long memory conditional volatility and asset allocation

International Journal of Forecasting 29 (2013) 258–273

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International Journal of Forecasting

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Long memory conditional volatility and asset allocationRichard D.F. Harris ∗, Anh NguyenXfi Centre for Finance and Investment, University of Exeter, Exeter, EX4 4PU, UK

a r t i c l e i n f o

Keywords:Conditional variance-covariance matrixLong memoryAsset allocation

a b s t r a c t

In this paper, we evaluate the economic benefits that arise from allowing for long memorywhen forecasting the covariance matrix of returns over both short and long horizons,using the asset allocation framework of Engle and Colacito (2006) In particular, wecompare the statistical and economic performances of four multivariate long memoryvolatility models (the long memory EWMA, long memory EWMA–DCC, FIGARCH-DCCand component GARCH-DCC models) with those of two short memory models (the shortmemory EWMA and GARCH-DCC models). We report two main findings. First, for longerhorizon forecasts, long memory models generally produce forecasts of the covariancematrix that are statistically more accurate and informative, and economically more usefulthan those produced by shortmemorymodels. Second, the twoparsimonious longmemoryEWMA models outperform the other models – both short and long memory – acrossmost forecast horizons. These results apply to both low and high dimensional covariancematrices and both low and high correlation assets, and are robust to the choice of theestimation window.© 2012 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

s. P

1. Introduction

It is awell-established fact that the covariancematrix ofshort horizon financial asset returns is both time-varyingand highly persistent. A number of multivariate condi-tional volatility models, including the multivariate Risk-Metrics EWMA model, multivariate GARCH models andmultivariate Stochastic Volatilitymodels, have been devel-oped for capturing these features. These models are nowused routinely in many areas of applied finance, includ-ing asset allocation, risk management and option pricing(see, for example, Engle & Colacito, 2006, Han, 2006). Inthe vast majority of conditional volatility models whichare used in practice, the elements of the conditional co-variance matrix are specified as weighted averages of thesquares and cross-products of past return innovations,with weights that decline geometrically, so that shocksto individual variances and covariances dissipate rapidly.

∗ Corresponding author.E-mail addresses: [email protected] (R.D.F. Harris),

[email protected] (A. Nguyen).

0169-2070/$ – see front matter© 2012 International Institute of Forecasterdoi:10.1016/j.ijforecast.2012.09.003

However, there is a mounting body of empirical evidencethat suggests that although the volatility is almost certainlystationary, the autocorrelation functions of the squares andcross-products of returns declinemore slowly than the ge-ometric decay rate of the EWMA, GARCH and StochasticVolatilitymodels, andhence volatility shocks aremoreper-sistent than these models imply (see, for example, Ander-sen, Bollerslev, Diebold, & Ebens, 2001, Ding, Granger, &Engle, 1993, Taylor, 1986). This ‘long memory’ feature isimportant not only for the measurement of the currentvolatility, but also for forecasts of the future volatility, es-pecially over longer horizons. In particular, in the GARCHand Stochastic Volatility frameworks, forecasts of the fu-ture volatility converge to the unconditional volatility atan exponential rate as the forecast horizon increases. Incontrast, in the EWMA framework, a volatility shock hasa permanent effect on the forecast volatility at all hori-zons, meaning that forecasts of the future volatility do notconverge at all, in spite the fact that it is a short memorymodel. If the volatility is indeed a long memory process, asthe empirical evidence suggests, the shortmemory EWMA,GARCH and Stochastic Volatility models are misspecified.

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R.D.F. Harris, A. Nguyen / International Journal of Forecasting 29 (2013) 258–273 259

Moreover, the errors in forecasting the elements of the co-variance matrix that arise from this misspecification arecompounded as the forecast horizon increases.

The empirical evidence on the volatility dynamics hasprompted the development of long memory models of theconditional volatility, and a number of approaches havebeen proposed in the univariate context. The FIGARCHmodel of Baillie, Bollerslev, and Mikkelsen (1996) intro-duces long memory through a fractional difference op-erator, which gives rise to a slow hyperbolic decay forthe weights on lagged squared return innovations, whilestill yielding a strictly stationary process. The HyperbolicGARCH (HYGARCH) model of Davidson (2004) is a gen-eralisation that nests the GARCH, FIGARCH and IGARCH(or EWMA) models, allowing for a more flexible dynamicstructure than the FIGARCH model, and facilitating testsof short versus long memory in the volatility dynamics.Breidt, Crato, and De Lima (1998) extended the Stochas-tic Volatility framework to allow for long memory, by in-corporating an ARFIMA process in the standard discretetime Stochastic Volatility model. Long memory can alsobe induced using a component structure for the volatilitydynamics. For example, the Component GARCH(CGARCH)model of Engle and Lee (1999) assumes that the volatil-ity is the sum of a highly persistent long run componentand a mean-reverting short run component, each of whichfollows a short memory GARCH process. Zumbach (2006)introduces a long memory model in which the dynamicprocess for volatility is defined as the logarithmicallyweighted sum of standard EWMA processes at differentgeometric time horizons. Like the short memory EWMAmodel of JP Morgan (1994) on which it is based, the longmemory EWMA model has a highly parsimonious specifi-cation, which facilitates its implementation in practice.

In the multivariate context, long memory volatil-ity modelling poses significant computational challenges,especially for the high dimensional covariance matricesthat are typically encountered in asset allocation and riskmanagement. Indeed, so far the literature on long mem-ory multivariate volatility modelling has generally re-stricted itself to the analysis of low dimensional covariancematrices, and has provided only limited evidence on therelative benefits of allowing for long memory in the mul-tivariate setting. For example, Teyssiere (1998) estimatesthe covariance matrix for three foreign exchange returnseries using both an unrestricted multivariate FIGARCHmodel and a FIGARCH model implemented with the Con-stant Conditional Correlation (CCC) structure of Bollerslev(1990). Similarly, Niguez and Rubia (2006) model the co-variance matrix of five foreign exchange series using anOrthogonal HYGARCHmodel, which combines the univari-ate HYGARCH long memory volatility model of Davidson(2004) with the multivariate Orthogonal GARCH frame-work of Alexander (2001). They show that the OrthogonalHYGARCH model outperforms the standard OrthogonalGARCH model in terms of 1-day-ahead forecasts of the co-variance matrix. Zumbach (2011) develops a multivariateversion of the univariate long memory EWMA model, inwhich elements of the covariance matrix are estimatedas the averages of the squares and cross products ofpast returns with predetermined logarithmically decayingweights.

In this paper, we evaluate the economic benefits of al-lowing for long memory when forecasting the covariancematrix of returns over both short and long horizons, us-ing the asset allocation framework of Engle and Colacito(2006). In so doing, we compare the performances of anumber of long and short memory multivariate volatil-ity models. While many alternative volatility models havebeen developed in the literature, our choice reflects theneed for parsimonious models that can be used for fore-casting high dimensional covariance matrices. We em-ploy four long memory volatility models: the multivariatelong memory EWMA model of Zumbach (2006), and threemultivariate long memory models implemented using theDynamic Conditional Correlation (DCC) framework of En-gle (2002). These are the univariate long memory EWMAmodel of Zumbach (2006), the component GARCH modelof Engle and Lee (1999) and the FIGARCH model of Bail-lie et al. (1996). While the models that are based on theDCCdecomposition allow for longmemory only in the vari-ances, the long memory EWMA model captures the longmemory behaviour in both the variances and covariances.We compare the four multivariate long memory modelswith twomultivariate shortmemorymodels. These are thevery widely used RiskMetrics EWMA model of JP Morgan(1994), and the DCCmodel implemented with the univari-ate GARCH model.

We use these six multivariate conditional volatilitymodels to forecast the covariance matrices for the threesets of assets employed by Engle and Colacito (2006). Thesecomprise a high correlation bivariate system (the S&P500and DJIA indices), a low correlation bivariate system (theS&P500 and 10-year Treasury bond futures), and a moder-ate correlation high dimensional system (21 internationalstock indices and 13 international bond indices). In addi-tion, we also consider another high dimensional system,namely the components of the DJIA index. The analysis isconducted using data over the period 1 January 1988–31December 2009, and considers forecast horizons of up tothree months. For the two bivariate systems, we first eval-uate the forecasts of the models using a range of statisticalcriteria that measure the accuracy, bias and informationalcontent of the models’ forecasts over varying time hori-zons. For all four systems,we then employ Engle and Colac-ito’s (2006) approach to assess the economic value of theforecast covariance matrices in an asset allocation setting.We report two main findings. The first is that for longerhorizon forecasts, multivariate long memory models gen-erally produce forecasts of the covariance matrix that areboth statistically more accurate and informative, and eco-nomicallymore useful than those produced by shortmem-ory volatility models. The second is that the two longmemory models that are based on the Zumbach (2006)univariate model outperform the other models – bothshort memory and long memory – across most forecasthorizons. These results apply to all four datasets and arerobust to the choice of the estimation window.

The remainder of this paper is organised as follows.Section 2 provides details of the six multivariate condi-tional volatility models used in the empirical analysis.Section 3 describes the methods used to evaluate the fore-cast performance for the six models. The data are sum-marised in Section 4. In Section 5, we report the empirical

260 R.D.F. Harris, A. Nguyen / International Journal of Forecasting 29 (2013) 258–273

results of our analysis. Section 6 discusses the robustnesstests, while Section 7 offers some concluding commentsand some suggestions for future research.

