Global COE Hi-Stat Discussion Paper Series 037

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

Multivariate Realised Kernels:
Consistent Positive Semi-Definite Estimators of
the Covariation of Equity Prices with Noise and
Non-Synchronous Trading

Hi-Stat Discussion Paper

Ole E. Barndorff-NielsenPeter Reinhard Hansen
Asger Lunde
Neil Shephard

March 2009

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

Multivariate realised kernels: consistent positive semi-definite
estimators of the covariation of equity prices with noise andnon-synchronous trading∗
O LE E. BARNDORFF -N IELSENThe T.N. Thiele Centre for Mathematics in Natural Science,
Department of Mathematical Sciences,
University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark& CREATES, University of Aarhus
oebn@imf.au.dk
P ETER R EINHARD H ANSEN
Department of Economics, Stanford University, Landau Economics Building,579 Serra Mall, Stanford, CA 94305-6072, USA
peter.hansen@stanford.edu
A SGER L UNDE
Department of Marketing and Statistics, Aarhus School of Business,
University of Aarhus, Fuglesangs Alle 4, DK-8210 Aarhus V, Denmark& CREATES, University of Aarhus
alunde@asb.dk
N EIL S HEPHARD
Oxford-Man Institute, University of Oxford,
Blue Boar Court, Alfred Road, Oxford OX1 4EH, UK& Department of Economics, University of Oxford
neil.shephard@economics.ox.ac.uk
July 25, 2008

Abstract
We propose a multivariate realised kernel to estimate the ex-post covariation of log-prices. We
show this new consistent estimator is guaranteed to be positive semi-definite and is robust to measurement noise of certain types and can also handle non-synchronous trading. It is the first estimatorwhich has these three properties which are all essential for empirical work in this area. We derive
the large sample asymptotics of this estimator and assess its accuracy using a Monte Carlo study. We
implement the estimator on some US equity data, comparing our results to previous work which has
used returns measured over 5 or 10 minutes intervals. We show the new estimator is substantiallymore precise.

Keywords: HAC estimator, Long run variance estimator; Market frictions; Quadratic variation; Realised variance.
∗ The second and fourth author are also affiliated with CREATES, a research center funded by the Danish National Research
Foundation. The Ox language of Doornik (2001) was used to perform the calculations reported here. We thank Torben Andersen,
Tim Bollerslev, Ron Gallant, Xin Huang, Oliver Linton and Kevin Sheppard for comments on a previous version of this paper.

11 Introduction
The last seven years has seen dramatic improvements in the way econometricians think about time-varying
financial volatility, first brought about by harnessing high frequency data and then by mitigating theinfluence of market microstructure effects. Extending this work to the multivariate case is challenging as
this needs to additionally remove the effects of non-synchronous trading while simultaneously requiringthat the covariance matrix estimator be positive semi-definite. In this paper we provide the first estimatorwhich achieves all these objectives. This will be called the multivariate realised kernel, which we willdefine in equation (1).
′
We study a d-dimensional log price process X = X (1), X (2) , ..., X (d) . These prices are observed

irregularly and non-synchronous over the interval [0, T ]. For simplicity of exposition we take T = 1
throughout the paper. These observations could be trades or quote updates. The observation times forthe i-th asset will be written as t1(i) , t2(i) , . . .. This means the available database of prices is X (i) (t (i)j ), forj = 1, 2, ..., N (i) (1), and i = 1, 2, ..., d. Here N (i) (t) counts the number of distinct data points available
for the i-th asset up to time t.
X is assumed to be driven by Y , the efficient price, abstracting from market microstructure effects.
The efficient price is modelled as a Brownian semimartingale (Y ∈ BSM) defined on some filteredprobability space (, F, (Ft ) , P),Z t
Z t
Y (t) =
a(u)du +σ (u)dW (u),00where a is a vector of elements which are predictable locally bounded drifts, σ is a càdlàg volatility
matrix process and W is a vector of independent Brownian motions. For reviews of the econometrics ofthis type of process see, for example, Ghysels, Harvey & Renault (1996). If Y ∈ BSM then its ex-postcovariation, which we will focus on for reasons explained in a moment, is
Z 16(u)du, where 6 = σ σ ′ ,[Y ](1) =0

where[Y ](1) = plim

n→∞

n
X


′
Y (t j ) − Y (t j −1) Y (t j ) − Y (t j −1) ,j =1(e.g. Protter (2004, p. 66–77) and Jacod & Shiryaev (2003, p. 51)) for any sequence of deterministic
synchronized partitions 0 = t0 < t1 < ... < tn = 1 with sup j {t j +1 − t j } → 0 for n → ∞. This is thequadratic variation of Y .
The contribution of this paper is to construct a consistent, positive semi-definite estimator of [Y ](1)from our database of asset prices.

The challenges of doing this are three fold: (i) there are market

microstructure effects U = X − Y , (ii) the data is irregularly spaced and non-synchronous, (iii) themarket microstructure effects are not statistically independent of the Y process.
2

Quadratic variation is crucial to the economics of financial risk. This is reviewed by, for example,
Andersen, Bollerslev & Diebold (2008) and Barndorff-Nielsen & Shephard (2007), who provide very
extensive references. The economic importance of this line of research has recently been reinforced by
the insight of Bollerslev, Tauchen & Zhou (2008) who have showed that expected stock returns seemwell explained by the variance risk premium (the difference between the implied and realised variance)
and this risk premium is only detectable using the power of high frequency data. See also the papers by
Drechsler & Yaron (2008), Fleming, Kirby & Ostdiek (2003) and de Pooter, Martens & van Dijk (2008).
Our analysis builds upon earlier work on the effect of noise on univariate estimators of [Y ](1) by,
amongst others, Zhou (1996), Andersen, Bollerslev, Diebold & Labys (2000), Bandi & Russell (2008),
Zhang, Mykland & Aı̈t-Sahalia (2005), Bandi & Russell (2006a), Andersen, Bollerslev & Meddahi(2006), Hansen & Lunde (2006), Kalnina & Linton (2008), Zhang (2006), Renault & Werker (2008),
Barndorff-Nielsen, Hansen, Lunde & Shephard (2008a) and Jacod, Li, Mykland, Podolskij & Vetter(2007). The case of no noise is dealt with in the same spirit as the papers by Andersen, Bollerslev, Diebold& Labys (2001) and Barndorff-Nielsen & Shephard (2002), Barndorff-Nielsen & Shephard (2004), Mykland & Zhang (2006), Goncalves & Meddahi (2008), Mykland & Zhang (2008) and Jacod & Protter(1998).
A distinctive feature of multivariate financial data is the phenomenon of non-synchronous trading or
nontrading. These two terms are distinct. The first refers to the fact that any two assets rarely trade at
the same instant. The latter to situations where one assets is trading frequently over a period while someother assets do not trade. The treatment of non-synchronous trading effects dates back to Fisher (1966).
For several years researchers focused mainly on the effects that stale quotes have on daily closing prices.
Campbell, Lo & MacKinlay (1997, chapter 3) provides a survey of this literature. When increasing thesampling frequency beyond the inter-hour level several authors have demonstrated a severe bias towards
zero in covariation statistics. This phenomenon is often referred to as the Epps effect. Epps (1979)
found this bias for stock returns, and it has also been demonstrated to hold for foreign exchange returns,
see Guillaume, Dacorogna, Dave, Müller, Olsen & Pictet (1997). This is confirmed in our empirical
work where realised covariances computed using high frequency data, over specified fixed time periods
such as 15 seconds, dramatically underestimate the degree of dependence between assets. Some recent
econometric work on this topic includes Malliavin & Mancino (2002), Reno (2003), Martens (2003),
Hayashi & Yoshida (2005), Bandi & Russell (2005), Voev & Lunde (2007), Griffin & Oomen (2006),
Large (2007) and Zhang (2005). We will draw ideas from this work.
The form of multivariate realised kernel we propose is, in the univariate special case, subtly different
from that studied in the univariate paper by Barndorff-Nielsen et al. (2008a). Their flat-top kernel, which
has the advantage of being unbiased and fully efficient, is not guaranteed to be non-negative. It also could
not directly deal with non-synchronous data. This is essential in the multivariate case, which motivates3the specific form of the multivariate realised kernel proposed here. We discuss in some detail the differences between these estimates in Section 6.1. The change to our preferred estimator means the rate ofconvergence, bandwidth choice and asymptotic distribution of our new estimator differs from the flat-top
version. In particular, our theory can be used to tune the bandwidth selection for estimating particular
correlations, betas, inverse covariance matrices or just covariances.
The structure of the paper is as follows. In Section 2 we synchronize the timing of the multivariate
data using what we call Refresh Time. This allows us to refine high frequency returns and in turn themultivariate realised kernel. Further we make precise the assumptions we make use of in our Theorems
to study the behaviour of our statistics. In Section 3 we give a detailed discussion of the asymptotic
distribution of realised kernels in the univariate case. The analysis is then extended to the multivariatecase. Section 4 contains a summary of a simulation experiment designed to investigate the finite sample
properties of our estimator. Section 5 contains some results from implementing our estimators on some
US stock price data taken from the TAQ database. This is followed by a Section on extensions and further
remarks, while the main part of the paper is finished by a Conclusion. This is followed by an Appendix
which contains the proofs of various theorems given in the paper, and an Appendix with results related toRefresh Time sampling. More details of our empirical results and simulation experiments are given in a
web Appendix which can be found at http://www.hha.dk/˜alunde/BNHLS/BNHLS.htm.

