On the Benefits of Equicorrelation for Portfolio Allocation · A portfolio allocation problem is used to compare forecasting methods. The global minimum variance portfolio and Model

NCER Working Paper SeriesNCER Working Paper Series

On the Benefits of Equicorrelation for Portfolio Allocation

Adam ClementsAdam Clements Ayesha ScottAyesha Scott Annastiina SilvennoinenAnnastiina Silvennoinen Working Paper #99Working Paper #99 June 2014June 2014

On the Benefits of Equicorrelation for Portfolio Allocation

Adam Clements∗, Ayesha Scott†, and Annastiina Silvennoinen‡

School of Economics and Finance, Queensland University of Technology§

First version: August 2010This version: June 2014

Abstract

The importance of modelling correlation has long been recognised in the field of portfolio man-agement with large dimensional multivariate problems are increasingly becoming the focus ofresearch. This paper provides a straightforward and commonsense approach toward investigat-ing a number of models used to generate forecasts of the correlation matrix for large dimensionalproblems. We find evidence in favour of assuming equicorrelation across various portfolio sizes,particularly during times of crisis. During periods of market calm however, the suitability ofthe constant conditional correlation model cannot be discounted especially for large portfolios.A portfolio allocation problem is used to compare forecasting methods. The global minimumvariance portfolio and Model Confidence Set are used to compare methods, whilst portfolioweight stability and relative economic value are also considered.

Keywords

Volatility, multivariate GARCH, portfolio allocation

JEL Classification Numbers

C22, G11, G17

∗e-mail: [email protected]†Corresponding author, e-mail: [email protected]‡e-mail: [email protected]§Postal address: GPO Box 2434, Brisbane, QLD, 4001We would like to thank Christopher Coleman-Fenn, Mark Doolan, Joanne Fuller, Stan Hurn, Andrew McClel-

land and Daniel Smith for their suggestions and advice. The technical support of High Performance Computing& Research Support at QUT is gratefully acknowledged. The responsibility for any errors and shortcomings inthis paper remains ours.

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1 Introduction and Motivation

The importance of modelling the volatility of asset returns for the purpose of portfolio alloca-

tion has been the subject of extensive research. Despite this significant body of work, less is

understood about the most appropriate method of handling large portfolios and subsequently

generating useful volatility forecasts. Increasingly, portfolios of hundreds or thousands of assets

are becoming the focus of research and many methods have been proposed to aid in dealing

with the issue of dimensionality. Surveys of the multivariate generalised autoregressive condi-

tional heteroskedasticity (MGARCH) models literature include Bauwens, Laurent, and Rom-

bouts (2006) and Silvennoinen and Terasvirta (2009). This paper considers the performance of

a number of models to generate forecasts of the correlation matrix across a number of portfolio

sizes, and evaluates the economic value of using these models in the context of asset allocation.

The main methods used to generate forecasts of the correlation matrix of interest here are the

Dynamic Equicorrelation (DECO) model of Engle and Kelly (2012), the consistent Dynamic

Conditional Correlation (cDCC) model of Aielli (2013) and Constant Conditional Correlation

(CCC) model of Bollerslev (1990), as well as semi-parametric models. DECO assumes pairwise

correlations to be equal at a point in time, whilst changing over time. This method is based

upon the cDCC model, however is tractable for large dimensions through a simplified likelihood

specification. Another notable advantage of equicorrelation is reduced estimation error due to its

use of the history of all assets in the portfolio for each element of the correlation matrix, in severe

contrast to cDCC where the history of only a given pair of assets is used to forecast the element

of the correlation matrix pertaining to that particular pair. This advantage is highlighted in

the analysis presented later in this paper. As alluded to above, the cDCC likelihood equation

can be burdensome to estimate in large dimensions however the composite likelihood approach

of Engle, Shephard, and Sheppard (2008) is used here to make the cDCC model more suitable

for large portfolios.

In this paper an empirical portfolio allocation exercise is used to compare various correlation

forecasting techniques. Minimum variance portfolios are formed and the out-of-sample perfor-

mance of the methods compared, using the Model Confidence Set (MCS) of Hansen, Lunde, and

Nason (2003). Out-of-sample periods are divided into subsamples based on the relative level of

volatility and performance again compared as before. The benefits of the choice of the MCS as

an evaluation tool are twofold: it does not require a benchmark model to be specified; and is a

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statistical test of equivalence of a given set of forecasting methods with respect to a particular

loss function. In this case, the loss function will be the variance of returns from the global min-

imum variance portfolios based on each of the forecasts. In terms of the out-of-sample period,

investigation of the forecasting performance of models under differing volatility conditions is

becoming more popular in the literature (see Luciani and Veredas (2011) for a recent example)

and is of great interest here given the market turbulence of recent years. In addition, the eco-

nomic value of the methods is discussed including the stability of the resulting portfolio and the

incremental value of switching from a particular method to another. The various models are

again compared across a number of portfolio sizes and for both a full out-of-sample application

as well as periods of relatively high and low levels of market volatility.

Recent work along the lines of this paper include Laurent, Rombouts, and Violante (2012) and

Caporin and McAleer (2012). Both papers focus on the evaluation of multivariate GARCH-

type models suitable for large dimensional problems, although the former studies a portfolio of

just 10 assets and this is not considered ‘large’ in the context of this paper. The latter does

not consider DECO but rather includes an exponentially weighted moving average (EWMA)

specification in its comparison of methods and so is relevant to here. This paper differs from

Laurent, Rombouts, and Violante (2012) and Caporin and McAleer (2012) in two important

ways: firstly, the sole use of daily data as opposed to intraday allows scope for larger dimensional

portfolios (the largest number of assets here is 100). Daily data allows us to circumvent a number

of issues posed by the use of high frequency data for large dimensional problems, such as stock

liquidity problems. Any problems with the positive definitiveness of the covariance matrix are

also easily avoided. Secondly, a wider range of non-MGARCH based methods are considered

here, shifting the focus to a more practical, less GARCH-orientated study. This is of interest

as Laurent, Rombouts, and Violante (2012) found less complex assumptions, such as that of

constant conditional correlation, can be appropriate in times of market tranquility and thus

simpler forecasting techniques as well as the MGARCH methods are used here.

