Modeling Conditional Correlations of Asset Returns: A Smooth Transition Approach

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Modeling Conditional Correlations of Asset Returns: ASmooth Transition ApproachAnnastiina Silvennoinen a & Timo Tersvirta ba School of Economics and Finance , Queensland University of Technology , Brisbane ,Queensland , Australiab CREATES, Department of Economics and Business , Aarhus University , Aarhus , DenmarkAccepted author version posted online: 30 Jul 2014.Published online: 14 Oct 2014.

To cite this article: Annastiina Silvennoinen & Timo Tersvirta (2015) Modeling Conditional Correlations of Asset Returns: ASmooth Transition Approach, Econometric Reviews, 34:1-2, 174-197, DOI: 10.1080/07474938.2014.945336

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Econometric Reviews, 34(12):174197, 2015Copyright Taylor & Francis Group, LLCISSN: 0747-4938 print/1532-4168 onlineDOI: 10.1080/07474938.2014.945336

Modeling Conditional Correlations of Asset Returns:A Smooth Transition Approach

Annastiina Silvennoinen1 and Timo Tersvirta21School of Economics and Finance, Queensland University of Technology,

Brisbane, Queensland, Australia2CREATES, Department of Economics and Business, Aarhus University,

Aarhus, Denmark

In this paper we propose a new multivariate GARCH model with time-varying conditionalcorrelation structure. The time-varying conditional correlations change smoothly betweentwo extreme states of constant correlations according to a predetermined or exogenoustransition variable. An LMtest is derived to test the constancy of correlations and LM- andWald tests to test the hypothesis of partially constant correlations. Analytical expressions forthe test statistics and the required derivatives are provided to make computations feasible.An empirical example based on daily return series of ve frequently traded stocks in theS&P 500 stock index completes the paper.

Keywords Constant conditional correlation; Dynamic conditional correlation; MultivariateGARCH; Return comovement; Variable correlation GARCH model; Volatility modelevaluation.

JEL Classication C12; C32; C51; C52; G1.

1. INTRODUCTION

During major market events, correlations change dramatically Bookstaber(1997)

Financial decision makers usually deal with many nancial assets simultaneously.Modeling individual time series separately is thus an insufcient method as it leaves outinformation about comovements and interactions between the instruments of interest.Investors are facing risks that affect the assets in their portfolio in various ways, whichencourages them to nd a position that allows hedging against losses. In practice, thisis often done by trying to diversify across many stock markets. When optimizing aportfolio, correlations among, say, international stock returns are needed to determine

Address correspondence to Annastiina Silvennoinen, School of Economics and Finance, QueenslandUniversity of Technology, GPO Box 2434, Brisbane QLD 4001, Australia; E-mail: [email protected]

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MODELING CONDITIONAL CORRELATIONS 175

gains from international portfolio diversication, and also the calculation of minimumvariance hedge ratio needs updated correlations between assets in the hedge. Evidencethat the correlations between national stock markets increase during nancial crises butremain more or less unaffected during other times can be found for instance in King andWadhwani (1990), Lin et al. (1994), de Santis and Gerard (1997), and Longin and Solnik(2001). As further examples, options depending on multiple underlying assets are verysensitive to correlations among those assets, and asset pricing models as well as some riskmeasures need measures of covariance between the assets in a portfolio. It is clear thatthere is a need for a exible and accurate model that can incorporate the information ofpossible comovements between the assets.

Volatility in multivariate nancial data has been typically modeled by applyingthe concept of conditional heteroskedasticity originally introduced by Engle (1982);see Bauwens et al. (2006) and Silvennoinen and Tersvirta (2009) for recent reviewson multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH)models. In the multivariate context, one also has to model the conditional covariances,not only the conditional variances. One possibility is to model the former directly andanother is to do that through conditional correlations. One of the most frequently usedmultivariate GARCH models is the Constant Conditional Correlation (CCC) GARCHmodel of Bollerslev (1990). In this model comovements between heteroskedastic timeseries are modeled by allowing each series to follow a separate GARCH process whilerestricting the conditional correlations between the GARCH processes to be constant.The estimation of parameters of the CCCGARCH model is relatively simple and themodel has thus become popular among practitioners.

In practice, the assumption of constant conditional correlations has often been foundtoo restrictive. In order to mitigate this problem, Tse and Tsui (2002) and Engle(2002) dened dynamic conditional correlation GARCH models (VCGARCH andDCCGARCH, respectively) that impose GARCH-type dynamics on the conditionalcorrelations as well as on the conditional variances. These models are exible enough tocapture many kinds of heteroskedastic behavior in multivariate series. However, due totheir structure, these models have limited capability to explain what drives correlations.

Pelletier (2006) proposed a model with a regime-switching correlation structure drivenby an unobserved state variable that follows a K-dimensional rst-order Markov chain.The regime-switching model asserts that the correlations remain constant in each regimeand the change between the states is abrupt and governed by transition probabilities.There exists a parsimonious version of this model that contains two (extreme) correlationmatrices, one of which has all correlations equal to zero. In order to compensate for thatrestriction, the number of states can be made large so that the correlations are describedby as many linear combinations of the two extreme correlation matrices as there arestates. This model is motivated by the empirical nding that the correlations among assetreturns tend to increase during periods of distress whereas the series behave in a moreindependent manner in tranquil periods.

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176 A. SILVENNOINEN AND T. TERSVIRTA

In this paper we introduce another way of modeling comovements in the returns.The Smooth Transition Conditional Correlation (STCC) GARCH model allows theconditional correlations to change smoothly from one state to another as a functionof a transition variable. This continuous variable may be a combination of observablestochastic variables, or a function of lagged error terms.

The model has the appealing feature that it provides a framework in which constancyof the correlations, and thus the adequacy of the model, can be tested in a straightforwardfashion. Implications of the STCCGARCH correlation structure for the effects ofnews on the covariances can be considered through news impact surfaces, introducedby Kroner and Ng (1998). This concept can be easily adapted to the STCCGARCHcontext.

