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The Factor–Spline–GARCH Model for High and LowFrequency CorrelationsJosé Gonzalo Rangel a & Robert F. Engle ba Global Investment Research, Goldman Sachs Group, Inc., 200 West Street, New York,NY, 10282-2198b Department of Finance, Stern School of Business, New York University, 44 West FourthStreet, Suite 9-62, New York, NY, 10012-1126

Available online: 22 Feb 2012

To cite this article: José Gonzalo Rangel & Robert F. Engle (2012): The Factor–Spline–GARCH Model for High and LowFrequency Correlations, Journal of Business & Economic Statistics, 30:1, 109-124

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The Factor–Spline–GARCH Model for Highand Low Frequency Correlations

Jose Gonzalo RANGELGlobal Investment Research, Goldman Sachs Group, Inc., 200 West Street, New York, NY 10282-2198(josegonzalo.rangel@gs.com)

Robert F. ENGLEDepartment of Finance, Stern School of Business, New York University, 44 West Fourth Street, Suite 9-62,New York, NY 10012-1126 (rengle@stern.nyu.edu)

We propose a new approach to model high and low frequency components of equity correlations. Ourframework combines a factor asset pricing structure with other specifications capturing dynamic propertiesof volatilities and covariances between a single common factor and idiosyncratic returns. High frequencycorrelations mean revert to slowly varying functions that characterize long-term correlation patterns. Weassociate such term behavior with low frequency economic variables, including determinants of marketand idiosyncratic volatilities. Flexibility in the time-varying level of mean reversion improves both theempirical fit of equity correlations in the United States and correlation forecasts at long horizons.

KEY WORDS: DCC; Dynamic correlation; Factor models; Idiosyncratic volatility; Long-term forecastevaluation; Time-varying betas.

1. INTRODUCTION

This article introduces a new approach to characterize andforecast high and low frequency variation in equity correla-tions. By separating short-term from long-term components,our method not only facilitates the economic interpretation ofchanges in the correlation structure but achieves improvementsover leading methods in terms of fitting and forecasting equitycorrelations.

A number of multivariate time series models have been pro-posed over the last two decades to capture the dynamic proper-ties in the comovements of asset returns. Multivariate versionsof the well-known univariate Generalized Autoregressive Con-ditional Heteroscedasticity (GARCH) and Stochastic Volatility(SV) models guided the initial specifications (e.g., Bollerslev,Engle, and Woodridge 1988 and Harvey, Ruiz, and Shephard1994), but they had limitations because they were heavily param-eterized and/or difficult to estimate. Simplified versions, such asconstant conditional correlation models (e.g., Bollerslev 1990;Alexander 1998; Harvey et al. 1994), were also unattractive be-cause they had problems in describing the empirical features ofthe data. Bauwens, Laurent, and Rombouts (2006) and Shep-hard (2004) presented extended surveys of this literature. Engle(2002) introduced the Dynamic Conditional Correlation (DCC)model as an alternative approach to achieve parsimony in the dy-namics of conditional correlations, thus maintaining simplicityin the estimation process. However, none of the aforementionedmodels associate correlation dynamics with features of funda-mental economic variables. Moreover, since they return to aconstant mean in the long run, their forecasting implicationsfor long horizons do not take into account changing economicconditions. Thus, they produce the same long-term forecast atany point in time.

Financial correlation models, on the other hand, have onlyrecently introduced dynamic patterns in correlations (e.g., Ang

and Bekaert 2002; Ang and Chen 2002; Bekaert, Hodrick, andZhang 2009). Although these models are linked to asset pricingframeworks, which facilitate the association of correlation be-havior with financial and economic variables, a substantial partof the variation in the correlation structure remains unexplainedand their implementation for forecasting correlations appearsdifficult.

This article presents a new model that captures complex fea-tures of the comovements of financial returns and allows em-pirical association of economic fundamentals with the dynamicbehavior of variances and covariances. Following recent devel-opments in time series methods, this model incorporates, withina single framework, the attractive features of the two approachesmentioned above. Based on the correlation structure suggestedby a simple one-factor Capital Asset Pricing Model (CAPM)specification, and short- and long-term dynamic features of mar-ket and idiosyncratic volatilities, we derive a correlation modelthat allows conditional (high frequency) correlations to meanrevert toward smooth time-varying functions, which proxy thelow frequency component of correlations. This property notonly represents a generalization of multivariate GARCH mod-els that show mean reversion to a constant covariance matrix,but also provides flexibility to the long-term level of correla-tions in adapting to the changing economic environment. (Factormodels in which second moments mean revert to constant levelswere presented by Engle, Ng, and Rothschild 1990; Diebold andNerlove 1989; and Chib, Nardari, and Shephard 2006, amongothers.)

To achieve this goal, the semiparametric Spline–GARCH ap-proach of Engle and Rangel (2008) is used to model high and

© 2012 American Statistical AssociationJournal of Business & Economic Statistics

January 2012, Vol. 30, No. 1DOI: 10.1080/07350015.2012.643132

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110 Journal of Business & Economic Statistics, January 2012

low frequency dynamic components of both systematic andidiosyncratic volatilities. As a result, a slow-moving low fre-quency correlation part is separated from the high frequencypart. Effects of time-varying betas and unspecified unobserv-able factors are incorporated into the high frequency correlationcomponent by adding dynamic patterns to the correlations be-tween the market factor and each idiosyncratic component andbetween each pair of idiosyncratic risks. (A range of modelswith time-varying betas have been studied by Bos and New-bold 1984; Ferson and Harvey 1991, 1993; and Ghysels 1998;among many others.) The high frequency patterns are modeledusing a DCC process. Therefore, the resulting “Factor–Spline–GARCH” (FSG-DCC) model blends Spline–GARCH volatilitydynamics with DCC correlation dynamics within a factor assetpricing framework.

From an empirical perspective, this study analyzes high andlow frequency correlation patterns in the U.S. market by con-sidering daily returns of stocks in the Dow Jones IndustrialAverage (DJIA) during a period of 17 years. We find that, inaddition to the recently documented economic variation in mar-ket volatility at low frequencies (e.g., Engle and Rangel 2008;Engle, Ghysels, and Sohn 2008), average idiosyncratic volatilityshows substantial variation in its long-term component, whichis highly correlated with low frequency economic variables in-cluding an intersectoral Employment Dispersion Index given byLilien (1982). Since this variable measures the intensity of shiftsin product demand across sectors, we use it to proxy changes inthe intensity of idiosyncratic news. For instance, technologicalchange or any other driver of demand shifts can induce largemovements of productive factors from declining to growing sec-tors, and lead to increases in the intensity of firm-specific news.Consistent with this intuition, we find that this variable is pos-itively related to idiosyncratic volatility. Since the intensity ofsectoral reallocation is associated with the same sources thatlead to variation in productivity (or profitability) across firmsand sectors, our empirical findings are also consistent with thepositive relationship between idiosyncratic volatility and thevolatility of firm profitability suggested by Pastor and Veronesi(2003).

We also investigate the forecast performance of the FSG-DCC model focusing on long horizons (four to six months).Based on an economic loss function and following the approachof Engle and Colacito (2006), we perform a sequential out-of-sample forecasting exercise that compares this model witha broad set of competing models. We find significant evidencethat the FSG-DCC outperforms its competitors at long horizons.A complementary in-sample exercise presented in the onlineAppendix A5 also supports this finding.

Overall, by combining Spline–GARCH dynamics with DCCdynamics within a factor framework that separates systematicand idiosyncratic terms, our model provides flexibility to thelevel of mean reversion of asset correlations and captures eco-nomic variation of its components. These features explain theimprovements in terms of empirical fit and log-term correla-tion forecasts shown in this paper and lead us to new empiricalresults on the economic drivers of idiosyncratic volatility andcorrelations.

The article is organized as follows: Section 2 provides a de-scription of a number of correlation specifications associatedwith different assumptions in the factor setup. Section 3 in-

troduces the FSG–DCC model. Section 4 presents an empiricalanalysis of correlations in the U.S. market, empirical evidence ofeconomic variation in aggregate idiosyncratic volatility, and anempirical evaluation of correlation specifications derived fromdifferent factor models. Section 5 examines the forecast perfor-mance of the FSG–DCC model, and Section 6 concludes.

2. A SINGLE-FACTOR MODEL AND RETURNCORRELATIONS

In this section, we use a simple one-factor version of the arbi-trage pricing theory (APT) asset pricing model of Ross (1976)and describe how modifying its underlying assumptions changesthe specification of the correlation structure of equity returns.Let us assume that there is a single market factor that enterslinearly in the pricing equation as in the Sharpe (1964) CAPMmodel. Under this specification, and measuring returns in ex-cess of the risk-free rate, the excess return of asset i is generatedby

rit = αi + βirmt + uit, (1)

where rmt denotes the market excess return. The first termcharacterizes asset i’s systematic risk and the second term de-scribes its idiosyncratic component. Absence of arbitrage as-sures αi = 0 and E(rit) = λβi , where λ denotes the risk pre-mium per unit of systematic risk. The standard APT structureassumes constant betas and the following conditions:

E(rmtuit) = 0, ∀i, (2)

E(uitujt) = 0, ∀i �= j . (3)

The assumptions in the factor structure thus impose a restric-tion in the covariance matrix of returns. Assuming that momentrestrictions in Equations (2) and (3) hold conditionally, we ob-tain a specification for conditional correlations, which dependsexclusively on dynamic patterns in the conditional variances ofmarket and idiosyncratic risks. Furthermore, if the restriction inEquation (2) holds conditionally, then the betas are constant andcorrectly estimated from simple time series regressions of excessreturns on the market factor. Restriction (3) rules out correlationbetween idiosyncratic innovations, which precludes the possi-bility of missing pricing factors in the model. As suggested byEngle (2007), these restrictions are empirically unappealing andlimit importantly the dynamic structure of correlations. Allow-ing temporal deviations from Equations (2) and (3) increasessubstantially the ability of the resulting correlation models tocapture empirical features of the data maintaining the economicessence of the factor pricing structure. The following proposi-tion characterizes these more flexible correlation models.

