753 −0.0885 −0.0833 −0.1024S.D. 0.0115 0.0125 0.0120 0.0119 0.0111 0.0201Skewness −0.2241 0.0296 0.0390 −0.2743 −0.1601 0.1755Kurtosis 8.4814 5.7336 6.9455 6.9495 8.1752 6.9132Q(16) 50.80 (0.00) 47.99 (0.00) 47.35 (0.00) 80.39 (0.00) 29.78 (0.01) 45.31 (0.00)Observations 3504 3504 3504 3504 3504 3504

Notes: The sample contains daily sector returns from January 1, 1992 to June 6, 2005. The total number of usableobservations is 3504. Q(16) is the Ljung-Box statistic for serial correlation. The values in parenthesis are the actualprobability values.

financial market participants use these indexes more than any others to follow movements ofindustry groups and are widely used for measuring sector performance. Consistent with earlierresearch, returns were used as all series in level form possessed a unit root. Table 1 gives descriptivestatistics for all daily sector return series used in the paper.

All series were found to be leptokurtic (i.e., fat tails) and therefore the mean equation in allcases were tested for the existence of autoregressive conditional heteroscedasticity using the testgiven by Engle (1982). The mean equation for all series exhibited evidence of ARCH effects andtherefore estimation of a GARCH model is appropriate. We found significant autocorrelation asdetected by the Ljung-Box statistic in all cases. Technology sector shows the largest standarddeviation which is consistent with the general impression that technology stocks are more volatilerelative to other sectors.

5. Empirical results

As discussed earlier, we have six sectors under investigation; thus, we precede with the esti-mation of two trivariate GARCH models each containing three sectors.10 The estimation resultsof the multivariate GARCH model with BEKK parameterization for each variance equation arereported in Table 2 and Table 3. The symbol h11,t describes the conditional variance (volatility)for the first sector at time “t” and h12,t shows the conditional covariance between the first andsecond sector in our model. The error term “ε” in each model represents the effect of ‘news’ (i.e.,unexpected shocks) in each model on different sectors. For instance, ε2

1,t , ε22,t , and ε2

3,t representthe deviations from the mean due to some unanticipated event in a particular sector. The crossvalues of error terms like ε1,tε2,t represent the “news” in the first and second sector in time period“t”.

The results for the model that includes consumer, financial, and technology sectors are reportedin Table 2. We will discuss only the significant terms starting from the first column onwards and

10 There are twenty different possible combinations of six sectors taken three at a time. We study and report the resultsfor two such combinations. We estimated all different combinations and found similar results of shock and volatilitytransmission. Thus our results are not sensitive to re-grouping of sectors. The complete results are not reported for thesake of brevity but are available upon request.


Table 2Trivariate GARCH model for consumer, financial and technology sectors

Independent variable h11,t+1 h22,t+1 h33,t+1

ε21,t

0.0453 (1.896) 0.0087 (1.001) 0.0050 (0.553)ε1,tε2,t 0.0581 (4.271) 0.0492 (2.449) −0.0513 (−0.964)ε1,tε3,t −0.0047 (−0.386) −0.0105 (−1.115) −0.0134 (−0.847)ε2

2,t0.0186 (1.262) 0.0692 (2.820) 0.1300 (2.874)

ε2,tε3,t −0.0030 (−0.476) −0.0297 (−2.608) 0.0681 (2.383)ε2

3,t1.25e−04 (0.211) 0.0032 (1.097) 0.0089 (1.437)

h11,t 1.0261 (21.909) 0.0036 (1.302) 0.1813 (3.975)h12,t 0.1087 (−3.405) 0.1073 (−2.549) 0.424 (−7.751)h13,t −0.0488 (−0.979) −0.0123 (−1.854) 0.8108 (9.978)h22,t 0.0028 (1.740) 0.7936 (33.508) 0.2489 (11.914)h23,t 0.0025 (1.003) 0.1824 (5.015) −0.9501 (−17.637)h33,t 5.81e−04 (0.498) 0.0104 (2.499) 0.9064 (18.227)

Notes: h11 denotes the conditional variance for consumer sector return series, h22 is the conditional variance for the financialsector return series, and h33 is the conditional variance for the technology sector return series. The corresponding t-valuesare given in parenthesis below each estimated coefficient. Our multivariate GARCH model uses BEKK parameterization.The sum of the ARCH and GARCH terms (i.e., volatility persistence) for the consumer, financial, and technology sectoris 0.98, 0.96, and 0.91, respectively.