2 Defining the multivariate realised kernel
2.1 Synchronizing data
2.1.1

Refresh timeNon-synchronous trading delivers fresh (trade or quote) prices at irregularly spaced times which differ
across stocks. Dealing with non-synchronous trading has been an active area of research in financial
econometrics in recent years, e.g. Hayashi & Yoshida (2005), Voev & Lunde (2007) and Large (2007).
Stale prices are a key feature of estimating covariances in financial econometrics as recognised at least
since Epps (1979), for they introduce cross-autocorrelation amongst asset price returns.
Write the number of observations in the i-th asset made up to time t as the counting process N (i) (t),and the times at which trades are made as t1(i) , t2(i) , .... We now define a time scale which will be key tothe construction of multivariate realised kernels.
Definition 1 Refresh Time1 for t ∈ [0, 1]. We define the first refresh time as τ1 = max t1(1) , ..., t1(d) ,and then subsequent refresh times as


(1)
(d)
τ j +1 = max t (1) , ..., t (d).
Nτ j +1

Nτ j +1

1 Refresh time was used in a cointegration study of price discovery by Harris, McInish, Shoesmith & Wood (1995). Martens(2003) used the same idea in the context of realised covariances, but his estimator is inconsistent.

4

The resulting Refresh Time sample size is N , while we write n (i) = N (i) (1).
The τ1 is the time it has taken for all the assets to trade, i.e. all their posted price have been updated.
τ2 is the first time when all the prices are again refreshed. This process is displayed in Figure 1 for d = 3.
Our analysis will now be based on this time clock {τ j }. Our approach will be to:• Assume the entire vector of up to date prices are seen at these refreshed times X (τ j ), which is notcorrect — for we only see a single new price and d − 1 stale prices2 .

• Show these stale pricing errors have no impact on the asymptotic distribution of the realised kernels.
This is shown in Appendix B.
bAsset 1b

Asset 2

Asset 3

b

b

b

b

τ1

b

b

τ2

b

b

b

b

b

b

b

b

τ3

τ4

b

b

b

b

b

b

b

b

τ5

b

b

τ6

b

τ7

Time

Figure 1: This figure illustrates Refresh Time in a situation with three assets. The dots represent the times{t (i)j }. The vertical lines represent the sampling times generated from the three assets with refresh timesampling. Note, in this example, N = 7, n (1) = 8, n (2) = 9 and n (3) = 10.
This approach to dealing with non-synchronous data converts the problem into one where the Refreshed Times’ sample size N is determined by the degree of non-synchronicity and n (1), n (2), . . . , n (d).3
The degree to which we keep data is measured by the size of the retained data over the original size of theP
database. For Refresh Time this is p = d N/ di=1 n (i) . For the data in Figure 1, p = 21/27 ≃ 0.78.
2 Their degree of staleness will be limited by their Refresh Time construction to a single lag in Refresh Time. The extension

to a finite number of lags is given in Section 6.6.
3 How does the expected sample size N change with d? Suppose trade times {t (i) } arrive as independent standard Poisson1(i)process with common intensity λ, so that E{N (i) (t)} = λt. Then t1

(1)

(2)

(d)

∼ exp(λ) and as τ1 = max{t1 , t1 , ..., t1 }, so,de.g. Embrechts, Kluppelberg & Mikosch (1997, pp. 125 & 176) τ1 / log d → λ−1 , or more refined Pr(τ1 − log d ≤ x) ={1 − exp(−λx)/d}d → exp {− exp(−λx)}. Hence the sample size from the refreshed analysis will fall with log d, the dimensionof the asset prices. The situation where the intensity varies across assets, i.e. E{N (i) (t)} = λi t, will not substantially change
this result.
The loss of observations is relatively cheap here, because the rate of convergence for our realised kernel will be n 1/5 , where
we will see in a moment that n is very close to N — the difference being due to end effects. In a standard situation where an
estimator converges at rate n 1/2 , one can expect confidence intervals to widen by about 100% when the sample size is reduced
by a factor of 4. When the rate of convergence is n 1/5 , confidence intervals only widen by about 32%.

5

2.1.2Jittering end conditions

Realised kernels are built out of n high frequency returns computed from synchronized vector prices
recorded at N times. It turns out that our asymptotic theory dictates we need to average m prices at thevery beginning and end of the day to define these returns4 . The theory behind this will be explained in
Section 6.5, where experimentation suggests m should be around 2 for the kind of data we see in thispaper, but for now we just define what we mean by jittering. Let n, m ∈ N, with n − 1 + 2m = N , then
set the vector observations X 0 , X 1 , ..., X n as X j = X (τ j +m ), j = 1, 2, ..., n − 1, and
m

X0 =


1 X
X τj
m j =1

m

and

Xn =

1 X
X (τ N−m+ j ).
m j =1So X 0 and X n are constructed by jittering initial and final time points. By allowing m to be moderately
large but very small in comparison with n, it means these observations record the efficient price without
much error, as the error is averaged away.
These prices allow us to define the high frequency vector returns as x j = X j − X j −1 , j = 1, 2, ..., n.

2.2 Realised kernel
Having synchronized the high frequency vector returns x j we can define our class of positive semi-

definite multivariate realised kernels (RK). It takes on the following form
K (X ) =

n
X

h=−n

k( Hh+1 )Ŵh .

(1)

Here the non-stochastic k(x) for x ∈ R is a weight function. The h-th realised autocovariance is
 Pn
′
h≥0

j =|h|+1 x j x j −h ,
Ŵh =
 Pn
′
h < 0.
j =|h|+1 x j −h x j ,We focus on the class of kernel functions, K, that is characterized by:(i) k(0) = 1, k ′ (0) = 0;(ii) k is twice differentiable with continuous derivatives;(iii) k•0,0, k•1,1 , k•2,2 < ∞, where k•0,0 =
(iv)

R∞

R1

2
1,1
0 k(x) dx, k• =−∞ k(x) exp(i xλ)dx ≥ 0 for all λ ∈ R.

R1

′
2
2,2
0 k (x) dx, k• =

R1

′′
2
0 k (x) dx;

The assumption k(0) = 1 means Ŵ0 gets unit weight, while k ′ (0) = 0 means the kernel gives close
to unit weight to Ŵh for small values of |h|. Condition (iv) guarantees K (X ) to be positive semi-definite,(e.g. Bochner’s theorem and Andrews (1991)).
4 This kind of averaging appears in, for example, Jacod et al. (2007).

6

The multivariate realised kernel has the same form as a standard heteroskedasticity and autocorrelated(HAC) covariance matrix estimator familiar in econometrics (e.g. Gallant (1987), Newey & West (1987),
and Andrews (1991)). But there are a number of important differences. For example, the sums that define
the realised autocovariances are not divided by the sample size and k ′ (0) = 0 is critical in our framework.
Unlike the situation in the standard HAC literature, an estimator based on the Bartlett kernel will not be
consistent for the ex-post variation of prices, measured by quadratic variation, in the present setting. Laterwe will recommend using the Parzen kernel (its form is given in Table 1) instead.

2.3 Assumptions about the noise and refresh time
Having defined the positive semi-definite realised kernel, we will now write out our assumptions about
the market microstructure effects U and the τ j which govern the properties of the vector returns x jand so K (X ).

The assumptions about the noise are stated in observations time — that is we only model the noise atexactly the times where there are trades or quote updates. This type of assumption is familiar from thework of, for example, Zhou (1998), Bandi & Russell (2005), Zhang et al. (2005), Barndorff-Nielsen et al.
(2008a) and Hansen & Lunde (2006). We define
U j = X (τ j ) − Y (τ j ),j = 0, 1, ..., N,

which is noise associated with X (τ j ), the observation at time τ j .
Assumption 1 Suppose that, conditional on {Y }, {U j } is covariance stationary (U ∈ CS) with E(U j ) =P
0 and h |h h | < ∞, where h = cov(U j , U j −h ). Let M = max{|i − j |, |h − l|}. For h, l ≥ 0, thereP′iexists ̺ M such that, Cov(Ui Ui−h
, U j U ′j −l ) ≤ ̺ M , where ∞i=1 ̺i (1 + ǫ) < ∞, for some ǫ > 0.
A key quantity in our analysis is the, so-called, long-run variance:
=

∞
X

h ,

h=−∞

which is a non-stochastic d × d matrix.
On occasions we refer to a white noise assumption about the U process (U ∈ WN ) which means we
assume it has E U j = 0, Var U j =  and Ui ⊥⊥ U j , for all i 6 = j . This white noise assumption isunsatisfactory from a number of viewpoints (e.g. Phillips & Yu (2008) and Kalnina & Linton (2008)) andwill not be used to derive our limit theorems.
Remark. Naturally, one can write the joint law L(Y, U ) as L(U |Y )L(Y ). So Assumption 1, which
concerns L(U |Y ), does not rule out dependence between Y and U.
Remark. Assumption 1 can be relaxed by redefining h to be the “average covariance” defined as thePprobability limit of (n − h)−1 j cov(U j , U j −h ). Furthermore, the autocovariances of the noise, h ,7

h = 0, ±1, ±2, . . . , and hence , may depend on Y. This could create the sort of heteroskedastic noise
documented in Bandi & Russell (2006b).
Remark. Assumption 1 can accommodate the sort of noise that arises when the sequence of prices is outof order. This occurs when prices are recorded with an incorrect time stamp.
Throughout the paper we follow Barndorff-Nielsen et al. (2008a) in making this assumption about
the times that we have Refresh Time data5 .
Assumption 2 Define T (t) =

Rt

0 τ

2

(u)du, where τ (u) is strictly positive, càdlàg univariate process.