2. Multivariate long memory conditional volatilitymodels

Motivated by the need for parsimonious models thatcan be used to forecast high dimensional covariancematri-ces, we first consider two simple multivariate long mem-ory conditional volatility models based on the univariatelong memory volatility model of Zumbach (2006). Thefirst is the multivariate long memory EWMA (LM–EWMA)model of Zumbach (2011), which is a simple multivariateextension of the univariate long memory EWMA model,in which both the variances and covariances are gov-erned by the same long memory process; this modelis thus the long memory analogue of the short mem-ory multivariate RiskMetrics EWMA model of JP Morgan(1994). In the second, we employ the Dynamic Condi-tional Correlation framework of Engle (2002) to modelthe dynamic processes of the correlations directly, usingthe univariate long memory EWMA model for the indi-vidual variances. This is the long memory EWMA–DCC(LM–EWMA–DCC)model.We compare the two longmem-ory EWMA models with the multivariate FIGARCH(1, d, 1)and component GARCH(1, 1) (CGARCH) long memorymodels, both implemented using the DCC framework.The long memory models that are based on the DCCdecomposition allow for long memory behaviour in thevariances, but not in the covariances. In contrast, theLM–EWMA model captures long memory behaviour inboth the variances and covariances. In order to evaluatethe relative benefits of allowing for long memory whenforecasting the covariance matrix, we compare the fourlongmemorymultivariate models with two short memorymultivariate volatility models. These are the multivariateRiskMetrics EWMA model of JP Morgan (1994) and theGARCH(1, 1) model implemented using the DCC frame-work. In this section, we give details of each of these sixmodels.

2.1. The multivariate LM–EWMA model

Consider an n-dimensional vector of returns rt = (r1t ,r2t , . . . , rnt)′ with a conditional mean of zero and aconditional covariance matrix Ht :

rt = H12t εt (1)

where εt is i.i.d. with E(εt) = 0 and var(εt) = In. Zumbach(2011) considers the class of conditional covariance matri-ces that are the weighted sum of the cross products of pastreturns:

Ht+1 =

∞i=0

λ(i)rt−ir ′

t−1 (2)

with

λ(i) = 1. In the RiskMetrics EWMA model ofJP Morgan (1994), the weights λ(i) decay geometrically,yielding a short memory process for the elements of thevariance–covariance matrix. The long memory conditional

covariance matrix is defined as the weighted average of Kstandard (short memory) multivariate EWMA processes:

Ht+1 =

Kk=1

wkHk,t , (3)

where

Hk,t = µkHk,t−1 + (1 − µk) rtr ′

t . (4)

The decay factor µk of the kth EWMA process is defined bya characteristic time τk such that µk = exp(−1/τk), withgeometric time structure τk = τ1ρ

k−1 for k = (1, . . . , K).Zumbach (2006) sets the value of ρ to

√2. The memory

of the volatility process is determined by the weights wk,which are assumed to decay logarithmically:

wk =1C

1 −

ln (τk)

ln (τ0)

, (5)

with the normalization constant C = K −

kln(τk)ln(τ0)

, suchthat

k wk = 1. The conditional covariance matrix is

therefore defined parsimoniously as a process with justthree parameters: τ1 (the shortest time scale at which thevolatility is measured, i.e. the lower cut-off), τK (the uppercut-off, which increases exponentially with the number ofcomponents K ), and τ0 (the logarithmic decay factor). Forthe univariate case, Zumbach (2006) sets the optimal pa-rameter values at τ0 = 1560 days = 6 years, τ1 = 4 daysand τK = 512 days, which is equivalent to K = 15.

The EWMA process in Eq. (4) can also be expressed as

Hk,t = (1 − µk)

∞i=0

µikrt−ir ′

t−1. (6)

Hence, the LM–EWMAmodel can be written in the form ofEq. (2):

Ht+1 =

∞i=0

Kk=1

wk (1 − µk) µikrt−ir ′

t−1

=

∞i=0

λ(i)rt−ir ′

t−1, (7)

with λ(i) =

k wk (1 − µk) µik and

i λ(i) = 1. When

K = 1, the LM–EWMA process reduces to the shortmemory RiskMetrics EWMA process. Note that since Hk,tis a positive definite matrix (see JP Morgan, 1994), Ht+1,which is a linear combination ofHk,t with positiveweights,will also be positive definite. Since the LM–EWMA covari-ance matrix is the sum of EWMA processes over increas-ing time horizons, forecasts of the covariance matrix arestraightforward to obtain using a recursive procedure (seeZumbach, 2006, for details of the univariate case). The1-step-ahead forecast of the covariance matrix is alreadygiven by Eq. (7). Under the assumption of serially uncorre-lated returns, the h-step-ahead cumulative forecast of thecovariance matrix given the information set Ft at time t isequal to:

Ht+1:t+h = hT

i=0

λ(h, i)rt−ir ′

t−1, (8)


with the weights λ (h, i) being given by

λ (h, i) =

Kk=1

1h

h−1j=1

wj,k(1 − µk)

1 − µTk

µik, (9)

where T is the cut-off time,wj,k is the kth element of vectorwj = w ′

M + (ι − µ)w ′

j,w = (w1, w2, . . . , wK )′ , µ

is the vector of µk, M is the diagonal matrix consisting ofµk, and ι is the unit vector.1 Since

k wk = 1, we obtain

λ(h, i) = 1. Also note that when K = 1, then w = 1,and therefore the LM–EWMA forecast function reduces toa standard short memory EWMA forecast function withforecast weights λ (h, i) = (1 − µk) µi

k/1 − µT

k

, inde-

pendent of the forecast horizon. Since the weights λ (h, i)are specified a priori, without reference to the data, theforecast in equation (8) is straightforward to compute. Aswith the standard EWMAmodel, the LM–EWMAmodel cir-cumvents the computational burden of other multivariatelong memory models, and indeed, can be implemented ina spreadsheet easily.

2.2. The multivariate LM–EWMA–DCC model

In the dynamic conditional correlation (DCC) modelof Engle (2002), the conditional covariance matrix isdecomposed as follows:

Ht = DtRtDt (10)

Rt = diag Qt−

12 Qtdiag Qt

−12 (11)

Qt = Ω + αεt−1ε′

t−1 + βQt−1 (12)

where Rt is the conditional correlation matrix, Dt is adiagonal matrix with the time varying standard deviationshi,t on the main diagonal, i.e., Dt = diag

hi,t,Qt is

the approximation of the conditional correlationmatrixRt ,and Ω = (1 − α − β)R, with R being the unconditionalaverage correlation R =

1T

Tt=1 εtε

′t . The positive semi-

definiteness ofQt is guaranteed ifα andβ are positivewithα + β < 1 and the initial matrix Q1 is positive definite.

Here, we estimate the conditional volatility Dt by em-ploying the univariate long memory volatility model ofZumbach (2006). We divide the returns by their con-ditional volatility and use the standardized, zero-meanresiduals εt = D−1

t rt to compute the quasi-conditionalcorrelation matrix Qt . As the diagonal elements of Qt areequal to unity only on average, Qt is rescaled to obtain theconditional correlation matrix Rt = diagQt

−1/2QtdiagQt

−1/2. The conditional volatility Dt and conditional cor-relations Rt are then combined to estimate the conditionalcovariance matrix Ht .

The h-step-ahead conditional covariance matrix isgiven by

Ht+h = Dt+hRt+hDt+h. (13)

1 Zumbach (2011) suggests that for many practical applications, thememory length T is of the order of one to two years (T = 260 to T = 520).Here, we choose T equal to the estimationwindow length (T = 252 in themain tests, and T = 504, 1260 and 2520 in the robustness tests).

The forecast of each volatility inDt+h is estimated using therecursive procedure, as in Eq. (8) for the univariate case.Since Rt is a non-linear process, the h-step-ahead forecastof Rt cannot be computed using a recursive procedure.However, assuming for simplicity that Et

εt+1ε

′

t+1

≈

Qt+1, Engle and Shephard show that the forecasts of Qt+hand Rt+h are given by

Qt+h

=

h−2j=0

(1 − α − β)Q(α + β)j + (α + β)h−1Qt+1, (14)

and

Rt+h = diag Qt+h−

12 Qt+hdiag Qt+h

−12 . (15)

2.3. The FIGARCH(1, d, 1)–DCC model

Baillie et al. (1996) propose the fractionally integratedGARCH (FIGARCH) model, in which long memory isintroduced through a fractional difference operator, d. Thismodel incorporates a slow hyperbolic decay for laggedsquared innovations in the conditional variance, whilestill letting the cumulative impulse response weights tendto zero, thus yielding a strictly stationary process. Inthe FIGARCH(1, d, 1) model, the conditional volatility ismodelled as:

ht = ω + [1 − βL − (1 − φL) (1 − L)d]r2t + βht−1, (16)

with L being the lag operator. Baillie et al. (1996) show thatfor 0 < d ≤ 1, the FIGARCH process does not have afinite unconditional variance, and is not weakly stationary,a feature shared with the IGARCH model. However, by adirect extension of the corresponding proof for the IGARCHmodel, they show that the FIGARCH model is strictlystationary and ergodic. The FIGARCH process reduces tothe GARCH process when d = 0. The h-step-ahead forecastof the FIGARCH (1, d, 1) model is given by

ht+h = ω (1 − β)−1+ [1 − (1 − βL)−1

× (1 − φL) (1 − L)d]r2t+h−1. (17)

To implement the FIGARCH(1, d, 1) model in the multi-variate context, we use the DCC approach described above,with the same forecast functions for Qt+h and Rt+h.