A special case of the STCCGARCH model was independently introduced by Berbenand Jansen (2005). Their model is bivariate, and the variable controlling the transitionbetween the extreme regimes is simply the time.

The paper is organized as follows. In Section 2 the model is introduced and theestimation of its parameters by maximum likelihood considered. Section 3 is devotedto tests of constant correlations. An application to illustrate the capabilities of themodel can be found in Section 4. Section 5 concludes. Technical derivation of the teststatistics in the paper and other relevant tests is available in Additional Material(AM) at http://econ.au.dk/research/research-centres/creates/research/research-papers/supplementary-downloads/.

2. THE SMOOTH TRANSITION CONDITIONAL CORRELATION GARCH MODEL

2.1. The General Multivariate GARCH Model

Consider the following stochastic N -dimensional vector process with the standardrepresentation label:

yt = E[yt |t1] + t, t = 1, 2, ,T , (1)

where t1 is the sigma-eld generated by all the information until time t 1. Each ofthe univariate error processes has the specication

it = h1/2it zit,

where the errors zit form a sequence of independent random variables with mean zeroand variance one, for each i = 1, ,N . The conditional variance hit follows a univariateGJRGARCH process of Glosten et al. (1993),

hit = i0 +q

j=1ij

2i,tj +

pj=1

ijhi,tj +q

j=1ij(

i,tj)

2, (2)

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where it = min(it, 0), with the non-negativity and stationarity restrictions imposed onthe parameters. Other GARCH models may also be considered as the GJRGARCHmodel offers just one way of introducing asymmetry in the conditional variance.

The second conditional moment of the vector zt = (z1t, , zNt) is given by

E[ztzt |t1] = Pt (3)

Since zit has unit variance for all i, Pt = [ij,t]i,j=1,,N is the conditional correlation matrixfor the t. The correlations ij,t are allowed to be time-varying in a manner that will bedened later on. In this paper it will be assumed, however, that Pt t1, but extensionsare possible.

To establish the connection to the approach often used in context of conditionalcorrelation models, let us denote the conditional covariance matrix of t as

E[tt |t1] = Ht = StPtSt,

where Pt is the conditional correlation matrix as in Eq. (3) and St = diag(h1/21t , ,h1/2Nt )with elements dened in (2). For the positive deniteness of Ht, it is sufcient to requirethe correlation matrix Pt to be positive denite. The total number of parameters in (2)and (3) equals N (p + 2q + 1) + N (N 1)/2.

The individual GARCH processes (2) contain a component that allows asymmetricvolatility, which enables us to account for potential leverage effects. This is importantin modeling stock returns. Another asymmetry that has recently attracted attention isthe asymmetry of correlations. This may mean that the correlation between a pair ofindividual returns increases more after a negative shock to the system than it does whenthe shock is positive and of the same size; see the discussion in Cappiello et al. (2006).(In the latter case, the correlation need not change at all.) Thorp and Milunovich (2007)recently provided empirical evidence suggesting that accounting both for asymmetricvolatility and asymmetric correlations in a multivariate GARCH model can improve theaccuracy of volatility forecasts. It will be seen that our GARCH model is eminentlysuitable for modeling conditional correlations with an asymmetric response to shocks. Infact, our model allows much more exible cases of asymmetry than the simple examplegiven here.

2.2. Smooth Transitions in Conditional Correlations

In order to complete the denition of the model, we have to specify the time-varyingstructure of the conditional correlations. We propose the STCCGARCH model, inwhich the conditional correlations are assumed to change smoothly over time dependingon a transition variable. In the simplest case, there are two extreme states of nature withstate-specic constant correlations among the variables. The correlation structure changes

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smoothly between the two extreme states of constant correlations as a function of thetransition variable. More specically, the conditional correlation matrix Pt is dened asfollows:

Pt = (1 Gt)P(1) + GtP(2), (4)

where P(1) and P(2) are positive denite correlation matrices. Furthermore, Gt is atransition function whose values are bounded between 0 and 1. This structure ensures Ptto be positive denite with probability one, because it is a convex combination of twopositive denite matrices.

The transition function is chosen to be the logistic function

Gt = (1 + e(stc))1, > 0, (5)

where st is the transition variable, c determines the location of the transition, and > 0the slope of the function, that is, the speed of transition. Increasing increases the speedof transition from 0 to 1 as a function of st, and the transition between the two extremecorrelation states becomes abrupt as . For simplicity, the parameters c and areassumed to be the same for all correlations. This assumption may sometimes turn outto be restrictive, but letting different parameters control the location and the speed oftransition in correlations between different series may cause conceptual difculties. This isbecause then P(1) and P(2) being positive denite does not imply the positive denitenessof every Pt. The difference between this model and that of Pelletier (2006) is that in thisone, the variable controlling the transitions is continuous and observable. In Pelletiers,the corresponding variable is latent and discrete.

The choice of transition variable st in (5) depends on the process to be modeled. Animportant feature of the STCCGARCH model is that the investigator can choose st tot the research problem. In some cases, economic theory proposals may determine thetransition variable, in others the available empirical information may be used for thispurpose. Possible choices include time as in Berben and Jansen (2005), or functions ofpast values of one or more of the return series. Yet another option would be to use anexogenous variable, which is a natural idea, for example when co-movements of individualstock returns are linked to the behavior of the stock market itself. In that case, st couldbe a function of lagged values of the whole index. The Chicago Board Options Exchangeindex VIX constitutes an example. One could use the past conditional variance of indexreturns, which Lanne and Saikkonen (2005) suggested when they constructed a univariatesmooth transition GARCH model.