Proposition 1. Consider the model specification in Equations(1), (2), and (3), and let �t denote the set of current and pastinformation available in the market.

(a) If Et−1(uitujt) �= 0, for some i �= j, andEt−1(rm,tuit) =0,∀ i, then the conditional correlations will be given by

ρi,j,t = βiβjVt−1(rmt)+Et−1(uitujt)√β2i Vt−1(rmt)+Vt−1(uit)

√β2j Vt−1(rmt)+Vt−1(ujt)

.

(4)

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Rangel and Engle: Factor–Spline–GARCH Model 111

(b) If Et−1(uitujt) �= 0, for some i �= j, and Et−1(rm,tuit) �=0, for some i, then the conditional correlations will begiven by

ρi,j,t = [βiβjVt−1(rmt ) + βjEt−1(rmtuit ) + βiEt−1(rmtujt )

+Et−1(uitujt )]

× [β2i Vt−1(rmt ) + Vt−1(uit ) + 2βiEt−1(rmtuit )

]−1/2

× [β2j Vt−1(rmt ) + Vt−1(ujt ) + 2βjEt−1(rmtujt )

]−1/2

(5)

where βi = cov(rit,rmt)V (rmt)

.

(c) Under the assumptions in b), an equivalent representationincorporates time-varying conditional betas

(i)

rit = βitrmt + uit, βit ≡ covt−1 (rit, rmt)

Vt−1 (rmt), ∀i.

(6)

(ii) The conditional correlation can then be equiva-lently expressed as

ρi,j,t

= βi,tβj,tVt−1 (rmt) + covt−1(uit, ujt)√(β2

i,tVt−1 (rmt) + Vt−1 (uit)) (β2

j,tVt−1 (rmt) + Vt−1(ujt)) .

(7)

The proof is given in Appendix A1. The constant terms areomitted without loss of generality. Equation (4) describes a casewhere the factor loadings are constant, but latent unobservedfactors (omitted from the specification and embedded in the errorterm) can have temporal effects on conditional correlations.Parts b) and c) of Proposition 1 consider the case of time-varyingbetas. The representation in Equation (6) satisfies assumption(2) (see Appendix A1) and can be rewritten as

rit = βirmt + uit, βi = cov (rit, rmt)

V (rmt),

uit = uit + (βit − βi) rmt, ∀i, (8)

where this uit term also satisfies Equation (2) and the equivalencebetween the two representations follows.

Hence, Equation (5) provides a more general specificationfor conditional correlations which, under some assumptions,simultaneously incorporates the effects of time variation in thebetas and latent unobserved omitted factors. For this reason, itis the basis of our econometric approach.

3. AN ECONOMETRIC MODEL FOR A FACTORCORRELATION STRUCTURE

3.1 The Factor–Spline–GARCH Model

Equity volatilities show different patterns at high and lowfrequencies. Short-term volatilities are mainly determined byfundamental news arrivals, which induce price changes at very

high frequencies. Longer term volatilities show patterns gov-erned by slow-moving structural economic variables. Engle andRangel (2008) found economically and statistically significantvariation in the low frequency market volatility in the UnitedStates as well as in most developed and emerging countries(Schwert 1989 and Engle et al. 2008 also examine the economicvariation of market volatility at low frequencies). We introducethis effect in our factor correlation model by including an equa-tion that describes the dynamic behavior of this low frequencymarket volatility.

Incorporating the low frequency variation of idiosyncraticvolatilities into the correlation structure is also appealing be-cause there is empirical evidence of long-term dynamics insuch volatilities. For instance, Campbell, Lettau, Malkiel, andXu (2001) found evidence of a positive trend in idiosyncraticfirm-level volatility during the 1962–1997 period and no trend inmarket volatility, suggesting a declining long-term pattern in thecorrelations of individual stocks over this period. Theoretical ex-planations of the upward trend in idiosyncratic volatilities havebeen associated with different firm features such as the varianceof total firm profitability, uncertainty about average profitabil-ity, age, institutional ownership, and the level and variance ofgrowth options available to managers (see Pastor and Veronesi2003; Wei and Zhang 2006; Cao, Simin, and Zhao 2008). Atthe aggregate level, Guo and Savickas (2006) found high cor-relation between idiosyncratic volatility and the Consumption–Wealth Ratio (CAY) proposed by Lettau and Ludvigson (2001).Their results suggest that idiosyncratic volatility could measurean omitted risk factor or dispersion of opinion. Overall, theseempirical and theoretical results motivate our approach of incor-porating long-term patterns of both systematic and idiosyncraticvolatilities into a model for correlation dynamics.

From an econometric standpoint, the Spline–GARCH modelof Engle and Rangel (2008) provides a semiparametric frame-work to separate high and low frequency components of volatil-ities. Following this approach, we model the market factor inEquation (1) as

rmt = αm + √τmtgmtε

mt , where εmt |�t−1 ∼ (0, 1),

gmt =(

1 − θm − φm − γm

2

)+ θm

(rmt−1 − αm)2

τmt−1

+ γm (rmt−1 − αm)2

τmt−1Irmt−1<0 + φmgmt−1,

τmt = cm exp

(wm0t +

km∑i=1

wmi ((t − ti−1)+)2

), (9)

where gmt and τmt characterize the high and low frequencymarket volatility components, respectively, (t − x)+ = (t − x)if t > x, and, otherwise, is zero, and Irmt<0 is an indicatorfunction of negative market returns, but different from Engleand Rangel (2008), the high frequency component is mod-eled as an asymmetric unit GARCH process following Glosten,Jagannathan, and Runkle (1993). We thus capture the well-documented leverage effect (see Black 1976; Christie 1982;Campbell and Hentschel 1992), where bad news increase the fu-ture high frequency volatility more than good news. The term τmt

approximates the unobserved low frequency market volatility

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112 Journal of Business & Economic Statistics, January 2012

that responds to low frequency fundamental variables, such asmacroeconomic aggregates, which characterize the variation inthe economic environment. As in Engle and Rangel (2008),this component is modeled using an exponential quadraticspline with equally spaced knots that are selected according tothe Schwarz Criterion (BIC or Bayesian information criterion).The term gmt describes transitory volatility behavior that, de-spite its persistence, does not have long-term impacts on marketvolatility.

Similarly, we model the idiosyncratic part of returns in (1) as

uit = √τitgitεit, where εit|�t−1 ∼ (0, 1),

git =(

1 − θi − φi − γi

2

)+ θi

(rit−1 − αi − βirmt−1)2

τit−1

+ γi (rit−1 − αi − βirmt−1)2

τit−1Irit−1<0 + φigit−1,

τit = ci exp

(wi0t +

ki∑r=1

wir ((t − tr−1)+)2

),∀i. (10)

The git’s characterize the high frequency component ofidiosyncratic volatilities associated with transitory effects,whereas the τit’s describe low frequency variation in idiosyn-cratic volatilities. The Ir<0’s are indicator functions of neg-ative returns that allow for firm-specific leverage effects. Asbefore, we take a multiplicative error model to describe interac-tions between firm-specific news arrivals and low frequencystate variables measuring firm- and industry-specific condi-tions. The intuition here is that a firm-specific news event willhave a larger effect, for example, when the firm is close tobankruptcy or when a major technological change is affectingthe firm’s industry. In sum, the τit’s approximate nonparamet-rically the unobserved long-term idiosyncratic volatilities thatare functions of low frequency economic variables, which af-fect the magnitude and intensity of high frequency idiosyncraticshocks.

Following the discussion in the previous section andin Proposition 1, we incorporate time-varying correlationsbetween the market factor and idiosyncratic returns, andamong idiosyncratic terms themselves. Specifically, we as-sume that the vector of innovations in Equations (9) and (10),(εmt , ε1t , ε2t , . . . , εNt )′, follows the stationary DCC model ofEngle (2002). All elements in this vector have unit conditionalvariance. Thus, from the second DCC estimation stage, wehave

ρεi,j,t = qi,j,t√qi,i,t

√qj,j,t

,

qi,j,t = ρεi,j + aDCC(εi,t−1εj,t−1 − ρεi,j

)+ bDCC

(qi,j,t−1 − ρεi,j

), ∀i, j ∈ {1, . . . , N} ,

qm,i,t = ρεm,i + aDCC(εmt−1εi,t−1 − ρεm,i

)+ bDCC

(qm,i,t−1 − ρεm,i

), ∀i ∈ {m, 1, . . . , N} ,

(11)

where ρεi,j = E(εitεjt) and ρεi,i = 1, for all i = 1, 2, . . . , N. Weadd mean reverting features to the betas described in Propo-

sition 1 and assume that ρεm,i = 0, for all i = 1, 2, . . . , N.The other intercepts are estimated by “correlation targeting”(see Engle 2002; Engle and Sheppard 2005a). The abovespecifications, together with the factor structure presented inSection II, constitute the full Factor–Spline–GARCH (FSG-DCC) model. The next proposition describes its correlationstructure.