Then we assume Refresh Times occur at τ j = T ( j/n). We also assume that τ is adapted to F. When bothconditions hold we write τ ∈ T .

3 Asymptotic results
3.1 Consistency
We first give a consistency result for the multivariate realised kernel K (X ), which can be written K (X ) =PP
K (Y ) + K (Y, U ) + K (U, Y ) + K (U ) , where K (Y, U ) = nh=−n k( Hh+1 ) j y j u ′j −h , with y j = Y j −Y j −1 and u j = U j − U j −1 .

pp

Theorem 1 Let k ∈ K and n → ∞. If K (U ) → 0 and K (Y ) → [Y ] thenp

K (X ) → [Y ].
p

If H ∝ n η with η ∈ (0, 1) and τ ∈ T then K (Y ) → [Y ]. If H ∝ n η with η ∈ (1/2, 1) , U ∈ CS, andp

m → ∞, then K (U ) → 0. Further, if K (Y ) − [Y ] = O p (n −ǫ ) and K (U ) = O p (n −2ǫ ) for some ǫ > 0,then K (X ) − [Y ] = O p (n −ǫ ).

ppNote in particular that, whatever the relationship between Y and U , if K (U ) → 0 and K (Y ) → [Y ]pthen K (X ) → [Y ] . Hansen & Lunde (2006) have shown that endogenous noise is empirically important,particularly for mid-quote data. The above theorem is comparatively clean, it means endogeneity does
not matter for consistency. What matters is that the realised kernel applied to the noise process would
converge to zero as n → ∞.6
5 This means that Z(t) = Y (T (t)) is a Brownian semimartingale with [Z] = [Y ]1
T (1) and spot volatility λ(t) =τ (t) {σ ◦ T (t)}. The point of this assumption is that Z( j/N) = Y (T ( j/N)) = Y (τ j ), where τ j = T ( j/N). So irregularly

spaced data on Y can be thought of as equally spaced on Z.
(i)(i)
An implication of this is that sup j {τ j +1 − τ j } = O p (n −1 ) for n → ∞, which means that supi, j {t j +1 − t j } = O p (n −1 ) byconstruction of refresh time.
6 Of course, if U had a component V, which evolved in calendar time, e.g. V is an Ornstein-Uhlenbeck process, then U ∈/ CS
and K (U ) would not vanish in probability.

8

3.2 Central limit theory
3.2.1

Univariate asymptotic analysis of realised kernels

Before introducing the results on the multivariate case, it is helpful to consider the univariate case
Z tj
Z tj
X (t j ) =a(u)du +σ (u)dW (u) + U j .
00

In order to present the results for the univariate case, we write ω2 in place of , so ω2 =P∞

h=−∞ E(Ui Ui−h ).

Proposition 1 Let k ∈ K, τ ∈ T , H = c0 n 3/5, U ∈ CS, Y ⊥⊥ U and m −1 = o(n −1/5 ). Then


Z 1

Ls
n 1/5 K(X ) −σ 2 (u)du → MN c0−2 k ′′ (0) ω2 , 4c0 k•0,0 IQ ,0

where IQ =R1

40 λ (u)du is the integrated quarticity, andλ(t) = τ (t) {σ ◦ T (t)} .
Ls

The notation → MN means stable convergence to a mixed Gaussian distribution. The notion of stable
convergence is important for the construction of confidence intervals and the use of the delta method.
The reason is that IQ is random, and stable convergence guarantees joint convergence that is needed here.
Stable convergence is discussed, for example, in Mykland & Zhang (2006) and Mykland & Zhang (2008),
who also provides extensive references. The presence of λ in the limit theory is due to the irregularly
R1
spaced nature of the data, if it was equally spaced then τ (t) = 1 and T (t) = t, so IQ = 0 σ 4 (u)du as
usual.

Remark. The asymptotic distribution in Proposition 1 has a non-zero asymptotic mean which implies
that the upward asymptotic bias of the realised kernel is roughly n −1/5 c0−2 k ′′ (0) ω2 . Having an asymptotic bias term in the asymptotic distribution is familiar from kernel density estimation with the optimal
bandwidth. Here the situation is slightly easier for in principle the bias term can be estimated from thedata.
We now explain why Proposition 1 is the most interesting to us. The rest of this subsection can beskipped on first reading if the reader is not interested in these background results.
To start this consider first some moments of various quantities.
Proposition 2 Let k ∈ K and U ∈ CS. Then E {K(U )} = Hn2 |k ′′ (0)|ω2 + O(m −1 ) + o(n/H 2) and if,
additionally τ ∈ T , the asymptotic variance of K (Y ) and K (U ) are given byH
4k•0,0 IQnandn
4k•2,2 ω4 .
H3

(2)

Remark. The second term in E {K (U )} highlights the need for the averaging at the end-points. The
O(m −1 ) term roughly equals 2m −1 ω2 , so we need m → ∞ for the bias to vanish. Empirically ω2 is tiny

so 2m −1 ω2 will be small even with m = 1, but theoretically this is an important observation.
9Remark. The result shows that estimators in this class of realised kernels are generally biased due tothe kernels not being entirely flat-top, but the bias is modest so long as H increases at a faster rate than√n. For a weight function with k ′′ (0) = 0 we could take H ∝ n 1/2 which would result in a faster rateof convergence. However, no weight function with k ′′ (0) = 0 can guarantee a positive semi-definite
estimate, see Andrews (1991, p. 832, comment 5).

Remark. If m −1 = o(n −1/5 ), then the mean square error optimal rate for H is H ∝ n 3/5, equalising therate of the squared bias and the variance. All but the first term in (2) vanish as n → ∞ when H ∝ n 3/5 .
Note that the asymptotic bias is tied to k ′′ (0) whereas the asymptotic variance is tied to k•0,0 .

Remark. This result looks rather weak compared to the corresponding result for the flat-top kernel
K F (X ) introduced by Barndorff-Nielsen et al. (2008a) with k ′ (0) = 0. They had the nicer result that7
n
o Ls
n
o
R1
R1
n 1/4 K F (X ) − 0 σ 2 (u)du → MN 0, 4ck•0,0 IQ + 8c k•1,1 ω2 0 σ 2 (u)du + c43 k•2,2 ω4 ,when H = cn 1/2 , under the assumption that U ∈ WN . Hence the use of non-flat top kernels comes at
an asymptotic cost, but ensures positive semi-definiteness. Section 6.1.2 also shows that K (X ) is morerobust to endogeneity and serial dependence in U than K F (X ).
3.2.2

Choosing the bandwidth H and weight function

Next we turn to the optimal (mean square error) choice for the bandwidth parameter H .

1/5
Proposition 3 Let k ∈ K, τ ∈ T , U ∈ CS and set H = c∗ ξ 4/5 n 3/5, where c∗ = k ′′ (0)2 /k•0,0and√ξ 2 = ω2 / I Q, then


Z 1
Ls
1/5
{ωI Q}2/5 .
n 1/5 K(X ) −σ 2 (u)du → MN(κ, 4κ 2 ), where κ = k ′′ (0)(k•0,0 )20

The relative efficiency of different realised kernels in this class are determined solely by the constantk ′′ (0)(k•0,0 )2

1/5

and so can be universally determined for all Brownian semimartingales and noise pro-

cesses. This constant is computed for a variety of kernel weight functions in Table 1. This shows that
the Quadratic Spectral (QS), Parzen and Fejér weight functions are attractive in this context. The optimal weight function minimizes, k ′′ (0)(k•0,0 )2

1/5, which is also the situation for HAC estimators, see

Andrews (1991). Thus, using Andrews’ analysis of HAC estimators, it follows from our results that the
QS kernel is the optimal weight function within the class of weight functions that are guaranteed to produce a non-negative realised kernel estimate. A drawback of the QS and Fejér weight functions is that
they, in principle, require n (all) realised autocovariances to be computed, whereas the number of realised
autocovariances needed for the Parzen kernel is only H — hence we advocate the use of Parzen weightfunctions. We will discuss estimating ξ 2 in Section 3.3.1.
7 See also Zhang (2006) who independently obtained a n 1/4 consistent estimator using a multiscale approach.

10

Kernel function, k(x)ParzenQuadratic Spectral
FejérTukey-Hanning∞
BNHLS (2008)

k•0,0

k ′′ (0)(k•0,0 )2

12

0.269

0.97

1/5

3π/5

0.93

2/3
π 2 /2

π/3
0.52

0.94
1.06

1

5/4

1.09


0 ≤ x ≤ 1/2
1 − 6x 2 + 6x 3
k P (x) = 2(1 − x)3
1/2 ≤ x ≤ 1

0
x >1


3
sin
x
x ≥0
k Q S (x) = 2
x − cos x ,x2k F (x) = sinx x ,x ≥0

kT H∞ (x) = sin2πk(x) = (1 + x) e

2
−x

x ≥0

exp (−x) ,x ≥0Table 1: Properties of some realised kernels. k ′′ (0)(k•0,0 )2of k ∈ K.
3.2.3|k ′′ (0)|

1/5

1/5

measures the relative asymptotic efficiency

Some multivariate notation

To start we define some terms. Let
Z 1
τ 2 (u) {6 ◦ T (u) ⊗ 6 ◦ T (u)} du,
9=0which is the d 2 × d 2 random matrix analog of integrated quarticity.