Another point worth considering in this context is the number of parameters.It increases rapidly with the number of series in the model, although the currentparameterization is still quite parsimonious. However, if one wishes to model the dynamicbehavior between the series, a very small number of parameters may not be enough.Simplications that are too radical are likely to lead to models that do not capture

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the behavior that is the objective of the modeling. It is possible to simplify the STCCGARCH model to some extent such that it may still be useful in certain applications.As an example, one may restrict one of the two extreme correlation states to be that ofcomplete independence (Pk = IN , k = 1 or 2). This is a special case of a model where Pk =[(k),ij] such that (k),ij = , i = j. Another possibility is to allow some of the conditionalcorrelations to be time-varying, while the others remain constant over time. Examples ofthis will be discussed both in connection with testing and in the empirical application.

2.3. Estimation of the STCCGARCH Model

For the maximum likelihood estimation of parameters, we assume joint conditionalnormality of the errors:

zt |t1 N (0,Pt)

Denoting by the vector of all the parameters in the model, the log-likelihood forobservation t is

lt() = N2 log(2) 12

Ni=1

loghit 12 log |Pt| 12ztP

1t zt, t = 1, ,T , (6)

and maximizingT

t=1 lt() with respect to yields the maximum likelihood estimator T .Asymptotic properties of the maximum likelihood estimators in the present case

remain to be established. Bollerslev and Wooldridge (1992) provided a proof ofconsistency and asymptotic normality of the quasi maximum likelihood estimators(QMLE) in the context of general dynamic multivariate models. The required conditionsfor their results to hold, however, have not yet been veried. Recently, Ling and McAleer(2003) considered a class of vector ARMAGARCH models and established strictstationarity and ergodicity as well as consistency and asymptotic normality of the QMLEunder reasonable moment conditions. Extending their results to the present situationwould be interesting. The STCCGARCH model is an inherently nonlinear model. Atthe moment, however, the asymptotic theory for nonlinear GARCH models only coversa class of univariate GARCH models. Meitz and Saikkonen (2008) have recently provenconsistency and asymptotic normality of maximum likelihood estimator for this classof models that includes the smooth transition GARCH model. Their results cannot,however, be immediately generalized to the STCCGARCH model. Proving asymptoticnormality for the maximum likelihood estimator for the parameters of the multivariate

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STCCGARCH model would therefore be a tedious task and is beyond the scope of thispaper.1

Nevertheless, since we nd our model useful and the simulation results do not provideany evidence that asymptotic normality does not hold, we simply proceed by assumingthat asymptotic normality holds, that is,

T(T 0) d N (0,1(0)),

where 0 is the true parameter and (0) the population information matrix evaluated at = 0.

Before estimating the STCCGARCH model, however, it is necessary to rst test thehypothesis that the conditional correlations are constant. The reason for this is that someof the parameters of the STCCGARCH model are not identied if the true modelhas constant conditional correlations. Estimating an STCCGARCH model without rsttesting the constancy hypothesis could thus lead to inconsistent parameter estimates. Thesame is true if one wishes to increase the number of transitions in an already estimatedmodel. Testing constancy of conditional correlations will be discussed in the next section.

Maximization of the log-likelihood (6) with respect to all the parameters at once canbe difcult due to numerical problems. For the DCCGARCH model, Engle (2002)proposed a two-step estimation procedure based on the decomposition of the likelihoodinto a volatility and a correlation component. The univariate GARCH models areestimated rst, independently of each other, and the correlations thereafter, conditionallyon the GARCH parameter estimates. This implies that the dynamic behavior of eachreturn series, characterized by an individual GARCH process, is not linked to the time-varying correlation structure. Under this assumption, the parameter estimates of theDCCGARCH model are consistent under reasonable regularity conditions; see Engle(2002) and Engle and Sheppard (2001) for discussion. For comparison, in the STCCGARCH model, the dynamic conditional correlations form a channel of interactionbetween the volatility processes. Parameter estimation accommodates this fact: theparameters are estimated simultaneously by conditional maximum likelihood.

Due to the large number of parameters in the model, estimation of the STCCGARCHmodel is carried out iteratively by concentrating the likelihood. The parameters are dividedinto three sets: parameters in the GARCH equations, correlations, and parameters of thetransition function, and the log-likelihood is maximized by sequential iteration over thesesets. After the rst completed iteration, the parameter estimates correspond to the estimatesobtained by a two-step estimation procedure. Even if the parameter estimates do not changemuch during the sequence of iterations, the iterative method increases efciency by yieldingsmaller standard errors than the two-step method. Furthermore, convergence is generallyachieved after a reasonably small number of iterations.

1It may be mentioned that at the moment asymptotic normality of maximum likelihood estimators ofcorrelation parameters remains unproven for other CCGARCH models such as the ones by Engle (2002),Tse and Tsui (2002), and Pelletier (2006) as well.

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It should be pointed out, however, that estimation requires care. The log-likelihoodmay have several local maxima, so estimation should be initiated from a set of differentstarting-values of the nonlinear parameters and c, and the obtained maxima comparedbefore settling for nal estimates.

3. TESTING CONSTANCY OF CORRELATIONS

The modeling of time-varying conditional correlations must begin by testing thehypothesis of constant correlations, as previously discussed. Tse (2000), Bera and Kim(2002), and Engle and Sheppard (2001) already proposed tests for this purpose. We shallpresent a Lagrange multiplier (LM)type test of constant conditional correlations againstthe STCCGARCH alternative. A rejection of the null hypothesis supports the hypothesisof time-varying correlations or other types of misspecication but does not imply that thedata have been generated from an STCCGARCH model. For this reason, our LMtypetest can also be seen as a general misspecication test of the CCCGARCH model. Aswe shall see, it is also useful in selecting an appropriate transition variable if it has notbeen chosen in advance.