Proposition 2. Given a vector of returns (r1t ,r2t , . . . , rNt )′ satisfying the factor structure in Equation(1), let us assume that the common market factor rmt isdescribed by Equation (9), the idiosyncratic term uit follows theprocess in Equation (10), for all i = 1, 2, . . .N, and the vectorof innovations (εmt , ε1t , ε2t , . . . , εNt )′ follows the DCC processin Equation (11) and its assumptions. The high frequency(conditional) correlation between rit and rjt is then given by

ρi,j,t = [βiβj τmtgmt + βi

√τmtgmt

√τjtgjtρ

εm,j,t

+βj√τmtgmt√τitgitρεm,i,t +√τitgit

√τjtgjtρ

εi,j,t

]× [β2i τmtgmt + τitgit + 2βi

√τmtgmtτitgitρ

εm,i,t

]−1/2

× [β2j τmtgmt + τjtgjt + 2βj

√τmtgmtτjtgjtρ

εm,j,t

]−1/2

(12)

and the low frequency component of this correlation is time-varying and takes the following form:

ρi,j,t = βiβj τmt + √τit

√τjtρ

εi,j√

β2i τmt + τit

√β2j τmt + τjt

. (13)

Assuming that Et (τk,t+h) = τk,t ,∀h > 0, k = 1, 2, . . . , N,(13) is then the long-term forecast of ρi,j,t

limh→∞

ρi,j,t+h|t = ρi,j,t. (14)

The proof is given in Appendix A2. Note that Equation (12)is a parameterized version of Equation (5) in Proposition 1.Equation (13) approximates the slow-moving component ofcorrelations, which can be associated with long-term correla-tion dynamics. Indeed, the high frequency correlation parsimo-niously mean reverts toward this time-varying low frequencyterm. This approximation may be improved by adding moreobservable factors or allowing for time variation in the ρεi,j ’s,or both. The first alternative can be easily implemented oncewe have selected the new factors (see Engle and Rangel 2009).The second extension would capture long-term effects of ex-cluded factors, but it is methodologically challenging because itrequires exploring alternative functional forms (or restrictions)to guarantee positive-definiteness of the covariance matrix. Inthe empirical part of this article, we focus on the simplest casegiven in Proposition 2.

Equation (13) also provides a simple approach to forecastlong-term correlations using economic variables. Specifically,we can use forecasts of low frequency market and idiosyn-cratic volatilities, which can be obtained from models thatincorporate economic variables. For instance, the results of

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Rangel and Engle: Factor–Spline–GARCH Model 113

Engle and Rangel (2008) allow construction of forecasts ofmarket volatility using macroeconomic and market information.From a pure time series perspective, Equation (14) presents auseful forecasting relation in which the time-varying low fre-quency correlation can be interpreted as the long run correlationforecast, assuming that the low frequency market and idiosyn-cratic volatilities remain constant during the forecasting period.

Another interesting case embedded in Proposition 2 oc-curs when low frequency variation is not allowed. This re-stricted version corresponds to the Factor DCC (FG-DCC)model of Engle (2007) and is derived by assuming τm,t = σ 2

m

and τi,t = σ 2i ,∀i, in Equations (9) and (10). The correspond-

ing conditional variance specifications for the factor and theidiosyncrasies (hmt and hit,∀i) become standard mean revertingasymmetric GARCH(1,1) processes. Thus, Equation (12) takesthe following form:

ρi,j,t = [βiβjhmt + βj

√hmt

√hitρ

εm,i,t

+βj√hmt

√hitρ

εm,j,t +

√hit√hjtρ

εi,j,t

]×[β2i hmt + hit + 2βi

√hmthitρ

εm,i,t

]−1/2

×[β2j hmt + hjt + 2βj

√hmthjtρ

εm,j,t

]−1/2(15)

and the low frequency correlation, which represents thelong-run correlation forecast associated with the FG-DCCmodel, is a constant given by the unconditional version ofEquation (15).

3.2 Estimation

To facilitate presenting our estimation approach, we rewritethe FSG-DCC model using matrix notation. The system of equa-tions in the factor setup can be written as

rt = α + But , (16)

where rt = (rmt, r1t , . . . , rNt )′ is the vector of excess returnsat time t, ut contains both the market factor and the idiosyn-

crasies, ut = (rmt, u1t , u2t , . . . , uNt )′, B =(

1 01×Nβ IN×N

), β =

(β1, β2, . . . , βN )′, 01 × N is an N-dimensional row vector of ze-ros, IN × N is the N-dimensional identity matrix, andα is a vectorof intercepts. The market factor is assumed to be weakly exoge-nous (see Engle, Hendry, and Richard 1983). In this setup, thecovariance matrix of ut can be written as

Hut = DtRtDt , (17)

where Dt = diag{√τkthkt }, for k = m,1, . . . , N, Rt =diag(Qt )1/2Qtdiag(Qt )1/2 is a correlation matrix, and the typ-ical element of Qt is defined by Equation (11). The standard-ized innovations in Proposition 2 are the elements of D−1

t ut =(εmt , ε1t , ε2t , . . . , εNt )′. Going back to the original vector of re-turns, we have that rt |�t−1 ∼ (α,Ht ), where the full covariancematrix is

var (rt ) = Ht = BDtRtDtB′, (18)

and the correlation matrix is Rt = diag{BDtRtDtB′}−1/2

BDtRtDtB′diag{BDtRtDtB

′}−1/2. As in the standard DCC

model, the estimation problem can be formulated follow-ing Newey and McFadden (1994) as a two-stage Gen-eralized Method of Moments (GMM) problem. We canwrite the vector of moment conditions as g(rt , ψ, η) =(m1(rt , η),m2(εt , ψ, η))′, where εt ≡ D−1

t ut is a vector of de-volatized returns, η is a vector of parameters containing thealphas, the unconditional betas, and the volatility parametersin Equations (9) and (10), and ψ is a vector containing theDCC parameters in Equation (11). The first vector of momentconditions, m1, has the scores of individual asymmetric Spline–GARCH models as its components. The second set of momentconditions, m2, contains the functions that involve the corre-lation parameters. If the first optimization problem involvingthe moment conditions in m1 provides consistent estimates ofthe volatility and mean parameters, the optimization of m2 in thesecond step will also provide consistent estimates.

In a Gaussian quasi-likelihood (QML) framework, assumingmultivariate normality leads to consistent estimates under mildregularity conditions as long as the mean and the covarianceequations are correctly specified (see Bollerslev and Wooldridge1992; Newey and Steigerwald 1997). Despite the convenienceof the Gaussian QML approach, choosing the number of knotsfor the spline functions introduces an additional procedure inthe estimation process that does not necessarily provide QMLestimates of such quantities. Inaccurate choices of the numberof knots might introduce some biases due to misspecification ofvolatilities. Since distributional assumptions might have an ef-fect on this part of the estimation, we depart from normality anduse distributional assumptions that more realistically describethe empirical features of returns. This can improve the accuracyof the knot selection criterion and reduce the misspecificationproblem. Therefore, for the first stage of the estimation process,we consider likelihoods from the Student-t distribution becausethis distribution is better able to capture fat tails and the effectof influential outliers. The following log-likelihoods correspondto the Asymmetric Spline–GARCH model with Student-t inno-vations (an extension of the Bollerslev 1987 model) and deter-mine the moment conditions associated with the first stage ofour GMM estimation:

L(ηm, vm) = log

(� ((vm + 1)/2)

� (vm/2) ((vm − 2)gmtτmt)1/2

)

− vm + 1

2log

(1 + (rmt − αm)2

τmtgmt(vm − 2)

),

L(ηi, vi) = log

(� ((vi + 1)/2)

� (vi/2) ((vi − 2)gitτit)1/2

)

− vi + 1

2log

(1 + (rit − αi − βirmt)2

τitgit(vi − 2)

),

i = 1, . . . , N, (19)

where � denotes the gamma function and the v’s refer to thecorresponding degrees of freedom. The resulting first stageestimates satisfy consistency under the identification conditionsof Newey and Steigerwald (1997).

Regarding the second stage, the estimation can be performedas a joint estimation of the whole correlation matrix using QMLmoment conditions. However, although DCC is very parsimo-nious in its parameterization, it is biased and slow for large

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114 Journal of Business & Economic Statistics, January 2012

covariance matrices. Alpha is biased downward and may ap-proach zero (see Engle and Sheppard 2005b). Since our empir-ical analysis includes a moderately large number of assets, weconsider two approaches that have recently been suggested inthe literature to reduce the bias problem: the MacGyver methodof Engle (2007), and the Composite Likelihood (CL) methodof Engle, Shephard, and Sheppard (2008). The first approach isappealing because it is easy to implement, although it has thedrawback of making it difficult to conduct inference. The secondmethod addresses this issue by defining a CL function as the sumof quasi-likelihoods of pairs of assets. In the following section,we present results using the standard multivariate method andthese two novel strategies. In Section 5, we perform a sequentialout-of-sample forecasting exercise using the CL method.