The results indicate that the assumption of equicorrelation offers stability (both from a portfolio

exposure perspective and in terms of minimising portfolio variance). It is conjectured the

reduced estimation error of the DECO methodology provides superior forecasts. On balance,

DECO outperforms cDCC for periods of both market tranquility and turbulence in the context

of minimising portfolio variance especially at larger dimensions, however the key difference

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between the two models is stability of portfolio weights. The equicorrelated model produces

forecasts that lead to comparatively stable portfolio exposures and is relatively immune to the

increase in change in weights seen for all other models over the forecast period as portfolio size

increases. This is evidence of what Christoffersen, Errunza, Jacobs, and Jin (2013) term the

‘noisy estimates of the correlation paths’ that cDCC can provide. In terms of the incremental

gain of switching from a particular model to another, DECO dominates the other models across

the various portfolio sizes. These findings further strengthen the economic value argument in

favour of equicorrelation.

The paper proceeds as follows. Section 2 outlines the forecasting methods compared and how

they will be implemented, along with details of the tools used for evaluation of forecasts. Sim-

ulation methodology and results are contained in Section 3. Section 4 details the dataset used

here. Section 5 contains analysis of results of the empirical application. Section 6 concludes.

2 Methodology

2.1 Generating forecasts of the correlation matrix

This section outlines the models used to forecast the conditional correlation matrix and discusses

parameter estimation. Engle (2002) popularised the decomposition of the covariance matrix,

Ht = DtRtDt (1)

where Ht is the conditional covariance matrix, Rt the conditional correlation matrix and Dt the

diagonal matrix of conditional standard deviations of the returns at time t. This decomposition

leads naturally to an estimation procedure that can be split into two stages. A univariate

volatility model is used to form Dt, which contains conditional standard deviations on the

diagonal. The conditional correlations, Rt, are then estimated using asset returns standardised

by volatility estimates in Dt. Consistent with the MGARCH literature, all correlation models

outlined here are estimated using this two-stage procedure whereby each series of returns is

standardised using a univariate GARCH-type model.

Consider the first stage of the two step estimation, Dt in (1). The asset return series can be

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expressed as the product of the conditional standard deviation and the disturbance term,

ri,t =√hi,t εi,t , i = 1, 2, ... , N (2)

where εi,t ∼ N (0, 1) and independent and identically distributed. The concept of volatility

persistence in equity returns coupled with the specification of ri,t in (2) is the basis of the

empirically successful autoregressive conditional heteroskedasticity (ARCH) family of models2.

The most commonly applied extension of ARCH is the Generalised ARCH (GARCH) model

of Bollerslev (1986), a successful predictor of conditional variances (see Hansen and Lunde

(2005) for an extensive study of the ARCH family of models). The GARCH (1,1) model is

mean reverting and conditionally heteroskedastic with a constant unconditional variance. It is

defined as

hi,t = ωi + αir2i,t−1 + βihi,t−1 (3)

where hi,t is the conditional variance of asset i and ωi, αi and βi parameters constrained to

ωi ≥ 0, αi ≥ 0, βi ≥ 0 for positivity of the variance and αi + βi < 1 (for stationarity).

Along with exhibiting persistence, volatility in equity returns are found to react asymmetrically

to past returns, that is negative returns have a larger influence on future volatility than positive

ones. Glosten, Jagannathan, and Runkle’s (1993) GJR–GARCH model addresses asymmetry

in volatility by including a dummy variable that takes the value 1 should the asset return be

negative and zero otherwise. The specification of the model is

hi,t = ωi + αir2i,t−1 + φir

2i,t−1Ii,t−1 + βihi,t−1 (4)

where Ii,t−1 is the indicator variable for asset i. The positivity and stationarity constraints now

become ωi ≥ 0, αi + (φi/2) ≥ 0, βi ≥ 0 and αi + (φi/2) + βi < 1.

Attention now turns to estimating the conditional correlations, Rt in (1). Increasingly over

recent years, researchers have been interested in portfolio allocation applications of volatility

timing that are not limited in terms of the number of assets modelled. Previously, work was

limited in scope: primarily due to the problems associated with the estimation of MGARCH

models with time-varying correlations. The two well publicised complications in developing

2First proposed by Engle in (1982), the ARCH model allows the conditional variance to vary over timedependent on the past squared forecast errors.

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these models are the statistical requirement that the covariance matrix be positive definite; and

that the model be parsimonious to avoid parameter proliferation when modelling the conditional

correlation of multiple time series.

The Constant Conditional Correlation (CCC) model of Bollerslev (1990) assumes that the con-

ditional correlations are constant over time, that is Rt = R in (1). Any variation over time in

the covariance, Ht, is a result of variation in Dt. The correlation matrix, R, is formed by calcu-

lating the sample correlations of the volatility-standardised returns, εi,t = ri,t/√hi,t, generated

by estimating univariate GARCH models for each series. The Dynamic Conditional Correlation

(DCC) family of models formally introduced by Engle (2002) is considered to be a parsimonious

MGARCH model and is an extension of the CCC. A so-called ‘pseudo’ time varying correlation

matrix Qt is estimated using εi,t−1. The specification used here is the consistent DCC (cDCC)

of Aielli (2013), given by

Qt = Q(1− a− b) + a diag(Qt−1)1/2εt−1ε

′t−1 diag(Qt−1)

1/2 + b Qt−1 (5)

where Q is the unconditional sample correlation of volatility-standardised returns; a and b

are parameters subject to the positivity constraints a > 0, b > 0 and a + b < 1; and εt−1

the vector of volatility-standardised returns. As the parameters here are scalar values, the

correlation dynamics are the same for all assets. The pseudo-correlation matrix Qt in (5) is

then transformed to the conditional correlation matrix, Rt, using

Rt = diag(Qt)−1/2Qtdiag(Qt)

−1/2 . (6)

The second stage of the cDCC log-likelihood is

logL = −1

2

T∑t=1

(log(|Rt|) + ε′tR

−1t εt

)(7)

and is included here to allude to the potential issue of inverting the correlation matrix Rt for this

type of estimator in large dimensions. For standard maximum likelihood optimisation routines,

this term will be computed for each time step t a number of times. For large N , inversion of this

matrix becomes numerically intensive and will impact on the practical implementation of the

model. Engle, Shephard, and Sheppard (2008) introduced a composite likelihood approach to

the estimation of correlation models, rendering the cDCC plausible for large-scale applications.

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This approach has been used successfully in the mathematics literature for some time, see

Lindsay (1988) and more recently, Varin and Vidoni (2005) for applications where standard

likelihood methods are infeasible. The composite likelihood is the sum of quasi-likelihoods

obtained by breaking the portfolio of assets into subsets. Each subset will yield a quasi-likelihood

estimator, which can then be added to the others to produce the composite likelihood. The

process avoids having to invert large covariance matrices, reducing computational burden and

thus is useful for our purposes, as it allows cDCC to be compared to the other models used here

that are more tractable for large dimensions. Engle, Shephard, and Sheppard (2008) consider

all unique pairs of data and this is the approach used here.