Our result will use the matrix normal distribution. For M ∈ Rq×q , M ∼ N (A, B) simply means thatvec(M) is Gaussian distributed with mean vec(A) and the covariance between a ′ Mb and c′ Md is given
′

′

′

′

′
by cov(a ′ Mb, c′ Md) = vab
Bvcd , with vab = vec( ab +ba) and vcd = vec( cd +dc
).
2
2

3.2.4

Multivariate central limit theorem

Theorem 2 Suppose H = c0 n 3/5, τ ∈ T , U ∈ CS, m −1 = o(n −1/5), Y ⊥⊥ U and k ∈ K then


Z 1
Ls
1/5
n
K (X ) −6(u)du → MN{c0−2 |k ′′ (0)|, 4c0 k•0,0 9}.
0This is the multivariate extension of Proposition 1, yielding a limit theorem for the consistent multivariate estimator in the presence of noise. The bias is determined by the long-run variance , the variancesolely by integrated quarticity.
Corollary 1 An implication of Theorem 2 is that for a, b ∈ Rd we have


Z 1

Ls
1/5 ′
′
n a K (X ) −6(u)du b → MN c0−2 |k ′′ (0)|a ′ b, 4c0 k•0,0 vab
9vab .
0

So once a consistent estimator for 9 is obtained, Corollary 1 makes it straightforward to compute a

confidence interval for any element of the integrated variance matrix.
Example 1 In the bivariate case we can write the results as
R1
K (X (i) ) − 0 6ii duR Ls
n 1/5  K (X (i) , X ( j )) − 01 6i j du  → MN (A, B) ,R
1
K (X ( j ) ) − 0 6 j j du11(3)

where
ii
A = c0−2 |k ′′ (0)|  i j jj

B = 2c0 k•0,0

and



26ii2
2
•
τ
0
•

Z 1

26ii 6i j
6ii 6 j j + 6i2j
•


26i2j
26ii 6 j i  ◦ T du,
26 2j jwhich has features in common with the noiseless case discussed in Barndorff-Nielsen & Shephard (2004,
eq. 18). By the delta method we can deduce the asymptotic distribution of the kernel based regression
and correlation. For example
K (X (i) , X ( j ))Ls
1/5(i, j )
→ MN (A, B) ,
n
−β
(
j
)
K (X )
where

and


c−2 |k ′′ (0)|A = R0 1
i j −  j j βi j ,0 6 j j duB = R

2c0 k•0,0
1
0 6 j j du

2 1, −β(i, j )

β(i, j )

R1

= R 01

6i j du,0 6 j j du

Z 1
0

τ

2



6ii 6 j j + 6i2j
26 j j 6i j

26ii 6 j i
26 2j j

◦ T du1−β (i, j ).

To produce the result (3) we notice the asymptotic variance consists of terms 4c0 k•0,0 ve′ i eh 9ve j ek whereei denotes the i-th unit vector in Rd . Consider, for simplicity, the case with equidistant sampling times, so
R1
that 9 = 0 6(u) ⊗ 6(u)du. Then
Z

′
′
′ ′
4vei eh 9ve j ek = vec(ei eh + eh ei )(6 ⊗ 6)du vec(e j ek′ + ek e′j )
Z
Z
′
′
′
′
= 2 tr{6ei eh 6ek e j + 6ei eh 6e j ek }du = 2 (6i j 6hk + 6ik 6 j h )du,and the result follows by using various combinations of (i, h, k, j ).

3.3 Some practical issues
3.3.1

Choice of H in practice

A main feature of multivariate kernels is that there is a single bandwidth parameter H which controls thenumber of leads and lags used for all the series. It must grow with n at rate n 3/5, the key question here is
how to estimate a good constant of proportionality — which controls the efficiency of the procedure.
If we applied the univariate optimal mean square error bandwidth selection to each asset price individ
√
1/5
4/5
ually we would get d bandwidths H (i) = c∗ ξi n 3/5, where c∗ = k ′′ (0)2 /k•0,0and ξi2 = ii / I Q ii ,√where 6ii (u) is the spot variance for the i-th asset. In practice we usually approximate I Q ii by
R1
R1
2
0 6ii (u)du and use ξi = ii / 0 6ii (u)du, which can be estimated relatively easily by using a low
12frequency estimate ofR1

0 6ii (u)du and one of many sensible estimators of ii which use high frequency

data. Then we could construct some ad hoc rules for choosing the global H , such as
Hmin = min(H (1), ..., H (d) ),Hmax = max(H (1), ..., H (d) ),orH̄ = d −1dX

H (i) ,

i=1

or many others. In our empirical work we have used H̄ , while our web Appendix provides an analysis of
the impact of this choice.
An interesting alternative is to optimise the problem for a portfolio, e.g. letting ι be a d-dimensional

vector of ones then d −2 ι′ K (X )ι = K d −1 ι′ X , which is like a “market portfolio” if X contains many
assets. This is easy to carry out, for having converted everything into Refresh Time one computes themarket (ι′ X/ι′ ι) return and then carry out a univariate analysis on it, choosing an optimal H for themarket. This single H is then applied to the multivariate problem.
From the results in Example 1 it is straightforward to derive the optimal choice for H, when theobjective is to estimate a covariance, a correlation, the inverse covariance matrix (which is important for2
portfolio choice) or β (i, j ) . For example, for β (1,2) the trade-off is between c0−4 |k ′′ (0)|2 12 − 22 β (1,2) ,and

2c0 k•0,0
3.3.2

Z 1
0


2
τ 2 611 622 + 612− 4β (1,2) 611 622 + 2β (1,2)2 622 ◦ T du.

Realised kernel based beta and correlation

A key reason for needing our realised kernel to be positive semi-definite is that elements of it can be
combined to consistently estimate the quadratic variation version of the beta and correlation between
assets i and j
β (i, j ) =[Y (i) , Y ( j )][Y ( j )]and[Y (i) , Y ( j )]
ρ (i, j ) = p,[Y (i) ][Y ( j )]where we have written [Y ] = [Y (i) , Y ( j )] i, j =1,2,... . The quantities β (i, j ) and ρ (i, j ) have been highlighted
in previous research by, for example, Andersen, Bollerslev, Diebold & Labys (2003), Barndorff-Nielsen& Shephard (2004) and Dovonon, Goncalves & Meddahi (2007), but their work was hampered by only
being able to use 5-15 minute returns due to the effect of noise and irregularly spaced data.
The realised kernel estimators of these quantities are straightforward and the asymptotic distribution
simply follows by the application of the delta method. In particular


(i)
( j)
(i)
( j)
K
X
,
X
K
X
,
X

β̂ (i, j ) =and ρ̂ (i, j ) = q ∈ [−1, 1],
K X( j)K X (i) K X ( j )where we have written the elements of the realised kernel matrix K (X ) as K (X ) = K X (i) , X ( j ) i, j =1,2,... .
13

4 Simulation Study
So far the analysis has been asymptotic, based on n → ∞. Here we reinforce this by carrying out asimulation analysis to assess the accuracy of the asymptotic predictions in finite samples. We simulate
over the interval t ∈ [0, 1].
The following multivariate factor stochastic volatility model is useddY (i) = µ(i) dt + dV (i) + dF (i) ,dV (i) = ρ (i) σ (i) dB (i) ,dF (i) =q

2
1 − ρ (i) σ (i) dW.

where the elements of B are independent standard
motions and W ⊥⊥ B. Here F (i) is theq Brownian
2common factor, whose strength is determined by 1 − ρ (i) .

This model means that each Y (i) is a diffusive SV model with constant drift µ(i) and random spot
volatility σ (i) . In turn the spot volatility obeys the independent processes σ (i) = exp β0(i) + β1(i) ̺(i) withd̺(i) = α (i) ̺(i) dt + dB (i) . Thus there is perfect statistical leverage (correlation between their innovations)(i)between V (i) andbetween Y (i) and ̺(i) is ρ (i) . The correlation between Y (1) (t)q σ , whileq the leverage22and Y (2)(t) is 1 − ρ (1)1 − ρ (2) .

The price process is simulated via an Euler scheme8 , and the fact that the OU-process have an exact

discretization (see e.g. Glasserman (2004, pp. 110)). Our simulations are based on the following configuration (the same for both processes) (µ(i) , β0(i) , β1(i) , α (i) , ρ (i) ) = (0.03, −5/16, 1/8, −1/40, −0.3), soR
1
that β0(i) = (β1(i) )2 /(2α (i) ). Throughout we have imposed that E 0 σ (i)2 (u)du = 1. The stationary dis-tribution of ̺(i) is utilised in our simulations to restart the process each day at ̺(i) (0) ∼ N (0, (−2α (i) )−1 ).
For our design we have that the variance of σ 2 is exp(−2(β1(i) )2 /α (i) ) − 1 ≃ 2.5. This is comparable tothe empirical results found in e.g. Hansen & Lunde (2005) which motivate our choice for α (i) .
We add noise simulated asi.i.d.
U (i)j |σ, Y ∼ N

0, ω

2



with

2

ω =ξ

2

r

N −1

XN

j =1

σ (i)4 ( j/N ),where the noise-to-signal ratio, ξ 2 takes the values 0, 0.001 and 0.01. This means that the variance ofthe noise increases with the volatility of the efficient price (e.g. Bandi & Russell (2006b)). The observed
process is then given by X ( j/N ) = Y ( j/N ) + U j , j = 0, . . . , N .
To model the non-synchronously spaced data we use two independent Poisson process sampling
n oschemes to generate the times of the actual observations t (i)to which we apply our realised kernel.
j

We control the two Poisson processes by λ = (λ1 , λ2 ), such that for example λ = (5, 10) means thaton average X (1) and X (2) is observed every 5 and 10 second, respectively. This means that the simulated
number of observations will differ between repetitions, but on average the processes will have 23400/λ1

and 23400/λ2 observations, respectively.
8 We normalize one second to be 1/23, 400, so that the interval [0, 1] contains 6.5 hours. In generating the observed price,

we discretize [0, 1] into a number N = 23, 400 of intervals.