2.4. The CGARCH(1, 1)–DCC model

An alternative way to capture the long memory featureis through a component structure for volatility. Engleand Lee (1999) propose the component GARCH(CGARCH)model, in which the long memory volatility process ht ismodelled as the sum of a long term trend component, qt ,and a short term transitory component, st . The CGARCH(1,1) model has the following specification:

ht − qt = αr2t−1 − qt−1

+ β(ht−1 − qt−1) (18)

qt = ω + ρqt−1 + φ(r2t−1 − ht−1), (19)

where st = ht − qt is the transitory volatility component.The volatility innovation r2t−1 − ht−1 drives both the trend


and the transitory components. The long run componentevolves over time following an AR process with ρ close to1, while the short run component mean reverts to zero at ageometric rateα+β . It is assumed that 0 < α+β < ρ < 1,meaning that the long run component is more persistentthan the short run component. Engle and Lee also showthat the component GARCH model is in fact a constrainedversion of the GARCH(2, 2) model. The h-step-ahead fore-cast of the CGARCH(1, 1) model is given by

ht+h = qt+h + (α + β)h−1 (ht − qt) (20)

qt+h =ω

1 − ρ+ ρh−1

qt −

ω

1 − ρ

. (21)

Aswith the FIGARCH(1, d, 1)model, in order to implementthe CGARCH (1, 1) model in the multivariate context, weuse theDCC approach described above,with the same fore-cast functions for Qt+h and Rt+h.

2.5. The RiskMetrics EWMA model

The short memory RiskMetrics EWMA covariancematrix is defined by

Ht = λHt−1 + (1 − λ) rt−1r ′

t−1, (22)

where λ is the decay factor 0 < λ < 1. The larger the valueof λ, the higher the persistence of the covariance matrixprocess and the lower the response of the volatility to re-turn shocks. It is straightforward to show that the h-stepcumulative forecast of the EWMAmodel is given by

Ht+1:t+h = h × Ht+1 (23)

(see, for example, JP Morgan, 1994). In the empirical anal-ysis, we set λ to the values of 0.94 and 0.97 for daily andweekly forecasts, respectively, as suggested by JP Morgan(1994).

2.6. The GARCH(1, 1)–DCC model

The short memory GARCH(1, 1) model of Bollerslev(1990) is given by

ht = ω + αr2t−1 + βht−1. (24)

The parameter α determines the speed at which the con-ditional variance responds to new information, while theparameter α + β determines the speed at which the con-ditional variance reverts to its long run average. In theGARCH(1, 1)model, theweights on past squared errors de-cline at an exponential rate. The h-step-ahead forecast ofthe GARCH(1, 1) model is given by

ht+h = σ 2+ (α + β)h−1 ht+1 − σ 2 , (25)

where σ 2 is the unconditional variance. In order to imple-ment the GARCH(1, 1) model in the multivariate context,we again use the DCC approach described above, with thesame forecast functions for Qt+h and Rt+h.

3. Forecast performance measurement

We evaluate the forecast performances of the sixconditional volatilitymodels using a range of statistical andeconomic measures. We begin by measuring the accuracy,bias and information content of the models’ forecasts foreach element of the covariance matrix, using the squaresand cross-products of daily returns as proxies for theactual variances and covariances. The forecast accuracy isevaluated using the root mean squared error (RMSE) andthe mean absolute error (MAE). These are given by

RMSEij =

1T

Tt=1

ri,t rj,t − σij,t

2 (26)

and

MAEij =1T

Tt=1

ri,t rj,t − σij,t . (27)

Patton (2011) shows that the RMSE is a robust lossfunction, in the sense that it yields the same ranking ofcompeting forecasts using an unbiased volatility proxyas would be obtained using the unobservable conditionalvolatility, while the MAE loss function may not be robustto the noise of the proxy. The forecast bias and informationcontent are then measured using the Mincer–Zarnowitz(MZ) regression, given by

ri,t rj,t = αij + βijσij,t + εij,t . (28)

A forecast is conditionally unbiased (i.e., weak-form effi-cient) if and only if αij = 0 and βij = 1. The residuals fromthe MZ regression above will generally be heteroskedastic(see Patton & Sheppard, 2009), and so we apply the trans-formedMZ regressionwith generalised least squares (GLS)estimation, in order to improve the power of the test. TheMZ-GLS regressions for the variances and covariances are

r2i,tσ 2i,t

= αi1

σ 2i,t

+ βi + ei,t (29)

and

ri,t rj,tσi,t σj,t

= αij1

σi,t σj,t+ βijρij,t + eij,t , (30)

respectively, with ρij,t being the forecast correlation be-tween assets i and j.

As Engle and Colacito (2006) noted, the statistical eval-uation of covariance matrix forecasts on an element-by-element basis has a number of drawbacks, particularly forhigh dimensional systems. In particular, direct compar-isons between two covariance matrices are difficult, be-cause the distance between them is not well specified.Indeed, the statistical approaches described above implic-itly assume that all elements of the covariance matrix areequally important (in the sense that the same error isequally costly in economic terms in each element), butthere is no a priori reason why this should necessarily bethe case. Moreover, the use of the low frequency realizedvolatility as a proxy for the true volatility introduces a con-siderable amount of noise that inflates the forecast errors


of the conditional volatility forecasts, substantially reduc-ing their explanatory power. This has prompted tests ofcovariance matrix forecast performance based instead oneconomic loss criteria. Such tests have shown that condi-tional volatility models perform better when the perfor-mance is measured using an economic loss function thanwhen it is based on traditional statisticalmeasures (see, forexample, Engle, Kane, & Noh, 1996, West, Edison, & Cho,1993).

In this paper, we employ the economic loss functiondeveloped by Engle and Colacito (2006), who studied theusefulness of forecasts of the conditional covariancematrixin an asset allocation framework. Assume that an investorallocates a fraction wt of his wealth to n risky assets andthe remainder

1 − w′

t ιto the risk-free asset, where ι is

the (n × 1) unit vector. In themean–variance optimizationframework, the investor solves the following optimizationproblem at time t:

minwt

w′

tHt+1wt , (31)

subject to

w′

tµ +1 − w′

t ιr ft = µ∗

p, (32)

where Ht+1 is the covariance matrix at time t + 1, µ isthe vector of expected returns, r ft is the risk-free rate, andµ∗

p is the target return. As µ is assumed to be constant,the optimal weight of each asset changes over time as aresult of changes in the covariance matrix. Since the truecovariance matrix Ht+1 is unobserved, the optimisationproblem is solved using a forecast of Ht+1 obtained froma multivariate conditional volatility model, to yield anapproximation to the true optimal portfolio. The investorchooses among competing forecasts of the conditionalcovariance matrix on the basis of the volatility of theresulting portfolio. Engle and Colacito (2006) show thatthe lowest volatility of the investor’s portfolio is obtainedwhen the forecast covariance matrix is equal to the truecovariancematrix, irrespective of both the expected excessreturn vectorµ and the target returnµ∗

p . This then yields astraightforward economic test of the relative performancesof competing covariance matrix forecasts based on thevolatility of the optimal portfolio.

Engle and Colacito (2006) also note that, in the bivariatecontext, the relative volatilities of portfolios depend onthe relative returns of the n risky assets, not on theirabsolute returns. Using polar coordinates, all possible pairsof relative expected returns can be expressed in the formµ =

sin π j

20 , cosπ j20

, for j ∈ 0, . . . , 10. When j = 5,

for example, the expected returns are identical, whichyields the global minimum variance portfolio. To obtain asingle summary vector of expected returns, we constructprior probabilities for different vectors of expected returnsusing the sample data and the quasi-Bayesian approachintroduced by Engle and Colacito (2006). We use theseprobabilities as weights for estimating a single weightedaverage vector of expected returns. In the empirical study,we assume a target excess return equal to 1.2

2 The choice of the target return is immaterial, in the sense that it doesnot affect the relative volatilities of the portfolios.

For each vector of expected returns, and for each pairof covariance matrix forecasts, we test whether the port-folio variances are equal using the Diebold and Mariano(1995) test. In particular, we consider the loss differential

ukt =

σ

1,kt

2−

σ

2,kt

2, where

σ

1,kt

2and

σ

2,kt

2are

the variances of portfolios 1 and 2, respectively, for the ex-pected return vectorµk. By regressing uk

t on a constant, andusing the Newey andWest (1987) adjusted covariancema-trix, the null hypothesis of equal variances is simply a testfor the mean of u being equal to zero. Engle and Colacitonote that because uk

t is itself heteroscedastic, a more effi-cient estimator can be obtained by dividing u by the truevariance. Since the true covariance matrix is unknown andtwo estimators are being compared, they suggest using thegeometric mean of the two variance estimators as the de-nominator. The improved loss differential is given by

vkt = uk

t

2µk′ H1

t

−1µk

µk′ H2t

−1µk−1

. (33)

We apply the Diebold andMariano tests to both the u and vseries. Joint tests for all vectors of expected returns are alsoconducted. To compare the forecast performances of allmodels, we then use the superior predictive ability (SPA)test of Hansen (2005) and the model confidence set (MCS)test of Hansen, Lunde, and Nason (2011). While the SPAtest examines whether any of the other competing modelsoutperform the benchmark LM–EWMA models, the MCSselects a set of the best models without the need for abenchmark. The SPA test is considered for both the u and vseries, where one of the models is the benchmark model.

4. Data description

The empirical analysis employs the datasets used byEngle and Colacito (2006). We begin by studying theforecast performances of the six conditional volatilitymodels in two bivariate systems. The low correlationsystem uses daily data for the S&P500 and 10-yearTreasury bond futures, while the high correlation systemuses daily data for the S&P500 and Dow Jones industrialaverage (DJIA) indices. All of the data are from Datastreamand cover the period 01 January 1988–31 December 2009.The returns are calculated as the log price difference overconsecutive days. We exclude from the sample all dayson which any of the markets were closed, yielding atotal of 5548 observations for each dataset. As the futurescontracts require no initial investment, the futures returnscan be interpreted as excess spot returns. The returns of theS&P500 andDJIA indices are converted to excess returns bysubtracting the daily 1-month T-Bill rate.3 Table 1 reportsthe descriptive statistics of the four return series. Thesample correlation of the stock and bond futures is veryclose to zero, while that for the S&P500 and DJIA indicesis close to one. For all four series, returns are negativelyskewed and leptokurtic.