A second approach of forecasting the correlation martix that is suitable for larger systems is the

DECO or equicorrelation approach of Engle and Kelly (2012). In the equicorrelated model the

conditional correlation matrix is defined as

Rt = (1− ρt)IN + ρt1 (8)

where ρt is the scalar equicorrelation measure; IN the N -dimensional identity matrix; and 1 a

N × N matrix of ones. All pairs of returns are restricted to have equal correlation on a given

day, and DECO averages the pairwise cDCC correlations to obtain

ρt =1

N(N − 1)

(1′RcDCC

t 1−N)

=2

N(N − 1)

∑i>j

qi,j,t√qi,i,tqj,j,t

(9)

where qi,j,t is the i, j th element of the pseudo-correlation matrix, Qt, using equation (5). This

assumption of equicorrelation leads to a much simpler likelihood equation

L = − 1

T

T∑t=1

[log([1− ρt]N−1[1 + (N − 1)ρt]

)+

1

1− ρt

N∑i=1

(ε2i,t)−ρt

1 + (N − 1)ρt

(N∑i=1

εi,t

)2 (10)

where ρt is given by (9). Comparing (7) with the DECO approach, one avoids the inversion of

the Rt matrix and so DECO is less burdensome and computationally quicker to estimate than

the cDCC framework.

In addition to using the MGARCH methods outlined, a number of simpler benchmark methods

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are used to capture correlations for multivariate systems, not assuming equicorrelation. These

models are of practical interest due to the ease with which they can be implemented. The

simplest forecasting tool is a simple moving average (SMA) as in

QSMAt =

1

K

K∑j=1

εt−j ε′t−j , (11)

where K is the moving average period (referred to as the ‘rolling window’), and εt−j ε′t−j the

jth lag of the cross product of the volatility-standardised returns. The requirement for positive

definiteness is N < K. This matrix is rescaled to give the correlation matrix, Rt, using (6).

A 252-day rolling window is used (this corresponds to a trading year) to ensure the symmetric

covariance matrix is positive definite. The use of a full trading year is also consistent with Value

at Risk (VaR) applications, in accordance with the Basel Committee on Banking Supervision

(1996).

The natural extension of this basic model is the exponentially-weighted moving average (EWMA)

which places a higher emphasis on more recent observations. J.P. Morgan and Reuters’s (1996)

RiskMetrics examine the exponential filter in detail and Fleming, Kirby, and Ostdiek (2001)

extend their specification to

QEWMAt = exp(−γ)QEWMA

t−1 + γ exp(−γ)εt−1ε′t−1 . (12)

The rate of decay, exp(−γ) is set using γ = 2/(K + 1) (refer to the RiskMetrics Technical

Document (1996) for further discussion regarding the length of the window) where K = 252

following the window length of the SMA described above.

The third method used is the MIxed DAta Sampling (MIDAS) model of Ghysels, Santa-Clara,

and Valkanov (2006). The MIDAS approach is shown as

QMIDASt = Q +

K∑j=0

b(j,θ)εt−j ε′t−j (13)

where parameters θ = [θ1, θ2]′ govern the beta density weighting scheme b(j,θ). The maximum

lag length K is again set to 252 days. The parameter θ1 is restricted to equal 1 and θ2 = 0.98,

implying a slow decay3. The MIDAS framework is becoming increasingly popular for a range

3See the RiskMetrics Technical Document (1996) for a comparison of optimal decay rates.

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of applications, however most focus on univariate implementation of the model. The use of the

MIDAS specification to estimate and forecast the conditional correlations of a large dimension

portfolio remains a largely open area of research.

Both moving average techniques and the MIDAS approach outlined here are simplistic in nature

and can be readily applied to a range of dimensions. As applied here, they require no optimisa-

tion at all and together with the first step of conditional volatility estimation, can be thought of

as semi-parametric approaches. The multivariate methods outlined in this section are used to

form forecasts of the conditional correlation matrix, Rt, and coupled with the diagonal matrix

of standard deviations, Dt, form the covariance matrix Ht in (1).

2.2 Evaluating forecasts

The previous section detailed the models used here to forecast the covariance matrix, Ht. This

section will discuss the methods of evaluating such forecasts. In the first instance, the volatility

of global minimum variance (GMV) portfolios is compared in order to evaluate forecasting

performance of the competing models and statistical significance of any differences examined

using the Model Confidence Set (MCS) of Hansen, Lunde, and Nason (2003). For completion,

an equally-weighted portfolio (EQ-W) is also generated as an example of a portfolio where

no forecasting or volatility timing is used. A measure of portfolio stability is generated by

considering the average absolute change in portfolio weights. This study of the economic value

of the correlation forecasting methods is furthered by quantifying and comparing the incremental

gain of switching from a particular model to another.

The benefits of utilising the GMV portfolio as the loss function for this problem center on not

needing to specify or make assumptions regarding the expected return of the portfolio. Both

Caporin and McAleer (2012) and Becker, Clements, Doolan, and Hurn (2014) employ the GMV

portfolio as a useful tool to compare correlation forecasts. The GMV portfolio with weights wt

satisfies

w∗′

t Ht w∗t = min

wt∈ RNw′tHt wt , (14)

which is a solution to the minimisation problem given,

w∗t =H−1t 1

1′H−1t 1. (15)

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Once the GMV portfolios are formed given the forecasts from each of the models, the annualised

percentage volatility of each is calculated and compared.

The Model Confidence Set (MCS) is used to evaluate the significance of any differences in

performance between models. Papers using the MCS in similar multivariate settings include

Becker, Clements, Doolan, and Hurn (2014) and Laurent, Rombouts, and Violante (2012),

among others. The premise of the MCS procedure is to avoid specifying a benchmark model,

rather it begins with a full set of candidate models M0 = 1, ...,m0 and sequentially discards

members of M0 leaving a set of models exhibiting equal performance. This MCS will contain

the best model with a given level of confidence (1− α). The loss function is defined as

L(Ht) = w∗′

t rtr′tw∗t (16)

and the loss deferential between models i and j at time t is specified as

dij,t = L(Hit)− L(Hj

t ) , i, j = 1, ...,m0 . (17)

The procedure involves testing the following

H0 : E(dij,t) = 0 , ∀ i > j ∈M (18)

for a set of models M⊂M0. The initial step sets M =M0. The t-statistic scales the average

loss differential of models i and j by var(dij) and these t-statistics are converted into one test

statistic, the range statistic, defined as

TR = maxi,j∈M

|dij |√var(dij)

(19)

with rejection of the null hypothesis occurring for large values of the statistic. The worst

performing model, determined by

i = arg maxi∈M

dij√var(dij)

, dij =1

m− 1

∑j∈M

dij,t (20)

is removed from M and the entire procedure repeated on the new, smaller set of models.