14

We vary λ through the following configurations (3, 6), (5, 10), (10, 20), (15, 30), (30, 60), (60, 120)
motivated by the kind of data we see in databases of equity prices.
Table 2: Simulation results
Panel A: Integrated Variance
Series A
RV15m
R.mse
R.mse
0.113
0.505
0.111
0.547
0.229
0.504

Series B
RV15m
R.mse
R.mse
0.122
0.436
0.114
0.450
0.227
0.517

bias
0.006
0.011
0.003

K (X)
R.mse
0.147
0.262
0.557

K (X)
bias
R.mse
0.003
0.134
0.011
0.224
0.001
0.490

ξ 2 = 0.001
λ = (3, 6)
λ = (10, 20)
λ = (60, 120)

1.509
1.432
1.013

0.654
0.660
0.559

0.040
0.041
0.014

0.253
0.359
0.557

1.417
1.318
0.636

0.488
0.492
0.554

0.033
0.035
0.013

0.215
0.295
0.551

ξ 2 = 0.01
λ = (3, 6)
λ = (10, 20)
λ = (60, 120)

14.39
14.01
8.893

1.531
1.452
1.222

0.096
0.106
0.077

0.410
0.568
0.611

13.67
13.15
5.386

1.168
1.305
1.322

0.084
0.081
0.080

0.351
0.424
0.776

Cov 1mbias R.mse-0.051 0.076-0.085 0.108-0.160 0.186-0.342 0.395-0.445 0.510

Cov 15mbias R.mse-0.004 0.183-0.006 0.183-0.011 0.186-0.038 0.188-0.071 0.203

K (X) Covarbias R.mse-0.007 0.062-0.009 0.076-0.009 0.097-0.021 0.142-0.034 0.189

K (X) Corr
bias R.mse-0.012 0.016-0.015 0.020-0.018 0.026-0.028 0.042-0.036 0.054

K (X) beta
bias R.mse-0.016 0.061-0.019 0.064-0.023 0.084-0.035 0.125-0.035 0.178-0.046-0.082-0.156-0.344-0.445

0.091
0.123
0.189
0.400
0.513

-0.005-0.006-0.010-0.039-0.074

0.191
0.186
0.195
0.187
0.206

-0.000-0.002-0.004-0.019-0.034

0.090
0.099
0.118
0.150
0.195

-0.027-0.029-0.032-0.039-0.044

0.032
0.036
0.040
0.052
0.060

-0.034-0.033-0.042-0.049-0.049

0.085
0.083
0.111
0.153
0.204

-0.027-0.073-0.139-0.354-0.451

0.398
0.431
0.407
0.486
0.561

-0.009-0.005-0.001-0.044-0.083

0.263
0.257
0.263
0.236
0.265

0.000
-0.002-0.005-0.017-0.032

0.123
0.133
0.153
0.180
0.222

-0.063-0.067-0.074-0.089-0.092

0.071
0.076
0.084
0.104
0.111

-0.072-0.082-0.099-0.119-0.120

0.132
0.149
0.198
0.242
0.310

RV1m

ξ 2 = 0.0
λ = (3, 6)
λ = (10, 20)
λ = (60, 120)

RV1m

Panel B: Integrated Covariance/Correlation
ξ 2 = 0.0#rets
λ = (3, 6)
3,121
λ = (5, 10)
1,921
λ = (10, 20)982
λ = (30, 60)332
λ = (60, 120) 166
ξ 2 = 0.001
λ = (3, 6)
3,121
λ = (5, 10)
1,921
λ = (10, 20)982
λ = (30, 60)332
λ = (60, 120) 166
ξ 2 = 0.01
λ = (3, 6)
3,121
λ = (5, 10)
1,921
λ = (10, 20)982
λ = (30, 60)332
λ = (60, 120) 166

Simulation results for the realised kernel using a factor SV model with non-syncronous observations and measurement noise. Panel A looks at estimating integrated variance using realised variance and the Parzen type realised
kernel K (X). Panel B looks at estimating integrated covariance and correlation using realised covariance andrealised kernel. Bias and root mean square error are reported.

For each simulated day we compute the observed the price process, X ( j/N ). In order to calcu(i)late K (X ) we need to select H . To do this we evaluate ω̂δ(i)2 = [X δ(i) ](1)/(2n) and [X 1/900](1), the15realised variance estimator based on 15 minute returns. These give us the following feasible values

2/5
(i)
](1)
. The results for Hmean are presented in Table 2.
Ĥi = cn 3/5 ω̂δ(i)2 /[X 1/900

Panel A of the table reports the univariate results of estimating integrated variance. We give the bias

and root mean square error (MSE) for the realised kernel and compare it to the standard realised variance.
In the no noise case of ξ 2 = 0 the RV statistic is quite a bit more precise, especially when n is large.

The positive bias of the realised kernel can be seen when ξ 2 is quite large, but it is small compared to theestimators variance. In that situation the realised kernel is far more precise than the realised variance.
None of these results are surprising or novel.
In Panel B we break new ground as it focuses on estimating the integrated covariance. We compare
the realised kernel estimator with a realised covariance. The high frequency realised covariance is a very
precise estimator of the wrong quantity as its bias is very close to its very large mean square error. In thiscase its bias does not really change very much as n increases.
The realised kernel delivers a very precise estimator of the integrated covariance. It is downward
biased due to the non-synchronous data, but the bias is very modest when n is large and its sampling variance dominates the root MSE. Taken together this implies the realised kernel estimators of the correlation
and regression (beta) are strongly negatively biased — which is due to it being a non-linear function of
the noisy estimates of the integrated variance. The bias is the dominant component of the root MSE in
the correlation case.

5 Empirical illustration
We analyze high-frequency stock prices for ten assets, namely Alcoa Inc. (AA), American International
Group Inc. (AIG), American Express Co. (AXP), Boeing Co. (BA), Bank of America Corp. (BAC),
Citygroup Inc. (C), Caterpillar Inc. (CAT), Chevron Corp.(CVX), General Electric Co. (GE), and Standard & Poor’s Depository Receipt (SPY). The SPY is an exchange-traded fund that holds all of the S&P
500 Index stocks and has enormous liquidity. The sample period runs from January 3, 2005 to June 29,
2007, delivering 626 distinct days. The data is the collection of trades and quotes recorded on the NYSE,
taken from the TAQ database through the Wharton Research Data Services (WRDS) system. We present
empirical results for both transaction and mid-quote prices.
Throughout our analysis we will estimate quantities each day, in the tradition of the realised volatility
literature following, for example, Andersen et al. (2001) and Barndorff-Nielsen & Shephard (2002). Thismeans the target becomes functions of [Y ]s = [Y ](s) − [Y ](s − 1), s ∈ N. The functions we will deal
with are covariances, correlations and betas.

165.1 Procedure for cleaning the high-frequency data
Careful data cleaning is one of the most important aspects of volatility estimation from high-frequency
data. Numerous problems and solutions are discussed in Falkenberry (2001), Hansen & Lunde (2006),
Brownless & Gallo (2006) and Barndorff-Nielsen, Hansen, Lunde & Shephard (2008b). In this paper wefollow the step-by-step cleaning procedure used in Barndorff-Nielsen et al. (2008b) who discuss in detailthe various choices available and their impact on univariate realised kernels. For convenience we briefly
review these steps.
All data: P1) Delete entries with a timestamp outside the 9:30 a.m. to 4 p.m. window when the exchange
is open. P2) Delete entries with a bid, ask or transaction price equal to zero. P3) Retain entries originatingfrom a single exchange (NYSE except for SPY for which all retained observations are from Pacific).
Delete other entries.
Trade data only: T1) Delete entries with corrected trades. (Trades with a Correction Indicator, CORR 6 =0). T2) Delete entries with abnormal Sale Condition. (Trades where COND has a letter code, except for“E” and “F”). T3) If multiple transactions have the same time stamp: use the median price. T4) Delete
entries with prices that are above the ask plus the bid-ask spread. Similar for entries with prices below
the bid minus the bid-ask spread.
Quote data only: Q1) When multiple quotes have the same timestamp, we replace all these with a single
entry with the median bid and median ask price. Q2) Delete rows for which the spread is negative. Q3)
Delete rows for which the spread is more that 10 times the median spread on that day. Q4) Delete rows forwhich the mid-quote deviated by more than 5 mean absolute deviations from a centered mean (excluding
the observation under consideration) of 50 observations. We note steps P2, T1, T2, T4, Q2, Q3 and Q4collectively reduce the sample size by less than 1%.

5.2 Sampling schemes
We applied three different sampling schemes depending on the particular estimator. The simplest one isthe estimator by Hayashi & Yoshida (2005) that uses all the available observations for a particular asset
combination. Following Andersen et al. (2003) the realised covariation estimator is based on calendertime sampling. Specifically, we consider 15 second, 5 minute, and 30 minute intraday return, aligned
using the previous tick approach. This results in 1560, 78 and 13 daily observations, respectively.
For the realised kernel the Refresh Time sampling scheme discussed in section 2.1.1 is used. Our
analysis first considers estimates for each of the 45 unique pairs of assets — delivering 45 distinct 2 × 2
covariance matrix estimates each day.