In order to derive the test, consider an N -variate case where we wish to test theassumption of constant conditional correlations against conditional correlations thatare time-varying with a simple transition of type (4) with a transition function denedby (5). For simplicity, assume that the conditional variance of each of the individualseries follows a GJRGARCH(1, 1) process, and let i = (i0, i, i, i) be the vectorof parameters for conditional variance hit. Generalizing the test to other types ofGARCH models for the individual series is straightforward. The STCCGARCH modelcollapses into a constant correlation model under the null hypothesis of = 0 in (5).When this restriction holds, however, some of the parameters of the model are notidentied. To circumvent this problem, we follow Luukkonen et al. (1988) and consideran approximation of the alternative hypothesis. It is obtained by a rst-order Taylorapproximation around = 0 to the transition function Gt:

Gt = (1 + e(stc))1 1/2 + (1/4)(st c) (7)Applying (7) to (4) linearizes the time-varying correlation matrix Pt as follows:

Pt = P(1) + stP(2),

where

P(1) =12(P(1) + P(2)) + 14c(P(1) P(2)),

P(2) =14(P(2) P(1)) (8)

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If = 0, then P(2) = 0 and the correlations are constant. Thus we construct an auxiliarynull hypothesis Haux0 : (2) = 0, where (2) = veclP(2).2

This null hypothesis can be tested by an LMtest. Note that when H0 holds, there is noapproximation error because then Gt 1/2, and the standard asymptotic theory remainsvalid. Let = (1, ,N , (1), (2)), where (j) = veclP(j), j = 1, 2, be the vector of allparameters of the model. Assuming asymptotic normality of the score, the LMstatistic

LMCCC = T1(

Tt=1

lt()(2)

)[T ()

]1((2) ,

(2)

)(

Tt=1

lt()(2)

), (9)

evaluated at the maximum likelihood estimators under the restriction (2) = 0, hasan asymptotic 2 distribution with N (N 1)/2 degrees of freedom. In expression (9),[T ()]1((2) ,(2)) is the south-east

N (N1)2 N (N1)2 block of the inverse of T , where T is a

consistent estimator for the asymptotic information matrix. For derivation and details ofthe statistic, as well as the suggested consistent estimator for the asymptotic informationmatrix, see AM.

A straightforward extension is to test the constancy of conditional correlations againstpartially constant correlations:

H0 : = 0 H1 : (1),ij = (2),ij for (i, j) N1

where N1 1, ,N 1, ,N . Under the null hypothesis, we again face theidentication problem which is solved by linearizing the transition function. For details,see AM.

These tests involve a particular transition variable. Thus a failure to reject the nullof constant correlations is just an indication that there is no evidence of time-varyingcorrelations, given this transition variable. Evidence of time-varying correlations may stillbe found in case of another indicator. In practice, it may be useful to consider severalalternatives unless restrictions implied by economic theory or other considerations makethe choice unique. If there is uncertainty about which transition variable to use, thestrongest rejection rule may be applied.

It should be mentioned that Berben and Jansen (2005) have in a bivariate contextcoincidentally proposed a test of the correlations being invariant with respect to calendartime. Their test is derived using an approach similar to ours, but they choose a differentestimator for the information matrix in (9). Based on our simulation experiments, theestimator they use is substantially less efcient than ours in nite samples, especially whenthe number of series in the model is large.

2The vecl operator stacks the columns of the strict lower diagonal (obtained by excluding the diagonalelements) of the square argument matrix.

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Finite-sample properties of the test of constancy of correlations have been studied bysimulation in a bivariate case and found satisfactory, see AM for details. Selected resultsof power simulations can be found in AM as well.

4. APPLICATION TO DAILY STOCK RETURNS

The data set of our application consists of daily returns of ve S&P 500 compositestocks traded at the New York Stock Exchange and the S&P 500 index itself. The maincriterion for choosing the stocks is that they are frequently traded and that the tradesare often large. The stocks are Ford, General Motors, Hewlett-Packard (HPQ), IBM,and Lockheed Martin (LMT), and the observation period begins May 25, 1984 andends November 23, 2009. As usual, closing prices are transformed into returns by takingnatural logarithms, differencing, and multiplying by 100, which gives a total of 6,432observations for each of the series. To avoid problems in the estimation of the univariateGARCH equations, the observations in the series are truncated such that extremelylarge positive (negative) returns are set to + () 4 standard deviation of the series.This is preferred to removing them altogether, because we do not want to remove theinformation in comovements related to very large negative returns.3 Descriptive statisticsof the original and truncated return series, including skewness and kurtosis, can be foundin AM.

4.1. Choosing the Transition Variable

We consider the possibility that common shocks affect conditional correlations betweendaily returns. The transition variable in the transition function is a function of laggedreturns of the S&P 500 index. As discussed in Section 2.2, several choices are available.A question frequently investigated, see for instance Andersen et al. (2001) and Chesnayand Jondeau (2001), is whether comovements in the returns are stronger during generalmarket turbulence than they are during more tranquil times. In that case, a laggedsquared or absolute daily return, or a sum of lags of either ones, would be an obviouschoice. Following Lanne and Saikkonen (2005), one could also consider the conditionalvariance of the S&P 500 returns. A model-based estimate of this quantity may be obtainedby specifying and estimating an adequate GARCH model for the S&P 500 return series.Yet another possibility would be VIX, the volatility index based on implied volatilities.

We restrict our attention to different functions of lagged squared and absolute returnsof the index. Specically, we consider up to ve-day lags of both the lagged squaredand lagged absolute returns, as well as equally weighted averages of both the laggedsquared and lagged absolute returns over periods ranging from two days, and one to fourweeks, and nally weighted averages of the same quantities with exponentially decaying

3The estimation of the correlation parameters is not affected by the truncation.

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FIGURE 1 The S&P 500 returns from May 25, 1984 to November 23, 2009. The top panel shows the returns(three observations falls outside the presented range), the middle panel shows the average of the absolute valuereturns over ten days (eight observations fall outside the presented range), and the bottom panel shows thelog of the price difference averaged over twenty days, or average return over four weeks (three observationsfall outside the presented range).

weights with the discount ratios 0.9, 0.7, and 0.3. The constant conditional correlationshypothesis is then tested using each of the 26 transition variables in the complete ve-dimensional model as well as in every one of its submodels. The strongest overall rejectionoccurs (these results are not reported here) when the transition variable is the equallyweighted ten-day average of lagged absolute returns. The graph of this transition variableis presented in the mid-panel of Fig. 1.