Iterations continue until the null hypothesis is not rejected, the resulting set of models is the

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MCS. The estimate of the variance of average loss differential, var(dij), can be obtained using

the bootstrap procedure in Hansen, Lunde, and Nason (2003).

In terms of analysing forecasts, transactions costs are an important practical issue. Without

assuming a specific form for the transactions costs, forecasts can be ranked in terms of the

stability of the portfolio weights they generate. This is measured by the absolute weight changes

for each portfolio. The equally-weighted portfolio is not included in this analysis. The absolute

percentage weight change at time t for a given asset i, is calculated as |∆wi,t| = |(wi,t −

wi,t−1)/wi,t−1|. Stability is measured by calculating the median absolute weight change for each

asset in a portfolio, i, and taking the mean across the N assets as

µMED =

N∑i=1

median(|∆wi,t|)/N . (21)

In addition to consideration of portfolio stability, the economic value of the correlation forecast-

ing methods is also compared using the methodology of Fleming, Kirby, and Ostdiek (2003).

That is, the relative economic benefit of each of the forecasts of the correlations is measured by

forming optimal portfolios and finding the constant δ that solves

T1∑t=T0

U(r1p,t) =

T1∑t=T0

U(r2p,t − δ) (22)

where r1p,t and r2p,t are the portfolio return series of two competing forecasting methods and T0

and T1, respectively mark the beginning and the end of the forecast period. The constant δ is

the incremental value of using the second method instead of the first and measures the maximum

average daily return an investor would forgo to switch to the second forecasting method. The

investor’s utility function is assumed to be negative exponential utility,

U(rp,t) = − exp(−λ rp,t) (23)

where λ is the investor’s risk aversion coefficient and rp,t is their return during the period to

time t and is standard throughout the literature. Following the method of Fleming, Kirby,

and Ostdiek (2003) block bootstrapping is used to generate artificial samples of returns to take

into account the uncertainty surrounding the expected returns required for the formation of

the optimal portfolios. These samples are 5000 observations in length (samples of 10000 were

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also used, however did not lead to substantive differences relative to the results presented in

this paper) and are generated by randomly selecting blocks of random length, with replacement,

from the original sample. A bootstrap is considered acceptable if the expected return is positive.

The bootstrapping procedure is repeated 500 times with δ calculated for each replication.

3 Costs & Benefits of DECO: A small simulation study

A simulation study is carried out to assess the behaviour of the forecasting methods when

the data generating process (DGP) is known to be either equicorrelated (DECO) or non-

equicorrelated (cDCC). Of particular interest here is the question of whether there is a cost

of incorrectly assuming equicorrelation for the purposes of portfolio allocation. Portfolios of

N = 5, 10, 25, 50, 100 are generated with T = 2500 and 1000 simulations are carried out in

each case4. The initial in-sample period is 2000 observations, giving 500 one-step ahead fore-

casts. The correlation model parameters are re-estimated every 20 observations, approximating

a trading month. Results presented here have been averaged over the 1000 simulations.

Table 1 contains the mean and standard deviations, averaged across the simulations, of the

correlation parameters in (5) for each estimation method. In general there appears to be a large

cost associated with using cDCC-CL to estimate an equicorrelated process, however the use of

DECO in the case of a non-equicorrelated process bears little if any cost in estimation accuracy

and in fact provides more accurate mean estimates for large values of N . This supports the

use of DECO regardless of the form the underlying correlation process takes, especially in large

multivariate systems. In the equicorrelated case, cDCC-CL provides poor estimates of the pa-

rameter α and appears to be approaching zero as portfolio size increases. Unsurprisingly, overall

DECO provides the better parameter estimates under the assumption of an equicorrelated pro-

cess. The DECO parameter estimates display a higher standard deviation than those of the

cDCC-CL, with the exception of the β parameter in the equicorrelated process. The results

of this study support the similar findings of Engle and Kelly (2012) and further highlights the

usefulness of equicorrelation.

4For each replication, samples of 4500 are generated and the first 2000 observations discarded as the lead-inperiod.

12

α = 0.05

DGP N cDCC-CL DECO cDCC-CL DECO

cDCC x s.d.5 0.0504 0.0510 0.0008 0.002010 0.0504 0.0512 0.0005 0.001925 0.0503 0.0507 0.0003 0.001650 0.0502 0.0503 0.0003 0.0015100 0.0502 0.0502 0.0003 0.0014

DECO5 0.0167 0.0503 0.0008 0.002010 0.0108 0.0504 0.0005 0.001725 0.0081 0.0499 0.0004 0.001250 0.0079 0.0497 0.0003 0.0009100 0.0074 0.0497 0.0003 0.0007

β = 0.9

DGP N cDCC-CL DECO cDCC-CL DECO

cDCC x s.d.5 0.8961 0.8932 0.0020 0.005410 0.8967 0.8940 0.0012 0.005125 0.8967 0.8960 0.0008 0.004050 0.8968 0.8970 0.0007 0.0036100 0.8970 0.8976 0.0006 0.0036

DECO5 0.9205 0.8939 0.0058 0.005110 0.9236 0.8955 0.0058 0.004525 0.9255 0.8980 0.0052 0.003150 0.9243 0.8987 0.0045 0.0022100 0.9262 0.8993 0.0042 0.0017

Table 1: Mean (x) and standard deviation (s.d.) of correlation parameter values for eachDGP and estimation method, averaged across the 1000 simulations.

4 Data

The portfolios used contain a selection of S&P500 stocks that continuously traded over the

period 03/01/1996 to 31/12/2012. The full dataset contains 100 stocks and 4271 observations.

All GICS sectors are represented across the dataset and the full list of stocks including their

ticker code, company name and sector is provided in the Appendix. Over 60% of the assets

contained in the dataset represent the Industrials, Consumer Staples, Consumer Discretionary

and Health Care sectors. Log returns are calculated using ri,t = log pi,t − log pi,t−1 where pi,t

denotes the daily closing price of asset i at time t, adjusted for stock splits and dividends.

13

Descriptive statistics for each stock are provided in the Appendix. The upper panel of Figure 1

shows the S&P500 returns series and the lower panel the squared index returns series. Of

note are the periods of relative high and low volatility over the sample, thus it is becoming

increasingly common for researchers to evaluate forecasting methods for sub-periods of differing

levels of volatility. The beginning of the sample is characterised by relatively low volatility,

followed by a higher overall level of volatility that continues until the end of the in-sample

period of 2000 observations. This high volatility spans a period from around mid 1997 until

late 2003. This period includes events such as the dot-com bubble and September 11. The

following three or so years are again a time of lower market volatility. From around March 2007

is a period of higher overall volatility corresponding to the onset of the Global Financial Crisis

(GFC). Finally, the last portion of the sample is a period of lower relative volatility. These

overall changes are of interest as we look into any possible effect the overall level of volatility

has on the relative performance of the forecasting methods.

