The Quarterly Review of Economics and Finance 47 (2007) 470–480 Multivariate GARCH modeling of sector volatility 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 States b 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 2006 Available online 19 April 2007 Abstract This paper employs a multivariate GARCH model to simultaneously estimate the mean and conditional variance 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 market participants to understand the volatility transmission mechanism over time and across sectors in order to make optimal portfolio allocation decisions. We find significant transmission of shocks and volatility among different sectors. These findings support the idea of cross-market hedging and sharing of common information by 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 financial market participants are interested in knowing how shocks and volatility are transmitted across markets 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 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 specific financial 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 this context; the first one is the cointegration analysis which was originally used by Kasa (1992) to investigate the transmission of shocks among stock prices and stock returns. This approach is normally adopted to study the co-movements between different international financial markets over a long period of time. The second line of research is to study the time path of volatility in stock prices and stock returns. Researchers have mostly used the autoregressive conditional heteroscedasticity (ARCH) to model time variant conditional variances. In recent years research has focused more on the persistence and transmission of volatility from one market to other markets. This paper combines elements of the two lines of research by examining the volatility and shock transmission mechanism among six US sector indexes, i.e., financial, industrial, consumer, health, energy, and technology sectors. Specifically, we employ a trivariate GARCH model to simultaneously estimate the mean and conditional variance using daily returns from January 1, 1992 to June 6, 2005. We find significant volatility transmission among the sectors under investigation. Our results are important for building accurate asset pricing models, forecasting volatility 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 financial market participants to understand the volatility transmission mechanism over time and across sectors in order to make optimal portfolio allocation decisions. We illustrate the usefulness of our results 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 discuss the 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., capital goods, financials, industrials, transportation, and utilities. Using monthly data from S&P stock indexes from January 1988 to July 1997, he finds that unanticipated ‘news’ or shocks in one sector have significant impact on other sector returns. Ewing, Forbes, and Payne (2003) studied the effects of macroeconomic shocks on five major S&P sector-specific stock market indexes for the post-1987 crash period. Using generalized impulse response analysis, they showed that individual asset prices are influenced more by unanticipated macroeconomic events as compared with some predictable events. Fornari, Monticelli, Pericoli, and Tivegna (2002) used a trivariate GARCH model to analyze the impact of political and economic ‘news’ on conditional volatility of several Italian financial variables. They found a significant regime shift and seasonal daily pattern in the unconditional variance of the variables under study. Bernanke and Kuttner (2005) documented how monetary policy affects equity prices and explore the economic sources of the total impact. They find that an unanticipated 25 basis point cut in the federal funds target rate will increase overall stock indexes by 1%. They also studied the impact of monetary policy on different sectors and found that monetary policy has less of an impact on individual sectors as compared with the broad indexes. 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 is transmitted to other foreign exchange markets like a “meteor shower”, while Ross (1989) showed that 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 increasing integration 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 attributed to cross-market hedging and change in common information, which may simultaneously alter expectations across markets.1 Another line of research to explain the mean and volatility spillover effects is through financial contagion. Financial contagion is defined as a shock to one country’s asset market that causes changes in asset prices in another country’s financial market. Kodres and Pritsker (2002) developed a multiple asset rational expectations model to explain financial market contagion. 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 of the financial contagion depends upon market sensitivities of shared macroeconomic risk factors and 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-frequency financial time series data.2 Multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) models have been commonly used to estimate the volatility spillover effects among different markets. Among others, Kearney and Patton (2000) used a multivariate GARCH model to document significant volatility transmission among different exchange rates in the European Monetary System. Poon and Granger (2003) surveyed the financial market volatility literature and showed how it affects asset pricing, risk management, and monetary policy. They argue that volatility in financial markets is predictable. In this paper, we use multivariate GARCH models to simultaneously estimate the mean and conditional variance of daily sector index returns, thus avoiding the generated regressor problem associated with the two-step estimation process found in many earlier studies (Pagan, 1984). In addition, we employ the BEKK parameterization of the multivariate GARCH model which does not impose the restriction of constant correlation among variables over time. Specifically, we use a trivariate GARCH model which allows us to study the volatility transmission among three different sectors simultaneously.3 Ewing and Malik (2005) have also used the BEKK parameterization of the multivariate GARCH model to study the volatility transmission mechanism between large and 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 common information 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 not converge. This is hardly surprising as multivariate GARCH models are notorious for convergence