Global COE Hi-Stat Discussion Paper Series 269

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Realized Beta GARCH:
A Multivariate GARCH Model with
Realized Measures of Volatility and Covolatility

Hi-Stat Discussion Paper

Peter Reinhard Hansen
Asger Lunde
Valeri Voev

December 2012

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Realized Beta GARCH:
A Multivariate GARCH Model with
Realized Measures of Volatility and CoVolatilityPeter Reinhard Hansena∗aAsger Lundeb

Valeri Voevb

European University Institute & CREATESb Aarhus University, Department of Economics and Business,

Fuglesangs Allé 4, 8210 Aarhus V, Denmark & CREATES

December 14, 2012

Abstract
We introduce a multivariate GARCH model that incorporates realized measures ofvolatility and covolatility. The realized measures extract information about the current
level of volatility and covolatility from high-frequency data, which is particularly useful forthe modeling of return volatility during periods with rapid changes in volatility and covolatility. When applied to market returns in conjunction with returns on an individual
asset, the model yields a dynamic model of the conditional regression coefficient that is
known as the beta. We apply the model to a large set of assets and find the conditional
betas to be far more variable than is usually found with rolling-window regressions based exclusively on daily returns. In the empirical part of the paper we examine the cross-sectional
as well as the time variation of the conditional beta series during the financial crises.

Keywords: Financial Volatility; Beta; Realized GARCH; High Frequency Data.
JEL Classification: G11, G17, C58∗
Corresponding author. All authors acknowledge financial support by the Center for Research in EconometricAnalysis of Time Series, CREATES, funded by the Danish National Research Foundation.

1

1

Introduction

Relatively accurate measurements of volatility and covolatility can be computed from highfrequency data, and such statistics are commonly referred to as realized measures. Incorporating
realized measures when modeling the dynamic properties of volatility, such as done with GARCH
models, is very beneficial. The reason is that returns yield very weak signals about latent
volatility, whereas realized measures provide accurate measurements. The latter is particularly
useful during times with rapid changes in volatility and covolatility.
In this paper we propose a multivariate GARCH-type model that utilizes and models realizedmeasures of volatility and covolatility. The model has hierarchical structure where the “market”
return is modeled with a univariate Realized GARCH model, see Hansen and Huang (2012)and Hansen et al. (2012). A multivariate structure is constructed by modeling “individual”
returns conditional on the past and contemporary market variables (return and volatility). The
resulting model has the structure of a conditional CAPM model that enables us to extract the“betas” and study their dynamic properties. Moreover, the model is complete in the sense that
all observables (returns, realized volatilities and realized correlations) are modeled. The latterenables us to infer the distribution of multi-period returns including the joint distribution of“market” returns and “individual” returns over longer horizons.
The main contributions of our paper are the following. We propose a flexible and tractable
framework that enables the modeling of a potentially large set of assets. Unlike conventional
multivariate GARCH models, which can suffer from the curse of dimensionality and estimation
issues, we avoid such issues by incorporating realized measures and the use of measurement
equations. Measurement equations tie realized measures to the latent volatility quantities,
which induces a useful regularization of the model. This particular structure was chosen for anumber of reasons. First, the model provides good empirical fit for the wide range of assets usedin our empirical study; Second, the structure of the model is amenable to a deeper analysis ofsecondary quantities such as betas; Third, the model is simple to estimate, which is particularly
important when a large set of assets are to be analyzed as is the case in our empirical analysis.
The proposed model structure has a hierarchical structure where the market return and acorresponding realized measure forms the core of the model. The model can be extended to anarbitrary large set of individual returns, by adding a conditional model for an individual return2and two realized measures, one being a realized measure of return volatility, the other being arealized measure of the correlation between the individual return and the market. This yielda flexible model with a dynamic covariance structure that is constantly revised by using theinformation contained in the realized measures.
The concept of realized betas is not new. Bollerslev and Zhang (2003) carry out a large scaleestimation of the Fama-French three-factor model using high-frequency (5-minute) data on 6,400
stocks over a period of 7 years. Their analysis showed that high-frequency data can improve thepricing accuracy of asset pricing models. Their approach differs from ours in important ways.
For instance, they model raw realized factor loadings and use simple time series processes toforecast these. So there is no explicit link between realized and conditional moments of returns
in their framework. Nor do they explicitly account for the measurement error (the sampling
error) in the realized quantities. Another related paper is Andersen et al. (2006) who study
the time variation in realized variances, covariances, and betas using daily returns to constructquarterly realized measures. They find evidence of long memory in the time series for varianceand covariances, while the realized beta time series is less persistent and seemingly a shortmemory process, which is indicative of fractional cointegration between realized volatility andrealized covariance. Other related studies include: Barndorff-Nielsen and Shephard (2004a)
who established asymptotic results for realized beta and Dovonon et al. (2013) who establishedthe theory for bootstrap inference. MSE-optimal estimation of realized betas was analyzed in
Bandi and Russell (2005) and Patton and Verardo (2012) studied the impact of news on betas.
The importance of separating jump and continuous component of returns in relation to betas
as highlighted in Todorov and Bollerslev (2010) and Tsay and Yeh (2011) allow the dynamic
beta to vary within the day.
The use of realized volatility measures in this context yields valuable insight about the degreeof time-variation in the betas, which has been up for debate in the literature. The studies by
Ferson and Harvey (1991, 1993), and Shanken (1990) specify parametric relationships between
betas and proxies for the state of the economy and find support for time-varying betas. Gomes
et al. (2003) provide a theoretical justification for a time-varying conditional beta specificationin the context of a dynamic general equilibrium production economy. Conditional betas havebeen modeled by means of conventional GARCH models by Braun et al. (1995) and Bekaert
and Wu (2000), among others. Lewellen and Nagel (2006) argue that variation in betas would
have to be “implausibly large” to explain important asset-pricing anomalies. In our empirical3

analysis we do find a substantial amount of time-variation in the conditional betas, this isparticularly the case during the global financial crises period. We find the variation in betas tobe substantial, even over short periods of time, such as a quarter. Figure 5 in this paper is agood illustration of this point.
The research devoted to high-frequency volatility measures was spurred by Andersen and
Bollerslev (1998), who documented that the sum of squared intraday returns, known as therealized variance, provides an accurate measurement of daily volatility. The theoretical foundation of realized variance was developed in Andersen, Bollerslev, Diebold and Labys (2001) and
Barndorff-Nielsen and Shephard (2002). Currently a large number of related estimators, such as
realized bipower variation, realized kernels, multiscale estimators, preaveraging estimators andMarkov chain estimators have been proposed to deal with issues such as jumps and market microstructure frictions, see Barndorff-Nielsen and Shephard (2004b), Barndorff-Nielsen, Hansen,
Lunde and Shephard (2008), Zhang (2006), Jacod et al. (2009), Hansen and Horel (2009) andreferences therein. The multivariate extensions of the concept of realized volatility is theoretically developed in Barndorff-Nielsen and Shephard (2004a). Estimators that are robust to noise
and/or asynchronous observations have been proposed by Hayashi and Yoshida (2005), Voev
and Lunde (2007), Griffin and Oomen (2011), Christensen et al. (2010), and Barndorff-Nielsen
et al. (2011). In this paper we will rely on the multivariate kernel estimator by BarndorffNielsen, Hansen, Lunde and Shephard (2011) that guarantees positive semi-definite estimatesof the realized variance-covariance matrices we need.
While volatility is unobservable, the use of realized measures allows us to construct precise
ex-post volatility proxies. Currently, a growing body of research investigates the extend to which
realized measures can be used to specify better volatility models and improve the accuracyof volatility forecasts. Hansen and Lunde (2010) categorize the existing approaches into two
broad classes: reduced-form and model-based. Reduced-form volatility forecasts are basedon a time series model for the series of realized measures, while a model-based forecast rests
on a parametric model for the return distribution. Model-based approaches effectively build
on GARCH models in which a realized measure is included as an exogenous variable in the
GARCH equation, see e.g. Engle (2002). A complete framework that jointly specifies models
for returns and realized measures of volatility was first proposed by Engle and Gallo (2006),
who refer to their model as the Multiplicative Error Model (MEM). A simplified MEM structure
was proposed in Shephard and Sheppard (2010), who estimated their referred to this model as

4the HEAVY model. The realized GARCH model by Hansen et al. (2012) involves a different
approach to the joint modeling of returns and realized volatility measures. A key componentof the Realized GARCH model is a measurement equation that links the realized measure withthe underlying conditional variance. This idea is generalized to the multivariate framework inthis paper, where we introduce measurement equations for the realized measures of correlations.
The rest of the paper is structured as follows. The model and the underlying theory ispresented in Section 2, and we discuss estimation of the model in Section 3. In section 4 weshow how multistep predictions of volatilities and correlations as well as forecasts of return
densities can be obtained with our model. Section 5 contains the empirical application of themodel, and Section 6 concludes.