Figure 1: Daily returns, rt, of the S&P500 index (Upper Panel) and squared daily returnsof the index (Lower Panel). Period spanning 03/01/1996 - 31/12/2012.

5 Empirical Results

This section contains the results of the empirical study, outlining the evaluation of the correlation

forecasts as described earlier. Portfolios used here contain N = 5, 10, 25, 50, 100 assets, randomly

chosen from the list of 100 stocks of the S&P500 (list available in the Appendix). The forecasting

14

horizon is one day. The in-sample period is 2000 observations, allowing for 2271 one-step-ahead

forecasts.

To avoid any potential cost associated with estimating the GJR–GARCH asymmetry parame-

ter φ unnecessarily, the significance of this parameter is tested. Nine stocks are found to have

insignificant (at the 5% level) asymmetry coefficients, φ, and thus their volatility processes are

estimated using GARCH, as suggested by Thorp and Milunovich (2007). The remaining 91

stocks’ univariate volatility processes are estimated using GJR–GARCH. One-step-ahead fore-

casts of the correlation matrix, Rt+1, are generated using the MGARCH and semi-parametric

approaches discussed in Section 2.1.

Results presented in Table 2 are the out-of-sample standard deviations of the GMV portfolio

returns described above, across the various portfolio sizes and models. As expected, the equally-

weighted portfolio results in a higher standard deviation across all portfolio sizes. The DECO

method provides the lowest measure of volatility across each of the portfolios with the exception

of the largest (N = 100), where CCC provides the lowest standard deviation. For the largest

portfolio DECO delivers the second lowest standard deviation after CCC, with cDCC producing

the third lowest. The cDCC method follows DECO for the small and moderate portfolio sizes,

providing the second-lowest standard deviations for N = 5, 10 and 25. Of the semi-parametric

methods, the EWMA method results in comparatively lower measures of volatility for all port-

folio sizes and is followed by MIDAS in each case, with the exception of N = 50 where SMA

follows EWMA. Although inferior to the MGARCH models, the semi-parametric methods per-

form relatively well for the small portfolios however the gap in performance widens as portfolio

size increases.

While the results in Table 2 provide simple rankings, the MCS is used to statistically distinguish

between the performance of the models and these results are presented in Table 3. The MCS

contains the best model(s) with a level of confidence of 95%. Unsurprisingly, the equally-

weighted portfolio is excluded from the MCS for all N . In the case of N = 5, all other models are

included in the MCS. For the moderate portfolios the cDCC is also contained in the MCS along

with DECO. In line with the volatility measure above, DECO is found to be the superior method

of forecasting across all portfolio sizes, with the exception of the largest portfolio. CCC is also

included for the large portfolios and is considered the superior method in the largest case. This

is similar to the results of Laurent, Rombouts, and Violante (2012), although this study takes

15

the analysis further in considering various portfolio sizes and indeed larger N . It is thought that

DECO exhibits less estimation error relative to the cDCC model as N increases and this may

account for its superior performance. In terms of the cDCC method, as portfolio size increases

the estimation error dominates due to the necessary estimation of the unconditional correlation

matrix, see Ledoit and Wolf (2004) for discussion of estimation error and the sample covariance

matrix. Equicorrelation has previously been found to be useful as a shrinkage target by Ledoit

and Wolf (2004) and the usefulness of the assumption of equicorrelation is also apparent here

in the portfolio allocation context.

N EQ-W MIDAS SMA EWMA CCC cDCC DECO

5 23.6612 17.7930 18.3648 17.5828 18.1969 17.3645 17.046010 28.6109 17.5249 18.1462 17.3154 17.8250 16.9898 16.981325 21.7895 13.8291 14.1532 13.7366 13.5186 13.2511 12.928350 20.6940 13.7027 13.6887 13.5402 12.8916 12.9794 12.7979100 21.1951 13.9067 14.0765 13.4768 12.2585 12.7043 12.5088

Table 2: Annualised percentage volatility of out-of-sample minimum variance portfolio re-turns for each volatility timing strategy. In-sample period of 2000, entire period spanning03/01/96 - 31/12/12.

N EQ-W MIDAS SMA EWMA CCC cDCC DECO

5 0.0220 0.0880* 0.0880* 0.1540* 0.0880* 0.1540* 1.0000*10 0.0070 0.0070 0.0070 0.0120 0.0120 0.9740* 1.0000*25 0.0001 0.0001 0.0001 0.0001 0.1200* 0.3510* 1.0000*50 0.0000 0.0000 0.0000 0.0000 0.8600* 0.7470* 1.0000*100 0.0000 0.0000 0.0130 0.0000 1.0000* 0.0380 0.5570*

Table 3: Empirical MCS of out-of-sample global minimum-variance portfolio. Range MCSp-values are used; * indicates the model is included in the MCS with 95% confidence.

To gain a deeper understanding of forecast performance, the out-of-sample period has been

split into periods of relatively high and low volatility. In this dataset, lower relative volatility is

seen during the beginning and end of the out-of-sample period. The period of higher volatility

approximately corresponds to the recent GFC of 2007/2008, beginning February 2007 with

a higher overall level of volatility through to the end of 2011. The annualised percentage

volatilities of GMV portfolio returns in Table 4 show overall patterns similar to the full sample

results in Table 2. The equally-weighted portfolio is inferior in all cases. The MGARCH methods

dominate the less complex models for all portfolios across each of the sub-samples of high and

16

low volatility, although it is the CCC forecasts that provide a lower variance for all portfolios,

except that of 10 assets, during the second period of low volatility.

During the first period of low volatility, the MGARCH forecasts provide the smallest variances

with cDCC and DECO outperforming the simpler methods across all portfolios, with the excep-

tion of N = 10. For this portfolio, DECO performs poorly in comparison to the other models,

however this is an isolated case. For CCC, the results are mixed across the various portfolios.