Table 3Trivariate GARCH model for energy, health and industrial sectors

Independent variable h11,t+1 h22,t+1 h33,t+1

ε21,t

0.0198 (4.656) 2.32e−04 (0.463) 4.74e−04 (1.050)ε1,tε2,t 0.0034 (0.717) −0.0055 (−0.905) 2.99e−04 (0.449)ε1,tε3,t 0.0165 (3.079) −0.0017 (−0.849) 0.0093 (2.176)ε2

2,t1.48e−04 (0.351) 0.0325 (5.224) 4.73e−05 (0.231)

ε2,tε3,t 0.0014 (0.754) 0.0209 (3.313) 0.0029 (0.482)ε2

3,t0.0034 (1.524) 0.0033 (1.611) 0.0461 (4.163)

h11,t 0.9751 (189.686) 4.54e−08 (0.031) 3.69e−05 (1.313)h12,t 0.0034 (−0.465) 4.18e−04 (−0.063) 3.27e−05 (0.789)h13,t −0.0190 (−2.634) 3.69e−06 (0.063) −0.0118 (−2.621)h22,t 3.11e−06 (0.232) 0.9612 (141.456) 7.23e−06 (0.423)h23,t 3.40e−05 (0.503) −0.0169 (−3.010) −0.00525 (−0.844)h33,t 9.32e−05 (1.316) 7.49e−05 (1.507) 0.9534 (82.208)

Notes: h11 denotes the conditional variance for energy sector return series, h22 is the conditional variance for the healthsector return series, and h33 is the conditional variance for the industrial sector return series. The corresponding t-valuesare given in parenthesis below each estimated coefficient. Our multivariate GARCH model uses BEKK parameterization.The sum of the ARCH and GARCH terms (i.e., volatility persistence) for the energy, health, and industrial sector is 0.99,0.99, and 0.99, respectively.

moving from top to bottom. Note that the consumer sector is significantly indirectly affected bynews generated from the financial sector (see the significant ε1,tε2,t coefficient term). Consumersector is directly affected by volatility generated by its own sector (see significant coefficientfor h11,t) and indirectly affected by volatility from the financial sector (see significant coefficientfor h12,t).11 Looking at the financial sector (second column), we see that financial sector is indi-

11 This is particularly interesting as univariate GARCH estimation was performed separately for all six sectors. Theresults (not reported here but available upon request) indicated that both GARCH and ARCH terms for each return


rectly affected by news in the consumer and technology sectors (see the significant ε1,tε2,t andε2,tε3,tcoefficient terms), and is directly affected by news generated within its own sector. However,the financial sector is also indirectly affected by volatility in the consumer and technology sector,and directly affected by the volatility in the technology sector. Examining the technology sector, wesee that it is affected directly and indirectly by news in the financial sector. Lastly, volatility in thetechnology sector is directly and indirectly affected by volatility from all sectors including its own.Overall, there is significant volatility transmission between all three sectors under investigation.

Likewise, the results for the energy, health, and industrial sectors are reported in Table 3.Energy sector is directly affected by news from its own sector and indirectly affected by newsfrom the industrial sector. The volatility in energy sector is directly affected by its own volatilityand indirectly affected by volatility in the industrial sector. The health sector is directly affectedby news from its own sector and indirectly affected by news in the industrial sector. The volatilityin energy sector is directly affected by its own volatility and indirectly affected by volatility inthe industrial sector. The industrial sector is indirectly affected by news from energy sector anddirectly affected by news in its own sector. The volatility in industrial sector is indirectly affectedby volatility in energy sector and directly affected by volatility by its own sector. Althoughsimilar to the first model, we find less volatility transmission among sectors but all sectors areaffected more by their own sector in terms of news and volatility. As explained in the introductionsection, this volatility transmission is usually attributed to cross-market hedging and changes incommon information, which may simultaneously alter expectations across sectors as suggestedby Fleming, Kirby, and Ostdiek (1998). Thus these results could be interpreted as an outcome ofcross-market hedging undertaken by financial market participants within these sectors. Althoughthe transmission of shocks from one sector to another sector returns was documented by Ewing(2002), the finding of spillover of shocks from one sector to other sectors’ variance is novel withwhole new set of implications.

6. Economic implications of the model

Decisions regarding asset pricing, risk management and portfolio allocation require accurateestimation of the time-varying covariance matrix. In order to understand the importance of thecovariance matrix regarding the above financial decisions, we follow the applications outlined byKroner and Ng (1998).