problems. S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 473 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 test described in Engle (1982) for the existence of ARCH. All estimated series exhibited evidence of ARCH effects.4 Since we are interested in the possibility of volatility transmission among different sectors, as well as persistence of volatility within each sector, we employ a variant of the multivariate GARCH model. The two popular parameterizations for the multivariate GARCH model used in the literature are the VECH and BEKK parameterizations.5 The traditional VECH parameterization technique was 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 conditional variance matrix. In our trivariate case, we have a total number of 78 estimated elements for our variance equation. In order to ensure a positive semi-definite covariance matrix, all elements must stay positive during estimation. A more practicable alternative is the BEKK6 model given by Engle and Kroner (1995). This model 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 BEKK parameterization 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 of parameters and shows how conditional variances are correlated with past squared errors. The elements of matrix A measure the effects of shocks or ‘news’ on conditional variances. B is also a 3 × 3 square matrix of parameters and shows how past conditional variances affect current levels of conditional variances. The total number of estimated elements for the variance equations for our trivariate case is 24. 4 Specifically, the test statistic is distributed as �2 with degrees of freedom equal to the number of restrictions. We found significant ARCH effects in each return series which suggests that past values of volatility can be used to predict current volatility. 5 Some researchers have used the constant correlation model, which by assuming constant correlations among variables over time significantly reduces the number of estimated parameters. Bollerslev (1990) and Karolyi (1995) have used this model. However, Longin and Solnik (1995) argue that in equity markets the assumption of constant correlations among variables does not hold over time. Additionally, the assumption does not permit volatility spillovers across markets. Our approach 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, and Kroner (1990). 474 S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 The conditional variance for each equation, ignoring the constant terms, can be expanded for the trivariate GARCH(1,1) as: h11,t+1 = a211ε21,t + 2a11a12ε1,tε2,t + 2a11a31ε1,tε3,t + a221ε22,t + 2a21a31ε2,tε3,t + a231ε23,t + b211h11,t + 2b11b12h12,t + 2b11b31h13,t + b221h22,t + 2b21b31h23,t + b231h33,t (5) h22,t+1 = a212ε21,t + 2a12a22ε1,tε2,t + 2a12a32ε1,tε3,t + a222ε22,t + 2a22a32ε2,tε3,t + a232ε23,t + b212h11,t + 2b12b22h12,t + 2b12b32h13,t + b222h22,t + 2b22b32h23,t + b232h33,t (6) h33,t+1 = a213ε21,t + 2a13a23ε1,tε2,t + 2a13a33ε1,tε3,t + a223ε22,t + 2a23a33ε2,tε3,t + a233ε23,t + b213h11,t + 2b13b23h12,t + 2b13b33h13,t + b223h22,t + 2b23b33h23,t + b233h33,t (7) Eqs. (5), (6), and (7) show how shocks and volatility are transmitted across sectors and over time.7 Since we have six sectors, we study the transmission mechanism by estimating two trivariate GARCH 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 obtain the 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 technology sectors 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. Kearney and 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 by Bollerslev and Wooldridge (1992). All calculations were performed using RATS version 5.01 (Regression Analysis of Time Series). 9 The use of daily data was motivated to be consistent with almost all earlier studies. Daily (rather than weekly or monthly) data give precise estimates as it yields more degrees of freedom per estimated parameter of the covariance matrix. Additionally, one can generate forecast at longer horizons (weekly or monthly) from daily data but the converse is not true. S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 475 Table 1 Descriptive statistics Consumer Energy Financial Health Industrial Technology Mean 0.0003 0.0003 0.0004 0.0003 0.0002 0.0004 Median 0.0002 0.0000 0.0001 0.0001 0.0001 0.0007 Maximum 0.0736 0.0792 0.0783 0.0760 0.0668 0.1624 Minimum −0.0926 −0.0702 −0.0753 −0.0885 −0.0833 −0.1024 S.D. 0.0115 0.0125 0.0120 0.0119 0.0111 0.0201 Skewness −0.2241 0.0296 0.0390 −0.2743 −0.1601 0.1755 Kurtosis 8.4814 5.7336 6.9455 6.9495 8.1752 6.9132 Q(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 usable observations is 3504. Q(16) is the Ljung-Box statistic for serial correlation. The values in parenthesis are the actual probability values. financial market participants use these indexes more than any others to follow movements of industry groups and are widely used for measuring sector performance. Consistent with earlier research, returns were used as all series in level form possessed a unit root. Table 1 gives descriptive statistics 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 all cases were tested for the existence of autoregressive conditional heteroscedasticity using the test given by Engle (1982). The mean equation for all series exhibited evidence of ARCH effects and therefore estimation of a GARCH model is appropriate. We found significant autocorrelation as detected by the Ljung-Box statistic in all cases. Technology sector shows the largest standard deviation which is consistent with the general impression that technology stocks are more volatile relative 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 results of the multivariate GARCH model with BEKK parameterization for each variance equation are reported 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 and second 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, ε21,t , ε22,t , and ε23,t represent the deviations from the mean due to some unanticipated event in a particular sector. The cross values 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 reported in 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 results for two such combinations. We estimated all different combinations and found similar results of shock and volatility transmission. Thus our results are not sensitive to re-grouping of sectors. The complete results are not reported for the sake of brevity but are available upon request. 