Of the semi-parametric models, EWMA provides the least variance followed by MIDAS for all

portfolio sizes except the largest. For the second low volatility period, representing the post-

GFC period, the results are mixed across the various portfolio sizes. For the small portfolio,

N = 5, CCC provides the smallest variance followed by cDCC and SMA. EWMA and MIDAS

are equivalent in terms of variance for N = 5. For the moderate portfolio sizes, the MGARCH

methods dominate the less complex models. Most notable for this post-GFC period of lower

relative volatility is the performance of the CCC for the large portfolios. The CCC appears

to provide an adequate forecast of correlation for the larger portfolios as it provides the lowest

variance across all methods. This is in contrast to the sometimes poorer performance of this

method for the pre-GFC period of lower volatility. Overall, CCC performs well during periods

of market tranquility across various portfolio sizes, as does cDCC.

During the period of high volatility DECO provides the lowest annualised percentage volatility

for all N , suggesting the assumption of equicorrelation may be of benefit during times of crisis.

As is the case for the total sample period (Table 2) the dominance of DECO appears to increase

with N . The CCC method performs comparatively badly to the other methods for the small

portfolio sizes, however the reverse is true for the large portfolios. In these cases, the EWMA

dominates SMA and MIDAS and the CCC model is superior to cDCC. Across the small and

moderate portfolios cDCC follows DECO.

Table 5 contains the MCS results for the high and low volatility sub-samples and they are

broadly consistent with the full sample results in Table 3. The size of the MCS differs between

that of the entire out-of-sample period and each sub-sample. For the smallest portfolio, N = 5,

all models with the exception of the equally-weighted portfolio are included in the MCS across

all sub-periods. The cDCC is included in the MCS along with CCC for the largest portfolio

in the first low volatility period, however is excluded from the MCS for N = 100 during the

second low volatility sub-sample. During this period the CCC model dominates for all portfolios

17

except N = 10. In line with previous discussion, DECO is considered superior during the high

volatility period across all N . These results broadly support those of Laurent, Rombouts, and

Violante (2012), although we find during periods of relative market tranquility the performance

of cDCC is sample specific, especially in the case of the largest portfolios. Indeed, during these

periods the CCC outperforms the other MGARCH specifications and this seems to confirm

similar findings of Laurent, Rombouts, and Violante (2012). We also find evidence supporting

the assumption of equicorrelation during periods of crisis, a method unexplored by Laurent,

Rombouts, and Violante (2012).

Period N EQ-W MIDAS SMA EWMA CCC cDCC DECO

Low 12001:2806 5 15.0031 13.0022 13.0952 12.9613 12.8362 12.8828 12.8636

10 15.9680 11.2053 11.8570 11.1706 11.3528 11.1181 11.211025 11.7705 9.7725 10.3012 9.5927 9.3855 9.2998 9.350050 10.8722 9.3769 9.7177 9.2217 8.8776 8.8158 9.0778100 10.5383 9.6070 9.3852 9.0828 8.2113 8.3618 9.0089

High2807:4019 5 29.3304 21.3296 22.1953 21.0212 22.0561 20.7357 20.2280

10 35.9006 21.6395 22.3349 21.3349 22.0780 20.8946 20.849525 27.5080 16.6972 16.9906 16.6209 16.4495 16.0252 15.505850 26.2766 16.5644 16.5242 16.4153 15.6727 15.7803 15.2207100 27.0432 16.6911 17.2704 16.3221 14.9781 15.5267 14.8674

Low 24020:4271 5 13.5321 10.9012 10.8591 10.9010 10.6220 10.7072 10.8686

10 18.8057 10.6127 10.8008 10.6055 10.2632 10.2466 10.239925 13.8606 8.7718 8.7683 8.7888 7.9360 8.3913 8.372850 12.5644 9.5454 8.5559 9.1692 7.9942 8.4713 9.9033100 13.1667 10.3757 8.3107 9.5653 7.6932 8.4336 9.3437

Table 4: Annualised percentage volatility of out-of-sample minimum variance portfolio re-turns for each volatility timing strategy, split into high and low volatility. In-sample periodof 2000, entire period spanning 03/01/96 - 31/12/12.

Table 6 contains the absolute change in portfolio weights, µMED in (21), for each model across

the entire out-of-sample period and is used here to measure the stability of the minimum vari-

ance portfolios formed using each of the various methods of correlation forecasts. The overall

trend is increasing instability of portfolio weights as N increases, although µMED drops slightly

across all methods as the portfolio size increases from N = 50 to N = 100. As N increases,

DECO performs comparatively better than all other models in terms of this measure of sta-

18

Period N EQ-W MIDAS SMA EWMA CCC cDCC DECO

Low 12001:2806 5 0.0230 0.0820* 0.0820* 0.1220* 1.0000* 0.8710* 0.9060*

10 0.0000 0.3080* 0.0000 0.4350* 0.0540* 1.0000* 0.4350*25 0.0000 0.0001 0.0000 0.0020 0.4880* 1.0000* 0.7350*50 0.0000 0.0000 0.0000 0.0000 0.4800* 1.0000* 0.3910*100 0.0000 0.0000 0.0000 0.0000 1.0000* 0.1520* 0.0000

High2807:4019 5 0.0240 0.0740* 0.0740* 0.1760* 0.0740* 0.1760* 1.0000*

10 0.0040 0.0040 0.0040 0.0130 0.0130 0.8530* 1.0000*25 0.0000 0.0000 0.0000 0.0000 0.0700* 0.3000* 1.0000*50 0.0040 0.0040 0.0040 0.0040 0.6980* 0.6980* 1.0000*100 0.0000 0.0000 0.0190 0.0000 0.8700* 0.0880* 1.0000*

Low 24020:4271 5 0.0000 0.0710* 0.0710* 0.0710* 1.0000* 0.3760* 0.0710*

10 0.0000 0.2070* 0.0170 0.2380* 0.9970* 1.0000* 0.9970*25 0.0000 0.0001 0.0001 0.0001 1.0000* 0.0400 0.0580*50 0.0000 0.0000 0.0190 0.0000 1.0000* 0.0230 0.0000100 0.0000 0.0000 0.0040 0.0000 1.0000* 0.0000 0.0000

Table 5: Empirical MCS of out-of-sample global minimum-variance portfolio. Range MCSp-values are used; * indicates the model is included in the MCS with 95% confidence.

bility. All other methods, including CCC and cDCC, are comparatively much more volatile

in terms of resulting portfolio weights over the forecast period as N increases. For the small

portfolios of N = 5 and 10 the SMA and CCC methods provide the smallest values of µMED

respectively. DECO provides a more stable portfolio in terms of asset weights for the moder-

ate and large portfolios, with cDCC providing relative instability. From an economic point of

view, the relative instability of the CCC and cDCC forecasts provide further evidence in favour

of equicorrelation. Christoffersen, Errunza, Jacobs, and Jin (2013) mention the dominance of

DECO can be attributed to the somewhat ‘noisy’ estimates of the correlations provided by

cDCC and this is confirmed here in terms of portfolio allocation.