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TABLE 1Test of Constant Conditional Correlation Against STCCGARCH Model for All Combinations of Assets.The Transition Variables are s(1)

t, the Lagged Absolute S&P 500 Index Returns Averaged Over Ten Days,

and s(2)t, the Lagged S&P 500 Index Average Return Over 20 Days

s(1)t s(2)t

LMCCC p-value LMCCC p-value

F GM 6.54 0.0106 11.05 0.0009F HPQ 17.77 2 105 33.64 7 109F IBM 31.18 2 108 21.73 3 106F LMT 13.32 0.0003 11.43 0.0007GM HPQ 19.40 1 105 26.90 2 107GM IBM 35.02 3 1010 25.51 4 107GM LMT 8.73 0.0031 11.82 0.0006HPQ IBM 43.19 5 1011 15.68 8 105HPQ LMT 25.44 5 107 37.96 7 1010IBM LMT 22.77 2 106 24.91 6 107F GM HPQ IBM LMT 102.04 2 1017 71.72 2 1011

Table 1 contains the p-values of the constancy test based on this transition variable

for all bivariate models and the full ve-variable CCCGARCH model. The test rejects

the null hypothesis of constant correlations at signicance level 0.01 for all models except

the Ford-General Motors one. The p-value for this model (0.0106) indicates that the

correlation dynamics of these two models are only weakly related to the level of volatility

in the markets. The rejections grow stronger as the dimension of the model increases, the

only exception being when moving from four-variate models to the full ve-variate one.

If the interest lies in nding out whether the direction of the price movement as well

as its strength affect conditional correlations, a function of lagged returns that preserves

the sign of the returns is an appropriate transition variable. In order to accommodate

this possibility, we consider the following three sets of lagged returns: rtj : j = 1, , 5and (1/j)

ji=1 rti : j = 2, 5, 10, 15, 20; note that

ji=1 rti = 100(pt1 pt(j+1)) where

pt is the log-price of the stock, and nally weighted averages of the lagged returns with

exponentially decaying weights with the discount ratios 0.9, 0.7, and 0.3. The constant

conditional correlations hypothesis is tested using these thirteen choices of transition

variables. The strongest rejection most frequently occurs (results not reported here) when

the transition variable is the lagged average four-week return 100 1/20(pt1 pt20). Inthis case, all CCCGARCH models are clearly rejected at the 0.01 level and the strength

of rejection again grows with the dimension of N (see Table 1). The transition variable is

depicted in the bottom panel of Fig. 1.

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4.2. Effects of Market Turbulence on Conditional Correlations

We shall rst investigate the case in which the conditional correlations are assumed touctuate as a result of time-varying market distress which is measured by the lagged ten-day averages of absolute S&P 500 returns. Three remarks are in order. First, we onlyconsider rst-order STCCGARCH models. In order to account for the leverage effectpresent in stock returns, the univariate volatilities are modeled using the GJRGARCHmodel. Second, the STCCGARCH model is only tted to data for which the constantcorrelations hypothesis is rejected at the 5% level. Third, estimation of parameters iscarried out either by the iterative full maximum likelihood (STCCGARCH) or the two-step method (DCCGARCH). The standard errors of the parameter estimates of theSTCCGARCH model are calculated using numerical second derivatives for all estimates

TABLE 2Estimation Results for Bivariate Models (Standard Errors in Parentheses) When the Transition Variable is

the Lagged Absolute S&P 500 Index Returns Averaged Over 10 Days

Model 0 (1) (2) c /sst

F 00196(00049)

00258(00038)

00089(00042)

09663(00035)

06369(00105)

05535(00109)

06046(00022)

1000()

GM 00350(00057)

00144(00035)

00363(00049)

09608(00040)

F 00259(00064)

00258(00045)

00148(00049)

09619(00044)

02794(00119)

04722(00387)

18036(00142)

30268(22452)

HPQ 00441(00099)

00222(00044)

00136(00051)

09630(00051)

F 00270(00070)

00300(00050)

00145(00052)

09580(00049)

00201(02399)

06932(02005)

13274(06966)

06557(04717)

IBM 00214(00047)

00291(00049)

00411(00070)

09443(00067)

F 00273(00070)

00271(00048)

00182(00054)

09588(00049)

01829(00133)

02725(00342)

11204(00139)

36329(26798)

LMT 00429(00113)

00516(00093)

00236(00093)

09242(00122)

GM 00442(00043)

00179(00075)

00452(00060)

09513(00052)

02577(00151)

04862(00477)

14941(01451)

75969(80501)

HPQ 00498(00047)

00229(00113)

00148(00052)

09606(00058)

GM 00426(00073)

00202(00044)

00455(00062)

09498(00050)

00839(01355)

05575(01052)

08377(05134)

11921(06568)

IBM 00222(00048)

00314(00051)

00390(00071)

09429(00067)

GM 00498(00083)

00185(00045)

00507(00067)

09471(00056)

01795(00149)

03439(00453)

17394(00717)

7696(11468)

LMT 00470(00132)

00553(00105)

00237(00093)

09193(00148)

HPQ 00621(00143)

00276(00056)

00129(00057)

09547(00070)

04140(00123)

06143(00251)

12169(00035)

1000()

IBM 00241(00050)

00268(00048)

00373(00064)

09465(00067)

HPQ 00425(00100)

00208(00044)

00169(00052)

09632(00052)

01570(00141)

03033(00395)

12280(00056)

1000()

LMT 00369(00097)

00472(00086)

00219(00077)

09312(00111)

IBM 00236(00051)

00299(00053)

00446(00078)

09413(00071)

00854(00381)

02050(00138)

04217(00027)

1000()

LMT 00438(00116)

00533(00093)

00224(00087)

09229(00125)

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with an occasional exception for the estimate of the velocity of transition parameter , fordetails see AM.