2

A Hierarchical Realized GARCH Framework

Broadly speaking, our objective is the same as that of existing multivariate GARCH models,
which is to model the conditional distribution of a vector of returns. But unlike conventional
GARCH models we also model the realized measures of volatility and covolatility and makeextensive use of these in the modeling of returns. The realized measures are highly informative
about local (in time) levels of volatility and covolatility which is the main reason these variables
are beneficial in this context. By tying all individual return series to the market return, we are
implicitly imposing a factor structure on the volatility, where the variation in the correlation
structure is driven by time-variation in the correlations between the market return and the individual assets. This keeps the model relatively simple and parsimonious, facilitates estimation,and makes it easy to relate key variables in the model to (dynamic) betas.
Our model has a hierarchical structure. The core of our framework is a marginal modelfor the market return and its realized measure of volatility. Individual returns, their realizedmeasures of volatility and covolatility (with the market) are then modeled conditionally onmarket variables. The marginal model we use for the market-specific time series is the Realized
EGARCH model by Hansen and Huang (2012), see also Hansen et al. (2012, section 6.3), whichshares certain features with the EGARCH model by Nelson (1991). The conditional model we
use in this paper is new.
Initially, we present the Realized Beta GARCH model in the simplest situation with a bivariate vector of returns (the market return and an individual asset return) and the corresponding
2×2 matrix realized volatility measures. Subsequently we discuss the straight forward extension
5

to an arbitrary number of individual assets.

2.1

Notation and Modeling Strategy

Let r0,t and x0,t denote the market return and a corresponding realized measure of volatility,
respectively. Similarly, we use the notation r1,t and x1,t for the same variables associatedwith an individual asset return, and use yi,t to denote a realized measure of correlation, whereyit ∈ (−1, 1).
In this context with two returns, two realized measures of volatility, and a realized measureof correlation we have five observable variables to model. The natural filtration is given by
Ft = σ(Xt , Xt−1 , . . .) with Xt = (r0,t , r1,t , x0,t , x1,t , y1,t )0 .

The structure of our model will take advantage of the simple decomposition of the conditional
density,f (r0,t , x0,t , r1,t , x1,t , y1,t |Ft−1 ) = f (r0,t , x0,t |Ft−1 )f (r1,t , x1,t , y1,t |r0,t , x0,t , Ft−1 ),(1)which serves to illustrate the hierarchical structure of our model. We will adopt the Realized
EGARCH model as our specification of the first term, f (r0,t , x0,t |Ft−1 ). The individual asset
variables (r1t , x1t , y1t ) will be modeled with a novel structure that conditions on contemporary
market variables. The specification for the second conditional density, f (r1,t , x1,t , y1,t |r0,t , x0,t , Ft−1 ),defines how the time series associated with the individual asset evolves conditional on contemporary market variables. Our specification of this conditional density has a structure that issimilar to that of the univariate Realized GARCH model, but has some important adaptations
for the modeling of the correlation structure. This structure is very convenient because it avoids
the need for introducing realized measures of the correlation measures between the individual
assets, because it is implicitly assumed that these correlations are characterized through thecorrelations between the individual returns and the market return. In our empirical analysis,
we investigate the validity of this assumption.
In practice, the estimation proceeds by first estimating the model for the market data(r0,t , x0,t ) and then estimating each conditional model for (ri,t , xi,t , yi,t ) separately for i =1, 2, . . . , n, where n is the number of assets. This can be done for a very large number ofassets. For instance, in the empirical analysis we estimated the Realized Beta GARCH model

6

for about 600 assets.

2.2

Realized EGARCH Model for Market Returns

The Realized EGARCH model for market returns and realized measures of volatility is givenby the following three equations

r0,t = µ0 +p
h0,t z0,t ,(2)

log h0,t = a0 + b0 log h0,t−1 + c0 log x0,t−1 + τ0 (z0,t−1 )(3)log x0,t = ξ0 + ϕ0 log h0,t + δ0 (z0,t ) + u0,t ,(4)

where model z0,t ∼ iidN (0, 1), and u0,t ∼ iidN (0, σu20 ). As is the case in conventional GARCH
models, h0,t , denotes a conditional variance, h0,t = var(r0,t |Ft−1 ), the key difference being thatthe information set, Ft , is richer than in the conventional framework. The normality of u0,t is
not critical for estimation, but can be motivated by findings in Andersen, Bollerslev, Dieboldand Labys (2001), Andersen, Bollerslev, Diebold and Ebens (2001) and Andersen et al. (2003),
who document that realized volatility is approximately log-normal. Furthermore, Andersen,
Bollerslev, Diebold and Ebens (2001) find that returns standardized by realized volatility areapproximately normally distributed.
The functions τ (z) and δ(z) are called leverage functions because they model aspects related
to the leverage effect, which refers to the dependence between returns and volatility. Hansen
et al. (2012) found that a simple second-order polynomial form provides a good empirical fit.
We will adopt this structure in our framework, and set τ (z) = τ1 z + τ2 (z 2 − 1) and δ(z) =δ1 z + δ2 (z 2 − 1). This leads to a GARCH equation that is somewhat similar to that of an
EGARCH model. An important difference is that we also utilize the realized measure xt−1 inthis equation to model the dynamic variation in volatility.
We refer to the first two equations, (2) and (3), as the return equation and the GARCH
equation, respectively. These two equations define a GARCH-X model, similar to those thatwere estimated by Engle (2002), Barndorff-Nielsen and Shephard (2007), and Visser (2011). See
also ? for additional variants of the GARCH-X model and some related models.
The third equation, (4) called the measurement equation, completes the model. Tying therealized measure, xt , to the conditional variance, ht , is motivated by the fact that the GARCH

7

equation trivially implies thatlog(rt − µ)2 = log ht + log zt2 .
Since the realized measure, xt , is similar to rt2 in the sense of being a measurement of volatility(just far more accurate), it is natural to expect that log xt ≈ log ht + f (zt ) + errort . Because we
may compute realized measures of volatility over a shorter period of time than the one spannedby the return (e.g., if we use only data from the trading session, which often excludes theovernight period), some flexibility in the specification may be required motivating the “intercept”
ξ0 and the “slope” ϕ0 . So long as x0,t is roughly proportional to h0,t , we should expect ϕ0 ' 1,and ξ0 < 0 , which is always the case empirically.
Note that we do not follow the conventional GARCH notation, because we want to reserve
the notation “β” for
β1,t = cov(r1,t , r0,t |Ft−1 )/var(r0,t |Ft−1 ),(5)

We are particularly interested in the dynamic properties and the cross-sectional variation of βi,t ,where i = 1, . . . , N with N being the number of individual assets in our analysis.

2.3

Conditional Model for Individual Asset Returns, Volatility, and Covolatility

To extend the framework to a joint model for the market returns/volatility and another asset’s
return/volatility and their correlation, we shall formulate a model for the time series associatedwith the individual asset, conditional on contemporaneous “market” variables, i.e., a specification
for f (r1,t , x1,t , y1,t |r0,t , x0,t , Ft−1 ). We utilize a further decomposition of this conditional density,
specificallyf (r1,t , x1,t , y1,t |r0,t , x0,t , Ft−1 ) = f (r1,t |r0,t , x0,t , Ft−1 )f (x1,t , y1,t |r1,t , r0,t , x0,t , Ft−1 ).
The first part, f (r1,t |r0,t , x0,t , Ft−1 ), is modeled with three equations. The first two have theRealized EGARCH structure as above,

r1,t = µ1 +p
h1,t z1,t ,(6)

log h1,t = a1 + b1 log h1,t−1 + c1 log x1,t−1 + d1 log h0,t + τ1 (z1,t−1 ).

8(7)The difference between the two GARCH equations, (3) for the market return and (7) for theasset return, is the presence of the term, d1 log h0,t . This term relates the market conditionalvariance to the conditional variance of the individual asset under consideration. Note that h0,t
is Ft−1 -measurable, so that h1,t can be interpreted as the conditional variance of r1,t w.r.t.
Ft−1 . The parameter d1 can be interpreted as a spillover effect that measures the extend towhich the market’s volatility affects the volatility of the individual asset while accounting forthe asset-specific volatility dynamics.
To capture the dependence between market returns and individual returns, we introduce theconditional covarianceρ1,t = cov(z0,t , z1,t |Ft−1 ).
It follows directly that ρ1,t is the conditional correlation between r0,t and r1,t , so that the beta
for asset 1 is given by
β1,t =

ρ1,tp
qh0,t h1,t= ρ1,t h0,t /h1,t .
h0,t

For the dynamic modeling of ρt we shall use of the Fisher transformation (also known as theinverse hyperbolic tangent, arctanh), ρ 7→ z(ρ) ≡ 12 log 1+ρ
1−ρ , which is a one-to-one mappingfrom (−1, 1) into R. The GARCH equation for the transformed correlations is given byz(ρ1,t ) = a10 + b10 z(ρ1,t−1 ) + c10 z(y1,t−1 ).
The measurement equations for the two realized measures in the conditional model are:log x1,t = ξ1 + ϕ1 log h1,t + δ1 (z1,t ) + u1,t ,(8)z(y1,t ) = ξ10 + ϕ10 z(ρ1,t ) + v1,t ,

(9)

and

and the covariance structure for the error terms in the three measurement equations is denoted
by








σu20

σu0 ,u1

σu0 ,v1


 

 
Σ = var  u1,t  =  •

 
v1,t
•

σu21



σu1 ,v1  .

2
σv1

u0,t

•

Hence, we allow the error terms to be correlated across measurement equations, which we find9

to be empirically important.