Similar results are obtained when taking into account periods of relatively high and low volatility

(Table 7). The advantage of assuming equicorrelation is evident as DECO provides the most

stable weights across the various portfolios, regardless of the sub-sample. The CCC method

provides mixed results, although it provides stability during periods of market calm for large

portfolio sizes. The cDCC method is broadly much more volatile than DECO in terms of

portfolio weights regardless of the sub-period. Of the semi-parametric methods, the results are

mixed although SMA appears more stable in terms of weights as N increases and this is the

19

case regardless of sub-sample. As N increases, DECO again appears more stable in comparison

to all other approaches and this is perhaps indicative of it containing less estimation error in

the forecasts of the correlation matrix.

N MIDAS SMA EWMA CCC cDCC DECO

5 0.1007 0.0777 0.0990 0.0814 0.0934 0.095310 0.1398 0.1223 0.1429 0.1185 0.1420 0.136525 0.1984 0.1589 0.1938 0.1736 0.1981 0.145750 0.2427 0.1912 0.2404 0.2157 0.2379 0.1510100 0.2521 0.2143 0.2442 0.2120 0.2356 0.1453

Table 6: Mean, µMED, of the median absolute change in portfolio weights across each modelfor the out-of-sample period. In-sample period of 2000, entire period spanning 03/01/96 -31/12/12.

Period N MIDAS SMA EWMA CCC cDCC DECO

Low 12001:2806 5 0.0797 0.0769 0.0800 0.0691 0.0728 0.0723

10 0.0959 0.1007 0.0997 0.0918 0.1055 0.094425 0.1638 0.1553 0.1567 0.1627 0.1834 0.141650 0.2120 0.1989 0.2133 0.2167 0.2229 0.1502100 0.2144 0.2156 0.2160 0.2090 0.2191 0.1542

High2807:4019 5 0.1167 0.0802 0.1176 0.0963 0.1100 0.1163

10 0.1864 0.1446 0.1897 0.1426 0.1767 0.173625 0.2327 0.1687 0.2338 0.1954 0.2165 0.171850 0.2753 0.1990 0.2718 0.2270 0.2610 0.1654100 0.2872 0.2235 0.2714 0.2263 0.2581 0.1564

Low 24020:4271 5 0.1232 0.0937 0.1302 0.0887 0.1334 0.1319

10 0.1581 0.1454 0.1693 0.1529 0.1756 0.171125 0.2068 0.1877 0.2077 0.1652 0.1948 0.130350 0.2429 0.1960 0.2361 0.2066 0.2295 0.1423100 0.2696 0.2151 0.2545 0.2067 0.2243 0.1321

Table 7: Mean, µMED, of the median absolute change in portfolio weights across eachmodel, split into periods of high and low volatility. In-sample period of 2000, entire periodspanning 03/01/96 - 31/12/12.

Tables 8 through 12 report the average value of the constant δ in (22), a measure of the relative

economic value of choosing a particular correlation forecasting method over another, for each

of the various portfolio sizes. Here a positive value represents the economic gain of choosing

20

the column method over that contained in the row, with the proportion of bootstraps where δ

is positive reported in small text underneath. Results reported assume an expected return of

6% and a risk aversion coefficient of λ = 2. Expected returns of 8% and 10%, as well as a risk

aversion coefficient of λ = 5, were also used however did not lead to any significant difference

in the results.

As expected the equally-weighted portfolio is inferior by all methods for all portfolio sizes.

Broadly in line with the evaluation presented previously, DECO dominates the other forecasts

in all cases. Overall, differences between models become more pronounced as the size of the

portfolio increases as the value of δ increases with N , although this is not the case for the largest

portfolio. The MGARCH models dominate the semi-parametric methods for all portfolio sizes.

For N = 5 there is a gain in moving to MIDAS from either of the moving average approaches

and EWMA is found to be superior to the SMA. It is worth noting that δ is relatively small

when moving from EWMA to MIDAS and indeed the superiority of MIDAS is reversed for the

moderate portfolio sizes N = 10, 25. For N = 25 there is a gain in switching from EWMA to

the SMA and this remains the case for the large portfolios of N = 50, 100.

As mentioned above, DECO outperforms all other methods by this measure across the various

portfolios. On balance, the results presented here favour the assumption of equicorrelation

especially for large portfolios. Despite the overall good performance of cDCC, the instability of

portfolio weights generated by this method reduce the gains of using this method to generate

forecasts of the correlation matrix and this is most evident for the large portfolio sizes of 50 and

100 assets.

21

N = 5

EQW MIDAS SMA EWMA CCC cDCC DECOEQW - 419.372

0.926439.547

0.950401.228

0.908516.890

0.986481.369

0.978463.088

0.976

MIDAS - −55.2420.250

−6.0950.478

56.5870.586

67.5130.642

68.9010.656

SMA - 18.8000.604

106.6230.762

108.6110.826

110.4880.830

EWMA - 60.5080.568

74.3140.614

76.5910.622

CCC - 8.3590.540

15.2230.770

cDCC - 8.3580.774

DECO -

Table 8: Estimated relative economic value gained from moving from the forecast in therow heading to that in the column heading, for µ = 6%, λ = 2. Each entry reports theaverage value of δ across 500 bootstraps and the proportion of bootstraps where δ is positive.Portfolio of 5 assets.

N = 10


0.9981198.317

0.9981155.391

0.9961428.740

1.0001415.826

1.0001430.802

1.000

MIDAS - −67.2890.278

6.1320.642

183.9080.714

222.0090.778

248.6840.828

SMA - 24.0400.650

250.7200.958

277.9560.982

307.5020.988

EWMA - 177.4680.696

217.3450.754

243.7640.798

CCC - 27.5490.816

62.5970.940

cDCC - 32.7970.862

DECO -


6 Conclusion

Methods of generating forecasts of the correlation matrix, tractable for large systems have been

the subject of increasing research over recent years, due to the importance of such forecasts

to applications such as asset allocation. Equicorrelation models have featured prominently in

this literature and this paper presents an empirical study of the DECO model in comparison to

other popular forecasting techniques in the context of economic value. Out-of-sample forecasting

performance is compared through the volatility of global minimum variance portfolio returns,

portfolio stability and the explicit economic value of switching from one method to another.