When the transition variable is a function of lagged absolute S&P 500 returns, positiveand negative returns of the same size have the same effect on the correlations, and theabsolute magnitude of the returns carries all the information of possible comovements inthe returns. Small ten-day averages of absolute returns are associated with the conditionalcorrelation matrix P(1), whereas the large ones are related to P(2). We begin by consideringthe bivariate models. Results from the ve-dimensional model are discussed thereafter.The estimation results are presented in Table 2. The close-up graphs of the transitionvariables for the high volatility period 20082009 appear in Fig. 2. The estimatedcorrelations during this period are plotted in Fig. 3.

In all bivariate models, with the exception of the F-GM one which failed to rejectthe constancy of correlations hypothesis at 1% level, the correlations increase with anincreased level of volatility, and most of them quite dramatically. The strength of thecorrelation between Ford and GM shows a slight decrease when the market volatilitypeaks. The transition between the two extreme states of correlations is a step-function inthe model for F and GM, and in the models that are a combination HPQ, IBM, or LMT

FIGURE 2 The close-up graphs of the two transition variables from May 2008 to May 2009. The upperpanel shows the average of the absolute value returns over ten days, and the lower panel shows the log ofthe price difference averaged over twenty days, or average return over four weeks.

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FIGURE 3 The estimated time-varying conditional correlations from the bivariate STCCGARCH modelswhen the transition variable is the lagged absolute S&P 500 index returns averaged over ten days, see Table 2.The time period covers the year from May 2008 to May 2009.

whereas all other models exhibit genuine smooth transition between the states. However,for most of the models, the transitions are quite rapid. For the models for IBM andeither of the two automotive companies F and GM, the correlations spend most of thetime between the states, and the slower the velocity of the transition, the less likely thecorrelations are to reach the extreme states. The correlation between IBM and LMT isuctuating around 0.2 about 83% of the time, and the correlations only decrease when themarkets are very calm. Quite the opposite happens in all other models (with the exceptionof the F-GM one). In those cases, reasonably turbulent market conditions are requiredto shift the correlation levels from low to high (the estimated location for GM-IBM is at70% quantile and for the rest the locations range between 86% and 96% quantiles of theobserved two-week returns).

Before combining the assets into a ve-dimensional model, we note that in the STCCGARCH model the location of the transition is governed by one parameter common toall assets regardless of the dimension of the model. Naturally, as the dimension increases,the point estimate of the location parameter will have to accommodate the different needsof each of the dynamics between any two asset returns. Hence the resulting estimatecan be seen as an average of the locations from the bivariate relations, weighted by

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TABLE 3Estimation Results for the Five-Variate STCCGARCH Model (Standard Errors in Parentheses) When the

Transition Variable is the Lagged Absolute S&P 500 Index Returns Averaged Over 10 Days

Model 0 c /sst

F 00191(00048)

00267(00038)

00059(00041)

09669(00034)

08579(00781)

16147(04788)

GM 00333(00055)

00164(00036)

00319(00047)

09614(00039)

HPQ 00452(00106)

00244(00047)

00095(00047)

09626(00055)

IBM 00169(00038)

00248(00043)

00283(00053)

09554(00056)

LMT 00404(00106)

00504(00090)

00195(00078)

09279(00118)

P(1) P(2)

F GM HPQ IBM F GM HPQ IBM

GM 06895(00314)

GM 04745(00337)

HPQ 02092(00412)

01629(00447)

HPQ 03830(00418)

04135(00447)

IBM 01910(00447)

01741(00475)

02694(00458)

IBM 04249(00461)

04712(00478)

06347(00425)

LMT 01433(00432)

01325(00416)

00862(00515)

01013(00492)

LMT 02592(00444)

02505(00428)

02768(00523)

02778(00510)

the relative strength of their dynamics. One could argue that having a single locationparameter for the correlation dynamics in a high-dimensional model is too coarse asimplication. However, the estimated location reveals the point around which the dataprovides strongest evidence of changing correlation dynamics, given the specic transitionvariable. Finer details of the dynamics can be obtained by studying the submodels.

The estimation results from the ve-variate model are presented in Table 3. In theestimated bivariate models, the estimated locations are scattered between the 17% and96% quantiles of the observed two-week returns, with most of them at the high end.With the above note in mind, it is not surprising that the estimated location in the ve-variate model is at the 72% quantile. This has further effects on the speed of the transitionwhose estimate from the ve-variate model is slightly higher than the slowest transitionsin the bivariate models, but dramatically lower than the estimates from the majority of thebivariate models. Now that the transition has changed location and speed, the estimatesof the extreme levels of correlations adjust accordingly. An interesting nding is that thesingle location seems to replicate the bivariate dynamics in the ve-dimensional modelfor all but one combination: the F-IBM model nds extreme levels of correlations thatare closer together than the corresponding correlation levels from the bivariate model.Furthermore, the direction of change in the correlations in the model for F and GM isopposite to all the other ones in the ve-dimensional, and also in every bivariate model,

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i.e., when the uctuations are small, the correlation between F and GM is higher thanduring turbulent times.

4.3. Effects of Shock Asymmetry on Conditional Correlations

As already mentioned, asymmetric correlations have recently attracted attention. We usea market indicator to represent price changes and study time-variation in correlations byagain assuming the transition variable to be a function of the S&P 500 index. Since theinterest lies in the direction and size of the price movements, we select the lagged averagefour-week return to be the transition variable as discussed in Subsection 4.1. The results ofthe constancy tests appear in Table 1. The tests of constant correlations reject constancyfor each model. An STCCGARCH(1, 1) model is thus estimated for all combinations.The S&P 500 twenty-day average returns below the estimated location imply a correlationstate approaching that of P(1), whereas the returns greater than the estimated locationresult in correlations closer to the other extreme state, P(2). The estimation results for thebivariate STCCGARCH models are presented in Table 4. The estimated correlations aredepicted in Figure 4 for the period 20082009. The transition variable during this periodis shown in Fig. 2.