3 This is the simple daily rate that compounds to the 1-month T-Billrate from Ibbotson and Associates, Inc., over the number of trading daysin the month.


Table 1Summary statistics for the two bivariate systems. The table reports descriptive statistics for the daily returns on stock and bond futures, and the dailyexcess returns on the S&P500 and DJIA indices. The means and standard deviations are annualised. The normality tests report the test statistics of theJarque–Bera test. The sample period is from 01 January 1988 to 31 December 2009. The table also reports the fractional difference operator, d, estimatedusing the FIGARCH, Geweke–Porter–Hudak (GPH) and Moulines–Soulier (MS) tests. The GPH and MS estimators are applied to both squared and absolutereturns.

Return series Mean (%) Std. dev. (%) Skewness Kurtosis Min (%) Max (%) Normality test Corr. dFIGARCH Squared returns Absolute returnsdGPH dMS dGPH dMS

Stock 6.83 19.06 −0.19 14.18 −10.40 13.20 28936 −0.038 0.403 0.357 0.373 0.511 0.406Bond 1.48 6.53 −0.28 6.63 −2.86 3.57 3123 0.355 0.410 0.190 0.441 0.209S&P500 2.80 18.34 −0.25 12.32 −9.47 10.95 20117 0.960 0.492 0.441 0.461 0.576 0.443DJIA 3.59 17.72 −0.20 11.62 −8.20 10.51 17194 0.487 0.396 0.417 0.557 0.427

We conduct tests to confirm the evidence of longmemory dynamics in the volatility, the results of which arealso reported in Table 1. The parametric FIGARCH modelis estimated for the whole sample, and the estimatedfractional difference operators range from 0.35 to 0.49.We also apply two semi-parametric tests of long memory,namely the narrow band log periodogram (GPH) estimatorof Geweke and Porter-Hudak (1983) and the broad bandlog periodogram (MS) estimator of Moulines and Soulier(1999). For estimating the GPH and MS operators, we usethe recommended bandwidth m equal to the square rootof the sample size (m = 77) and the Fourier term p equalto the log of the sample size (p = 4), respectively. TheGPHandMS tests are applied for both squared and absolutereturns. The tests suggest the presence of long memory inthe volatility for all four series, and also suggest that thestock return volatility has longer memory than the bondreturn volatility. The autocorrelations of absolute returnsare also consistently higher than those of squared returns, afeature first identified by Taylor (1986).We conduct a one-sided test of the hypothesis d = 0.5, against the alternatived < 0.5. Rejecting this hypothesis, we confirm that thevolatility processes of all four series are characterised bylong memory, but are nevertheless stationary.

Following Engle and Colacito (2006), we also considera moderate correlation high dimensional system. An in-ternational stock and bond portfolio is constructed from34 assets, comprising 21 stock indices from the FTSE All-World indices and 13 5-year average maturity bond in-dices. The 21 stock indices and 13 bond indices includeall of the major world stock and government bond mar-kets. All of the data are taken from Datastream and con-verted to prices in US dollars. Following Engle and Colacito(2006),we useweekly returns to avoid the problemof non-synchronous trading. Weekly returns are calculated as thelog price difference, using Friday to Friday closing prices.The dataset comprises 22 years of weekly returns, yieldinga total of 1147 observations from 01 January 1988 to 31December 2009. The descriptive statistics for the interna-tional dataset are given in Table 2. Again, the returns areleptokurtic, and in most cases, negatively skewed. The in-ternational stock markets are relatively highly correlated,as are the international bond markets. The average corre-lation coefficient is 0.54 among the 21 stock market re-turn series, and 0.61 among the bondmarket return series.However, the stock and bond markets as a whole havean average correlation coefficient of only 0.20. All 34 re-turn series show evidence of long memory in the volatil-ity. For all countries in which both stock and bond indices

are present, the stock index volatility is, again,more persis-tent than the bond index volatility. The average fractionaldifference operator for the stock indices is 0.44 with theparametric FIGARCH test, and 0.32 with the semiparamet-ric GPH tests. The corresponding values for the interna-tional bond indices are 0.30 and 0.25.

In addition, we also consider a higher frequency highdimensional system, comprising the components of theDow Jones industrial average (DJIA) index as of 31 De-cember 2009. Daily data are collected from the Center forResearch in Security Prices from 01 March 1990 to 31 De-cember 2009. We exclude Kraft, which was not listed untilJune 2001. Returns are calculated as the log price differ-ence over consecutive days. All days on which the marketwas closed are excluded from the sample, yielding 5001observations. Summary statistics for the 29 DJIA stocks arealso given in Table 2. The return series are again highlynon-normal, with a very high leptokurtosis. The averagecorrelation coefficient of the DJIA components is 0.34. Thesystem also exhibits long memory in the volatility. The av-erage estimated fractional difference orders are 0.37 withthe FIGARCH test and 0.42 with the GPH test.

For each series, the whole sample is divided into an ini-tial estimation period of 252 observations (one year for thedaily returns series and five years for the weekly returnsseries), and forecast periods of 5296, 895 and 4749 obser-vations for the two bivariate portfolios, the internationalstock and bond portfolio and the DJIA component portfo-lio, respectively. The initial estimation period is used toestimate each model in order to generate out-of-sampleforecasts of the covariance matrix for observation 253. Theestimationwindow is then rolled forward one observation,the models are re-estimated, and forecasts are made forobservation 254, and so on, until the end of the sample isreached. We initially estimate the conditional covariancematrix using all of the multivariate models described inSection 2, except for the FIGARCH(1, d, 1)–DCCmodel. Thismodel is excluded owing to the prohibitively short estima-tion period. In Section 6.1, we employ longer estimationperiods and consider all six models.

5. Empirical results

5.1. Low dimensional systems: the stock-bond and S&P500-DJIA portfolios

5.1.1. Statistical evaluationTable 3 reports the statistical evaluation of the accu-

racies of the five conditional volatility models using the


Table 2Summary statistics for the two multivariate systems. The table reportsdescriptive statistics for the weekly returns on 34 international stock andbond indices, and the daily excess returns on 29 components of the DJIAindex. Means and standard deviations are annualised. The sample periodfor the international stock and bond portfolio is from 01 January 1988 to31 December 2009, while the sample period for the DJIA portfolio is from01 March 1990 to 31 December 2009.

Return series Mean (%) Std. dev. (%) Skewness Kurtosis

Panel A. International stock and bond portfolio

International stocks

Australia 7.30 21.64 −1.77 21.33Austria 6.03 25.81 −1.52 18.70Belgium 5.45 20.98 −1.21 12.68Canada 7.58 20.68 −1.13 13.91Denmark 9.89 21.00 −1.31 13.35France 7.44 21.25 −0.90 10.94Germany 6.69 23.49 −0.80 8.93Hong Kong 9.22 25.37 −0.62 6.57Ireland 3.44 25.49 −1.72 19.88Italy 2.91 24.82 −0.60 8.85Japan −1.29 22.56 0.07 4.67Mexico 19.21 33.81 −0.33 7.66Netherlands 6.88 20.89 −1.44 17.48New Zealand −0.08 22.04 −0.63 7.44Norway 8.93 26.82 −0.84 10.37Singapore 6.99 26.29 −0.69 13.21Spain 6.92 22.28 −0.90 10.21Sweden 9.79 26.88 −0.52 7.73Switzerland 8.44 19.42 −0.70 11.14UK 4.44 19.13 −1.05 16.81US 7.03 16.81 −0.81 10.54

International bonds

Austria 0.92 10.58 −0.03 3.64Belgium 0.95 10.68 −0.02 3.47Canada 2.36 8.71 −0.51 6.53Denmark 1.60 10.92 0.00 3.84France 1.81 10.54 −0.02 3.47Germany 0.73 10.62 0.01 3.37Ireland 1.83 10.89 −0.25 4.19Japan 1.67 12.11 0.89 8.33Netherlands 0.55 10.64 −0.02 3.36Sweden 0.06 12.06 −0.18 3.84Switzerland 0.95 12.05 0.11 3.72UK 0.13 10.60 −0.24 4.93US 1.23 4.43 −0.19 3.82

Panel B. DJIA portfolio

AA 3.47 39.12 −0.02 11.23AXP 7.22 38.76 0.03 9.94BA 4.61 31.89 −0.33 9.73BAC 1.51 45.21 −0.29 30.90CAT 10.19 33.62 −0.08 7.18C 28.74 46.95 0.00 7.48CVX 7.62 25.60 0.13 12.63DD 2.76 29.39 −0.09 7.10DIS 6.43 32.11 0.00 10.40GE 5.44 29.93 0.01 11.17GM 13.54 35.10 −0.67 16.81HD 11.51 40.37 −0.08 9.21HPQ 8.14 30.53 0.04 9.76IBM 13.99 42.72 −0.38 8.26INTC 11.37 23.70 −0.19 9.75JNJ 8.01 42.10 0.26 13.11JPM 9.39 24.79 0.08 8.01KO 10.43 26.89 −0.04 6.98MCD 7.12 24.25 0.01 7.50MMM 5.82 29.95 −1.09 22.53MRK 19.06 35.23 0.01 7.94MSFT 10.01 29.62 −0.18 6.07

Table 2 (continued)

Return series Mean (%) Std. dev. (%) Skewness Kurtosis

PFE 10.26 25.33 −2.78 68.38PG 3.50 28.68 0.08 7.39T 6.01 30.34 0.34 16.22UTX 11.97 28.77 −1.13 28.55VZ 1.90 27.61 0.17 7.64WMT 11.39 29.21 0.13 5.83XOM 8.91 24.83 0.09 11.92

Table 3RMSE and MAE for the two bivariate systems. The table reports theRMSE and MAE for each element of the conditional covariance matrixestimated using five multivariate conditional volatility models over theforecast period. The squares and cross-products of daily returns are usedas proxies for the actual variances and covariances.