2.4

The Extensions to Multiple Individual Assets

We have specified the model structure for a market return and a single individual assets (alongwith their corresponding realized volatility variables). Next we discuss the extension to multiple
individual assets. Fortunately, the existing structure is amendable to this extension, albeit
some additional assumptions are needed before certain interpretations carry over to the generalcontext. First we need to redefine the natural filtration, Ft = σ(Xt , Xt−1 , . . .), to be defined by
the full set of variables,
Xt = (r0,t , r1,t , . . . , rN,t , x0,t , x1,t , . . . , xN,t y1,t . . . , yN,t )0 .
The conditional model for the individual asset is assumed to be invariant to this enhancement of
the information set. This implicitly assumes that the dynamic variation in correlations between
individual assets is fully explained by the individual assets correlation with the market return.
Put differently: The variation in the N + 1 × N + 1 conditional covariance matrix is fully
described by the N + 1 conditional variances and the N conditional correlations. This structure
has testable implications that we return to in our empirical section.
2.4.1

Variables for Model Diagnostics

For model diagnostics, in particular the validity of the single-factor structure, we define conditional studentized residuals
ŵi,t =

ẑi,t −ρ̂i,t ẑ0,tq, i = 1, . . . , N.
1−ρ̂2i,t

So far the model structure has been silent about the dependence structure across the populationequivalents of these residuals,

wi,t =

zi,t −ρi,t z0,tq
, i = 1, . . . , N,
1−ρ2i,tand the same is true for the conditional error terms, (ui,t , vi,t |u0,t ), across individual assets. We
cast light on this dependence structure in our empirical section.

10

3

Estimation

In this section, we define the quasi log-likelihood function and exploit its structure to simplify the estimation problem. We have five observed variables, (r0,t , x0,t , r1,t , x1,t , y1,t ), and we
consider their joint density conditional on past information, Ft−1 . Without loss of generality
we can decompose this “joint” density as stated in (1), and, for the purpose of estimation, we
adopt Gaussian specifications for the “marginal” and “conditional” densities, f (r0,t , x0,t |Ft−1 )and f (ri,t , xi,t , yi,t |r0,t , x0,t , Ft−1 ), i = 1, . . . , N . Moreover, we assume that the studentized
returns, (z0,t , z1,t ), are independent of the error terms in the three measurement equations,(u0,t , u1,t , v1,t ). This enables us to decompose the quasi log likelihood function into four terms
as we discuss below.

3.1

The Marginal Model for Market Variables

The marginal model is essentially that of Hansen et al. (2012), which implicitly entails a further
decomposition of the conditional density,f (r0,t , x0,t |Ft−1 ) = fr0 (r0,t |Ft−1 )fx0 |r0 (x0,t |r0,t , Ft−1 ).
The two densities are given from r0,t ∼ N (µ0 , h0,t ) and log x0,t ∼ N (ξ0 +ϕ0 log h0,t +τ0 (z0,t ), σu20 ),which leads to the following two contribution to (minus two times) the log-likelihood function,

`z0
`u0

=

=

T
X
t=1
T
X
t=1

3.2

(r0,t − µ0 )2
log h0,t +h0,tlog σu20 +=(log x0,t − ξ0 − ϕ0 log h0,t − τ0 (z0,t ))2
σu20

=

T
X
t=1
T
X
t=1

2
log h0,t + z0,t.

log σu20 +

u20,t
.
σu20

The Conditional Model for Individual Assets

Next we consider the likelihood contributions from the conditional model. The conditionalmodel also permits a further decomposition of the conditional density,f (r1,t , x1,t , y1,t |r0,t , x0,t , Ft−1 ) = fr1 |r0 ,x0 (r1,t |r0,t , x0,t , Ft−1 )×fx1 ,y1 |r1 ,r0 ,x0 (x1,t , y1,t |r1,t , r0,t , x0,t , Ft−1 ).

11The first term is the density of the individual asset return conditional on the contemporaneous
market variables (and the past). Due to the Gaussian specification we only need to derive theconditional mean and variance of r1,t in order to compute the appropriate likelihood term. The
assumed independence between (z0,t , z1,t ) and u0,t and the iid assumptions imply thatE[g(r1,t )|r0,t , x0,t , Ft−1 ) = E[g(r1,t )|z0,t , u0,t , Ft−1 ) = E[g(r1,t )|r0,t , Ft−1 ),for any function g for which the conditional mean is well defined. Hence,var(r1,t |r0,t , x0,t , Ft−1 ) = var(r1,t |r0,t , Ft−1 ) = h1,t − (ρ1,tsince cov(r1,t , r0,t |Ft−1 ) = ρ1,tp
h0,t h1,t )2 /h0,t = (1 − ρ21,t )h1,t ,ph0,t h1,t . Next the conditional mean of r1,t isE(r1,t |r0,t , x0,t , Ft−1 ) = µ1 + β1,t (r0,t − µ0 ) = µ1 +p
= µ1 + ρ1,t h1,t z0,t ,ρ1,tph0,t h1,t(r0,t − µ0 )h0,tSo that the contribution to (minus two times) the log-likelihood function from this conditional
density is,`z1 |z0 =TXlog[(1 − ρ21,t )h1,t ] +(r1,t −µ1 −ρ1,t

t=1

√

2

h1,t z0,t )(1−ρ21,t )h1,t.

The last likelihood term, `u1 ,v1 |u0 , which relates to the two measurement equations is associated with the conditional density, fx1 ,y1 |r1 ,r0 ,x0 (x1,t , y1,t |r1,t , r0,t , x0,t , Ft−1 ). First we note thatthe conditional distribution of (u1,t , v1,t ) given (u0,t , z0,t , z1,t ) is Gaussian with mean

σu1 ,u0 /σu20

σv1 ,u0 /σu20 u0,t ,and variance

Ω=

σu21

σu1 ,v1

•

σv21





−

σu1 ,u0
σv1 ,u0h 1
σu0 ,u1
σu20

σu0 ,v1i.

So it does not depend on (z0,t , z1,t ) due to the assumed independence. The implication is that
f,x1 ,y1 |r1 ,r0 ,x0 (x1,t , y1,t |r1,t , r0,t , x0,t , Ft−1 ) = f,x1 ,y1 |r1 ,r0 ,x0 (x1,t , y1,t |u0,t , Ft−1 ),12and that the last term in (minus two times) the log-likelihood is given by`u1 ,v1 |u0 =TX0log det Ω + U1,tΩ−1 U1,t ,t=1

where we have defined
U1,t = 

u1,t





−

v1,t

3.3

σu1 ,u0 /σu20
σv1 ,u0 /σu20 u0,t .

Simplification in Estimation

To simplify the estimation we can concentrate the likelihood function with respect to the covariance matrix of (u0,t, u1,t , v1,t ). Let û0,t , û1,t and v̂1,t be the residuals of the three measurement
equations. The Gaussian likelihood implies that the maximum likelihood estimators of thevariance-covariance parameters are given by1
σ̂u20 =
T

T
X

Tû20,t ,t=1

1X
σ̂u1 ,u0 =
û1,t û0,t ,
T
t=1

T

1X
σ̂v1 ,u0 =v̂1,t û0,t ,
T
t=1

and
Ω̂ =

T
1X

T


0
Û1,t Û1,t,where Û1,t = 

û1,t





−

v̂1,t

t=1

σ̂u1 ,u0 /σ̂u20
σ̂v1 ,u0 /σ̂u20 û0,t .

The reduces the number of free parameters that the likelihood has to be maximized over, toθ = (θ00 , θ10 )0 , whereθ0 = (µ0 , ω0 , a0 , b0 , c0 , τ01 , τ02 , ξ0 , ϕ0 , δ01 , δ02 , h0,1 )0 ,is the vector of (remaining) parameters in the market model, andθ1 = (µ1 , ω1 , a1 , b1 , c1 , d1 , τ11 , τ12 , ξ1 , ϕ1 , δ11 , δ12 , a10 , b10 , c10 , ξ10 , ϕ10 , h1,1 , ρ1,1 )0 ,is the vector of (remaining) parameters in the conditional model. Here we follow the convention
and threat the initial values for the latent variables, h0,1 , h1,1 , and ρ1,1 , as were they unknown
parameters.
These parameters are now estimated by maximizing`(θ) = −1`z0 (θ0 ) + `u0 (θ0 ) + `z1 |z0 (θ1 ) + `u1 ,v1 |u0 (θ1 ) ,2

13

where `z0 (θ0 ) =PT

2
2
t=1 [log h0,t (θ0 )+z0,t (θ0 )], `u0 (θ0 ) = T [log σ̂u0 (θ0 )+1], `u1 ,v1 |u0 (θ) = T [log det Ω̂(θ)+

2], andTX(z1,t (θ) − ρ1,t (θ)z0,t (θ))2log{[1 − ρ21,t ]h1,t (θ)} +1 − ρ21,t (θ)
t=1`z1 |z0 (θ) =!
.