The estimation results support the theory, see, e.g., Hong and Stein (2003), thatpessimistic market conditions lead to higher correlations than optimistic views do.However, the magnitude and sign of the four-week return required to alter thecorrelations varies across the models. For the models FIBM, GMIBM, HPQIBM,and GMHPQ, large negative four-week return on the S&P 500 index implies highcorrelations, whereas for the remaining models (FHPQ being an exception) correlationsdecrease only after large positive index returns. The model for F and HPQ has a weakpositive correlation that increases slightly after relatively large positive four-week indexreturn (the estimated location is at the 70% quantile of the observed four-week returns).

The transitions in the correlations are quite smooth for most of the models, whereasthe correlations seem to show genuine or close to regime switching behaviour for fourof the models. Comparison of the estimated correlations with the ones from the previoussubsection shows that they behave differently when the transition variable differentiatesthe direction of price movements from general market turbulence. This is to be expectedas times of distress are characterized by high volatility which results from large, positiveor negative, shocks. For instance, the correlation in the FIBM model was close to zeromost of the time and peaked at 0.7 with very high volatility in the previous subsection.However, when the transition variable allows the correlations to depend on the directionof change, they increase from a level of 0.3 to a somewhat higher one when the pricechange is sufciently negative and large in absolute value.

When combining the assets into a ve-variate model, the estimation results providesomewhat surprising outcomes. These results are presented in Table 5. While the estimatesfor the location of the transition in the bivariate models ranged from the 1% to 74%

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TABLE 4Estimation Results for Bivariate STCCGARCH Models (Standard Errors in Parentheses) When the

Transition Variable is the Lagged Average S&P 500 Index Return Over 20 Days

Model 0 (1) (2) c /sst

F 00177(00048)

00258(00039)

00112(00043)

09657(00036)

06219(00196)

05267(00415)

01558(01606)

71487(37453)

GM 00332(00054)

00141(00034)

00384(00050)

09607(00039)

F 00281(00068)

00253(00045)

00165(00049)

09610(00046)

01354(00130)

02062(00222)

01354(00038)

82158(60923)

HPQ 00456(00103)

00220(00044)

00141(00051)

09627(00053)

F 00290(00072)

00288(00049)

00156(00052)

09579(00050)

03944(00303)

02753(00129)

01534(00010)

1000()

IBM 00226(00051)

00293(00053)

00423(00074)

09432(00074)

F 00280(00069)

00263(00047)

00193(00052)

09590(00048)

02351(00185)

01367(00325)

01054(00770)

1990(32218)

LMT 00462(00117)

00527(00093)

00250(00086)

09213(00125)

GM 00450(00076)

00174(00043)

00455(00060)

09514(00052)

04113(00583)

01842(00649)

00817(01948)

45927(25045)

HPQ 00505(00114)

00222(00047)

00155(00052)

09608(00057)

GM 00440(00075)

00192(00043)

00457(00061)

09500(00051)

07127(01870)

02609(00368)

05874(02772)

41809(29079)

IBM 00240(00051)

00316(00051)

00410(00073)

09410(00069)

GM 00505(00082)

00182(00044)

00515(00065)

09469(00054)

02372(00165)

01406(00174)

00457(00070)

95086(70509)

LMT 00496(00127)

00555(00098)

00255(00089)

09174(00133)

HPQ 00651(00151)

00249(00053)

00152(00054)

09554(00072)

05898(00879)

02984(01299)

00340(03932)

19828(14297)

IBM 00268(00055)

00268(00048)

00401(00069)

09443(00071)

HPQ 00443(00107)

00205(00046)

00176(00052)

09627(00056)

02195(00160)

00808(00227)

01267(00014)

1000()

LMT 00428(00128)

00510(00109)

00236(00085)

09248(00148)

IBM 00233(00051)

00291(00051)

00457(00076)

09416(00070)

02396(00274)

01188(00282)

00522(00716)

1739(1553)

LMT 00458(00117)

00538(00095)

00231(00085)

09213(00125)

quantile of the twenty-day return distribution, in the ve-variate model the estimate forthe location parameter is close to the 75% quantile. While most of the models haveno signicant differences between the bivariate and ve-variate correlation estimates,some differences emerge. In fact, the correlation between GM and IBM is now deemedconstant. The direction of change in correlations, however, remains the same as in thebivariate models, except for the model for F and HPQ.

In theory, as a solution to the multilocation problem one could generalize the STCCGARCH model such that it would allow different slope and location parameters foreach pair of correlations. However, as already mentioned, such an extension entails thestatistical problem of ensuring positive deniteness of the correlation matrix at each pointof time.

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TABLE 5Estimation Results for the Five-Variate STCCGARCH Model (Standard Errors in Parentheses) When the

Transition Variable is the Lagged Average S&P 500 Index Return Over 20 Days

Model 0 c /sst

F 00183(00047)

00255(00037)

00084(00041)

09671(00034)

01610(00239)

5081(2126)

GM 00323(00053)

00154(00034)

00333(00047)

09619(00038)

HPQ 00496(00112)

00221(00043)

00098(00046)

09637(00055)

IBM 00204(00044)

00254(00044)

00304(00058)

09527(00062)

LMT 00472(00124)

00537(00097)

00211(00083)

09216(00131)

P(1) P(2)

F GM HPQ IBM F GM HPQ IBM

GM 06027(00091)

GM 05483(00200)

HPQ 03223(00132)

03032(00133)

HPQ 02013(00300)

02158(00287)

IBM 03119(00131)

03172(00257)

R 04542(00116)

IBM 02685(00293)

03172(00257)

R 03996(00236)

LMT 02183(00146)

02102(00147)

02106(00165)

02073(00152)

LMT 01382(00275)

01195(00290)

00711(00288)

01125(00287)

The superscript R indicates that the correlation is restricted to be constant.