EWMA GARCHDCC

LM–EWMA LM–EWMADCC

CGARCHDCC

Panel A. Root mean square error (RMSE)

Stock 4.7483 4.7953 4.7459 4.7459 4.7649Bond 0.3964 0.3978 0.3957 0.3957 0.3988Stock-bond 0.7442 0.7593 0.7432 0.7536 0.7575

S&P500 4.0336 4.0921 4.0434 4.0434 4.0646DJIA 3.6876 3.7295 3.6900 3.6900 3.7076S&P500-DJIA 3.7951 3.8402 3.8015 3.8015 3.8166

Panel B. Mean absolute error (MAE)


S&P500 1.4088 1.4251 1.4089 1.4089 1.4407DJIA 1.3079 1.3232 1.3077 1.3077 1.3357S&P500-DJIA 1.3284 1.3435 1.3296 1.3306 1.3583

RMSE and MAE measures for the two bivariate systems,namely the stock-bond and S&P500-DJIA portfolios. TheLM–EWMA and LM–EWMA–DCC models yield identicalRMSE and MAE measures for the variances, since the vari-ance forecasts in both models are based on the univari-ate long memory EWMA model. However, the LM–EWMAmodel performs better with respect to the covariance fore-casts. The LM–EWMA model also yields the lowest RMSEand MAE values for all elements in the stock-bond co-variance matrix, while the short memory EWMA modelperforms best in the S&P500-DJIA case, although the differ-ence between the EWMA and LM–EWMA models is small.The parsimony of the EWMA model may offset the pos-sible gains from the more correctly specified, yet morecomplex LM–EWMAmodel. Among the DCCmodels, how-ever, the LM–EWMA–DCC model dominates, suggestingthat there are potential benefits of allowing for long mem-ory in the volatility. The short memory GARCH-DCCmodelis the worst model in terms of forecast accuracy under thesymmetric RMSE and MAE measures.

The results of the standard and GLS-transformed Min-cer–Zarnowitz regressions for the two bivariate systemsare summarised in Table 4. The table reports the estimatedcoefficients of the regression, the R2 statistics, and thep-value for each element of the covariance matrix for thenull hypothesis of conditional unbiasedness. The numbersin parentheses are the estimated coefficients for the stan-dard Mincer–Zarnowitz regressions. The LM–EWMA and


Table 4Mincer–Zarnowitz regressions for the two bivariate systems. The table reports the estimated coefficients of theMZ–GLS regressions for the elements of thecovariancematrix. The p-values are for the tests of the joint hypothesis H0 : αij = 0 and βij = 1. The numbers in parentheses are the estimated coefficientsfor the standard MZ regressions.

Intercept Slope R2 p-value Intercept Slope R2 p-value

Panel A. EWMA Panel B. GARCH-DCC

Stock 0.163 0.877 0.003 0.000 0.086 0.957 0.001 0.032(0.138) (0.888) (0.175) (0.000) (0.084) (0.783) (0.169) (0.000)

Bond 0.034 0.839 0.007 0.000 0.027 0.809 0.002 0.005(0.010) (0.657) (0.037) (0.000) (0.012) (0.642) (0.030) (0.000)

Stock-bond 0.003 0.850 0.047 0.054 −0.028 0.855 0.039 0.000(0.008) (0.707) (0.047) (0.000) (0.015) (0.553) (0.022) (0.000)

S&P500 0.133 0.895 0.004 0.000 0.054 0.978 0.001 0.096(0.137) (0.906) (0.205) (0.001) (0.264) (0.803) (0.191) (0.000)

DJIA 0.159 0.858 0.004 0.000 0.074 0.953 0.001 0.036(0.145) (0.890) (0.179) (0.000) (0.246) (0.801) (0.167) (0.000)

S&P500-DJIA 0.141 0.871 0.116 0.000 0.057 0.964 0.149 0.124(0.134) (0.899) (0.195) (0.001) (0.233) (0.811) (0.181) (0.000)

Panel C. LM–EWMA Panel D. LM–EWMA–DCC

Stock 0.132 0.880 0.002 0.001 0.132 0.880 0.002 0.001(−0.002) (1.011) (0.173) (0.154) (−0.002) (1.011) (0.173) (0.154)

Bond 0.024 0.895 0.003 0.000 0.024 0.895 0.003 0.000(0.053) (0.702) (0.037) (0.000) (0.053) (0.702) (0.037) (0.000)

Stock-bond 0.001 0.916 0.053 0.627 −0.020 0.881 0.040 0.000(−0.013) (0.743) (0.047) (0.000) (−0.051) (0.648) (0.030) (0.000)

S&P500 0.105 0.901 0.002 0.000 0.105 0.901 0.002 0.000(0.001) (1.011) (0.199) (0.224) (0.001) (1.011) (0.199) (0.224)

DJIA 0.122 0.877 0.002 0.001 0.122 0.877 0.002 0.001(0.008) (1.001) (0.175) (0.039) (0.008) (1.001) (0.175) (0.039)

S&P500-DJIA 0.109 0.885 0.123 0.001 0.104 0.889 0.123 0.002(0.003) (1.008) (0.189) (0.120) (−0.005) (1.012) (0.189) (0.202)

Panel E. CGARCH-DCC

Stock 0.140 0.871 0.002 0.001(0.252) (0.802) (0.178) (0.000)

Bond 0.044 0.701 0.007 0.000(0.062) (0.599) (0.028) (0.000)

Stock-bond −0.031 0.712 0.029 0.000(−0.070) (0.636) (0.026) (0.000)

S&P500 0.200 0.835 0.201 0.000(0.077) (0.935) (0.001) (0.021)

DJIA 0.090 0.917 0.001 0.015(0.186) (0.833) (0.176) (0.000)

S&P500-DJIA 0.071 0.929 0.150 0.073(0.173) (0.841) (0.191) (0.000)

LM–EWMA–DCC models generally dominate for the stan-dard Mincer–Zarnowitz test. The unbiasedness hypothe-sis cannot be rejected at conventional significance levelsfor any of the stock variance forecasts, nor for the co-variance forecasts in the S&P500-DJIA system for theLM–EWMA and LM–EWMA–DCC models, but it is rejectedin all other cases. The EWMA model, despite evidently notbeing as efficient, performs slightly better in terms of itsexplanatory power, as measured by the R2 statistic. TheCGARCH-DCCmodel performs rather badly; indeed, it per-forms only marginally better than the GARCH-DCC model.However, when heteroskedasticity is taken into account inthe MZ-GLS test, the performances of the LM–EWMA andLM–EWMA–DCCmodels deteriorate dramatically. The un-

biasedness hypothesis is rejected in most cases for thesetwo long memory models.

5.1.2. Economic evaluationWe use the forecasts of the covariance matrix to con-

struct the minimum variance portfolios, subject to a tar-get excess return of 1. The relative conditional volatilitiesof the portfolios constructed using the different condi-tional covariance matrix estimators and all possible vec-tors of expected returns are compared in Table 5. The pairsof Bayesian prior weighted returns are obtained from non-overlapping consecutive subsamples of 63 days (3months)from the full datasets. Engle and Colacito (2006) showthat by considering unconditional mean-adjusted returns,


Table 5Comparison of conditional volatilities: bivariate portfolios. The table reports the average conditional volatilities for the two bivariate portfolios, constructedwith the objective ofminimizing the variance subject to the target excess return of 1. Each row in the table shows the results for the pair of expected returnsin the corresponding first two columns. The overall returns are the pair of weighted returns using the Bayesian prior probabilities. The lowest volatility ineach row is normalised to 100.

Panel A. Stock-bond portfolio

µStock µBond EWMA GARCH-DCC LM-EWMA LM-EWMA-DCC CGARCH-DCC Const.

0.00 1.00 100.247 102.156 100.000 101.909 104.206 105.9380.16 0.99 100.243 102.330 100.000 101.983 104.557 105.1480.31 0.95 100.402 102.580 100.000 102.077 105.025 104.4220.45 0.89 100.505 102.587 100.000 101.956 105.457 103.7540.59 0.81 100.580 102.465 100.000 101.827 105.655 103.0740.71 0.71 100.521 102.632 100.000 101.746 105.472 102.8400.81 0.59 100.390 102.705 100.000 102.017 104.974 103.5070.89 0.45 100.317 102.673 100.000 102.040 104.832 105.5640.95 0.31 100.237 101.994 100.000 101.301 104.207 108.9040.99 0.16 100.465 101.949 100.000 100.509 103.752 111.0381.00 0.00 100.385 103.277 100.000 102.335 105.200 106.434

Overall (Weighted) 100.208 102.097 100.000 101.496 104.307 108.365

Panel B. S&P500-DJIA portfolio

µSP500 µDJIA EWMA GARCH-DCC LM-EWMA LM-EWMA-DCC CGARCH-DCC Const.