In practice this amounts to the following procedure:
1. Given initial values for θ0 , the time series for z0,t and h0,t are computed iteratively. First,
pz0,1 = (r0,1 − µ0 )/ h0,1 , then for t = 2, . . . , T we computeh0,t (θ0 ) = exp {a0 + b0 log h0,t−1 + c0 log x0,t−1 + τ0 (z0,t−1 )} ,r −µand z0,t (θ0 ) = √0,t 0 . This produces the first term of the log-likelihood function,
h0,t (θ0 )PT
2 (θ ).
`z0 (θ0 ) = t=1 log h0,t (θ0 ) + z0,t0

2. Next, we compute u0,t (θ0 ) = log x0,t − ξ0 − ϕ0 log h0,t − τ0 (z0,t ) for t = 1, . . . , T , which
yields the second term of the log-likelihood function, `u0 (θ0 ) = T log σu20 (θ0 ) + 1 , whereP
σu20 (θ0 ) = T1 Tt=1 u20,t (θ0 ).
p
3. Now we turn to the conditional model. We compute z1,1 (θ1 ) = (r1,1 − µ1 )/ h1,1 and thenfor t = 2, . . . , T , we proceed with

h1,t (θ1 ) = exp {a1 + b1 log h1,t−1 + c1 log x1,t−1 + d1 log h0,t + τ1 (z1,t−1 )} ,r1,t −µ1
and z1,t (θ1 ) = √. The notation above suppress that h1,t , and hence z1,t , depend on
h1,t

the market parameters, θ0 (unless d1 = 0). This is implicit since h0,t = h0,t (θ0 ) depends
on θ0 , and a similar dependence on θ0 arises below through z0,t and u0,t . To make thisdependence explicit we shall add the argument, mθ0 , to the likelihood terms below, whichis short for the market variables, {z0,t (θ0 ), h0,t (θ0 ), u0,t (θ0 )}.
Independently of h1,t and z1,t , we can compute:ρ1,t (θ1 ) = z−1 {a10 + b10 z(ρ1,t−1 ) + c10 z(y1,t−1 )}recursively, for t = 2, . . . , T . So the third likelihood term is given by`z1 |z0 (θ1 ; mθ0 }) =TX

log{(1 − ρ21,t (θ1 ))h1,t (θ1 )} +

t=1

14

(z1,t (θ1 ) − ρ1,t (θ1 )z0,t )2.
1 − ρ21,t (θ1 )4. The last step involves the two measurement equations in the conditional model, whose
residuals are computed by

u1,t (θ1 ) = log x1,t − ξ1 − ϕ1 log h1,t − δ1 (z1,t ),v1,t (θ1 ) = z(y1,t ) − ξ1,0 − ϕ1,0 z(ρ1,t ).
Next we get σu1 ,u0 (θ1 ) = T −1PT

t=1 u1,t (θ1 )u0,t and σv1 ,u0 (θ1 ) =

T −1

PT

t=1 v1,t (θ1 )u0,t ,that is the sample covariances of the measurement errors (that also depend on θ0 throughu0,t = u0,t (θ0 )). This leads to the last likelihood term,`u1 ,v1 |u0 (θ1 ; mθ0 ) = T (log det Ω(θ1 ; mθ0 ) + 2),
where

T

Ω(θ1 ; mθ0 ) =1X
0U1,t U1,t
,
T
t=1

with

U1,t = U1,t (θ1 ; mθ0 ) = 

u1,t (θ1 )v1,t (θ1 )

3.4





−

σu1 ,u0 (θ1 )/σu20
σv1 ,u0 (θ1 )/σu20 u0,t .

Hierarchical Approach to Estimation of Large Systems

When estimating a large system, it is advantageous to use a two-step procedure that the hierarchical structure is well suited for. First we estimate the market model by maximizing
1
−
2

"
!
#)
( T
T
X
X

2
log h0,t (θ0 ) + z0,t(θ0 ) + T log T −1
u20,t (θ0 ) + 1
.
t=1

t=1

Then in a second step, where we take {(h0,t , z0,t , u0,t )} as given, which amount to dropping theargument mθ0 in the expressions of the previous section (steps 3 and 4). So with the two-stepprocedure, we estimate θi by maximizingT−(zi,t (θi ) − ρi,t (θi )z0,t )21X
log{(1 − ρ2i,t (θi ))hi,t (θi )} ++ T (log det Ω(θi ) + 2),
2
1 − ρ2i,t (θi )t=1

for each of the individual assets, i = 1, . . . , N .

15

4

Forecasting

In this section we discuss how multistep predictions of volatilities and correlations as well as
return density forecasts can be obtained with our model. Denote h̃0,t ≡ log h0,t , h̃i,t ≡ log hi,tand ρ̃i,t ≡ z(ρi,t ). Point forecasts turn out to be very easy to obtain owing to the fact thatthe vector (h̃0,t , h̃i,t , ρ̃i,t ) can be represented as a VARMA(1,1) system. Substituting each of themeasurement equations (4), (8) and (9) into the equations for the corresponding conditional
moments one obtains

h̃0,t+1 = a0 + c0 ξ0 + (b0 + c0 ϕ0 )h̃0,t + c0 δ0 (z0,t ) + τ0 (z0,t ) + c0 u0,th̃i,t+1 = ai + ci ξi + (bi + ci ϕi )h̃i,t + di h̃0,t+1 + ci δi (zi,t ) + τi (zi,t ) + ci ui,t .
ρ̃i,t+1 = ai0 + ci0 ξi0 + (bi0 + ci0 ϕi0 )ρ̃i,t + ci0 vi,t(10)

Let Vt = (h̃0,t , h̃i,t , ρ̃i,t )0 , then by substituting the equation for h̃0,t+1 into that for h̃i,t+1 , one
can show that
Vt+1 = C + AVt + Bεt ,where εt = (δ0 (z0,t ), τ0 (z0,t ), δi (zi,t ), τi (zi,t ), u0,t , ui,t , vi,t )0 and

a0 + c0 ξ0







C =  a1 + ci ξi + di (a0 + c0 ξ 0 )  ,
ai0 + ci0 ξi0c0b0 + c0 ϕ0

0

0







A =  di (b0 + c0 ) bi + ci ϕi
,
0


0
0
bi0 + ci0 ϕi0

1

0

0

c0

0

0







B =  di c0 di ci 1 di c0 c1 0  .


0
0 0 0
0
0 ci0
The innovation process, εt , is a martingale difference sequence but is slightly heterogeneous.
Time-variation in the distribution of εt arises from (and is fully described by) ρi,t = corr(z0,t , zi,t |Ft−1 ).
Pk−1
It follows that E(Vt+k |Vt ) = Ak Vt + j=0 Aj C which can be used to produce a k-step ahead
forecast of Vt+k . Forecast of the conditional distribution of Vt+k |Ft , which can be used to deduce
unbiased forecasts of the non-transformed variables, e.g., h0,t = exp(h̃0,t ), can be obtained by16

simulation or bootstrap methods. In the simulation approach, we first generatez 0,t 
 z̃i,t 
 





I2 0

 
 ,
ηt = 
 u0,t  ∼ N5 0,


0
Σ


 ui,t 


vi,t


t = 1, . . . , n.

Given an initialization for ρi,0 , one can produce the entire time series {ρ̃i,t } from {vi,t } usingq(10). Next one can define zi,t = ρi,t z0,t + 1 − ρ2i,t wi,t , which has the proper correlation with
z0,t , and thus finally εt can be computed.
Alternatively, a bootstrap approach can be used if the Gaussian assumption concerning
the distribution of ηt is questionable. From the estimated model we can obtain residuals,(η̂1 , . . . , η̂n ), from which we can draw resamples instead of sampling from the Gaussian distribution. Time series for Vt can now be generated from the bootstrapped residuals {η̂t∗ } in thesame manner as with simulated {ηt }.
To simulate the time series for larger systems is straight forward using the bootstrap of theresiduals from the estimated structure. Simulations would require one to take an explicit standabout the correlation structure of wi,t -variables and the correlation structure of the errors in
the various measurement equations.

5

Empirical Analysis

5.1

Data Description

The model is estimated for a large cross section of assets. We included any asset that was aconstituent of the S&P 500 index at some point between January 19, 2006 and June 25, 2010,
albeit excluding assets for which we had less than 1000 daily observations during our sample
period from January 3, 2002 to the end of 2009. This results in a total of 594 time series with
distinct PERMNO (see below). Thus our sample period spans a total of 2,008 trading days andthe sample size for each of the individual stocks ranges from 1,000 to 2,008 observations.
Our data were constructed by merging information from the TAQ dataset and the CRSP
daily stock files that were accessed through the WRDS research service. The former provides thehigh-frequency data used for our construction of realized measures of volatility, and the latter17

has the opening and closing prices that are properly adjusted for stock splits and dividends. The
TAQ database uses ticker symbols as stock identifiers which can be problematic for a comprehensive analysis such as this one. The reason is that about 10% of the companies in our sample
have traded under different ticker symbols during the sample period and, more importantly,
some ticker symbols represent very different companies at different point in time. Relying on
tickers as identifiers can result in data for two or more companies being mixed up. The CRSP
data identifies companies using the CRSP Permanent Company Numbers (PERMNOs) and
we can use these to track changes in companies ticker symbol, thus ensuring that the proper
high-frequency data are extracted from TAQ. This is achieved as follows: First, we match theticker symbols of the S&P 500 constituents to the CRSP dataset and obtain their PERMNOs.
Second, we extract the ticker symbols that were associated with each PERMNO over the sample period. This information is then used to extract high-frequency data from the TAQ, fromwhich our realized measures of volatility are constructed, and the the daily data from the CRSP
are appended to the time series of realized measures. The high-frequency transaction data arecleaned according to the filtering algorithm described in Barndorff-Nielsen et al. (2009), and themultivariate realized kernel by Barndorff-Nielsen et al. (2011) is used as our realized measuresof volatility and co-volatility. We use the exchange traded fund, SPY, as a proxy for the market
index in our empirical analysis, making the total number of assets in our analysis 595.

5.2

Empirical Results

A summary of the estimation results is presented in in Table 1 and Figure 1.1 The first row
in Table 1 contains the estimates for the marginal model for the market return, as defined byequations (2-4), and the rest of the table presents a summary of the estimation results for the
594 conditional models, each defined by equations: (6-9). To conserve space the cross-sectional
statistics for the estimates of Σ are omitted, but some selected estimates of Σ will be presented
in Table 2.
1
The results reported are the estimates when imposing the restrictions ϕi = 1, i = 0, 1, . . . , N , which did not
result a significant reduction of the log-likelihood function, see Hansen and Huang (2012) for a discussion on
this. The initial values for the latent variables, h0,t , hi,t and ρi,t , are treated as parameters, and their estimatedvalues are reported as h0,1 , hi,1 and ρi,1 , respectively.