4. HIGH AND LOW FREQUENCY CORRELATIONS INTHE U.S. MARKET

4.1 Data

We use daily returns on DJIA stocks from December 1988to December 2006. The data are obtained from the Center forResearch in Security Prices (CRSP). The index underwent somechanges during this period, including additions, deletions, andmergers. We include all stocks in the 2006 index and those inthe 1988 index that could be followed over the sample period(Chevron (CVX), Goodyear (GT), and International Paper (IP)),obtaining a sample of 33 stocks. Regarding the market factor,we use daily returns on the S&P 500 and the one-month T-billrate as the time-varying risk-free rate. Stock names and tickersare described in the online Appendix A3.

4.2 Description of Estimation Results

We estimate the FSG-DCC model following the two-stageGMM approach described in Section 3 and present the resultsin the online Appendix A3 (Table A3-2). The parameters esti-mated in the first step are those in Equations (9), (10), and (19).For reasons of space, we only present the optimal number ofknots selected for the spline terms. The second step involvesthe DCC parameters in Equation (11). The first column of thistable shows the alphas which, in general, are not significantlydifferent from zero, as suggested by the CAPM framework (theonly exceptions are Altria Group (MO), General Motors (GM),and Goodyear (GT)). The second column reports the estimatedunconditional market betas associated with each stock. All arehighly significant. The third column presents the estimates of theARCH volatility coefficients. In general, they are significant, ex-cept for the market factor and IBM, and their median is 0.05. Thefourth column presents the estimates of the GARCH volatilityeffects. They are all significant and their median is 0.84. The fifthcolumn shows estimates of the leverage effects, which exhibit amedian of 0.03. They are statistically significant in about half ofthe cases and also positive. These results are consistent with theleverage theory of Black (1976) and Christie (1982). However,this effect is substantially higher and significant for the marketfactor, which provides stronger support for the volatility feed-back hypothesis. The sixth column shows the degrees of freedomof the univariate Student-t distributions, which fluctuate between

4 and 10, and their median is 6. These values are in line with thetraditional evidence of nonnormality and excess of kurtosis inequity returns. The last column of Table A3-2 shows the optimalnumber of knots in the spline functions according to the BIC.These numbers reflect changes in the curvature of the long-termtrend of idiosyncratic volatilities. Examples of such patterns forindividual stocks (Intel (INTC), Exxon Mobil (XOM), and In-ternational Paper (IP)) are shown in the first column of graphs inFigure 1. The bottom part of Table A3-2 presents the estima-tion results of the second stage. The first column reports thestandard DCC estimates from the traditional approach and theirstandard errors. The second and third columns present the Mac-Gyver DCC and the CL estimates, respectively. The estimatorof aDCC increases from 0.0027, using the standard multivariatemethod, up to 0.004 (or 0.005) when using the CL (or the Mac-Gyver) method. This suggests that both methods deliver a biascorrection of the traditional DCC estimator.

The third column of graphs in Figure 1 illustrates the timeseries properties of the FSG-DCC, with a number of exam-ples for individual stocks that show the high and low frequencycorrelation components and the model-free (100-day) rollingcorrelations. The high frequency component mean reverts to-ward the slow-moving low frequency component. It is visuallyclear by looking at the model-free rolling correlations that theFSG-DCC model characterizes fairly well the trend behavior incorrelations. The second column of graphs in Figure 1 illustrateswith some examples the model’s ability to capture time variationin the betas. The graphs show the conditional betas implied bythe FSG-DCC model, which are computed using Equation (6)and the model-based volatilities and correlations in Equations(9), (10), and (11) that allow us to obtain the relevant condi-tional covariance terms. To further illustrate the flexibility ofthe conditional betas, the graphs also present model-free (100-day) rolling betas computed from rolling regressions using thespecification in Equation (1).

4.3 Aggregate Volatility Components and AverageCorrelations

The most distinguishing feature of the FSG-DCC model is itsability to characterize dynamic long-term correlation behaviorby exploiting the structure of a factor asset pricing model andthe low frequency variation in systematic and idiosyncraticvolatilities. Engle and Rangel (2008) and Engle et al. (2008)provide evidence that the low frequency market volatilityresponds to changes in the slow-moving macroeconomicenvironment. This subsection illustrates the effects of the lowfrequency volatility components on aggregate correlations andpresents evidence that the aggregate low frequency componentof idiosyncratic volatility systematically varies with economicvariables. This evidence also holds at the sectoral level (wepresent an analysis based on the 48 equally weighted industryportfolios of Fama and French (1997) in Appendix A4).

Cross-sectional aggregation facilitates illustrating the effectof our volatility components on correlations. We constructaggregates by taking the cross-sectional average of our dynamiccomponents at each point in time. Hence, the aggregate averagecorrelation associated with model m for a specific period p isdefined as: ρmTp = 1

N

∑Ni=1 { 1

Tp

1(N−1)

∑j �=i

∑Tpt=1 ρ

mi,j,t},where Tp

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Rangel and Engle: Factor–Spline–GARCH Model 115

Figure 1. Idiosyncratic volatilities, betas, and correlations. NOTES: The estimation uses daily returns on the DJIA stocks from December1988 to December 2006. The data are obtained from CRSP. Company names are referred to by their tickers (INTC = Intel, XOM = ExxonMobil, and IP = International Paper). Volatilities column: HFV stands for “High frequency idiosyncratic volatility” (see the second equationin Equation (10)) and LFV refers to “Low frequency idiosyncratic volatility” (see the third equation in Equation (10)). The last number in theseries labels denotes the optimal number of knots. Betas column: The solid line plots the FSG-DCC conditional beta and the dashed line plotsthe rolling betas based on 100-day rolling regressions. Correlations column: LFC refers to “Low frequency correlation”, HFC stands for “Highfrequency correlation”, and CR refers to 100-day “Rolling correlations.”

denotes the number of daily observations in such a period, andρmi,j,t is a time-varying pairwise correlation (from model m). Theaverage low frequency idiosyncratic volatility for period p isdefined as IvolTp = 1

Tp

1N

∑Tpt=1

∑Ni=1 τ

1/2it . Figure 2 shows av-

erages for biannual subperiods (from the entire sample) of lowfrequency market and idiosyncratic volatilities. Before 1997–1998, while the average low frequency idiosyncratic volatilityshowed an increasing pattern, the low frequency market volatil-ity declined. This is consistent with the findings of Campbellet al. (2001) and suggests a decline in correlations, which is con-firmed by both the aggregate model-free rolling correlations andthe aggregate FSG-DCC correlations in Figure 3. After 1997,market and idiosyncratic volatilities seem to move in a similar

fashion with opposite effects on correlations which, at the aggre-gate level, follow a nonmonotonic pattern. An interesting effectis observed in the last period where, although market volatilityreached historical lows, correlations decreased only moderatelydue to the low levels of idiosyncratic volatility.

We have illustrated that low frequency variation in idiosyn-cratic volatilities is not negligible and can have importanteffects on the level of correlations. We now analyze the eco-nomic sources of this variation, keeping the analysis at the ag-gregate level. As mentioned in Section 3, the trend behaviorin idiosyncratic volatility has gained relevance in the literaturefollowing the results of Campbell et al. (2001). At the microlevel, the theoretical framework of Pastor and Veronesi (2003)

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116 Journal of Business & Economic Statistics, January 2012

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

89-90 91-92 93-94 95-96 97-98 99-00 01-02 03-04 05-06

Vola

tilit

ies

Period

Average Idiosyncratic VolatilityAverage Market Volatility

Figure 2. Average low frequency market and idiosyncratic volatilities for two-year periods. NOTES: The average low frequency volatility forperiod p is defined as follows: Average Annualized Idiosyncratic Volatility of Asset i = 1

Tp

∑Tp

t=1 τ1/2it , where τ 1/2

it is the annualized low frequencyvolatility for asset i at time t, ∀i = m, 1, 2, . . . , N , and Tp is the number of daily observations in period p.

suggests a positive relationship between idiosyncratic volatilityand both the variance of firm profitability and uncertainty aboutthe average level of firm profitability. In this regard, changes inthe profits of a firm are associated with two main causes. Thefirst cause is changes in the firm’s productivity, which can beexplained by supply-side factors such as technological change(e.g., Jovanovic 1982) and/or demand-side factors such as prod-

uct substitutability (e.g., Syverson 2004). The other cause isprice variations (of products and factors), which are also relatedto interactions between idiosyncratic demand shocks and thelevel of competition within the relevant industries. Therefore,fluctuations in demand (which could be related to changes intaste, technological changes, new trade liberalization policies,and new regulations, among others) can have an impact on these

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

89-90 91-92 93-94 95-96 97-98 99-00 01-02 03-04 05-06

Co

rrel

atio

ns

Period

Average Correlations

Rolling Correlations

FSG-DCC Correlations

Figure 3. Average correlations. NOTES: The average rolling correlation for period p is defined as follows: ρRollingTp

=1N

∑N

i=1 { 1Tp

1(N−1)

∑j �=i∑Tp

t=1 ρRollingi,j,t }, where Tp denotes the number of daily observations in period p, and ρRolling

i,j,t is the rolling correlationbetween assets i and j at time t. Similarly, the average FSG-DCC correlations are constructed by using ρi,j,t in Proposition 2.