0.00 1.00 100.402 102.054 100.089 100.000 102.233 101.2060.16 0.99 100.491 101.925 100.076 100.000 102.227 101.4350.31 0.95 100.511 101.534 100.090 100.000 102.136 101.8650.45 0.89 100.308 100.836 100.022 100.000 101.695 102.7510.59 0.81 100.172 100.917 100.000 100.559 101.547 105.2440.71 0.71 100.191 102.031 100.000 100.658 102.946 102.0880.81 0.59 100.184 101.040 100.000 100.658 101.317 107.0190.89 0.45 100.377 101.256 100.000 100.586 101.988 106.4250.95 0.31 100.347 101.473 100.000 100.404 102.137 103.9280.99 0.16 100.256 101.681 100.000 100.219 102.119 102.5581.00 0.00 100.261 101.697 100.000 100.087 102.045 101.784

Overall (Weighted) 100.204 102.057 100.000 100.719 103.022 102.508

one can obtain a consistent estimator of the true condi-tional portfolio variance. The lowest conditional volatil-ity, corresponding to the best covariance matrix estimate,is normalised to 100. The ‘Const.’ portfolio is the fixedweight portfolio constructed with the ex-post constantunconditional covariance matrix. It is clear that the con-ditional covariance matrices generally outperform the un-conditional covariance matrix, highlighting the economicvalue of volatility timing strategies. The results for thetwo LM–EWMA models are favourable. For both the lowcorrelation stock-bond portfolio and the high correlationS&P500-DJIA portfolio, the LM–EWMA model consistentlyyields the lowest portfolio volatility. Incorporating longmemory into the EWMA structure therefore appears to im-prove the forecasts of the conditional covariance matrix ina way that is economically valuable. Among the DCC mod-els, the LM–EWMA–DCCmodel again dominates. Althoughthe CGARCH model is designed to capture the long mem-ory volatility, its high degree of parameterization, whichcan potentially yield a high estimation error, evidently de-tracts from its performance. It is also interesting to notethat the simple EWMA model outperforms more sophis-ticated models such as the GARCH-DCC and CGARCH-DCC models, and is even superior to the LM–EWMA–DCCmodel in most cases. In practice, investors may be moreconcerned with the out-of-sample realized volatility thanthe in-sample conditional volatility. We therefore com-pare the out-of-sample volatilities for the two bivariateportfolios. The results are similar (and hence are not re-ported), in that the LM–EWMA model consistently yields

the lowest out-of-sample portfolio volatility, followed bythe LM–EWMA–DCC and EWMAmodels.

Next, Diebold–Mariano tests are used to test for theequality of different models with each vector of expectedreturns. Joint tests are also carried out for all vectors of ex-pected returns, applying the GMM method with a robustHAC covariance matrix. To save space, we only report theresults of the joint tests here. The results for other vec-tors of expected returns, especially those which are closeto the sample mean, are similar. Table 6 shows the re-sults of both the standard and improved tests for the twobivariate portfolios. The numbers in parentheses are theestimated t-statistics for the standard tests. Each cell inthe table corresponds to the test of the hypothesis thatthe two models in the row and column are equal in termsof volatility forecasting, against the alternative that themodel in the row is better or worse than the model inthe column. A positive sign indicates that the model in therow is better than themodel in the column, and vice-versa.The Diebold–Mariano tests confirm our earlier results. Thestandard Diebold–Mariano test shows that the LM–EWMAmodel significantly dominates all other conditional volatil-ity models, for both short memory and long memory.With the improved version of the Diebold–Mariano test,the differences between the pairs of models are lessclearly marked, and the outperformance of the LM–EWMAmodel is not significant in some cases. However, theDiebold–Mariano statistics are still uniformly positive.

The results of the MCS and SPA tests are also consis-tent with the previous experiments. For robustness, the


Table 6Diebold–Mariano joint tests of the bivariate portfolios. The table reports the t-statistics of the Diebold–Mariano joint tests of all the of expected vectorsof returns for the two bivariate portfolios using the improved version of the test described by Engle and Colacito (2006). Panel A reports the results forthe stock-bond portfolio, and panel B the results for the S&P500- DJIA portfolio. The t-statistics for the standard version of the Diebold–Mariano test arereported in parentheses. A positive number indicates that the model in the row is better than that in the column, and vice-versa. The critical values for the1%, 5% and 10% significance levels are 2.576, 1.960, and 1.645, respectively.

EWMA GARCH-DCC LM-EWMA LM-EWMA-DCC CGARCH-DCC Const.


EWMA 2.001 −0.422 1.741 1.742 3.229(4.072) (−6.393) (5.806) (4.710) (7.849)

GARCH-DCC −2.001 −2.054 −1.944 0.171 0.044(−4.072) (−4.442) (−2.385) (4.719) (2.156)

LM–EWMA 0.422 2.054 1.807 1.726 3.629(6.393) (4.442) (6.624) (4.957) (8.451)

LM-EWMA-DCC −1.741 1.944 −1.807 1.462 2.209(−5.806) (2.385) (−6.624) (3.719) (5.003)

CGARCH-DCC −1.742 −0.171 −1.726 −1.462 −0.322(−4.710) (−4.719) (−4.957) (−3.719) (−0.134)

Constant −3.229 −0.044 −3.629 −2.209 0.322(−7.849) (−2.156) (−8.451) (−5.003) (0.134)


EWMA 2.671 0.338 1.246 3.179 2.598(5.166) (−3.329) (1.469) (6.374) (5.222)

GARCH-DCC −2.671 −3.033 −3.296 −0.368 2.130(−5.166) (−6.004) (−5.545) (1.307) (3.795)

LM–EWMA −0.338 3.033 1.412 3.674 2.769(3.329) (6.004) (2.813) (7.161) (5.455)

LM-EWMA-DCC −1.246 3.296 −1.412 2.611 2.848(−1.469) (5.545) (−2.813) (5.939) (5.438)

CGARCH-DCC −3.179 0.368 −3.674 −2.611 2.004(−6.374) (−1.307) (−7.161) (−5.939) (3.161)

Constant −2.598 −2.130 −2.769 −2.848 −2.004(−5.222) (−3.795) (−5.455) (−5.438) (−3.161)

MCS and SPA tests are conducted with different bootstrapblock lengths (10 and 21 days). The MCS test chooses theLM–EWMAmodel as the best model from among the com-peting models at the 95% confidence level for the stock-bond portfolio, and the LM–EWMA and EWMA models forthe S&P500-DJIA portfolio. The SPA tests confirm this re-sult. Since the p-values of the SPA test for the u series ofboth portfolios are close to unity, there is no statisticalevidence that the LM–EWMA benchmark model is outper-formed by any of the alternative volatility models. How-ever, aswith the improved version of the Diebold–Marianotest, the domination of the LM–EWMAmodel is not signifi-cant in the SPA test with the v series, in which the p-valuesof the SPA test for both bivariate portfolios are zero.

5.2. High dimensional systems: the international stock andbond portfolio and the DJIA portfolio

5.2.1. Economic evaluationIn practice, a portfoliomay comprise hundreds of assets,

and consequently an investor may want to examine theforecast performances of different conditional volatilitymodels in a higher dimensional framework. In an assetallocation problem, the investor needs to estimate boththe expected returns and the covariance matrix. However,since the number of possible expected return vectors

for the high dimensional portfolios is prohibitively large,we study the values of covariance matrix forecasts in arestricted case of hedging portfolios, in which one asset ishedged against all other assets in the portfolio. In so doing,we select the expected return vectors such that one entryis equal to one and all of the others are set to zero. Notethat the correctly specified covariance matrix will producethe portfolios with the lowest volatilities for any particularvector of expected returns. For the multivariate portfolios,we assume a risk free rate of 4%. Compared to the bivariateportfolios, the results for the multivariate portfolios aremarkedly different, in that the DCC models tend tooutperform the non-DCC models. The LM–EWMA–DCCmodel is the best performingmodel in 33 of the 34 hedgingportfolios of international stocks and bonds, and in 24 ofthe 29 portfolios of DJIA components. The superiority ofthe LM–EWMA model deteriorates significantly. Althoughit still produces better forecasts than the EWMA model,it is generally inferior to the GARCH-DCC and CGARCH-DCC models. The Diebold–Mariano joint tests are appliedfor all hedging expected returns, and the findings areconsistent with those of the relative volatilities (Table 7).The LM–EWMA–DCC model significantly outperforms allother models in both versions of the Diebold–Marianotests. The LM–EWMA performs badly, only significantlyoutperforming the EWMAmodel. In the DJIA portfolio, the


Table 7Diebold–Mariano joint tests: hedging multivariate portfolios. The table reports the t-statistics of the Diebold–Mariano joint tests for the hedgingmultivariate portfolios, using the improved version of the test described by Engle and Colacito (2006). Panel A corresponds to the international stockand bond portfolio, while Panel B corresponds to the DJIA portfolio. The t-statistics for the standard test are reported in parentheses. A positive numberindicates that the model in the row is better than the model in the column, and vice-versa. The critical values for the 1%, 5% and 10% significance levels are2.576, 1.960, and 1.645, respectively.

EWMA GARCH-DCC LM-EWMA LM-EWMA-DCC CGARCH-DCC Const.

Panel A. International stock and bond portfolio

EWMA −3.439 −3.619 −4.631 −3.484 −3.041(−7.124) (−9.129) (−10.639) (−7.392) (−2.606)

GARCH-DCC 3.439 2.441 −2.781 1.283 0.592(7.124) (4.366) (−4.879) 0.990 (5.059)

LM–EWMA 3.619 −2.441 −3.738 −2.578 −1.498(9.129) (4.366) (−7.506) (−4.533) (−0.367)

LM-EWMA-DCC 4.631 2.781 3.738 2.661 4.394(10.639) (4.879) (7.506) (4.478) (9.569)

CGARCH-DCC 3.484 −1.283 2.578 −2.66 0.451(7.392) (−0.990) (4.533) (−4.48) (3.971)

Constant 3.041 −0.592 1.498 −4.39 −0.45(2.606) (−5.059) (0.367) (−9.57) (−3.97)

Panel B. DJIA portfolio

EWMA −7.021 −9.806 −9.792 −9.555 −13.682(−37.746) (−40.434) (−36.787) (−38.156) (−32.086)

GARCH-DCC 7.021 1.773 −1.515 −0.999 0.861(37.746) (26.101) (−4.095) (1.275) (11.104)

LM–EWMA 9.806 −1.773 −8.790 −7.579 −6.572(40.434) (−26.101) (−27.930) (−28.704) (−12.425)

LM-EWMA-DCC 9.792 1.515 8.790 2.536 7.180(36.787) (4.095) (27.930) (6.316) (13.770)

CGARCH-DCC 9.555 0.999 7.579 −2.536 3.815(38.156) (−1.275) (28.704) (−6.316) (11.918)

Constant 13.682 −0.861 6.572 −7.180 −3.815(32.086) (−11.104) (12.425) (−13.770) (−11.918)

LM–EWMAmodel is even dominated by the unconditionalestimator. The outperformance of the LM–EWMA–DCCmodel is also confirmed by the MCS and SPA tests. TheMCS uniformly contains only the LM–EWMA–DCC modelat the 95% confidence level. The SPA test with both the uand v series and with the LM–EWMA–DCC model as thebenchmarkmodel consistently produce p-valueswhich areclose to unity, suggesting that the LM–EWMA–DCC modelcannot be outperformed by any of the alternative volatilitymodels. Another experiment with the global minimumvariance strategy, though not reported here, yields similarresults.