18

19

0.231

0.691
25.57

54.62

56.98
74.39
76.10
100.4
139.4
207.7

1%

5%
95%

99%

Max(-5)Max(-4)Max(-3)Max(-2)Max(-1)
Max

0.149
0.153
0.164
0.168
0.220
0.229

0.140

-0.0420.102-0.073-0.157-0.130-0.121-0.082-0.079-0.077

0.523
0.525
0.537
0.565
0.649
0.800

0.522

0.063
0.402

0.023

-0.024-0.011-0.000
0.006
0.017
0.023

0.208
0.193

ai

0.175

a0

0.762
0.763
0.769
0.789
0.828
0.887

0.756

0.455
0.701

0.402

0.350
0.354
0.381
0.382
0.391
0.397

0.585
0.592

bi

0.601

b0

0.447
0.447
0.449
0.459
0.473
0.483

0.442

0.256
0.407

0.204

0.093
0.146
0.166
0.176
0.187
0.204

0.335
0.340

ci

0.377

c0

0.164
0.166
0.173
0.175
0.190
0.203

0.163

-0.0020.125-0.012-0.023-0.019-0.016-0.014-0.013-0.012

0.045
0.034

di

-0.005-0.005-0.005
0.000
0.001
0.020

-0.006-0.065-0.016-0.076-0.084-0.080-0.079-0.078-0.078-0.076-0.041-0.041

τi1-0.092

τ01

0.036
0.037
0.037
0.038
0.039
0.048

0.035

-0.0050.028-0.009-0.019-0.018-0.014-0.012-0.010-0.010

0.010
0.009

τi2

0.020

τ02

-0.012-0.009-0.005-0.0020.056
0.082-0.016-0.577-0.094-0.759-0.963-0.895-0.880-0.860-0.779-0.760-0.346-0.352ξi-0.469

ξ0

0.019
0.019
0.023
0.026
0.026
0.037

0.016

-0.0600.002-0.079-0.100-0.091-0.088-0.085-0.080-0.080-0.030-0.032δi1-0.104

δ01

0.113
0.113
0.116
0.120
0.128
0.154

0.112

0.033
0.095

0.021

0.011
0.011
0.012
0.018
0.018
0.020

0.064
0.063

δi2

0.018

δ02

0.754
0.764
0.772
0.779
0.804
0.854

0.750

0.076
0.659

0.003

0.140
0.143
0.157
0.185
0.193
0.205

0.139

-0.0350.107-0.082-0.103-0.097-0.096-0.095-0.089-0.086

0.036
0.036

0.353
0.350
-0.180-0.114-0.113-0.054-0.042-0.018

ai0

ρi,1

0.890
0.905
0.925
0.939
0.958
0.963

0.887

0.601
0.834

0.562

0.536
0.544
0.549
0.556
0.558
0.559

0.708
0.701

bi0

Correlation Parameters

0.600
0.602
0.615
0.631
0.640
0.647

0.600

0.121
0.484

0.060

0.027
0.032
0.043
0.048
0.051
0.051

0.285
0.278

ci0

0.229
0.238
0.239
0.242
0.247
0.252

0.227

-0.4070.148-0.942-1.661-1.522-1.516-1.379-0.955-0.952-0.099-0.066

ξi0

2.560
2.599
2.897
2.988
3.331
3.956

2.539

0.603
1.463

0.529

0.420
0.445
0.488
0.505
0.514
0.518

0.982
0.915

φi0

A summary of the estimated parameters. First row has the estimates from the market return model, (2-4), and information about the cross section of estimates
for the 594 conditional models (6-9) are presented in the remaining rows.

0.010
0.024
0.119
0.120
0.166
0.221

Min
Min(1)
Min(2)
Min(3)
Min(4)
Min(5)

0.023
0.018

µi

hi,1

7.747
3.374

0.001

µ0

1.333

Mean
MedianSPY

h0,1

Volatility parameters

Table 1: Parameter estimates of the Realized GARCH Beta model.

bi

ci
0.1

0.1
0.05

0.05
0
0.2

0.3

0.4

0.5

0.6

0.7

0
0.1

0.8

0.2

0.3

0.1

0.1

0.05

0.05

0

0.05

0.1

0.15

0.2

0
−1

0.25

−0.8

−0.6

bi0
0.1

0.05

0.05

−0.2

0

0
0.6

0.7

0.8

0.9

1

0

0.1

0.2

ξi0

0.3

0.4

0.5

0.6

φi0
0.3

0.15
0.1

0.2

0.05

0.1

0
−1

−0.4

ci0

0.1

0
0.5

0.5

ξi

di

0
−0.05

0.4

0
−0.5

0

0.5

0

1

2

3

4

Figure 1: Histograms of the 594 parameter estimates for some selected parameters.
The parameter ci , which captures the effect of the lagged realized measure on the conditionalvariance, is large and significant while the GARCH parameter, bi , is much smaller than is usually
the case for conventional GARCH models. This reason for this is that the realized measure is
far more informative about volatility than the squared return, which makes the model far more
adaptive to abrupt changes in volatility, which in turn, leads to a better empirical fit andmore accurate forecast. The negative estimates of τi1 and positive estimates of τi2 indicate thepresence of a leverage effect, see Hansen et al. (2012) for the relation of these leverage functions
to the news impact curve. Examining the parameters of the measurement equation, we findthat ξi is negative. This is to be expected because the realized measures is computed over
the open-to-close period, which only capture a fraction of daily (close-to-close) volatility. The
conditional model for the individual stocks has the additional parameter, di , in the GARCH
equation. This parameter measures the spillover effect from market volatility to individual stockvolatility. The mean and the median of this coefficient is positive, and so is the vast majority

20

of the individual estimates. Altogether this shows that market volatility tend to have a positive
contemporaneous effect on individual asset volatility.2
The cross sectional variation of parameter estimates are presented in the histogram plots in
Figure 1. Both Table 1 and Figure 1 show that the parameter estimates are quite stable in ourcross-section of stocks. Only φ̂i0 is estimated to have an extreme value some cases, but evenin these cases, we have verified that the estimated conditional variances and correlations are inagreement with their corresponding realized measures.
Table 2 presents the estimates of Σ for six selected assets. The upper left element in thesematrices is the estimated variance of u0,t in the measurement equation for the realized measuresassociated with the market return. This point estimate varies slightly across the six matrices due
to variation in the sample period for the six assets. Note that the measurement error variancefor the individual assets to be larger than that of the market. This is to be expected because
the realized measure for the market return is based on a larger number of high-frequency data.
Also note that there is substantial correlation across the measurements error, in particular forthe realized measures of volatility.
Table 2: Selected Point Estimates of the Measurement Error Variance.
CVX

UTX



0.112

 0.071
0.016


0.121
0.011



0.015

0.113

 0.060
0.016


0.224
0.001


0.161
0.003





0.112


 0.055 0.197

0.015 0.000 0.017





0.017



SNV

EK


0.112

 0.051
0.015

WMB





0.018

MSFT



0.114

 0.078
0.018

0.217
0.010



0.025


0.112


 0.076 0.138

0.017 0.010 0.014

The estimated measurement error variance matrix, Σ = var(u0,t , ui,t , vi,t ), for six selected assets.

In Figure 2 we present the realized variance of CVX and SPY against the model-implied
conditional variance.3 Clearly, the conditional variance tracks the realized series closely but hasless high-frequency variation. Naturally, this relation is largely imposed by the models structure, because the measurement equation implies a (noisy) relationship between the conditionalvariance and realized measure. The apparent downward bias of the realized measure is due to2

Note that a row in Table 1 does not present the estimates for a particular stock. For example, the 1%
quantile of b0 and c0 may not be estimates for the same asset.
3
The plots for all stocks are available from the authors upon request.

21

the fact that it is computed over a fraction of the day (the about 6.5 hours where assets areactively traded). This aspect of the realized measures explains that the coefficients ξ0 and ξ1
are negative.

hCVX,tRK CVX

200
100
30
20
10
3
2
1

2007

2008

100

2009

2010

2009

2010

hSPY,tRK SPY

20
10

2
1

0.2
0.1

2007

2008

Figure 2: Realized kernel (RK) variance and conditional variance of CVX (upper panel) andSPY (lower panel) over the period 2007 – 2009.
We turn next to the model-implied betas, given byβ̂t = ρ̂tqĥ1,t /ĥ0,t ,(11)

where ρ̂t = ρt (θ̂) and ĥi,t = hi,t (θ̂), i = 0, 1, denote the estimated quantities. The time series
can be contrasted to the realized betasβ̃t = y1,tq

x1,t /x0,t ,(12)that are computed exclusively from high-frequency data on day t.
The model-implied betas take into account the presence of measurement error in the realized
quantities as well as the dynamic linkages between realized measures and conditional moments.
To get an idea of the time variation of β̂t in our model compared to its raw realized counterpart,
we continue with our previous example, and present graphic results for the realized and the
22