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Rangel and Engle: Factor–Spline–GARCH Model 117

.1

.2

.3

.4

.5

.0004

.0006

.0008

.0010

.0012

.0014

.0016

90 92 94 96 98 00 02 04 06

Average Rolling Idiosyncratic VolatilityAverage Low frequency Idiosyncratic VolatilityEmployment Dispersion Index (EDI)

Figure 4. Employment dispersion index and aggregate idiosyncraticvolatilities. NOTES: The graph shows quarterly aggregates of the fol-lowing variables: (1) Average Rolling Idiosyncratic Volatility, defined

as ARIVt = 1N

∑N

i=1 ( 1100

∑100k=1 u

2i(t−k))

1/2, where uit is the daily id-

iosyncratic return from Equation (1). (2) Average Low FrequencyVolatility defined as Ivolt = 1

N

∑N

i=1 τ1/2it , where the τ ’s are daily

low frequency idiosyncratic volatilities (see Equation (10)). (3) Em-ployment Dispersion Index, which follows the definition as in Lilien(1982): EDIk = {∑11

i=1xikXk

(� log xik −� logXk)2}1/2,where xik is em-ployment in industry i (among 11 industry sectors) at quarter k, and Xk

is aggregate employment.

two fundamentals, as well as on their volatility and on the in-tensity of both firm- and industry-specific news.

At a more aggregate level, the theory of sectoral reallocationthat followed the work of Lilien (1982) explains how randomfluctuations in sectoral demand associate with sectoral shiftsin the labor market. The Employment Dispersion Index (EDI)suggested by Lilien (1982) proxies the intensity of these sec-toral shifts and can therefore be used as an indicator of id-iosyncratic news intensity. For example, a technological changecan induce important movements of labor and other productionfactors from declining to growing sectors. This sectoral reallo-cation of resources can be accompanied by a higher intensity ofidiosyncratic shocks and therefore by increases in firm-specificvolatility. Following this intuition, we associate the measureof Lilien with low frequency variation of aggregate idiosyn-cratic volatility. We also control for the economic variablesthat Guo and Savickas (2006) relate to the aggregate behav-ior of idiosyncratic volatility, such as the CAY of Lettau andLudvigson (2001), the market volatility, and the market liquid-ity. As in this study, all variables are aggregated at a quarterlyfrequency. The S&P 500 excess-returns volatility is used toproxy the market volatility. As in Chordia, Roll, and Subrah-manyam (2001), we use the average quoted spread (QSPR) tomeasure market liquidity.

We use two model-based measures of aggregate low fre-quency idiosyncratic volatility: 1) the cross-sectional averageof low frequency idiosyncratic volatilities, aggregated at a quar-terly level, and 2) the cross-sectional average of rolling moving

averages (based on a 100-day window) of squared idiosyncraticreturns from Equation (10), aggregated at a quarterly level.Figure 4 shows a graph of these two measures of aggregateidiosyncratic volatility and Lilien’s EDI from 1990 to 2006. Thevisual high correlation is confirmed in Table 1, which reports thesample Pearson’s correlation across our idiosyncratic volatilitymeasures and the economic explanatory variables mentionedabove. As expected, idiosyncratic volatility is positively corre-lated with the EDI and the market volatility. The rolling measureshows correlation coefficients of 0.48 and 0.72, respectively,whereas the spline measure shows correlations of 0.34 and 0.64,respectively. Moreover, idiosyncratic volatility is negatively cor-related with both CAY and market liquidity. Overall, the signsof these correlations are consistent with the results of Guo andSavickas (2006), and with the expected effects of sectoral real-location on the volatility of firms’ fundamentals. To explore fur-ther these relationships, we project separately the two measuresof idiosyncratic volatility on the explanatory variables over oursample period using a linear regression framework. Due to thenature of the idiosyncratic volatility aggregates, especially thespline measure, the regressions will be affected by a severe se-rial correlation problem in the residuals. To lessen this problemand address endogeneity issues associated with simultaneouscausality, we use the GMM with robust Newey and West (1987)standard errors, and four lags of the explanatory variables as in-struments. Table 2 reports the estimated coefficients associated

Table 1. Sample correlations: Idiosyncratic volatilities and economicvariables

Average idiosyncraticvolatilities (quarterly

frequency)a

Economic variablesd

AverageRolling

IdiosyncraticVolatilityb

Average LowFrequency

IdiosyncraticVolatilityc

Employment Dispersion Index (EDI) 0.48 0.34

Consumption–Wealth Ratio (CAY) −0.40 −0.39

Market factor volatility 0.72 0.64

Illiquidity (QSPR) −0.30 −0.23

NOTES: a) Average idiosyncratic volatilities are based on daily DJIA stock returns.Measures aggregated at a quarterly frequency for the 1990–2006 period.b) The Average Rolling Idiosyncratic Volatility is defined as

ARIVt = 1

N

N∑i=1

(1

100

100∑k=1

u2i(t−k)

)1/2

,

where uit is the daily idiosyncratic return from Equation (1).c) The Average Low Frequency Idiosyncratic Volatility is defined as

Ivolt = 1

N

N∑i=1

τ1/2it ,

where the τ ’s are the daily low frequency volatilities estimated from Equation (10).d) The EDI follows the definition of Lilien (1982):

EDIk =⎧⎨⎩

11∑i=1

xik

Xk(� log xik −� logXk)

2

⎫⎬⎭

1/2

,

where xik is employment in industry i (among 11 industry sectors) at quarter k, and Xk isaggregate employment; the CAY is defined as in Lettau and Ludvigson (2001); the marketfactor volatility is estimated according to Equation (9); and, the average quoted spread(QSPR) follows the definition as in Chordia et al. (2001).

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118 Journal of Business & Economic Statistics, January 2012

Table 2. Idiosyncratic volatility GMM regressions

Average RollingIdiosyncratic Volatility

Average LowFrequency

Idiosyncratic Volatility

Constant 0.082 0.080(0.037) (0.076)

EDI 104.581 117.298(53.314) (67.166)

CAY −0.154 −0.031(0.562) (0.699)

Market volatility 0.615 0.643(0.233) (0.218)

QSPR −0.083 −0.002(0.073) (0.087)

R2 0.533 0.365J-statistic 0.051 0.049

NOTES: The Average Rolling Idiosyncratic Volatility is defined as

ARIVt = 1

N

N∑i=1

(1

100

100∑k=1

u2i(t−k)

)1/2

,

where uit is the daily idiosyncratic return from Equation (1). The Average Low FrequencyIdiosyncratic Volatility is defined as

Ivolt = 1

N

N∑i=1

τ1/2it ,

where the τ ’s are the daily low frequency volatilities estimated from Equation (10). Theseaverages are based on daily DJIA stock returns. Aggregating the variables at a quarterlyfrequency from 1989 to 2006, the two measures of idiosyncratic volatility are regressed onthe following set of variables: the EDI of Lilien (1982), the CAY of Lettau and Ludvigson(2001), the market factor volatility estimated from Equation (9), and the average quotedspread (QSPR) of NYSE stocks (as defined in Chordia et al. (2001)). The regressions areestimated using the Generalized Method of Moments (GMM) with Newey–West standarderrors and four lags of the regressors as instruments (Standard errors are reported inparentheses, bold font indicates statistical significance at the 10% level).

with the two linear projections. The two regressions suggest thesame effects and, as in the sample correlation analysis, the es-timated coefficients show the expected sign. Nevertheless, onlythe EDI and the market volatility are statistically significant. It isimportant to mention that the EDI series is highly noisy beforethe mid-1980s. Structural breaks should therefore be taken intoaccount for analyses incorporating longer sample periods.

4.4 Empirical Fit of Factor Correlation Models

This subsection evaluates a range of one-factor models withvarying dynamic components in terms of their empirical fit usingour sample of 33 Dow stocks. The process follows a simple-to-general strategy. We start from the simplest case, labeled FC-C,where factor and idiosyncratic volatilities are constant over thesample period (at high and low frequencies), and restrictions inEquations (2) and (3) apply. We estimate the return correlationsfrom this model and use a Gaussian metric to compute the quasi-likelihood. We then consider subsequent models that relax oneor more assumptions of the initial model, and obtain their quasi-likelihoods. The last step involves comparing the empirical fitof this range of factor correlation models based on this Gaussianmetric. This allows us to assess which restrictions in the factorstructure are the most important to describe correlation behavior.The assumptions to be weakened are: 1) constant volatility ofthe market factor, 2) constant volatility of the idiosyncratic com-ponents of returns, 3) no omitted factors, and 4) constant betas.

By adding high frequency variation to the volatility of thefactor through a standard GARCH process, and keeping the id-

iosyncratic volatilities constant, we obtain a specification calledFG-C. Similarly, when high and low frequency spline–GARCHdynamics are added to the market factor volatility holding theidiosyncratic volatilities constant, we obtain the FSG-C model.These models and their correlations, along with a range of spec-ifications derived from adding dynamics to the previous as-sumptions, are described in Table 3. Their quasi-likelihoods areconstructed from the general factor structure in Equation (16) as-suming that rt |�t−1 ∼ N (0,Ht ). A mapping of each correlationspecification in Table 3 with a specific covariance matrix pro-vides the inputs to compute the quasi-likelihood of each model.We estimate the models in the first panel of Table 3 using QML.For the other models, we follow a two-stage GMM approachwith Gaussian moment conditions.