These results show consistently that incorporating longmemory in the volatility dynamics improves the forecastsof the covariance matrix. The LM–EWMA model generallyoutperforms the EWMAmodel, while the LM–EWMA–DCCmodel always yields the best results among the DCC mod-els. Our results also reveal an important difference in therelative forecasting powers of the DCC and non-DCC mod-els in low dimensional and high dimensional systems, re-spectively. In particular, the greater flexibility that arisesfrom estimating the volatility and correlation separately isevidently beneficial in the high dimensional case. This is-sue deserves attention in future research.

5.3. Longer horizon forecasts

Practical problems often require forecasts over longerhorizons than the 1-step-ahead forecasts consideredabove. In this section, we evaluate the forecast perfor-mances of different conditional volatility models, both sta-tistically and economically, for horizons of up to threemonths. Table 8 reports the RMSEs of different conditionalvolatility models for 1-week-, 1-month- and 1-quarter-ahead forecasts. The benchmarks are the true variancesand covariances, proxied by the sum of squares andcross products of daily returns over the forecast hori-zons. The longmemory volatility models generally outper-form the short memory models, with the LM–EWMA andLM–EWMA–DCCmodels consistently yielding the smallestforecast errors, although the standard EWMA model againproves itself a simple yet statistically accurate model. TheMAE results are similar, and hence are not reported.

We also estimate the Mincer–Zarnowitz regressionfor the longer horizons. Compared to the 1-step-aheadforecasts, the forecasts for longer horizons have a higherinformation content, which may be attributable to theuse of more accurate proxies of the true variances andcovariances. Again, the two LM–EWMA models dominatethe other short and long memory conditional volatility


Table 8RMSE for longer horizon forecasts: bivariate systems. The table reportsthe RMSEs for each element of the forecast conditional covariance matrixover the forecast period. The benchmarks are the realised variances andcovariances, proxied by the sum of squares and cross products of returnsover the forecast horizons, respectively.

EWMA GARCHDCC

LMEWMA

LM–EWMADCC

CGARCHDCC

Panel A. One week (5-step) ahead forecasts


S&P500 10.308 10.676 10.411 10.411 10.039DJIA 9.611 9.892 9.638 9.638 9.286S&P500-DJIA 9.783 10.062 9.851 9.858 9.460

Panel B. One month (21-step) ahead forecasts


S&P500 42.384 47.684 41.015 41.015 44.696DJIA 38.029 40.468 36.789 36.789 38.583S&P500-DJIA 39.655 43.093 38.330 38.350 40.755

Panel C. One quarter (63-step) ahead forecasts


S&P500 133.140 151.082 129.748 129.748 142.108DJIA 113.798 128.904 111.416 111.416 120.125S&P500-DJIA 121.775 137.591 118.879 118.890 128.746

models at all forecast horizons, especially in the standardversion of the test. They are the only two models thatgenerally yield conditionally unbiased forecasts of theelements of the covariance matrix.

The economic usefulness of alternative covariance ma-trix estimators is assessed for both low and high dimen-sional portfolios over longer investment horizons. We letthe investor rebalance his portfolios weekly, monthly andquarterly. In practice, these rebalancing frequencies coverthe situations of most investors, at least approximately,from a day trader to a mutual fund. The results are sum-marised in Table 9.We report the out-of-sample volatilitiesfor the bivariate portfolios using the overall Bayesian priorweighted returns, and for the multivariate global mini-mum variance portfolios, where all of the expected returnsare assumed to be equal. The performance of the in-sampleconditional volatilities is similar, and hence is not reported.The gains from using the conditional volatility models aresmaller for a trader who rebalances at lower frequenciesthan for a day trader. The two long memory EWMA mod-els still consistently produce better forecasts than the shortmemory and constant volatilitymodels over horizons of upto three months. The LM–EWMA model generally outper-forms the EWMAmodel, while the LM–EWMA–DCCmodeldominates the other short and long memory DCC models.The CGARCH-DCC model also generates more favourableresults than the GARCH-DCC model. As with the dailyrebalanced portfolios, the DCC models outperform thenon-DCC models in the high dimensional DJIA portfolio.Under the hedging strategy, the DCC models even domi-nate in both high dimensional portfolios, with all vectors

Table 9Comparison of out-of-sample volatilities: long forecast horizons. Thetable reports the out-of-sample volatilities of the four bivariate andmultivariate portfolios for longer forecast horizons, with the objectiveof minimizing the variance, subject to the target excess return of 1. Thebivariate portfolios are constructed using the Bayesian prior weightedreturns, while the multivariate portfolios follow the global minimumvariance strategywhere all expected returns are assumed to be equal. Thelowest volatility in each row is normalised to 100.

EWMA GARCHDCC

LMEWMA

LM–EWMADCC

CGARCHDCC

Const.

Panel A. Bivariate portfolios

A1. Stock-bond portfolio

1-week 100.360 101.099 100.000 100.049 102.178 102.9131-month 100.000 100.907 100.291 100.485 101.357 102.5341-quarter 100.727 103.749 100.336 100.000 102.098 109.219

A2. S&P500-DJIA portfolio


Panel B. Multivariate portfolios

B1. International stock and bond portfolio

1-month 102.884 123.706 100.000 104.699 110.065 105.7361-quarter 106.339 128.935 100.000 105.900 113.939 111.520

B2. DJIA portfolio


of expected returns, and across all rebalancing frequencies,where the LM–EWMA–DCC model consistently yields themost economically useful forecasts. A similar conclusion,though not reported here, follows from the MCS, DM andSPA tests of the predictive ability of the different models’forecasts at different forecast horizons. The short memoryconditional volatility models either revert to the uncondi-tional volatility rapidly, at an exponential rate, or, in thecase of the EWMA model, do not converge at all, and con-sequently have relatively uninteresting long-run forecasts.With slowly decaying autocorrelations, the long memoryvolatility models are able to exploit past information bet-ter, and consequently yield more accurate forecasts overlonger horizons.

6. Robustness tests

6.1. Sensitivity analysis for different estimation windows

The forecast performance can potentially be affectedby the size of the rolling window used to estimate theconditional volatility models. We therefore re-evaluatethe forecast performances of the multivariate conditionalvolatility models using estimation windows of two, fiveand ten years of daily returns. In the cases of the 5- and10-year rolling windows, we also estimate the condi-tional covariance matrix using the FIGARCH-DCC model.We do not estimate the FIGARCH-DCC model with 1- and2-year rolling windows, since the estimation of the FI-GARCH model requires a prohibitively high upper lag cut-off. Following standard practice in the literature, we set


Table 10Comparison of out-of-sample volatilities: 5-year estimation window. Thetable reports the out-of-sample volatilities for the bivariate portfolios,with the objective of minimizing the variance subject to the target excessreturn of 1, using a 5-year estimation window of returns. The bivariateportfolios are constructed using the Bayesian prior weighted returns. Thelowest volatility in each row is normalised to 100.

EWMA GARCHDCC

LMEWMA

LM–EWMADCC

CGARCHDCC

FIGARCHDCC


1-day 100.21 102.01 100.00 101.30 103.45 115.111-week 100.93 101.91 100.00 102.15 100.08 116.191-month 100.00 103.42 100.36 102.71 100.65 111.091-quarter 100.00 105.15 101.13 102.98 106.73 125.05


1-day 100.12 101.21 100.00 100.45 101.40 112.271-week 101.57 101.81 101.25 100.00 102.99 109.951-month 101.45 101.61 100.00 100.33 104.32 102.711-quarter 102.91 100.00 101.19 102.65 105.67 102.95

the truncation lag for the FIGARCH model equal to 1000.The FIGARCH-DCC model is also excluded from the inter-national stock and bond portfolio due to the short samplesize.

The outperformance of the two parsimonious longmemory EWMA models reported above is found to be in-sensitive to the choice of the estimation window length, inboth the low dimension and high dimension cases. To savespace, Table 10 only reports the economic evaluation forthe two bivariate portfolios with a 5-year estimation win-dow. The long memory LM–EWMA and LM–EWMA–DCCmodels generally produce forecasts that are more accurateand informative than other short and long memory mod-els, as well as being economically more useful. The sim-ple EWMA model, although not as good as the LM–EWMAmodel, performs quite well. Indeed, it consistently gen-erates better results than the more sophisticated DCCmodels. The long memory FIGARCH model is the worstperforming model, which may be attributable to the com-plexity of its specification. The DCC models, however, out-perform the non-DCC models in the multivariate hedgingportfolios. The LM–EWMA–DCC model consistently yieldsthe most favourable results for all vectors of expected re-turns and across all rebalancing frequencies. The explana-tory power of the non-DCC drops significantly, althoughthe LM–EWMA model still outperforms its short memoryEWMA counterpart.

6.2. Comparison with a component correlation model with along run specification

In this section, we compare the forecast performancesof the longmemory LM–EWMAand LM–EWMA–DCCmod-els with that of another benchmark model, the componentcorrelation DCC–MIDASmodel of Colacito, Engle, and Ghy-sels (2011). The DCC–MIDAS model combines the originalDCC structure with the GARCH–MIDAS component spec-ification of Engle, Ghysels, and Sohn (2008) that allowsus to extract the long run component of correlation viamixed data sampling. While the DCC structure only cap-tures short memory correlations, the DCC–MIDAS allowsboth short run and long run factors to affect correlations.