1.00

realized correlation

model correlation

0.75

0.50

0.25

0.00
2002

2003

2004

2005

2006

realized beta

2007

2008

2009

2010

2008

2009

2010

model beta

2.0

1.5

1.0

0.5

2002

2003

2004

2005

2006

2007

Figure 3: Realized and conditional correlation (upper panel) and beta (lower panel) of CVX(perm number 14,541) for the period 2002 – 2010.
conditional beta and correlation of CVX in Figure 3. The correlation changes rapidly during
the sample period, which carries over to the the systematic risk of CVX, as defined by its beta.
In fact the beta for CVX ranges from about 0 to more than 1.5 over this period.
In Figure 4 we present quantile time series plots of the cross sectional variation in the conditional correlation and beta during the financial crisis, where the time of the collapse of Lehman
Brothers is clearly identified. It is, perhaps, the period leading up to the collapse of Lehman
Brothers that stands out the most. On July 15th the SEC temporarily prohibited naked shortselling in the securities of Fannie Mae and Freddie Mac. The time of this announcement coincides with some major changes in the cross sectional distribution of correlations and betas,
although we do not claim any causal relation in this matter. In the subsequent period correlations decreased (on average) and the cross sectional distribution became increasingly left-skewed
correlations. This might suggest that assets became somewhat more susceptible to idiosyncratic
shocks and less to market-wide shocks. So it is perhaps surprising that the distribution of conditional betas became more right-skewed and increased. The explanation is that individual
asset volatility increased relatively more than market volatility, to an extend that the relative
increase more than offset the reduction in average correlation. The mechanics of this is easily
23

understood from the definition of the conditional beta, βt = ρi,tp
hi,t /h0,t . After this initialchaotic period correlations started to increase and the variation in betas decreased. Eventually,correlations peaked around mid-November with a median value of over 70% well above the 55%
value at the beginning of June.
1

Sep. 15: Lehman files for bankruptcy

Conditional Correlations

0.8

0.6

0.4

0.2
25−75%
1−10% & 90−99%
10−25% & 75−90%0
Apr−08Jul−08Oct−08Jan−09Date

May−09Aug−09Nov−09Aug−09Nov−095
Sep. 15: Lehman files for bankruptcy

Conditional Betas

4

3

2

1

0
Apr−08Jul−08Oct−08Jan−09Date

May−09Figure 4: Quantile time series plot of conditional realized GARCH correlations for the period
06.2008 – 12.2008.
It is important to understand that the high degree of variation that we find in the betas
cannot simply be attributed to variation in the realized quantities. In fact, the main sourceof this variation is driven by daily returns. The reason is simply that it is time variation in
the dependence structure in daily returns that causes the realized measures to be found to
be useful predictors in the GARCH equations. Variation of this magnitude would be close toimpossible to obtain with standard approaches using rolling window OLS techniques based ondaily returns. Based on rolling window estimated betas of this kind, Lewellen and Nagel (2006)24

concluded that time variation in beta is insufficient to explain certain asset pricing “anomalies”.
Given the large variation we observed in the systematic risk of individual companies, it could

5

5

4.5

4.5

4

4

3.5

3.5

Conditional Betas

3
2.5
2
1.5

3
2.5
2
1.5

1

1

0.5

0.5

0

0

−0.5
Jun 1st−0.5
Jun 1stJul 1stAug 1stSep 1stOct 1stNov 1stUnited Technologies Corp. (NYSE:UTX)

Dec 1st

5

5

4.5

4.5

4

4

3.5

3.5

3

Conditional Betas

Conditional Betas

Conditional Betas

be interesting to revisit this question.

2.5
2
1.5

1

0

0

−0.5
Jun 1st−0.5
Jun 1stDec 1stJul 1stAug 1stSep 1stOct 1stNov 1stSynovus Financial Corp. (NYSE:SNV)

Dec 1st

2
1.5

0.5

Sep 1stOct 1stNov 1stEastman Kodak Co. (NYSE:EK)

Dec 1st3

1

Aug 1stAug 1stSep 1stOct 1stNov 1stWilliams Companies, Inc. (NYSE:WMB)

2.5

0.5

Jul 1stJul 1stFigure 5: Quantile time series plot of conditional realized GARCH betas for the period 06.2008– 12.2008.
In Figure 5 we have replaced the median line of Figure 4 with the time series of conditional
betas for four selected stocks, and the four panels covers the second half of. Our objective isto demonstrate the substantial variation that some betas display. An interesting example isWilliams Companies (WMB) that moved from the lower 10% to the upper 10% in the fall of2008. The example of SNV shows how some financial companies got relatively more risky as
the financial crises approached in the early fall of 2008. Finally, we included UTX and EK toshow that the betas of some companies are relatively stable.

25

5.3

Residual Correlations and Test for Constant Correlations

The Realized Beta GARCH model implies that the correlation between the individual studentized returns, zit and zjt , is time varying. Recall the decomposition
zi,t = ρi,t z0,t + zi,t − ρi,t z0,t = ρi,t z0,t +q
1 − ρ2i,t wi,t ,where wi,t and z0,t are uncorrelated, both have mean zero and unit variance, and in the likelihood
analysis we modeled both as standard Gaussian random variables. It follows thatcorr(zi,t , zj,t ) = ρi,t ρj,t +q(1 − ρ2i,t )(1 − ρ2j,t )E(wi,t wj,t ),which is time varying unless E(wi,t wj,t ) behaves in a rather unlikely way that offsets the variationin ρi,t and ρj,t . We have not stated explicit assumptions about the correlation, E(wi,t wj,t ), which
induces additional dependence between zi,t and zj,t , beyond that inherited from their correlations
with the market return. This additional channel for dependence is ignored in our estimation (inorder to make the estimation of large systems feasible). A non-zero correlation between wi,t andwj,t is evidence that the Realized Beta GARCH model does not fully characterize the complete
system, so that the estimated model will need to be enhanced to capture such effects. It would
also suggest that the estimation is inefficient to some extent, albeit this is to be expected witha relatively simple estimation procedure in a highly complex model.
In this section we study the magnitude of E(wi,t wj,t ) and the potential evidence of timevariation in this correlations. Since our model implies time variation in the correlation between
zi,t and zj,t we shall evaluate the empirical evidence of this.
First we consider a test for constant correlation that is based on the general theory by
Nyblom (1989). This is the underlying framework of several test for parameter constancy
including that of Hansen (1992) (linear regression models) and that of Hansen and Johansen(1999) (cointegration VAR).
Consider a bivariate process (xt , yt ) of studentized variables, E(xt ) = E(yt ) = 0 and E(x2t ) =E(yt2 ) = 1; So that the correlation is given byρt = E(xt yt ).

We are to construct tests for constant correlation and zero correlation. The maintained hypoth-26

esis is that the partial sumbuT c

WT (u) ≡ T

− 12

X

(xs ys − ρs ),u ∈ [0, 1],

s=1satisfies a functional central limit theorem, so that WT (u) ⇒ σW B(u) where B(u) is a standard
2 is the long-run variance of x y − ρ .
Brownian motion, and σWt ttUnder the null hypothesis, H0 : ρt = ρ (constant correlation) it follows that

NBc =

T −1

2
P
ˆ 1
T −1/2 ts=1 (xs ys − ρ̄)d→Bb (u)2 du,
2
σ̂W
0

PT

t=1

where Bb (u) = B(u) − uB(1) is a standard Brownian bridge, ρ̄ = T −1

PT2
t=1 xt yt and σ̂Wis

2 . Under the null hypothesis H : ρ = 0 (zero correlation) wesome consistent estimator of σW
0
t

have
NB0 =

T −1

PT

t=1

T −1/22σ̂WPt

s=1 xs ys

2

d

ˆ 1
B(u)2 du,→0p

2 → σ 2 . In the absence of serial dependence we can use the estimator σ̂ 2 =where σ̂W
WW
P
2 under both null hypotheses. The 5% critiT −1 Tt=1 (xt yt − ρ̄)2 , which is consistent for σW

cal values of these limit distributions are 0.462 and 1.656, respectively, see Nyblom (1989).
In our application we shall apply the test for constant correlation to zi,t zj,t and wi,t wj,t , andwe apply the test for zero correlation to wi,t wj,t .
5.3.1

Empirical Results Concerning Residual Correlation StructureWith 594 stock in our cross section there are 176,121 distinct correlation series to look at. To
handle this we aggregate the correlation estimation and test results by industrial segmentation.
We employ the sector definition given by the Global Industry Classification Standard (GICS)
that is the industry taxonomy developed by Morgan Stanley Capital International (MSCI) andStandard & Poor’s. The GICS structure consists of 10 sectors, 24 industry groups, 68 industries
and 154 sub-industries and it is used as a basis for S&P and MSCI financial market indexes.
To make our analysis as clear as possible we aggregate to sector level.
To match our stocks to the ten GICS sector we pair TAQ with Standard & Poor’s CapitalIQ
database that contains continuously updated GICS classifications for a large set of publicly
listed companies assigned by S&P’s analysts. These GICS classifications reflect those usedby many wealth and investment managers and financial institutions. To match CUSIP and27

Ticker identifiers from TAQ to the GICS identifiers, TAQ stock identifiers are first matched by
CUSIP, and then double checked for a matched with company names from CapitalIQ. In cases
without a match from this procedure, Ticker’s are used. If this procedure does not provide amatch, CapitalIQ Equity Listings report is used to check for inactive listings and these are again
matched according to exchange tickers. If none of the above procedures achieve a positive match,
CapitalIQ’s business description is used to identify company name changes and a final match is
attempted. The above series of matching procedures match all considered TAQ identifiers withavailable GICS classifications. The 10 sectors are listed in Table 3 along with the number ofcompanies and summery statistics for their betas within each of the sectors.
Table 3: Sector Statistics.
SectorEnergyMaterials
Industrials
Consumer Discretionary
Consumer Staples
Healthcare
Financials
Information Technology
Telecommunication Services
Utilities

Company Counts

Min beta

MedianMax Beta

46
37
63
103
46
64
101
86
9
40

0.299 (0.174)
0.382 (0.104)
0.264 (0.133)
0.163 (0.156)
0.336 (0.127)
0.358 (0.147)
0.309 (0.127)
0.335 (0.156)
0.727 (0.166)
0.284 (0.140)

1.009 (0.145)
0.873 (0.117)
0.931 (0.144)
0.912 (0.136)
0.970 (0.135)
1.072 (0.158)
1.037 (0.141)
0.944 (0.145)
1.082 (0.186)
0.988 (0.175)

2.396 (0.510)
2.347 (0.836)
2.293 (0.427)
2.395 (0.664)
2.188 (0.422)
2.586 (0.632)
2.633 (0.665)
2.496 (0.734)
1.620 (0.307)
2.332 (0.579)

The table gives summary statistics of the sectoral aggregation. The third to fifth columns give the timeseries average of the minimum, median and maximum beta for each sector. Standard deviations are
given in parenthesis.