Table 4 presents the log quasi-likelihoods of the factor mod-els in Table 3, along with likelihood ratio tests that compareeach model with the biggest FSG-DCC model. The results in-dicate that the FSG-DCC model dominates the other specifi-cations. Close to this model is the FSG-IDCC model, whichassumes constant betas, but allows for aggregate effects of un-specified unobservable factors and both high and low frequencyvariation in market and idiosyncratic volatilities. The FG-DCCmodel follows in the list. The three biggest models show thebest empirical fit even when they are penalized by the BIC (seeTable 4). In contrast, the specification with poorest empiricalfit is the constant correlation model (FC-C). Thus, we find thatmodels with low frequency dynamics dominate those with onlyhigh frequency dynamics. The results also show large improve-ments in the quasi-likelihoods when we relax the assumptionsof “constant idiosyncratic volatilities” and “no omitted factors.”This confirms that, besides the importance of modeling marketbehavior, adding dynamic features to the second moments of id-iosyncratic terms largely improves the empirical fit of this classof one-factor CAPM models.

5. FORECAST PERFORMANCE OF THEFACTOR–SPLINE–GARCH MODEL

Our forecast comparison follows the approach of Engle andColacito (2006) by using an economic loss function to assessthe performance of each model. However, different from them,we use forward portfolios based on forecasts of the covariancematrix associated with each of the models to be compared.Specifically, we focus on a portfolio problem where an investorwants to optimize today his/her forward asset allocation givena forward conditional covariance matrix. In the classical meanvariance setup, this problem can be formulated as

minwt+k|t

w′t+k|tHt+k|twt+k|t s.t. w′

t+k|tµ = µ0, (20)

where µ is the vector of expected excess returns, µ0 is therequired return,wt+k|t denotes a portfolio at time t + k that wasformed using the information at time t, and Ht+k|t is a k-stepahead forecast (at time t) of the conditional covariance matrixof excess returns. The solution to Equation (20) therefore is

wt+k|t = H−1t+k|tµ

µ′H−1t+k|tµ

µ0, (21)

and represents optimal forward portfolio weights given the in-formation at time t. Each covariance forecast Ht+k|t implies a

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Rangel and Engle: Factor–Spline–GARCH Model 119

Table 3. Correlation models from factor assumptions

Panel 1: Dynamic volatility componentsa

Constant volatilitiesComponents with high and low frequency dynamics

Factor component Factor and idiosyncratic components

FC-C: Constant factor andidiosyncratic volatilities

FG-C: GARCH factor volatility FG-G: GARCH factor andidiosyncratic volatilities

ρFG−Ci,j,t = βiβj hmt√

β2ihmt+σ 2

i

√β2jhmt+σ 2

j

ρFG−Gi,j,t = βiβj hmt√

β2ihmt+hit

√β2jhmt+hjt

ρFC−Ci,j = βiβj σ

2m√

β2iσ 2m+σ 2

i

√β2jσ 2m+σ 2

j

FSG-C: Spline–GARCH factor volatility FSG-SG: Spline–GARCH factorand idiosyncratic volatilities

ρFSG−Ci,j,t = βiβj τmt gmt√

β2iτmt gmt+σ 2

i

√β2jτmt gmt+σ 2

j

ρFSG−SGi,j,t = βiβj τmt gmt√

β2iτmt gmt+τit git

√β2jτmt gmt+τjt gjt

Panel 2: Other dynamic componentsb

Idiosyncratic correlations (latent unobserved factors in the error term) All components

FG-IDCC: FG-G with latent omitted factors FG-DCC

ρFG−IDCCi,j,t = βiβj hmt+

√hit

√hjt ρ

ui,j,t√

β2ihmt+hit

√β2jhmt+hjt

See Equation (15)

FSG-IDCC: FSG-SG with latent omitted factors FSG-DCC

ρFSG−IDCCi,j,t = βiβj τmt gmt+√

τit git√τjt gjt ρ

ui,j,t√

β2iτmt gmt+τit git

√β2jτmt gmt+τjt gjt

See Equation (12)

NOTES: a) σ denotes constant (C) volatilities, h and g refer to GARCH (G) and Spline–GARCH (SG) variances, respectively.b) These models are parameterizations of Equations (4) and (5) in Proposition 1.

particular forward portfolio wt+k|tgiven a vector of expectedreturns. However, an important issue arises when the true vec-tor of expected returns is unknown. Engle and Colacito (2006)mentioned that a direct comparison of optimal portfolio volatil-ities can be misleading if a particular estimate of the expectedexcess returns, such as their realized mean, is used to com-pute such volatilities. Their framework isolates the effect ofcovariance information by using a wide range of alternativesfor the vector of expected excess returns and the asymptoticproperties of sample standard deviations of optimized portfo-lio returns. We follow their approach and consider differentvectors of expected excess returns associated with a variety ofmultivariate hedges that are constructed by holding one asset

for return and using the other assets for hedge. We then com-pare the standard deviations of returns on long-term forwardhedge portfolios formed with each model’s covariance matrixforecast.

5.1 Out-of-Sample Evaluation: A SequentialForecasting Exercise

We examine the out-of-sample forecast performance of theFSG-DCC model using our group of 33 Dow stocks. Our anal-ysis implements a sequential exercise that is described in Fig-ure 5 and can be characterized by iterations of the followingprocedure: 1) a set of models is estimated based on an initial

Table 4. Evaluation of factor correlation models

Assumptiona Model Quasi-log- likelihood Parameters Quasi-likelihood ratioc BIC

FC-C −426550 67 45980∗ −186.961 FG-C −427110 69 44860∗ −187.20

FSG-C −427160 75 44760∗ −187.212 FG-G −440670 135 17740∗ −193.03

FSG-SG −442820 383 13440∗ −193.513 FG-IDCC −446930 663 5220∗ −194.80

FSG-IDCC −449100 911 880∗ −195.294b FG-DCC −447330 696 4420∗ −194.91

FSG-DCC −449540 944 −195.42

NOTES: a) This column corresponds to the assumptions that are weakened in the factor specifications:1. Constant volatility of the market factor.2. Constant volatility of the idiosyncratic components.3. No omitted factors.4. Constant betas.The models are described in Table 3 and the sample includes the 33 DJIA stocks described in Appendix A3 (Table A3-1).b) The last step is associated with the full FG-DCC and FSG-DCC models.c) The quasi-likelihood ratios compare the models on each row with the FSG-DCC model (see last row).∗) Indicates that the quasi-likelihood ratios are above the 1% critical value of a chi-square distribution with degrees of freedom given by the difference between 944 and the number ofparameters associated with each model.

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120 Journal of Business & Economic Statistics, January 2012

Figure 5. Sequential forecasting exercise. NOTES: This figure describes the iterations of an out-of-sample sequential forecasting exercise.In the first iteration, a number of covariance models are estimated using data from 12/01/1988 to 12/31/1995. Then, multistep out-of-sampleforecasts are constructed for each model during a six-month period (from 01/01/1996 to 06/30/1996). In iteration 2, the estimation period isextended six months (from 12/01/1988 to 06/30/1996) and a new set of multistep out-of-sample forecasts are generated for the following sixmonths. At each iteration, the estimation period incorporates the previous iteration’s forecasting period and new out-of-sample forecasts aregenerated. Twenty-two iterations are completed, ending at 12/31/2006. The forecasting periods do not overlap.

period with T0 daily observations, and multihorizon covarianceforecasts from 1 to 126 days ahead are computed out of the sam-ple, 2) the 126-day forecasting period is incorporated into a newestimation period with T1 = T0 + 126 daily observations, themodels are re-estimated and, finally, a new set of out-of-sampleforecasts is constructed for the following six months. We iter-ate these two steps several times starting with a sample periodfrom December 1988 to June 1995. The process is repeated 22times (up to December 2006) and none of the out-of-sampleforecast blocks overlap (see Figure 5). The exercise consid-ers the FSG-DCC specification and five competing models: 1)the stationary DCC, 2) the FG-DCC, 3) the sample covariance(SCOV), 4) the static one-factor beta covariance (BCOV), and5) the optimal shrinkage covariance of Ledoit and Wolf (2003)(LCOV). The last three models are analyzed in Jagannathan andMa (2003). All models are re-estimated at each iteration. Forthe FSG-DCC model, we extrapolate the spline functions bykeeping them constant at their last values during the forecastingperiods and restricting them to have a zero-slope in the last ob-servation of the sample. By providing a conservative approachthat underestimates the behavior of the series near the end ofthe sample, the zero-slope condition helps to reduce potentialanomalies caused by outliers near the right boundary (e.g., Sil-verman 1984 and Nychka 1995 illustrate the boundary issuesassociated with smoothing splines).