17

Table 3: Summary statistics for the refresh sampling scheme, 2×2 case2 × 2 case
AA
AIG
AXP
BA
BAC
C
CAT
CVX
GE
SPY

AA
0.673
0.665
0.662
0.681
0.687
0.647
0.680
0.686
0.678

AIG
0.601
0.670
0.667
0.691
0.700
0.648
0.690
0.699
0.696

AXP
0.597
0.600
0.663
0.678
0.681
0.650
0.671
0.677
0.658

BA
0.594
0.602
0.600
0.673
0.678
0.646
0.670
0.675
0.665

BAC
0.601
0.624
0.602
0.599
0.717
0.655
0.707
0.719
0.721

C
0.594
0.628
0.585
0.592
0.634
0.657
0.719
0.733
0.747

CAT
0.587
0.590
0.590
0.590
0.592
0.582
0.649
0.653
0.633

CVX
0.570
0.603
0.552
0.568
0.605
0.624
0.560
0.726
0.743

GE
0.596
0.625
0.585
0.592
0.628
0.642
0.584
0.620

SPY
0.568
0.603
0.548
0.569
0.604
0.627
0.562
0.620
0.619

0.762

Average over daily number of high frequency observations available before the Refresh Time transformation
TradesQuotesAA
4,124
11,222

AIG
4,789
11,738AXP
3,528
10,482

BA
4,057
10,717BAC
4,757
12,562

C
5,687
13,393

CAT
4,039
9,937

CVX
6,292
13,573

GE
5,460
14,189SPY
6,554
18,587

Summary statistics for the refresh sampling scheme. In the two upper panels P
we present averages over the daily
data reduction induced by the refresh sampling scheme, measured by p = d N/ di=1 n (i) . The upper panel display
this in the 2×2 case. The upper diagonal is based on transaction prices, whereas the lower diagonal is based on
mid-quotes. In the lower panel we average over the daily number of high frequency observations.

The amount of data we discard by constructing Refresh Time is recorded in Table 3. It records theaverage of the daily p statistics defined in Section 2.1.1 for each pair. It emerges that we rarely lose morethat half the observations for most frequently traded assets. For the least active assets we typically lose
between 30% to 40% of the observations.
We will also apply the realised kernel to the full 10 × 1 vector of returns. Here the data loss ismore pronounced. Still, even in the worst case more that 20 percent of the observations remain in thesample. For transaction data the average number of Refresh Time observations in 1, 222, whereas thecorresponding number is 3, 942 for the quote data. So in most cases we have an observation on average
more often than every 8 seconds for quote data and 20 seconds for trade data.

5.3 Analysis of the covariance estimators: CovsK , CovsHY , CovsOtoC and Covs1m
Throughout this subsection the target which we wish to estimate is [Y (i) , Y ( j )]s , i, j = 1, 2, ..., d, s ∈ N.
In what follows the pair i, j will only be referred to implicitly. All kernels are computed with Parzen
weights.
We compute the realised kernel for (all possible) pairs of assets and for the full 10-dimensional vectorof assets, and the resulting estimates of [Y (i) , Y ( j )]s are denoted by CovsK 2×2 and CovsK 10×10 , respectively.
The two estimators differ in a number of ways, such as the bandwidth selection and the sampling times(due to the construction of Refresh Time).
18

To provide useful benchmarks for these estimators we also compute: CovHY
s , the Hayashi & Yoshida(2005) covariance estimator. Cov1
s , the realised covariance based on intraday returns that span a intervalof length 1, e.g. 5 or 30 minutes (the previous-tick method is used). CovOtoC, the outer products ofsthe open to close returns, which when averaged over many days provide an estimator of the averagecovariance between asset returns.
Table 4: Average high frequency realised covariance
Average of realised covariances
AA
AIG
AXP
BA
BAC
C
CAT
CVX
GE
SPY

AA
2.370
0.212
0.193
0.230
0.182
0.209
0.249
0.251
0.161
0.203

AIG
0.212
1.126
0.178
0.201
0.171
0.197
0.194
0.209
0.147
0.178

AXP
0.194
0.175
0.901
0.185
0.160
0.186
0.190
0.194
0.133
0.163

BA
0.230
0.192
0.173
1.287
0.168
0.197
0.227
0.218
0.150
0.190

BAC
0.182
0.171
0.158
0.162
0.807
0.189
0.171
0.186
0.136
0.150

C
0.213
0.198
0.182
0.195
0.192
0.924
0.203
0.215
0.157
0.179

CAT
0.236
0.180
0.178
0.215
0.162
0.194
1.450
0.228
0.152
0.192

CVX
0.247
0.201
0.185
0.211
0.184
0.213
0.214
1.648
0.163
0.212

GE
0.167
0.150
0.138
0.155
0.139
0.161
0.151
0.165
0.887
0.130

SPY
0.201
0.172
0.156
0.176
0.151
0.183
0.174
0.205
0.138
0.300

Average of Hayashi-Yoshida covariances (all times)
AA
AIG
AXP
BA
BAC
C
CAT
CVX
GE
SPY

AA
2.842
0.116
0.112
0.127
0.096
0.111
0.140
0.131
0.088
0.087

AIG
0.185
1.318
0.110
0.122
0.103
0.115
0.122
0.119
0.089
0.083

AXP
0.182
0.163
1.017
0.112
0.091
0.106
0.120
0.115
0.076
0.079

BA
0.208
0.179
0.168
1.390
0.093
0.110
0.137
0.123
0.087
0.089

BAC
0.160
0.153
0.144
0.150
1.096
0.103
0.099
0.105
0.078
0.065

C
0.177
0.172
0.163
0.170
0.161
1.211
0.120
0.118
0.087
0.078

CAT
0.215
0.170
0.170
0.208
0.154
0.170
1.471
0.131
0.090
0.093

CVX
0.207
0.171
0.164
0.185
0.158
0.173
0.197
1.718
0.093
0.094

GE
0.154
0.141
0.132
0.146
0.129
0.142
0.149
0.147
1.655
0.054

SPY
0.175
0.141
0.142
0.152
0.126
0.143
0.154
0.156
0.120
0.292

AXP
0.350
0.356
0.867
0.315
0.378
0.421
0.428
0.340
0.307
0.375

BA
0.456
0.268
0.323
1.371
0.278
0.293
0.550
0.321
0.301
0.363

BAC
0.264
0.287
0.377
0.277
0.524
0.391
0.302
0.257
0.264
0.295

C
0.307
0.322
0.422
0.297
0.394
0.660
0.327
0.265
0.304
0.315

CAT
0.664
0.351
0.435
0.559
0.301
0.330
1.585
0.533
0.340
0.427

CVX
0.618
0.268
0.344
0.326
0.256
0.270
0.539
1.447
0.185
0.398

GE
0.227
0.256
0.304
0.302
0.260
0.305
0.342
0.188
0.532
0.261

SPY
0.405
0.283
0.360
0.355
0.287
0.318
0.437
0.401
0.262
0.349

Open-to-close covariance
AA
AIG
AXP
BA
BAC
C
CAT
CVX
GE
SPY

AA
1.637
0.259
0.347
0.453
0.265
0.311
0.656
0.612
0.232
0.409

AIG
0.259
0.871
0.353
0.265
0.288
0.321
0.350
0.265
0.257
0.283

HY
The upper panel presents average estimates for Cov15s
s and the middle and lower panels display these for Covs
OtoC
and Covs , respectively. In all panels the upper diagonal is based on transaction prices, whereas the lower
diagonal is based on mid-quotes. The diagonal elements are the average of IV estimates based on transactions.
Outside the diagonals numbers are boldfaced if the bias is significant at the 1 percent level.

19

The empirical analysis of our estimators of the covariance is started by recalling the main statistical
impact of market microstructure and the Epps effect. Table 4 contains the time series average covariance
HYcomputed using the 15-second realised covariance Cov15s
s , the Hayashi & Yoshida (2005) estimator Covs

and the open to close estimator CovOtoC. Quite a few of these types of tables will be presented and theys
all have the same structure. The numbers above the leading diagonal are results from trade data, the
HY
OtoC
numbers below are from mid-quotes. Both Cov15s.
s and the Covs are typically much lower than Covs

The numbers which are bolded are statistically significantly different from the CovOtoC
numbers at thesone percent level. This assessment is carried out in the following way.
For a given estimator, e.g. CovsK 2×2 , we consider the difference ds = CovsK 2×2 − CovOtoC
, and

 s
P
q
h
γh , where
compute the sample bias as d̄ and robust (HAC) variance as e2 = γ0 + 2 h=1 1 − q+1

P
n−h
1
2/9
γh = T −h s=1 ηs ηs−h . Here ηs = dt − d̄ and q = int 4(T /100). The number is boldfaced if√
T d̄/e > 2.326. The results in Table 4 indicate the Cov1/4m
is severely downward bias, while CovHYss
is even more distorted. In both cases nearly every covariance estimator for every pair of assets for bothtrades and quotes seem statistically significantly biased.
KK

5.4 Results for Covs 2×2 , Covs 10×10 and Covs5m
We now move on to more successful estimators. The upper panel of Table 5 presents the time series
average estimates for CovsK 2×2 , the middle panel for CovsK 10×10 , and the lower panel give results for Cov5ms .
The diagonal elements are the estimates based on transactions. Off-diagonal numbers are boldfaced ifthey are significantly biased (compared to CovOtoC) at the 1 percent level.
s
These results are quite encouraging for all three estimators. The average levels of the three estimators
are roughly the same. CovsK 2×2 has three failures. CovsK 10×10 has four failures while Cov5ms is rejected fivetimes. All three estimators reject for the SPY/AXP combination, both for trades and quotes.
A much tougher comparison is to replace the noisy ds = CovsK − CovOtoC
with ds = CovsK 2×2 −s

CovsK 10×10 . Our tests will then ask if there is a significant difference in the average. The results reported

in our web Appendix suggest very little difference in the level of these two estimators. When we computethe same test based on ds = CovsK 2×2 − Cov5ms we get consistent rejection of no difference between theseestimators — now the levels of the CovsK 2×2 is judged to be above the corresponding result for Cov5ms —particularly for GE and SPY stocks. The same thing happens when CovsK 10×10 is compared to Cov5ms .
The result in that analysis is reinforced by the information in the summary Table 6, which shows
results averaged over all asset pairs for both trades and quotes. The results are not very different for mostestimators as we move from trades to quotes, the counter example is CovHY
s which seems sensitive to this.