Next we turn to the constancy of correlations within and and across sectors. These results
are presented in Tables 4-6. Table 4 gives the sample average of the unconditional correlations
for residuals sorted by industry classification (GICS). The upper panel is for the studentized
returns, ẑi,t and ẑj,t , and serve as a benchmark measure. It is interesting to compare theseto the numbers in the lower panel that are based on ŵi,t and ŵj,t . The difference between
these numbers show us how much of the correlations between individual stocks that could be
attributed to our market factor. The immediate impression that one gets is that the market
in most cases account for most of the co-variation between the stocks we consider. That iscertainly the case for all the cross sector combinations (the off diagonals of the lower panel areclose to zero). However, for stocks in the same sector we see from the numbers on the diagonalthat a substantial amount of correlation is left unexplained by the market (the diagonal entries28

of the lower panel). The implication is that we need to look for more factors when modelingthe co-variation between stocks in the same sector. For now we will simply investigate somestatistical properties of the residual co-variation and leave the modeling for future work.

Heal

Infor
mati
on
Tech
nolog
y

Telec
omm
Servi un.
ces

Utili
ties

0.180 0.231

0.217

0.196

0.298

Materials

0.417 0.376

0.308

0.240

0.242 0.341

0.297

0.273

0.285

Industrials

0.400

0.330

0.258

0.262 0.356

0.320

0.287

0.286

0.322

0.234

0.232 0.326

0.281

0.251

0.238

0.260

0.207 0.260

0.203

0.213

0.232

Healthcare

0.260 0.258

0.227

0.215

0.221

Financials

0.429

0.301

0.289

0.300

0.356

0.267

0.229

Consumer Staples

Fina

Mate
rials

Consumer Discretionary

ncial
s

Cons
umer
Stap
les

0.158

thcar
e

Cons
u
Discr mer
etion
ary

0.203

trialsEnergy 0.598 0.324 0.276

Ener
gy

Indu
s

Table 4: Unconditional Correlations (Sorted by GICS)

Information Technology
Telecommun. Services

0.369

0.251

Telec
omm
Servi un.
ces

Utili
tie

-0.022-0.026

0.099

Materials

0.160 0.082

0.042

0.008

0.004

0.023

0.018

0.006

0.021

Industrials

0.104

0.058

0.021

0.020

0.026

0.036

0.012

0.013

0.098

0.032

0.020

0.049

0.033

0.010

-0.005

0.105

0.036

0.021

-0.015

0.016

0.040

0.096

0.012

0.009

0.010

0.022

0.145

0.002

0.011

0.024

0.130

0.023

-0.023

0.187

0.029

Healthcare
Financials
Information Technology
Telecommun. Services
Utilities

Fina

Indu

Consumer Staples

thcar

rials
Mate

Ener

Consumer Discretionary

s

Infor
mati
on
Tech
nolog
y

-0.018 -0.056

ncial
s

Heal

-0.044

e

Cons
umer
Stap
les

-0.035ls

Energy 0.483 0.091 0.015

gy

Cons
u
Discr mer
etion
ary

0.487

stria

Utilities

0.341

Average sample correlations for residuals sorted by industry classification (GICS). Upper panel is forẑi,t and ẑj,t and the numbers in the lower panel are based on ŵi,t and ŵj,t .

29

Cons
umer
Discr
etion
a ry

Cons

umer
Stap
les

Heal
thcar
e

Fina
ncial
s

Infor
mati
on
Tech
nolog
y

Telec
omm
un.
Servi
ces

Utili
ties

0.636

0.589

0.423 0.718

0.794

0.679

0.570

Materials

0.619 0.676

0.619

0.470

0.341 0.612

0.609

0.616

0.588

Industrials

0.767

0.665

0.616

0.415 0.623

0.628

0.626

0.683

0.645

0.470

0.387 0.647

0.575

0.595

0.536

0.507

0.453 0.516

0.552

0.577

0.674

Healthcare

0.367 0.409

0.369

0.410

0.475

Financials

0.738

0.575

0.570

0.439

0.682

0.562

0.513

0.778

0.597

Mate
rials

trialsEnergy 0.649 0.781 0.854

Ener
gy

Indu
s

Table 5: Testing for Constant Correlations (Sorted by GICS)

Consumer Discretionary
Consumer Staples

Information Technology
Telecommun. Services

0.474

Fina
ncial
s

Infor
mati
on
Tech
nolog
y

0.090

0.244

0.354

Materials

0.386 0.221

0.207

0.150

0.125 0.260

0.135

0.072

0.125

Industrials

0.147

0.159

0.110

0.094 0.168

0.132

0.063

0.142

0.182

0.107

0.079 0.130

0.128

0.137

0.142

0.173

0.153 0.141

0.204

0.143

0.183

Healthcare

0.166 0.107

0.093

0.113

0.130

Financials

0.382

0.131

0.086

0.411

0.260

0.087

0.094

0.444

0.097

Information Technology
Telecommun. Services
Utilities

Utili

Indu

Consumer Staples

thcar

rials
Mate

Consumer Discretionary

ties

Heal

0.286 0.608

Telec
o

umer
Stap
les

0.248

e

Cons

0.373

stria
ls

Energy 0.722 0.435 0.200

Ener
gy

Cons
umer
Discr
etion
ary

mmu
n.
Servi
ces

Utilities

0.268

Rejection frequencies for the NBc test for constant correlation using a 5% significance level. The upper
table are the results for ẑi,t ẑj,t and the lower table are those for ŵi,t ŵj,t .

In Table 5 we apply the NBc test for constant correlation to our residual series. We report therejection frequencies for a 5% significance level. In the upper panel we present the frequencies
for the product of the studentized returns, ẑi,t ẑj,t . For example, in the case of the 46 energy
companies there are 1,035 tests and the null hypothesis of constant correlation is rejected foralmost 65% of these test. In fact the constant correlation of across studentized returns is30

frequently rejected across the board. Once we account for the market factor, and test the
hypothesis of constant correlation for the idiosyncratic studentized returns, wit , we observe thatthe rejection frequencies are much smaller, these frequencies are presented in the lower panel ofTable 5. One exception of the energy sector, where non-constant correlation is quite prevalent.
Both for the correlation with other energy stocks and with assets from other sectors. That is
the what we get from the rejection frequencies for the NBc applied to the ŵi,t ŵj,t series reported
in the lower panel. Especially, for the cross-sector combinations we have that the market factor
in many cases could fully account for the time-varying correlation between individual stocks.
Still, it seems that within some sectors (such as Energy, Materials, Financials, and Telecom)
there is a need for a sector factor that is allowed to correlate in a time-varying fashion with theindividuals stocks.
One key message to take away from Table 5 is that the evidence of time-varying correlations
across sectors is greatly reduced by accounting their associations with market returns. Within
sectors there is substantial residual time-variation and, evidently, there is a need for additional
factors, such as sector specific factors if a larger fraction of the time-variation is to be accounted
for.

Heal

Fina
ncial

Infor
mati
on
Tech
nolog
y

Telec
omm
un.
Servi
ces

Utili

0.277

0.326

0.868

Materials

0.943 0.746

0.454

0.223

0.183 0.344

0.176

0.186

0.226

Industrials

0.744

0.545

0.236

0.210 0.298

0.335

0.145

0.142

0.692

0.299

0.189 0.427

0.259

0.149

0.170

0.736

0.283 0.259

0.298

0.130

0.435

Healthcare

0.706 0.157

0.270

0.123

0.162

Financials

0.816

0.238

0.111

0.443

0.853

0.265

0.281

0.889

0.267

Consumer Staples

Information Technology

thcar

rials
Mate

Ener

Consumer Discretionary

Telecommun. Services
Utilities

ties

umer
Stap
les

0.153 0.484

s

Cons

0.389

e

Cons
umer
Discr
etion
ary

0.343

ls

Energy 1.000 0.605 0.304

gy

Indu
stria

Table 6: Testing for Zero Correlations (Sorted by GICS)

1.000

Rejection frequencies for the NB0 test for zero correlation applied to ŵi,t ŵj,t .

Table 6 presents the tests for zero correlation. The table reports the rejection frequencies for31the NB0 test applied to the ŵi,t ŵj,t series. Given the results in Table 5 it is not surprising thatthe test is frequently rejected for assets belonging in the same sector. Across sectors the zerocorrelation is also frequently rejected. By introducing sector specific factors it may be possible
to explain the correlation structure of stocks within the same sector. Since additional factors
would change the definition of the residual studentized returns, it is also plausible that sector
specific factors could mitigate the residual correlation we find for assets in different sectors. We
shall pursue this issue in future research.