We focus on long-horizon covariance forecasts to form op-timized forward hedge portfolios according to (21), using avariety of vectors of expected returns associated with differenthedges and a required return normalized to 1. From the multi-horizon forecasts generated at each iteration, we only considerthe last 40 days (i.e., out-of-sample forecast from 87 to 126 daysahead). We compute standard deviations of returns on these out-of-sample portfolios as

σ(j )p,OS =

√√√√∑22i=1

∑126k=87

{w

′(j )p,Ti+k|Ti (rTi+k − r)

}2

22 × 40,

j = FSG, FG-DCC, DCC, SCOV,BCOV, LCOV,p = 1, 2, . . . , 33,

(22)

where Ti is the last day of the estimation period associated withiteration i, r denotes the sample mean of daily excess returns,rTi+k is the vector of one-day excess returns at day Ti + k, andw

(j )p,Ti+k|Ticorresponds to a Ti + k-day forward hedge portfolio

constructed from covariance model j, where asset p is hedgedagainst the other assets.

Table 5 reports these standard deviations for each model andhedge portfolio. The FSG-DCC specification produces smallestvolatilities for 15 hedges; the sample covariance is preferred forfive hedges; the shrinkage covariance and the FG-DCC modelsdominate in four cases each; the DCC model is preferred inthree cases; and, the static beta covariance model dominates inonly one case. These results suggest that the FSG-DCC modelis the preferred model in most of the cases. Furthermore, theaverage across all hedges (shown in the last row of the table)also indicates that the best performer is the FSG-DCC specifica-tion. We also performed an in-sample forecasting exercise andobtained the same conclusions (see online Appendix A5). Sincethe differences across models appear small, we examine furtherthe significance of these results by performing joint Dieboldand Mariano (1995) style tests following Engle and Colacito(2006). The purpose is to test the equality of the FSG-DCCmodel with respect to each of its competitors. This approach isbased on statistical inference about the mean of the differencebetween square returns on optimized portfolios generated bythe FSG-DCC specification and a competitor model m. For eachiteration i (with last observation = Ti), a vector of differenceseries associated with hedge p is defined as

up,m

Ti={(w′FSG−DCCp,Ti+k|Ti (rTi+k − r)

)2

− (w′mfh,Ti+k|Ti (rTi+k − r)

)2, k = 87, . . . , 126

}, (23)

where m = FG-DCC, DCC, SCOV, LCOV, BCOV, and p =1, 2, . . . , 33. Using these hedge difference vectors, we constructa joint difference vector that stacks all of them as follows:

UFSG−DCC,mTi

=(u

1,mTi, u

2,mTi, . . . , u

33,mTi

), i = 1, 2, . . . , 22.

(24)

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Rangel and Engle: Factor–Spline–GARCH Model 121

Table 5. Out-of-sample evaluation: Standard deviations of optimizedforward hedge

Portfolios at multiple long horizons

Forwardhedgeportfolios FSG-DCC FG-DCC DCC BCOV SCOV LCOV

µAA 0.2573 0.2569 0.2569 0.2787 0.2549 0.2550µAIG 0.1914 0.1947 0.1917 0.1949 0.1925 0.1923µAXP 0.2130 0.2138 0.2137 0.2293 0.2151 0.2152µBA 0.2690 0.2702 0.2697 0.2744 0.2715 0.2714µC 0.1979 0.1992 0.1985 0.2182 0.1979 0.1983µCAT 0.2445 0.2461 0.2452 0.2532 0.2455 0.2451µCVX 0.1593 0.1586 0.1575 0.2065 0.1584 0.1592µDD 0.1997 0.2009 0.2013 0.2119 0.1981 0.1980µDIS 0.2718 0.2695 0.2708 0.2708 0.2726 0.2723µGE 0.1921 0.1947 0.1918 0.1874 0.1913 0.1909µGM 0.3035 0.3061 0.3055 0.3069 0.3043 0.3040µGT 0.3509 0.3504 0.3508 0.3645 0.3547 0.3546µHD 0.2704 0.2707 0.2705 0.2837 0.2703 0.2704µHON 0.2714 0.2756 0.2719 0.2812 0.2718 0.2718µHPQ 0.3688 0.3735 0.3677 0.3925 0.3665 0.3668µIBM 0.2363 0.2428 0.2367 0.2506 0.2388 0.2386µINTC 0.3325 0.3394 0.3347 0.3794 0.3351 0.3357µIP 0.2292 0.2296 0.2287 0.2531 0.2279 0.2283µJNJ 0.1792 0.1808 0.1797 0.1972 0.1803 0.1797µJPM 0.2169 0.2183 0.2190 0.2427 0.2187 0.2189µKO 0.2161 0.2163 0.2149 0.2232 0.2150 0.2150µMCD 0.2522 0.2541 0.2525 0.2531 0.2525 0.2523µMMM 0.1937 0.1965 0.1938 0.1977 0.1925 0.1924µMO 0.3172 0.3204 0.3174 0.3210 0.3176 0.3176µMRK 0.2156 0.2162 0.2156 0.2417 0.2142 0.2141µMSFT 0.2474 0.2510 0.2485 0.2711 0.2485 0.2481µPFE 0.2439 0.2482 0.2438 0.2710 0.2440 0.2439µPG 0.1827 0.1826 0.1824 0.1991 0.1820 0.1820µT 0.2139 0.2161 0.2145 0.2522 0.2158 0.2168µUTX 0.2158 0.2204 0.2166 0.2195 0.2157 0.2154µVZ 0.1924 0.1953 0.1913 0.2237 0.1901 0.1904µWMT 0.2351 0.2370 0.2349 0.2500 0.2364 0.2366µXOM 0.1517 0.1506 0.1513 0.1966 0.1518 0.1528All portfolios 0.2374 0.23757 0.2393 0.2545 0.23764 0.2377

NOTES: Sample standard deviations of returns on optimized forward hedge portfoliossubject to a required return of 1 and based on 22 iterations of out-of-sample covarianceforecasts at horizons from 87 to 126 days ahead. The forecasts are constructed fromFSG-DCC, FG-DCC, DCC, BCOV (static one-factor beta covariance), SCOV (samplecovariance), and LCOV (optimal shrinkage covariance of Ledoit and Wolf (2003)) models,respectively. The 22 sequential sample periods are described in Figure 5 and include the 33DJIA stocks described in Appendix A3 (Table A3-1). The stock in the corresponding rowis hedged against the rest of the stocks.

The null of equality of covariance models tests that themean of UFSG−DCC,m

t equals zero. For each comparison, thetest is performed by running a regression of UFSG−DCC,m

t ona constant. The regressions are estimated by GMM usingrobust heteroscedasticity and autocorrelation consistent (HAC)covariance matrices. Table 6 reports the t-statistics for theDiebold–Mariano tests (each column names a competingmodel). A negative value suggests that the FSG-DCC modeldominates over the column model. At a 5% confidence level,the results indicate once more that the FSG-DCC dominates itscompetitors at long horizons.

Table 6. Joint Diebold–Mariano tests to compare the long-termforecast performance of the FSG-DCC model relative to competing

models

Competing models

FSG-DCCvs. columnmodel DCC FG-DCC LCOV SCOV BCOV

t-statistics −3.66 −2.96 −2.14 −2.07 −5.72

NOTES: This table reports t-statistics for joint Diebold–Mariano tests that evaluate theforecast performance of the FSG-DCC model in relation to the following competitors:DCC, FG-DCC, BCOV (static one-factor beta covariance), SCOV (sample covariance),and LCOV (optimal shrinkage covariance of Ledoit and Wolf (2003)). Each t-statistic isderived from estimating a regression of a vector of differences of squared realized returnsof two models on a constant. The vector of differences is constructed using squared realizedforward hedge portfolio returns associated with the FSG-DCC model and the competingcolumn model. The forward hedge portfolios are constructed using the sample periodsdescribed in Figure 5 and the 33 DJIA stocks in Appendix A3 (Table A3-1). We includeonly long-term forward hedge portfolios (87–126 days forward). The vector regressionsare estimated using the GMM with a HAC covariance matrix. A negative value indicates abetter performance of the FSG-DCC model.

6. CONCLUDING REMARKS

This article develops a new model for asset correlations thatcharacterizes dynamic patterns at high and low frequenciesin the correlation structure of equity returns. By exploitinga factor asset pricing structure and the dynamic propertiesof low frequency volatilities associated with systematic andidiosyncratic terms, we introduce a slow-moving correlationcomponent that proxies low frequency changes in correlations.Our semiparametric approach generalizes DCC and othermultivariate GARCH models by allowing the high frequencycorrelation component to mean revert toward a time-varyinglow frequency component. This framework allows the levelto which conditional correlations mean revert to adapt to thevarying economic conditions.

At high frequencies, our model incorporates dynamic effectsthat arise from relaxing assumptions in the standard one-factorCAPM model. These effects account for time-varying betasand missing pricing factors. At low frequencies, the long-termtrends of the market and the idiosyncratic volatilities govern thedynamics of low frequency correlations. We provide evidencethat, in addition to the recently documented economic variationof market volatility at low frequencies, average idiosyncraticvolatility exhibits substantial variation in its long-term trend. Wefind that this variation is highly correlated with low frequencyeconomic variables including an intersectoral employment dis-persion index that proxies the intensity of sectoral reallocationof resources in the economy and, since these movements aremainly driven by shocks that are specific to either individualfirms or sectors, it also serves as an indicator of idiosyncraticnews intensity.