20Table 5: Averages for alternative integrated covariance estimators
Average of Parzen covariances (2×2)
AA
AIG
AXP
BA
BAC
C
CAT
CVX
GE
SPY

AA
2.278
0.310
0.352
0.390
0.328
0.355
0.566
0.535
0.308
0.401

AIG
0.307
0.999
0.284
0.254
0.297
0.306
0.313
0.221
0.256
0.282

AXP
0.351
0.286
0.833
0.277
0.320
0.331
0.339
0.246
0.261
0.289

BA
0.388
0.247
0.275
1.207
0.272
0.288
0.419
0.264
0.261
0.310

BAC
0.326
0.299
0.323
0.267
0.681
0.373
0.326
0.253
0.264
0.291

C
0.357
0.310
0.341
0.285
0.380
0.778
0.348
0.272
0.286
0.311

CAT
0.576
0.308
0.341
0.417
0.324
0.347
1.684
0.399
0.306
0.389

CVX
0.560
0.212
0.239
0.256
0.245
0.267
0.401
1.660
0.229
0.361

GE
0.308
0.258
0.264
0.264
0.263
0.291
0.309
0.225
0.639
0.270

SPY
0.402
0.281
0.289
0.305
0.292
0.314
0.387
0.361
0.274
0.325

BA
0.405
0.234
0.296
1.215
0.267
0.282
0.446
0.234
0.250
0.291

BAC
0.327
0.288
0.352
0.271
0.645
0.398
0.321
0.240
0.254
0.285

C
0.357
0.310
0.355
0.281
0.394
0.705
0.342
0.247
0.285
0.304

CAT
0.649
0.283
0.370
0.462
0.328
0.349
1.622
0.398
0.302
0.379

CVX
0.619
0.188
0.243
0.241
0.235
0.238
0.420
1.563
0.182
0.338

GE
0.275
0.251
0.268
0.248
0.249
0.282
0.306
0.173
0.585
0.252

SPY
0.396
0.259
0.292
0.295
0.283
0.300
0.388
0.334
0.247
0.296

C
0.356
0.292
0.323
0.284
0.361
0.790
0.342
0.267
0.275
0.303

CAT
0.539
0.300
0.327
0.401
0.305
0.342
1.657
0.378
0.298
0.376

CVX
0.526
0.219
0.240
0.260
0.246
0.268
0.377
1.658
0.223
0.350

GE
0.303
0.239
0.251
0.252
0.246
0.275
0.297
0.222
0.644
0.254

SPY
0.397
0.269
0.281
0.303
0.276
0.303
0.373
0.349
0.256
0.324

Average of Parzen covariances (10×10)
AA
AIG
AXP
BA
BAC
C
CAT
CVX
GE
SPY

AA
2.168
0.292
0.343
0.381
0.324
0.351
0.628
0.599
0.280
0.391

AIG
0.289
0.943
0.295
0.238
0.294
0.317
0.282
0.194
0.257
0.264

AXP
0.346
0.294
0.838
0.287
0.350
0.355
0.353
0.235
0.269
0.289

Average of 5 min realised covariance (pre-tick times)
AA
AIG
AXP
BA
BAC
C
CAT
CVX
GE
SPY

AA
2.315
0.310
0.342
0.380
0.322
0.358
0.538
0.527
0.303
0.393

AIG
0.312
0.996
0.275
0.253
0.273
0.294
0.300
0.219
0.243
0.269

AXP
0.347
0.274
0.833
0.275
0.306
0.323
0.322
0.244
0.250
0.280

BA
0.378
0.254
0.272
1.239
0.265
0.283
0.405
0.263
0.249
0.303

BAC
0.318
0.272
0.309
0.264
0.686
0.361
0.307
0.246
0.247
0.274
K

K

The upper panel presents average estimates for Covs 2×2 , the middle panel for Covs 10×10 , and the lower panel gives
results for Cov5ms . In both panels the upper diagonal is based on transaction prices, whereas the lower diagonal
is based on mid-quotes. The diagonal elements are the average of IV estimates based on transactions. Outside thediagonals numbers are boldfaced if the bias is significant at the 1 percent level.

The Table shows CovsK 2×2 and CovsK 10×10 have roughly the same average value, which is slightly below
CovOtoC. CovsK 2×2 has a nine times smaller variance than CovOtoC, which shows it is a lot more precise. Ofsscourse integrated variance is its self random so nine underestimates the efficiency gain of using CovsK 2×2 .
2

If volatility is close to being persistent then CovsK 2×2 is at least 0.3341.067
2 (1−ac f ) ≃ 17 times more informative
1

than the cross product of daily returns. The same observation holds for mid-quotes.
21

HY
5m
K 2×2
Cov15s,s and Covs are very precise estimates of the wrong quantity. Covs is quite close to Covs

the two measures have a correlation of 0.92.
Table 6: Summary statistics across all asset pairs
Transaction prices
Estimator
Average HAC
Summary stats for covariances

Stdev

Bias

cor(.,K)

acf1

acf2

acf3

acf4

acf5

acf10

Cov K 2×2
Cov K 10×10
CovHY
Cov1/4m
Cov5m
Cov30mCov Ot oC

0.334
0.447
0.099
0.167
0.334
0.471
1.067

-0.026-0.029-0.242-0.158-0.036-0.051

1.000
0.787
0.706
0.764
0.924
0.646
0.288

0.40
0.23
0.58
0.60
0.35
0.15
0.03

0.37
0.20
0.50
0.52
0.36
0.11
0.01

0.27
0.16
0.42
0.41
0.24
0.10
0.02

0.24
0.16
0.32
0.33
0.22
0.10
0.03

0.20
0.11
0.30
0.29
0.18
0.10
0.02

0.23
0.13
0.32
0.28
0.20
0.10
0.05

1.000
0.653
0.528
0.851
0.517

0.30
0.11
0.58
0.24
0.07

0.26
0.10
0.52
0.21
0.06

0.21
0.08
0.47
0.17
0.05

0.19
0.08
0.43
0.15
0.05

0.18
0.07
0.41
0.14
0.06

0.14
0.07
0.34
0.12
0.04

0.3180 [ 0.023]
0.3148 [ 0.026]
0.1026 [ 0.008]
0.1864 [ 0.013]
0.3082 [ 0.022]
0.2930 [ 0.025]
0.3435 [ 0.046]

Summary stats for correlations
Corr K 2×2
Corr K 10×10
Corr1/4m
Corr5mCorr30m

0.3273 [ 0.010]
0.3438 [ 0.013]
0.1758 [ 0.007]
0.3177 [ 0.010]
0.3358 [ 0.015]

0.155
0.264
0.084
0.165
0.315

Average unconditional Open-to-Close correlation = 0.3974
Mid-quotes
Estimator
Average HAC
Summary stats for covariances

Stdev

Bias

cor(.,K)

acf1

acf2

acf3

acf4

acf5

acf10

Cov K 2×2
Cov K 10×10
CovHY
Cov1/4m
Cov5m
Cov30mCov Ot oC

0.347
0.463
0.136
0.162
0.333
0.467
1.067

-0.026-0.027-0.181-0.161-0.036-0.052

1.000
0.767
0.743
0.733
0.921
0.668
0.299

0.37
0.19
0.57
0.62
0.36
0.16
0.03

0.36
0.17
0.50
0.53
0.36
0.12
0.02

0.26
0.14
0.41
0.42
0.25
0.10
0.02

0.23
0.15
0.33
0.34
0.22
0.10
0.03

0.19
0.10
0.30
0.30
0.18
0.10
0.02

0.22
0.13
0.31
0.29
0.20
0.10
0.05

1.000
0.653
0.519
0.838
0.571

0.26
0.09
0.56
0.24
0.07

0.22
0.08
0.49
0.20
0.06

0.18
0.06
0.44
0.17
0.05

0.16
0.07
0.40
0.15
0.05

0.15
0.06
0.37
0.13
0.06

0.13
0.06
0.30
0.11
0.04

0.3183 [ 0.023]
0.3171 [ 0.026]
0.1628 [ 0.010]
0.1829 [ 0.013]
0.3080 [ 0.022]
0.2918 [ 0.024]
0.3447 [ 0.046]

Summary stats for correlations
Corr K 2×2
Corr K 10×10
Corr1/4m
Corr5mCorr30m

0.3330 [ 0.010]
0.3460 [ 0.014]
0.1735 [ 0.006]
0.3194 [ 0.010]
0.3351 [ 0.015]

0.170
0.297
0.078
0.165
0.317

Average unconditional Open-to-Close correlation = 0.4035
Summary statistics across all asset pairs. The first column identify the estimator, and the second gives the average
estimate across all asset combinations, followed by the average Newey-West type standard error. The fourth gives
the average standard deviation of the estimator. The fifth is the average bias. Next is average sample correlation
with our realised kernel. The remaining columns give average autocorrelations. The upper panel is based ontransaction prices, whereas the lower panel is based on mid-quotes. The sub panels give first results for covarianceestimates followed by correlation results.