476 S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 Table 2 Trivariate 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) ε22,t 0.0186 (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) ε23,t 1.25e−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 financial sector return series, and h33 is the conditional variance for the technology sector return series. The corresponding t-values are 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 sector is 0.98, 0.96, and 0.91, respectively. Table 3 Trivariate 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) ε22,t 1.48e−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) ε23,t 0.0034 (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 health sector return series, and h33 is the conditional variance for the industrial sector return series. The corresponding t-values are 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 by news generated from the financial sector (see the significant ε1,tε2,t coefficient term). Consumer sector is directly affected by volatility generated by its own sector (see significant coefficient for h11,t) and indirectly affected by volatility from the financial sector (see significant coefficient for 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. The results (not reported here but available upon request) indicated that both GARCH and ARCH terms for each return S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 477 rectly affected by news in the consumer and technology sectors (see the significant ε1,tε2,t and ε2,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, we see that it is affected directly and indirectly by news in the financial sector. Lastly, volatility in the technology 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 news from the industrial sector. The volatility in energy sector is directly affected by its own volatility and indirectly affected by volatility in the industrial sector. The health sector is directly affected by news from its own sector and indirectly affected by news in the industrial sector. The volatility in energy sector is directly affected by its own volatility and indirectly affected by volatility in the industrial sector. The industrial sector is indirectly affected by news from energy sector and directly affected by news in its own sector. The volatility in industrial sector is indirectly affected by volatility in energy sector and directly affected by volatility by its own sector. Although similar to the first model, we find less volatility transmission among sectors but all sectors are affected more by their own sector in terms of news and volatility. As explained in the introduction section, this volatility transmission is usually attributed to cross-market hedging and changes in common information, which may simultaneously alter expectations across sectors as suggested by Fleming, Kirby, and Ostdiek (1998). Thus these results could be interpreted as an outcome of cross-market hedging undertaken by financial market participants within these sectors. Although the 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 with whole new set of implications. 6. Economic implications of the model Decisions regarding asset pricing, risk management and portfolio allocation require accurate estimation of the time-varying covariance matrix. In order to understand the importance of the covariance matrix regarding the above financial decisions, we follow the applications outlined by Kroner 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. If we 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 sector portfolio 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 the importance of the role played by the interdependence of different sectors. 478 S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 We use the financial and the technology sector in this illustration.12 The term h11,t denotes the conditional 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 between the financial and technology sectors. The above equation shows that the weight w12,t is a function of the conditional variances of financial and technology sectors for each time period. The average w12,t for our model is 0.66, which implies that the optimal portfolio holding for the financial sector should be 66 cents to a dollar. The optimal portfolio holdings for the technology sector would therefore be 1 − w12,t or 0.34. This example shows how results of multivariate GARCH models could 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 by using the results of our multivariate GARCH models as shown by Kroner and Sultan (1993). In order 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 is the conditional variance for the technology sector in time period “t”. The risk minimizing optimal hedge ratio value by using our multivariate GARCH model is 0.64. This value implies that for every dollar that is long in the financial sector the investor should short 64 cents of the technology sector. One caveat should be mentioned in using the multivariate GARCH models for projecting future estimates. GARCH model might give inaccurate forecasts if the underlying process which generates asset prices undergoes a structural break. Thus the real challenge for the researcher is to find an optimal sample size which captures all the main features of the data generating process but 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 using daily 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 significant transmission 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 over time. By uncovering the hidden dynamics of transmission channels among sectors, this research has shown that sectors do interact with each other in terms of shocks and volatility. This finding points to the presence of cross-market hedging and sharing of common information by investors in 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 and portfolio weights for two sectors. 12 For the sake of brevity we show the analysis for a simple two variable case based on certain assumptions including that the weights cannot be negative. Jagannathan and Ma (2003) show that these are valid assumptions and document in detail how to accurately estimate portfolio weights with more than two variables. S.A. Hassan, F. Malik / The Quarterly Review of Economics and Finance 47 (2007) 470–480 479 Our results are important for building accurate asset pricing models, forecasting future sector return volatility, 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 financial market participants to understand the volatility transmission mechanism over time and across sectors in order to make optimal portfolio allocation decisions. Acknowledgments The authors thank Thomas Steinmeier, Benjamin Keen, and Mark Thompson for helpful comments. The usual disclaimer applies. References Baba, Y., Engle, R. F., Kraft D., & Kroner. K. (1990). Multivariate simultaneous generalized ARCH. 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