The ability of our correlation model to incorporate non-parametrically low frequency features improves not only theempirical fit of equity correlations and their association witheconomic conditions, but the forecasting of correlations as well.Indeed, in-sample and out-of-sample forecasting experimentsindicate that, at long horizons, this new model with time-varyinglong-term trends outperforms standard models that mean revert

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122 Journal of Business & Economic Statistics, January 2012

to fixed levels. This result is explained by the model’s flexibilityin adjusting the level of mean reversion to the varying economicconditions.

APPENDIX A1

Proof of Proposition 1. Part a): Consider the specification inEquation (1), and assumptions (2) and (3). If Et−1(rm,tuit) = 0,∀i, but Et−1(uitujt) �= 0, for some i �= j , then covt−1(ri, rj ) =βiβjVt−1(rmt) + Et−1(uitujt), and Equation (4) follows from thedefinition of conditional correlation.

Part b): Given that Et−1(uitujt) �= 0, for some i �= j,andEt−1(rm,tuit) �= 0, for some i, then the relevant covariance andvariance terms are

covt−1(ri, rj ) = βiβjVt−1(rmt) + βiEt−1(rmtujt)

+βjEt−1(rmtuit) + Et−1(uitujt), (A.1)

and

Vt−1(rkt ) = β2k Vt−1(rmt) + Vt−1(ukt )

+ 2βkEt−1(rmtukt ), k = i, j. (A.2)

Equation (5) is thus obtained by combining Equations (A.1)and (A.2) with the definition of conditional correlation.

Part c): Under the representation in Equation (6), the condi-tional covariance between market returns and the idiosyncraticresidual will be zero. In fact, using the definition of conditionalbeta:

covt−1(rit, rmt) = βitVt−1(rmt), (A.3)

then, substituting the return equation in Equation (6) into thisexpression gives

covt−1(βitrmt + uit, rmt) = βitVt−1(rmt) + covt−1(uit, rmt).

(A.4)

Thus, from equalizing Equations (A.3) and (A.4), we have

covt−1(uit, rmt) = 0, (A.5)

and Equation (2) is satisfied. Now, rewriting Equation (6) as

rit = βirmt + uit, βi = cov (rit, rmt)

V (rmt),

uit = uit + (βit − βi) rmt, ∀i, (A.6)

and using Equations (A.3) and (A.5), it is clear that the new errorterm in Equation (A.6) also satisfies assumption (2). Moreover,combining Equations (A.5) and (A.6) yields the following con-ditional moments:

covt−1(uit, rmt) = (βit − βi)Vt−1(rmt), ∀i, (A.7)

covt−1(uit, ujt) = covt−1(uit, ujt)

+ (βit − βi)(βjt − βj )Vt−1(rmt). (A.8)

The numerator in (5) can be rewritten as

covt−1(ri, rj ) = βiβjVt−1(rmt) + βicovt−1(rmt, ujt)

+βjcovt−1(rmt, uit) + covt−1(uit, ujt), (A.9)

and by substituting Equations (A.7) and (A.8) into this equation,the numerator in Equation (7) is obtained.

Similarly, the expressions in the denominator of Equation (5)can be rewritten as

Vt−1(rkt ) = β2k Vt−1(rmt) + Vt−1(ukt )

+ 2βkcovt−1(rmt, ukt ), k = i, j. (A.10)

Now, from Equations (A.5) and (A.6), Vt−1(ukt ) =Vt−1(ukt ) + (βkt − βk)2Vt−1(rmt), k = i, j . Hence, pluggingthese expressions and Equation (A.7) into Equation (A.10) leadsto the denominator in (7).

APPENDIX A2

Proof of Proposition 2. Consider the following vectors of re-turns, factor loadings, and innovations: rt = (r1t , r2t , . . . , rNt )′,β = (β1t , β2t , . . . , βNt )′, and ut = (u1t , u2t , . . . , uNt )′. Given avector Ft of common factor(s) and omitting the constant terms,without loss of generality, we can rewrite the model in Equation(1) as

rt = βFt + ut . (A.11)

Given the t−1 information set �t−1, the conditional covari-ance matrix therefore is

Et−1(rt r′t ) = βEt−1(FtF

′t )β

′ + βEt−1(Ftu′t )

+Et−1(utF′t )β

′ + Et−1(utu′t ). (A.12)

In particular, for the one-factor CAPM case, Ft = rmt andEquation (A.12) takes the following form:

Et−1(rt r′t ) = Vt−1(rmt)ββ

′ + βEt−1(rmtu′t )

+Et−1(utrmt)β′ + Et−1(utu

′t ). (A.13)

From Equation (9), the typical (i, j) element of the first termon the right-hand side (RHS) of Equation (A.13) is

βiβjVt−1(rmt) = βiβj τmtgmt. (A.14)

Similarly, from Equations (9)–(11), the typical (i, j) elementof the second term is

βiEt−1(rmtujt) = βi√τmtgmt

√τjtgjtρ

εm,j,t , (A.15)

the typical (i,j) element of the third term is

βjEt−1(rmtuit) = βj√τmtgmt

√τitgitρ

εm,i,t, (A.16)

and, the typical (i,j) element of the last term is

Et−1(uitujt) = √τitgit

√τjtgjtρ

εi,j,t. (A.17)

Equation (12) follows by plugging these conditional expec-tations and Vt−1(ukt ) = τktgkt , k = i, j, into Equation (6).

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Rangel and Engle: Factor–Spline–GARCH Model 123

The unconditional version of Equation (5) is then used toderive the low frequency correlation

ρi,j = [βiβjV (rmt ) + βjE(rmtuit ) + βiE(rmtujt ) + E(uitujt )

]× [β2i V (rmt ) + V (uit ) + 2βiE(rmtuit )

]−1/2

× [β2j V (rmt ) + V (ujt ) + 2βjE(rmtujt )

]−1/2

Under the assumption that both the factor(s) and the idiosyn-crasies are unconditionally uncorrelated, we have

ρi,j = βiβjV (rmt)+E(uitujt)√β2i V (rmt)+V (uit)

√β2j V (rmt)+V (ujt)

, (A.18)

and, from (9), (10), and the law of iterated expectations (LIE):

V (rmt) = τmtE(gmt) = τmt,

and

V (uit) = τitE(git) = τit, ∀i = 1, 2, . . . , N.

Also,

E(uitujt)√τit

√τjt

= E(g

1/2it εitg

1/2jt εjt

)√E(git)

√E(gjt)

= corr(g

1/2it εit, g

1/2jt εjt

) ≡ ρεi,j.

Note that ρεi,j,t = ρεi,j,t, ∀t , thus we approximate ρεi,j with thesample correlation, ρεi,j , from Equation (11). Plugging the pre-vious expressions into Equation (A.18), we obtain the time-varying low frequency correlation in Equation (13).

Moreover, if we assume Et (τk,t+h) = τk,t ,∀h > 0, k =1, 2, . . . , N, then the long-horizon forecast of Equation (12)can be constructed using the mean reversion properties of theGARCH and DCC equations. Indeed, the GARCH dynamics im-ply lim

h→∞gk,t+h|h = 1, ∀k = m, 1, 2, . . . , N . The long-horizon

correlation forecasts associated with the vector of innovationsare also given by the terms targeting correlations (see Equation(11)):

limh→∞

ρεi,j,t+h = limh→∞

qi,j,t+h|t√qi,i,t+h|t

√qj,j,t+h|t

= ρεi,j√ρεi,i

√ρεj,j

= ρεi,j , (A.19)

∀i, j ∈ {1, 2, . . . , N}. Also, from our assumption that the id-iosyncrasies are unconditionally uncorrelated with the fac-tor, ρεm,i = 0 and lim

h→∞ρm,i,t+h|t = 0, ∀i = 1, . . . , N . Hence, by

substituting the long-run forecasts of each term in Equation (12),we obtain

limh→∞

ρi,j,t+h|t = βiβj τmt + √τit

√τjtρ

εi,j√

β2i τmt + τit

√β2j τmt + τjt

, (A.20)

which coincides with the low frequency correlation.

SUPPLEMENTAL MATERIALS

This appendix expands on the material presented in three sec-tions of the paper: 1) it provides a more detailed description of

the stocks used in our empirical analyses and the FSG-DCC es-timates described in Subsection 4.2, 2) it presents an empiricalexercise that extends the results in Subsection 4.3 by exploringthe association between sectoral idiosyncratic volatilities andeconomic variables, and 3) it presents results from an in-sampleforecasting evaluation that complements the out-of-sample re-sults presented in Subsection 5.1.

ACKNOWLEDGMENTS

We thank Siddhartha Chib, Stephen Figlewski, Eric Ghy-sels, Massimo Guidolin, Andrew Karolyi, Andrew Patton, JeffRussell, Kevin Sheppard, and seminar/conference participantsat the Federal Reserve Bank of New York, LSE, NYU Stern,Ohio State University, the University of Arizona, Cambridge,Chicago, Manchester, Warwick, the 2007 LAMES conference,the 2007 Multivariate Volatility Models Conference, the 2008Frank Batten Young Scholar Conference in Finance, the 2008Stanford University SITE Conference, and the 2009 SoFiE Con-ference for comments and suggestions. All errors are the respon-sibility of the authors. The final draft of this paper was writtenwhile Jose Gonzalo Rangel was working at the Central Bank ofMexico. The opinions expressed in this article are solely thoseof the authors and do not necessarily reflect the views of theCentral Bank of Mexico or Goldman Sachs Group, Inc.

[Received May 2009. Accepted May 2011.]

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