4.4. Comparison

We conclude this section with a brief informal comparison of the time-varyingcorrelations implied by the STCC and DCCGARCH models. The DCCGARCHmodel is chosen because it is the most frequently applied conditional correlation GARCHmodel. To keep the comparison transparent, we only consider bivariate models, andfocus on the similarities and differences in the correlation dynamics implied by the twomodeling approaches. These aspects can clearly be seen by looking at the specic timeperiods.

A fundamental difference between the two models is that in the DCCGARCH model,the correlation dynamics only uses the past returns of the series to be modeled. Onthe other hand, the STCCGARCH models make use of the two transition variablesdiscussed in the previous subsections. One can therefore expect the dynamics impliedby the two models to be somewhat different. The bivariate DCCGARCH(1, 1) modelsare estimated using the two-step estimation method of Engle (2002). The estimatedGARCH equations in the DCCGARCH model differ slightly from the ones in theSTCCGARCH models due to the two-step procedure, and the correlation dynamics arevery persistent (for conciseness we do not present the estimation results). The estimatedcorrelations from the bivariate DCCGARCH models are shown in Fig. 5 for the period20082009.

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By comparing Figs. 3, 4, and 5, it is seen that the correlations from the two families ofmodels are quite different. The STCCGARCH model nds evidence for the correlationsto increase quite rapidly. The DCCGARCH model suggests that the correlationsrespond very slowly to the turbulence: the correlations merely show an upward slopingtrend. Interestingly, the correlations from a few of the DCCGARCH models show asudden decrease right before the crisis hit. The STCCGARCH counterparts tend toindicate an increase in the correlations at the same time frame.

The bivariate GM-IBM model deserves a closer look as it offers an example ofdifferences that have to do with the fundamental structure of the models. When theSTCCGARCH model uses the lagged absolute S&P 500 returns averaged over a two-week period, the estimated correlations are similar to the ones obtained from the DCCGARCH model. The major difference is the rate at which the correlations revert backtowards the pre-crisis levels. This points at the fact that the transition variable in questionresponds to general market turbulence, as does the DCCGARCH model. The situationis quite different when one considers the estimated correlations from the STCCGARCHmodel that uses the lagged S&P 500 return over twenty days. They show a clear increase,albeit short-lived, in the correlations to levels that the other two models could notproduce. This is due to sufciently large negative shocks during the crisis.

FIGURE 4 The estimated time-varying conditional correlations from the bivariate STCCGARCH modelswhen the transition variable is the lagged average S&P 500 index return over twenty days, see Table 4. Thetime period covers the year from May 2008 to May 2009.

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FIGURE 5 The estimated conditional correlations from the bivariate DCCGARCH model. The time periodcovers the year from May 2008 to May 2009.

These two approaches thus lead to rather different conclusions about the conditionalcorrelations between the return series. Since the correlations cannot be observed, it isnot possible to decide whether the results from the STCCGARCH models are closerto the truth than the ones from the DCCGARCH model or vice versa. In theory,testing the models against each other may be possible but would be computationallydemanding. These models may also be compared by investigating their out-of-sampleforecasting performance, which is left for the future. In practice, nancial decisions thatbenet from analysing correlations are linked to market conditions. For this reason, theSTCCGARCH model can be found useful as it enables one to investigate the correlationdynamics with respect to their response to different variables.

5. CONCLUSIONS

We propose a new multivariate conditional correlation model with time-varyingcorrelations, the STCCGARCH model. The conditional correlations are changingsmoothly between two extreme states according to a transition variable that can beexogenous to the system. The transition variable controlling the time-varying correlationscan be chosen quite freely, depending on the modeling problem at hand. The STCCGARCH model may thus be used for investigating the effects of numerous potential

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factors, lagged predetermined as well as exogenous, on conditional correlations. In thisrespect the model differs from most other dynamic conditional correlation models suchas the ones proposed by Tse and Tsui (2002), Engle (2002), and Pelletier (2006).

The STCCGARCH model is applied to up to a ve-variable set of daily returns offrequently traded stocks included in the S&P 500 index. When using the two-week laggedaverage of the daily absolute return of the index as the transition variable we nd thatthe conditional correlations are substantially higher during periods of high volatility thanotherwise. Asymmetric response of correlations to shocks is examined using the one-daylag of the four-week average index returns. In that case market optimism weakens theconditional correlations between the asset returns.

In its present form the model allows for a single transition with location andsmoothness parameters common to all series. In theory this restriction can be relaxed, butnding a useful way of doing it is left for future work. The model may be further renedby allowing specications of the univariate GARCH equations beyond the standardGJRGARCH(1, 1) model. For example, the transition between the regimes could bemade smooth or nonstationarities could be introduced. A point worth considering isincorporating higher frequency data into the model. Recent research has emphasized theimportance of information present in the high-frequency data but lost in aggregation. Onecould use the realized volatility or bipower variation of stock index returns over a day ora number of days as the transition variable in a model for stock returns. This possibilityis left for future research.

ACKNOWLEDGMENTS

We would like to thank Robert Engle, Tim Bollerslev, W. K. Li and Y. K. Tse foruseful discussions and encouragement, as well as Markku Lanne and Michael McAleerfor their comments. Part of the research was done while the authors were visiting Schoolof Finance and Economics, University of Technology, Sydney, whose kind hospitalityis gratefully acknowledged. Our very special thanks go to Tony Hall for making thevisit possible. The rst author also wishes to thank CREATES, Aarhus University, foran opportunity to do research there. We are grateful to Mika Meitz for programmingassistance. The responsibility for any errors and shortcomings in this paper remains ours.

FUNDING

This research has been supported by the Danish National Research Foundation, the JanWallander and Tom Hedelius Foundation, Grants No. J0341 and P200533:1, OP BankGroup Research Foundation, The Foundation for Promoting Finnish Equity Markets,and Yrj Jahnsson Foundation.

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Modeling Conditional Correlations of Asset Returns: A Smooth Transition Approach

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