6

Conclusion

In this paper we propose a multivariate GARCH model that utilizes realized measures of volatility and correlation, and entails a complete modeling of their dynamic properties. The modelbuilds on a self-contained system of equations that link realized measures to the appropriate
population quantities of volatility and covolatility. The structure implies a dynamic model ofthe conditional betas, that are popular measures of risk in finance. The proposed framework
permits for leverage effects and spillover effects between the assets and the market volatility.
In this respect the model combines the flexibility of the GARCH modeling framework with thestatistical precision in volatility measurement resulting from the use of high-frequency data.
Importantly, the Realized Beta GARCH model has a hierarchical structure that makes it
easy to apply to vast number of assets. The model has a structure where the entire correlation
structure is driven by time-varying volatilities and time-varying correlation between each of theassets and the market return.
Our empirical study revealed some interesting features of the cross-sectional variation of
the conditional betas, as well as their time-series variation. In particular, we find that thebetas exhibit far more variation at a daily frequency – variation that is largely concealed in therolling-window estimates of β that one can obtain with regression methods using daily returns.
We have proposed Nyblom-type test for constant and zero correlation. In our empiricalanalysis we found that the Realized Beta GARCH model explains a great deal of the time
variation in the correlation structure, but the Nyblom tests revealed significant residual variationin the correlation structure, in particular between assets within the same sector. For this reason,
it will be interesting to consider a generalize structure where additional sector specific correlation
factors used. We shall pursue this generalization in future research.

32

References
Andersen, T., Bollerslev, T., Diebold, F. and Wu, G. (2006), Realized beta: Persistence andpredictability, in T. Fomby and D. Terrell, eds, ‘Advances in Econometrics: EconometricAnalysis of Economic and Financial Time Series’, Vol. 20, Elsevier Science.
Andersen, T. G. and Bollerslev, T. (1998), ‘Answering the skeptics: Yes, standard volatility
models do provide accurate forecasts’, International Economic Review 39(4), 885–905.
Andersen, T. G., Bollerslev, T., Diebold, F. X. and Ebens, H. (2001), ‘The distribution of stock
return volatility’, Journal of Financial Economics 61, 43–76.
Andersen, T. G., Bollerslev, T., Diebold, F. X. and Labys, P. (2001), ‘The distribution of
exchange rate volatility’, Journal of the American Statistical Association 96(453), 42–55.
Correction published in 2003, volume 98, page 501.
Andersen, T. G., Bollerslev, T., Diebold, F. X. and Labys, P. (2003), ‘Modeling and forecasting
realized volatility’, Econometrica 71(2), 579–625.
Bandi, F. M. and Russell, J. R. (2005), Realized covariation, realized beta, and microstructurenoise. Working paper, Graduate School of Business, The University of Chicago.
Barndorff-Nielsen, O. E., Hansen, P., Lunde, A. and Shephard, N. (2009), ‘Realized kernels inpractice: Trades and quotes’, Econometrics Journal 12/3, C1–C32.
Barndorff-Nielsen, O. E., Hansen, P., Lunde, A. and Shephard, N. (2011), ‘Multivariate realised
kernels: consistent positive semi-definite estimators of the covariation of equity prices withnoise and non-synchronous trading’, Journal of Econometrics 162, 149 – 169.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2008), ‘Designing realised
kernels to measure the ex-post variation of equity prices in the presence of noise’, Econometrica
76, 1481–536.
Barndorff-Nielsen, O. E. and Shephard, N. (2002), ‘Econometric analysis of realised volatility
and its use in estimating stochastic volatility models’, Journal of the Royal Statistical Society
B 64, 253–280.

33

Barndorff-Nielsen, O. E. and Shephard, N. (2004a), ‘Econometric analysis of realised covariation: High frequency based covariance, regression and correlation in financial economics’,
Econometrica 72, 885–925.
Barndorff-Nielsen, O. E. and Shephard, N. (2004b), ‘Power and bipower variation with stochastic
volatility and jumps (with discussion)’, Journal of Financial Econometrics 2, 1–48.
Barndorff-Nielsen, O. E. and Shephard, N. (2007), Advances in Economics and Econometrics.
Theory and Applications, Ninth World Congress, Econometric Society Monographs, Cambridge University Press, chapter Variation, jumps and high frequency data in financial econometrics, pp. 328–372.
Bekaert, G. and Wu, G. (2000), ‘Asymmetric volatility and risk in equity markets’, Review of
Financial Studies 13(1), 1–42.
Bollerslev, T. and Zhang, B. Y. B. (2003), ‘Measuring and modeling systematic risk in factor
pricing models using high-frequency data’, Journal of Empirical Finance 10(5), 533–558.
Braun, P. A., Nelson, D. B. and Sunier, A. M. (1995), ‘Good news, bad news, volatility, andbetas’, Journal of Finance 50(5), 1575–1603.
Christensen, K., Kinnebrock, S. and Podolskij, M. (2010), ‘Pre-averaging estimators of the
ex-post covariance matrix in noisy diffusion models with non-synchronous data’, Journal of
Econometrics 159/1, 116 – 133.
Dovonon, P., Goncalves, S. and Meddahi, N. (2013), ‘Bootstrapping realized multivariate volatility measures’, Journal of Econometrics 173, 49–65.
Engle, R. F. (2002), ‘New frontiers of ARCH models’, Journal of Applied Econometrics 17, 425–
446.
Engle, R. F. and Gallo, G. (2006), ‘A multiple indicators model for volatility using intra-daily
data’, Journal of Econometrics 131, 3–27.
Ferson, W. E. and Harvey, C. R. (1991), ‘The variation of economic risk premiums’, Journal of
Political Economy 99(2), 385–415.
Ferson, W. E. and Harvey, C. R. (1993), ‘The risk and predictability of international equity
returns’, Review of Financial Studies 6(3), 527–66.
34

Gomes, J., Kogan, L. and Zhang, L. (2003), ‘Equilibrium cross section of returns’, Journal of
Political Economy 111(4), 693–732.
Griffin, J. E. and Oomen, R. C. A. (2011), ‘Covariance measurement in the presence of nonsynchronous trading and market microstructure noise’, Journal of Econometrics 160(1), 58–
68.
Hansen, B. E. (1992), ‘Testing for parameter instability in linear models’, Journal of Policy
Modeling 14, 517–533.
Hansen, H. and Johansen, S. (1999), ‘Some tests for parameter constancy in cointegrated VARmodels’, Econometrics Journal 2, 306–333.
Hansen, P. and Lunde, A. (2010), Forecasting volatility using high frequency data, in M. P.
Clements and D. F. Hendry, eds, ‘Oxford Handbook of Economic Forecasting’, Oxford University Press. forthcoming.
Hansen, P. R. and Horel, G. (2009), ‘Quadratic variation by Markov chains’, working paper .
Hansen, P. R. and Huang, Z. (2012), ‘Exponential garch modeling with realized measures ofvolatility’, working paper .
Hansen, P. R., Huang, Z. and Shek, H. (2012), ‘Realized GARCH: A joint model of returns andrealized measures of volatility’, Journal of Applied Econometrics 27, 877–906.
Hayashi, T. and Yoshida, N. (2005), ‘On covariance estimation of non-synchronously observed
diffusion processes’, Bernoulli 11, 359–379.
Jacod, J., Li, Y., Mykland, P. A., Podolskij, M. and Vetter, M. (2009), ‘Microstructure noise in
the continuous case: The pre-averaging approach’, Stochastic Processes and their Applications
119(7), 2249 – 2276.
Lewellen, J. and Nagel, S. (2006), ‘The conditional CAPM does not explain asset-pricing anomalies’, Journal of Financial Economics 82(2), 289–314.
Nelson, D. B. (1991), ‘Conditional heteroskedasticity in asset returns: A new approach’, Econometrica 59(2), 347–370.
Nyblom, J. (1989), ‘Testing for the constancy of parameters over time’, Journal of the American
Statistical Association 84, 223–230.
35

Patton, A. and Verardo, M. (2012), ‘Does beta move with news? Firm-specific information
flows and learning about profitability’, Review of Financial Studies 25, 2789–2839. Working
paper, Duke University.
Shanken, J. (1990), ‘Intertemporal asset pricing : An empirical investigation’, Journal of Econometrics 45(1-2), 99–120.
Shephard, N. and Sheppard, K. (2010), ‘Realising the future: Forecasting with high frequency
based volatility (HEAVY) models’, Journal of Applied Econometrics 25, 197–231.
Todorov, V. and Bollerslev, T. (2010), ‘Jumps and betas: A new framework for disentangling
and estimating systematic risks’, Journal of Econometrics 157(2), 220–235.
Tsay, R. S. and Yeh, J.-H. (2011), ‘Random aggregation with applications in high-frequency
finance’, Journal of Forecasting 30, 72–103.
Visser, M. P. (2011), ‘GARCH parameter estimation using high-frequency data’, Journal of
Financial Econometrics 9, 162–197.
Voev, V. and Lunde, A. (2007), ‘Integrated covariance estimation using high-frequency data inthe presence of noise’, Journal of Financial Econometrics 5, 68–104.
Zhang, L. (2006), ‘Efficient estimation of stochastic volatility using noisy observations: a multiscale approach’, Bernoulli 12, 1019–1043.

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