DECO provides the lowest variance and was the superior method in the model confidence

22

N = 25


0.900381.746

0.972290.702

0.896431.637

1.000438.389

1.000466.123

1.000

MIDAS - 20.4260.592

−0.7590.468

70.7960.618

102.3090.668

134.8500.722

SMA - −48.1960.316

42.3320.590

68.8510.672

103.0060.758

EWMA - 71.7530.622

104.0520.680

136.3870.720

CCC - 26.2990.818

63.3390.978

cDCC - 35.0790.852

DECO -


N = 50


0.556169.587

0.86026.4320.570

191.7030.950

214.9430.936

267.2530.984

MIDAS - 94.3420.862

11.3640.702

126.6230.850

170.5580.952

231.7700.982

SMA - −104.2810.100

23.6120.606

63.6720.760

126.2270.888

EWMA - 114.7500.832

159.2900.932

220.4270.980

CCC - 39.3790.884

105.4810.984

cDCC - 63.7750.918

DECO -


set, with the exception of the largest portfolio where CCC is considered superior. It also

delivers the most stable portfolio in terms of asset weights of the techniques compared across

the various portfolio sizes. The incremental economic value of moving from another method to

equicorrelation is positive. The out-of-sample period is also broken into sub-periods of high and

low volatility to further evaluate the forecasts. DECO is found to be superior during the crisis

period across the various portfolios and CCC performs well during the second period of market

calm (post-GFC).

This paper details the advantages of the equicorrelation framework and provides evidence that

such methods are of use in the context of forecasting large correlation matrices. As a developing

literature, there is much still to be investigated regarding forecasting the correlation of large

23

N = 100


0.712166.719

0.91692.8880.718

204.3470.988

258.3570.990

272.9011.000

MIDAS - 29.2550.652

1.1560.506

67.3360.742

133.4320.938

155.1780.940

SMA - −53.6170.230

26.1050.638

89.1820.884

112.9650.932

EWMA - 66.0410.740

132.3400.956

153.9660.962

CCC - 62.0160.982

88.0350.998

cDCC - 23.5740.746

DECO -


systems and techniques utilising equicorrelation are of continuing interest.

24

References

Aielli, G. P. (2013): “Dynamic Conditional Correlations: On Properties and Estimation,”Journal of Business & Economic Statistics, 31(3), 282–299.

Basel Committee on Banking Supervision (1996): “Amendment to the Cap-ital Accord to Incorporate Market Risks,” Bank for International Settlements,http://www.bis.org/publ/bcbs128.htm, Accessed October 2011.

Bauwens, L., S. Laurent, and J. V. K. Rombouts (2006): “Multivariate GARCH models:A survey,” Journal of Applied Econometrics, 21, 79–109.

Becker, R., A. Clements, M. Doolan, and S. Hurn (2014): “Selecting volatility fore-casting models for portfolio allocation purposes,” International Journal of Forecasting, Inpress.

Bollerslev, T. (1986): “Generalized autoregressive conditional heteroskedasticity,” Journalof Econometrics, 31, 307–327.

(1990): “Modelling the coherence in short-run nominal exchange rates: A multivariategeneralized ARCH model,” The Review of Economics and Statistics, 72(3), 498–505.

Caporin, M., and M. McAleer (2012): “Robust ranking of multivariate GARCH models byproblem dimension,” Computational Statistics and Data Analysis.

Christoffersen, P., V. R. Errunza, K. Jacobs, and X. Jin (2013): “Correlation Dy-namics and International Diversification Benefits,” CREATES Research Paper #2013–49.

Engle, R. F. (1982): “Autoregressive conditional heteroscedasticity with estimates of thevariance of United Kingdom inflation,” Econometrica, 50(4), 987–1006.

(2002): “Dynamic conditional correlation: A simple class of multivariate generalizedautoregressive conditional heteroskedasticity models,” Journal of Business and EconomicStatistics, 20(3), 339–350.

Engle, R. F., and B. Kelly (2012): “Dynamic Equicorrelation,” Journal of Business &Economic Statistics, 30(2), 212–228.

Engle, R. F., N. Shephard, and K. Sheppard (2008): “Fitting and testing vast dimensionaltime-varying covariance models,” Oxford University Working Paper.

Fleming, J., C. Kirby, and B. Ostdiek (2001): “The Economic Value of Volatility Timing,”The Journal of Finance, 56(1), 329–352.

(2003): “The economic value of volatility timing using ‘realized’ volatility,” Journal ofFinancial Economics, 67, 473–509.

Ghysels, E., P. Santa-Clara, and R. Valkanov (2006): “Predicting volatility: gettingthe most out of return data sampled at different frequencies,” Journal of Econometrics, 131,59–95.

Glosten, L. W., R. Jagannathan, and D. E. Runkle (1993): “On the relation betweenthe expected value and the volatility of the nominal excess return on stocks,” Journal ofFinance, 48, 1779–1801.

Hansen, P. R., and A. Lunde (2005): “A forecast comparison of volatility models: Doesanything beat a GARCH(1,1)?,” Journal of Applied Econometrics, 20, 873–889.

25

Hansen, P. R., A. Lunde, and J. M. Nason (2003): “Choosing the Best Volatility Models:The Model Confidence Set Approach,” Oxford Bulletin of Economics and Statistics, 65, 839–861.

Laurent, S., J. V. K. Rombouts, and F. Violante (2012): “On the Forecasting Accuracyof Multivariate GARCH Models,” Journal of Applied Econometrics, 27, 934–955.

Ledoit, O., and M. Wolf (2004): “Honey, I Shrunk the Sample Covariance Matrix,” TheJournal of Portfolio Management, 31(1), 110–119.

Lindsay, B. G. (1988): “Composite Likelihood Methods,” Contemporary Mathematics, 80,221–239.

Luciani, M., and D. Veredas (2011): “A Simple Model for Vast Panels of Volatilities,”ECORE Discussion Paper 2011/80.

Morgan, J. P., and Reuters (1996): “RiskMetrics - Technical Document,” Risk-Metrics Group, http://www.riskmetrics.com/publications/techdocs/rmcovv.html, Ac-cessed June 2009.

Silvennoinen, A., and T. Terasvirta (2009): “Multivariate GARCH models,” in Handbookof Financial Time Series, ed. by T. G. Andersen, R. A. Davis, J.-P. Kreiss, and T. Mikosch.Springer, New York.

Thorp, S., and G. Milunovich (2007): “Symmetric versus asymmetric conditional covarianceforecasts: Does it pay to switch?,” Journal of Financial Research, 30(3), 355–377.

Varin, C., and P. Vidoni (2005): “A note on composite likelihood inference and modelselection,” Biometrika, 92(3), 519–528.

26

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27

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28

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onnextpage

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Tab

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he

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per

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span

nin

g03

/01/

96-

31/1

2/12

.

30

Documents

On the Benefits of Equicorrelation for Portfolio Allocation · A portfolio allocation problem is used to compare forecasting methods. The global minimum variance portfolio and Model