The DCC–MIDAS model for the correlation is analogous tothe CGARCH model for the variance.

In the DCC–MIDAS model, the conditional covariancematrix is decomposed as follows:

Ht = DtRtDt (34)

Rt = diag Qt−

12 Qtdiag Qt

−12 (35)

Qt = (1 − a − b)Rt(ωr) + αεt−1ε′

t−1 + βQt−1. (36)

Here, the short run correlation component is capturedby the autoregressive dynamic structure of the DCC spec-ification, with the intercept of the latter, Rt(ωr), being aslowly moving process that reflects the long-run correla-tion component:

Rt(ωr) =

Kcl=1

Φl(ωr) ⊗ Ct−l

Ct−l =

v1,t 0 0...

. . . 00 · · · vn,t

−

12

tk=t−Nc

εkε′

k

×

v1,t 0 0...

. . . 00 · · · vn,t

−

12

vi,t =

tk=t−Nc

ε2i,k, ∀i = 1, . . . , n

(37)

whereΦlωr

= ϕl

ωr

ιι′ and⊗ stands for theHadamard

product. The MIDAS weighting scheme helps us to extractthe slowly moving long run component with the numberof lags Kc for historical correlations, the lag lengths Nc forcomputing the historical correlations, and the weight vec-tor ωr of ω following a beta lag structure:

ϕlωr

=

(1 − k/Kc)ωr−1

Kcj=1

(j/Kc)ωr−1

, (38)

with ωr > 1. Here ωr , Kc and Nc are assumed to be identi-cal for all assets.

In this section, we employ a short memory volatil-ity GARCH(1, 1) structure for estimating the variancesand embedding the obtained standardised residuals inthe long memory component DCC–MIDAS model in or-der to estimate the covariance matrix for the economicevaluation. In particular, we evaluate the forecast perfor-mances of four multivariate volatility models: the longmemory volatility and covariance LM–EWMA model, thelong memory volatility and short memory correlationLM–EWMA–DCC model, the short memory volatility andlong memory correlation GARCH(1, 1)–DCC–MIDASmodeland the short memory volatility and correlation GARCH(1,1)–DCC model. The empirical analysis uses the bivariatestock-bond and multivariate international stock and bondportfolios, with the sample up to December 2007 for esti-mating the models’ coefficients and the sample from Jan-uary 2008 toDecember 2009 as the forecast period.We fol-low the procedure suggested by Engle and Colacito (2006)


for selecting Kc and Nc . In particular, we compare a profileof different MIDAS lags and choose the smallest number oflags at which the log likelihood functions reach their max-imum values. We choose Kc and Nc equal to 252 and 21(days) for the stock-bond portfolio and 153 and 13 (weeks)for the international stock and bond portfolio, respectively.

The forecast evaluation of the four models in both port-folios is reported in Table 11. Here, we let the investor up-date his portfolios monthly. The results suggest that theLM–EWMA and LM–EWMA–DCC models outperform theGARCH-DCC andGARCH-DCC–MIDASmodels in the bivari-ate stock-bond portfolio, while the LM–EWMA–DCCmodelalone dominates in the multivariate international stockand bond portfolio in terms of both out-of-sample andconditional volatilities. TheDiebold–Mariano tests confirmthis result, though the outperformance of the LM–EWMAand LM–EWMA–DCCmodels is not significant in the stock-bond portfolio.With the stock-bond portfolio, theMCS testcannot identify the best model, even though the SPA testrejects the hypothesis that either the LM–EWMA or theLM–EWMA–DCC model is outperformed by any compet-ingmodels. The result for themultivariate portfolio ismoreclear-cut, with the Diebold–Mariano, MCS and SPA tests allbeing in favour of the LM–EWMA–DCC model at conven-tional significance levels. Interestingly, the long memorycomponent correlation GARCH-DCC–MIDAS model doesnot performwell in these experiments. Indeed, it even per-forms worse than the short memory correlation GARCH-DCCmodel in both portfolios. Again, the DCCmodel provesto have an efficient structure, especially in themultivariateportfolio, where it outperforms both the non-DCC cross-product and the DCC–MIDAS structures. The results hereand in the previous sections suggest that the correlationstructure is of the utmost importancewhen estimating andforecasting the covariance matrix. Though the non-DCCcross-product and theDCC–MIDAS structuresmaybe spec-ified correctly, their rigid structure (such as in the non-DCCcross-product models) or their high degree of parameter-ization (such as in the DCC–MIDAS model) may generateless accurate forecasts, especially in high dimensional sys-tems. Once an appropriate correlation structure has beenchosen, the long memory volatility models generally out-perform the short memory volatility models.

7. Conclusion

In this paper, we evaluate the economic benefits thatarise from allowing for long memory in forecasting thecovariancematrix of returns over both short and long hori-zons, using the asset allocation framework of Engle and Co-lacito (2006). In so doing, we compare the performancesof a number of long memory and short memory multivari-ate volatilitymodels. Incorporating the longmemory prop-erty improves the forecasts of the conditional covariancematrix. In particular, we find that long memory volatilitymodels generally dominate short memory and uncondi-tional models on the basis of both statistical and economiccriteria, especially at longer horizons. Moreover, the rela-tively parsimonious long memory EWMA models outper-form the more complex multivariate long memory GARCH

Table 11Comparison with the component correlation DCC–MIDAS model. Thetable evaluates the forecast performances of the GARCH(1, 1)–DCC,LM–EWMA and LM–EWMA–DCCmodels, relative to that of the GARCH(1,1)–DCC–MIDASmodel. Panel A reports the results for the bivariate stock-bond portfolio, and panel B those for the multivariate international stockand bond portfolio, both with monthly rebalancing.

GARCHDCC

LM–EWMA LM–EWMADCC

GARCH-DCCMIDAS


A1. Comparison of volatilities (with the Bayesian priorweighted returns)

Out-of-samplevolatilities

102.738 100.739 100.000 103.078

Conditionalvolatilities

102.691 100.330 100.000 101.300

A2. Diebold–Mariano joint test (for all vectors of returns)

GARCH-DCC −1.4767 −1.6246 0.5878(−1.1228) (−1.1041) (0.7861)

LM–EWMA 1.4767 1.0092 0.6813(1.1228) (−0.8061) (1.0267)

LM–EWMADCC

1.6246 −1.0092 0.5935(1.1041) (0.8061) (1.0583)

GARCH-DCCMIDAS

−0.5878 −0.6813 −0.5935(−0.7861) (−1.0267) (−1.0583)

Panel B. International stock and bond portfolio

A1. Comparison of volatilities (average volatilities for allhedging vectors of returns)

Out-of-samplevolatilities

103.383 114.291 100.000 115.300

Conditionalvolatilities

102.782 112.941 100.000 104.012

A2. Diebold–Mariano joint test (for all hedging vectors ofreturns)

GARCH-DCC 1.2259 −1.7587 1.8205(1.6493) (−2.3406) (1.6012)

LM–EWMA −1.2259 −1.4084 −0.6371(−1.6493) (−1.8793) (−1.2545)

LM–EWMADCC

1.7587 1.4084 1.8936(2.3406) (1.8793) (2.2795)

GARCH-DCCMIDAS

−1.8205 0.6371 −1.8936(−1.6012) (1.2545) (−2.2795)

models. The high level of parameterization of the Com-ponent GARCH and FIGARCH models evidently generateslarge estimation errors that are detrimental to their perfor-mance. The results are consistent across different datasets,and are robust to different investment horizons and esti-mation windows. The findings of the paper are consistentwith those in the univariate volatility literature.

The results suggest the importance of choosing an ap-propriate correlation structure when estimating and fore-casting the covariance matrix. The non-DCC conditionalcovariance matrix estimators (such as the EWMA modelwith exponential weights and the LM–EWMA model withlogarithmic weights) impose the same dynamic structureon all elements of the covariance matrix, which facilitatestheir implementation in high dimensional systems, but ata cost in terms of accuracy. In a high dimensional system,employing a potentially less correctly specified but more


flexible DCC structure may yield better results. It would beinteresting to investigate this issue in greater detail.

The use of the long memory conditional covariancematrix produces optimal portfolios with a lower realisedvolatility than the static unconditional covariance matrix.However, since our aim is simply to evaluate the fore-casts of alternative conditional covariance matrices, andto choose the estimator that produces the lowest portfo-lio volatility, we do not consider realised portfolio returnsexplicitly. In particular, it does not follow that the portfo-lio with the lowest volatility is necessarily the best port-folio in terms of portfolio performance measures such asthe Sharpe ratio. Thus, it would also be of interest to inves-tigate further the economic value of long memory volatil-ity timing in the asset allocation framework, allowing fordifferences in returns as well as risk, and for the effect oftransaction costs. Finally, while the economic evaluationof the various models employs the covariance matrix asa whole, the statistical evaluation of the low-dimensionalcase is conducted on an element-by-element basis. Follow-ing recent research (see, for example, Patton & Sheppard,2009), a useful robustness check would be to use an ap-propriate multivariate loss function for covariance matri-ces for the high-dimensional case.

Acknowledgments

We are grateful to the associate editor, two referees andGilles Zumbach for their useful comments and suggestions,which have undoubtedly helped to improve the paper.

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Richard Harris is a Professor of Finance at the Xfi Centre for Finance andInvestment, University of Exeter, UK.

AnhNguyen is a doctoral candidate in finance at theXfi Centre for Financeand Investment, University of Exeter, UK.

http://dx.doi.org/doi:10.1080/14697688.2011.589399

long memory conditional volatility and asset allocation

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