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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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http://dx.doi.org/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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MODELING CONDITIONAL CORRELATIONS 177
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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178 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 179
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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180 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 181
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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182 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 183
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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184 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 185
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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186 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 187
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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188 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 189
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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190 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 191
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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192 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 193
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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194 A. SILVENNOINEN AND T. TERSVIRTA
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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MODELING CONDITIONAL CORRELATIONS 195
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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196 A. SILVENNOINEN AND T. TERSVIRTA
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