Let us consider the problem of computing the optimal fully invested portfolio holdings. Port-folio managers are often faced with this issue when deriving their optimal portfolio holdings. Ifwe assume that expected returns are zero, then the risk minimizing portfolio weight is given as

w12,t = h22,t − h12,t

h11,t − 2h12,t + h22,t

where “w12,t” is the portfolio weight for first sector relative to the second sector at time “t”.Assuming a mean-variance utility function, the optimal portfolio holdings of the financial sectorportfolio are:

w12,t = 0 if w12,t < 0, w12,t if 0 ≤ w12,t ≤ 1, and 1 if w12,t > 1.

series were significant at the 1% level. Thus, the finding of (weak) ARCH effects in the trivariate setting underscores theimportance of the role played by the interdependence of different sectors.


We use the financial and the technology sector in this illustration.12 The term h11,t denotes theconditional variance of the financial sector in time period “t” and the term h22,t shows the condi-tional variance in the technology sector. The term h12,t shows the conditional covariance betweenthe financial and technology sectors. The above equation shows that the weight w12,t is a functionof the conditional variances of financial and technology sectors for each time period. The averagew12,t for our model is 0.66, which implies that the optimal portfolio holding for the financial sectorshould be 66 cents to a dollar. The optimal portfolio holdings for the technology sector wouldtherefore be 1 − w12,t or 0.34. This example shows how results of multivariate GARCH modelscould be used by financial market participants for making optimal portfolio allocation decisions.

As another example, we estimate the risk-minimizing hedge ratios for these two sectors byusing the results of our multivariate GARCH models as shown by Kroner and Sultan (1993). Inorder to minimize the risk of some portfolio that is $1 long in first sector, the investor should short$β of the second sector. The risk minimizing hedge ratio is given as:

βt = h12,t

h22,t

where h12,t is the conditional covariance between the financial and technology sectors, and h22,t isthe conditional variance for the technology sector in time period “t”. The risk minimizing optimalhedge ratio value by using our multivariate GARCH model is 0.64. This value implies that forevery dollar that is long in the financial sector the investor should short 64 cents of the technologysector.

One caveat should be mentioned in using the multivariate GARCH models for projectingfuture estimates. GARCH model might give inaccurate forecasts if the underlying process whichgenerates asset prices undergoes a structural break. Thus the real challenge for the researcher isto find an optimal sample size which captures all the main features of the data generating processbut avoid periods of structural breaks that one suspects based on a priori information.

7. Concluding remarks

This paper examined the transmission of volatility and shocks among major sectors usingdaily data from January 1, 1992 to June 6, 2005. The sectors used in the analysis were financial,industrial, consumer, health, energy, and technology. Generally speaking, our results show signif-icant interaction between second moments of the US equity sector indexes. There is significanttransmission of shocks and volatility among all of these sectors.

While sector index investing has gained tremendous popularity over the last decade or so,investors continue to pick certain sectors and pay less attention on how other sectors behave overtime. By uncovering the hidden dynamics of transmission channels among sectors, this researchhas shown that sectors do interact with each other in terms of shocks and volatility. This findingpoints to the presence of cross-market hedging and sharing of common information by investorsin these sectors. This implies that investors should keep a close eye on all sectors because a‘news’ impacting a certain sector will eventually impact all sectors through their interdependence.We further demonstrate the importance of our empirical results by calculating hedge ratios andportfolio weights for two sectors.

12 For the sake of brevity we show the analysis for a simple two variable case based on certain assumptions includingthat the weights cannot be negative. Jagannathan and Ma (2003) show that these are valid assumptions and document indetail how to accurately estimate portfolio weights with more than two variables.


Our results are important for building accurate asset pricing models, forecasting future sectorreturn volatility, and will further our understanding of the equity markets. Additionally, sincedifferent financial assets are traded based on these sector indexes, it is important for financialmarket participants to understand the volatility transmission mechanism over time and acrosssectors in order to make optimal portfolio allocation decisions.

Acknowledgments

The authors thank Thomas Steinmeier, Benjamin Keen, and Mark Thompson for helpfulcomments. The usual disclaimer applies.

References

Baba, Y., Engle, R. F., Kraft D., & Kroner. K. (1990). Multivariate simultaneous generalized ARCH. Unpublishedmanuscript, University of California-San Diego.

Bernanke, B. S., & Kuttner, K. N. (2005). What explains the stock market’s reaction to federal reserve policy. Journal ofFinance, 60, 1221–1257.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics, 31, 307–327.

Bollerslev, T. (1990). Modeling the coherence in short run nominal exchange rates: A multivariate generalized ARCHapproach. Review of Economics and Statistics, 72, 498–505.

Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1988). A capital asset pricing model with time varying covariance.Journal of Political Economy, 96, 116–131.

Bollerslev, T., & Wooldridge, J. M. (1992). Quasi-maximum likelihood estimation and inference in dynamic models withtime-varying covariances. Econometric Reviews, 11, 143–172.

Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of the U.K. inflation.Econometrica, 50, 987–1008.

Engle, R. (2002). New frontiers for ARCH models. Journal of Applied Econometrics, 17, 425–446.Engle, R., Ito, T., & Lin, W. (1990). Meteor showers or heat waves?: Heteroscedasticity intra-daily volatility in the foreign

exchange markets. Econometrica, 58, 525–542.Engle, R., & Kroner, K. (1995). Multivariate simultaneous generalized ARCH. Econometric Reviews, 11, 122–150.Engle, R. F., & Susmel, R. (1993). Common volatility in international equity markets. Journal of Business and Economic

Statistics, 11, 167–176.Ewing, B. T. (2002). The transmission of shocks among S&P indexes. Applied Financial Economics, 12, 285–290.Ewing, B. T., Forbes, S. M., & Payne, J. E. (2003). The effects of macroeconomic shocks on sector-specific returns.

Applied Economics, 35, 201–207.Ewing, B. T., & Malik, F. (2005). Re-examining the asymmetric predictability of conditional variances: The role of sudden

changes in variance. Journal of Banking and Finance, 29, 2655–2673.Fornari, F., Monticelli, C., Pericoli, M., & Tivegna, M. (2002). The impact of news on the exchange rate of the lira and

long-term interest rates. Economic Modelling, 19, 611–639.Fleming, J., Kirby, C., & Ostdiek, B. (1998). Information and volatility linkages in the stock, bond, and money markets.

Journal of Financial Economics, 49, 111–137.Hamao, Y., Masulis, R. W., & Ng, V. (1990). Correlations in price changes and volatility across international stock markets.

Review of Financial Studies, 3, 281–307.Jagannathan, R., & Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints help. Journal

of Finance, 58, 1651–1683.Karolyi, A. (1995). A multivariate GARCH model of international transmission of stock returns and volatility. Journal

of Business and Economic Statistics, 13, 11–25.Kasa, K. (1992). Common stochastic trends in international stock markets. Journal of Monetary Economics, 29, 95–124.Kearney, C., & Patton, A. J. (2000). Multivariate GARCH modeling of exchange rate volatility transmission in the

European monetary system. Financial Review, 41, 29–48.King, M., Sentana, A., & Wadhwani, S. (1994). Volatility and links between national stock markets. Econometrica, 62,

901–933.


King, M. A., & Wadhwani, S. (1990). Transmission of volatility between stock markets. Review of Financial Studies, 3,5–33.

Kodres, L. E., & Pritsker, M. (2002). A rational expectations model of financial contagion. Journal of Finance, 57,768–799.

Kroner, F. K., & Ng, V. K. (1998). Modeling asymmetric comovements of asset returns. Review of Financial Studies, 11,817–844.

Kroner, K. F., & Sultan, J. (1993). Time varying distributions and dynamic hedging with foreign currency futures. Journalof Financial and Quantitative Analysis, 28, 535–551.

Lin, W., Engle, R. F., & Ito, T. (1994). Do bulls and bears move across borders? International transmission of stock returnsand volatility. Review of Financial Studies, 7, 507–538.

Longin, F., & Solnik, B. (1995). Is correlations in international equity returns constant: 1960–1990? Journal of Interna-tional Money and Finance, 14, 3–26.

Pagan, A. (1984). Econometric issues in the analysis of regressions with generated regressors. International EconomicReview, 25, 221–247.

Poon, S. H., & Granger, C. W. (2003). Forecasting volatility in financial markets: A review. Journal of Economic Literature,61, 478–539.

Ross, S. A. (1989). Information and volatility: The no-arbitrage martingale property to timing and resolution irrelevancy.Journal of Finance, 44, 1–17.

multivariate garch modeling of sector volatility transmission

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