22

The corresponding results for correlations are less good. All the estimates are biased, which is no
surprise due to it being a non-linear transform of roughly unbiased and somewhat noisy observations.
CorrsK 2×2 looks like the most effective estimate.
In our web appendix we give time series plots and autocorrelogram for the various estimates of realised covariance for the AA-SPY assets combination using trade data. They show CovsK 2×2 performing
much better than the 30 minute realised covariance but there not being a great deal of difference between
the statistics when the realised covariance is based on 5 minute returns. The web appendix also presents
scatter plots of estimates based on transaction prices (vertical axis) against the same estimate based on
mid-quotes (horizontal axis) for the same days. These show a remarkable agreement between estimates
30m
HY
K 2×2
based on CovsK 2×2 , Cov5mand Cov5ms and Covs , while once again Covs struggles. Overall Covss

behave in a similar manner, with CovsK 2×2 slightly stronger. CovsK 10×10 estimates roughly the same levelas CovsK 2×2 but is discernibly noisier.

5.5 Analysis of the correlation estimates(i, j )In this subsection we will focus
q on estimating ρs(i, j )K

correlation ρ̂s(i, j )

= Ks

( j, j )/ K s(i,i) K sp= [Y (i) , Y ( j )]s / [Y (i) ]s [Y ( j )]s by the realised kernel

and the corresponding realised correlation ρ̂sX m .
KKA table in our web Appendix average estimates for ρ̂s 2×2 , ρ̂s 10×10 and ρ̂s5m . It shows the expected
KK

result that ρ̂s 2×2 is more precise than ρ̂s 10×10 . Both have average values which are quite a bit below
the unconditional correlation of the daily open-to-close returns. This is not surprising. All the threeK(i, j )K

ingredients of the ρ̂s 2×2 are measured with noise and so when we form ρ̂s

it will be downward bias.

5.6 Analysis of the beta estimates(i, j )

Here we will focus on estimating βs(i, j )K= [Y (i) , Y ( j )]s /[Y ( j )]s , by the realised kernel beta βs=(i, j )( j, j )K s /K s . Figure 2 presents scatter plots of beta estimates based on transaction prices (vertical axis)
Kagainst the same estimate based on mid-quotes (horizontal axis). The two estimators are βs 2×2 to βs5m .
The results are not very different in these two cases.
Figure 3 compares the fitted values from ARMA models for the kernel and 5 minute estimates ofrealised betas for the AA-SPY assets combination. These are based on the model estimates for the daily
kernel based realised betasK
βsK = 1.20 + 0.923βs−1+ u s − 0.726u s−1 ,(0.06)(0.027)(0.048)adj−R 2 = 0.213,

and for 5 minute based realised betas
5 min
βs5 min = 1.16 + 0.950βs−1+ u s − 0.821u s−1 ,(0.06)(0.024)(0.039)adj−R 2 = 0.145.

Both models have a significant memory, with autoregressive roots well above 0.9 and with large
23

moving average roots. The fit of the realised kernel beta is a little bit better than that for the realised

Realised Beta Transactions

3.5 Slope = 1.088 (0.013)
3.5 Slope = 0.961 (0.009)
3.0 const = -0.082 (0.016)
3.0 const = 0.042 (0.012)
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
Assets: AA-SPY
Assets: AA-SPY
-1.0
-1.0
-1.5
-1.5
-1.5
-0.5
0.5
1.5
2.5
3.5
-1.5
-0.5
0.5
1.5
2.5
3.5

RKern. Beta Transactions

RKern. Beta mid quotes

3.5 Slope = 1.059 (0.012)
3.0 const = -0.038 (0.011)
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-1.5
-0.5
0.5
1.5
RKern. Beta mid quotes

refresh time

Assets: C-SPY

2.5

Realised Beta mid quotes

3.5 Slope = 0.986 (0.012)
3.0 const = 0.008 (0.012)
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
3.5
-1.5
-0.5
0.5
1.5
Realised Beta Transactions

RKern. Beta Transactions

beta.

refresh time

Realised Beta mid quotes

5min Cal. time

Assets: C-SPY

2.5

3.5

5min Cal. time

Figure 2: Scatter plots for daily realised kernel betas for the AA and SPY asset combination.

fitted BetaK

BetaK

Assets: AA−SPY

2

0
2005
fitted Beta5m

2006

2007

2006

2007

2006

2007

Beta5m

2

0
2005
2.0

fitted BetaK

fitted Beta5m

1.5
1.0
2005

Figure 3: ARMA(1,1) model for transaction based realised kernel betas for the AA and SPY combination.

24

We also calculate the encompassing regressions. The estimates for the realised kernel betas areK
5 min
βsK = 0.084 + 0.858βs−1+ 0.074βs−1+ u s − 0.726u s−1 ,(0.031)(0.053)(0.043)(0.044)adj−R 2 = 0.215,

with the corresponding 5 minute based realised betas
5 minK
βs5 min = 0.056 + 0.879βs−1+ 0.069βs−1+ u s − 0.822u s−1 ,(0.026)(0.047)(0.040)(0.035)adj−R 2 = 0.150.

This shows that either estimator dominates the other in terms of encompassing, although the realised
kernel has a slightly stronger t-statistic.

5.7 A scalar BEKK
5.7.1

Econometric framework

An important use of realised quantities is to forecast future volatilities and correlations of daily returns.
The use of reduced form has been pioneered by Andersen et al. (2001) and Andersen et al. (2003). One
useful way of thinking about the forecasting problem is to fit a GARCH type problem with lagged realised
quantities as explanatory variables, e.g. Engle & Gallo (2006). Here we follow this route, fitting abivariate GARCH model with E(rs |Fs−1 ) = 0, Cov(rs |Fs−1 ) = Hs , where rs is the 2 × 1 vector of dailyclose to close returns on the i-th and j -th asset, Fs−1 is the information available at time s − 1 to predictPT
rs . A standard Gaussian quasi-likelihood − 12 s=1
log |Hs | + rs′ Hs−1rs is used to make inference. The
model we fit is a variant on the scalar BEKK (e.g. Engle & Kroner (1995))
K′
2×2
Hs = C ′ C + β Hs−1 + αrs−1rs−1+ γ Ks−1,α, β, γ ≥ 0.

The question will be if γ is estimated to be statistically different from zero, for if it is not then the highfrequency data enhances the forecast of future covariation.
5.7.2

Empirical results

Our results will be based on a relatively short time series of 2.5 years of daily measures, which is achallenging environment for GARCH type models.
The results in Table 7 suggest that lagged daily returns are no longer significant for this multivariate
GARCH models once we have the realised kernel covariance. This is even though the realised kernel
covariance misses out the overnight effect — the information in the close-to-open returns. An interestingK

2×2
feature of the series is that in most cases including Ks−1
reduces the size of the estimated Hs−1 term.

25

Table 7: Scalar BEKK models for close-to-close bivariate returns
SeriesHs−1

rs−1 (rs−1 )′

2×2
Kt −1

K

RVs−12×2

5m

log L

AA-BA

0.62540.00000.2409–-1057.3

0.64040.0000–0.2222-1059.0

0.84860.0250––-1070.4

0.6254–0.24090.0000-1057.3

0.62590.00000.1774–-547.67

0.64300.0000–0.1606-549.110.87460.0239––-555.700.6259–0.17740.0000-547.67

0.91520.00000.0897–-1318.2

0.91540.0000–0.0898-1321.9

0.69110.1519––-1340.1

0.9152–0.08970.0000-1318.20.65070.00000.2975–-435.82

0.63920.0000–0.3162-433.440.85540.0345––-456.06

0.6392–0.00000.3161-433.440.28310.00630.4555–-91.0930.28470.0115–0.4426-91.502

0.84680.0368––-104.04

0.2872–0.31220.1624-90.937–(0.1530)–(0.1387)(0.1043)(0.0141)(0.1528)

AA-SPY(0.0099)(0.1583)(0.0722)(0.0405)(0.0501)(0.0719)–(0.1155)(0.0573)(0.0130)(0.0926)(0.2114)–(0.0249)(0.2197)(0.0145)(0.2012)–(0.0207)–(0.0861)(0.2327)(0.1273)(0.0250)(0.0690)(0.0665)(0.0938)–(0.0968)–(0.0206)–(0.0158)

GE-SPY(0.1873)–(0.0158)(0.0843)(0.0720)–(0.0459)

BA-SPY(0.1703)–(0.1556)(0.1389)

AIG-CAT(0.0912)(0.2689)(0.1231)(0.2662)

Likelihood ratio summary

K 2×2
Ks−1
5m
RVs−12×2

Mean

Std

5%

25%

Median

75%

95%

24.94

16.41

2.068

11.98

23.24

34.49

53.39

22.23

15.88

3.254

10.80

17.13

29.00

57.17

Estimation results for scalar BEKK models for close-to-close bivariate returns.

26

6 Additional remarks
6.1 Relating K (X ) to the flat-top realised kernel K F (X )6.1.1

Positivity

In the univariate case the realised kernel
K (X ) =

n
X

h=−n

k( Hh+1 )Ŵh ,H = c0 n

3/5

Ŵh =

,

n
X

j =|h|+1

x j x j −h

is at first sight very similar to the unbiased flat-top realised kernel of Barndorff-Nielsen et al. (2008a)
F

K (X ) = Ŵ0 +

n
X
h=1

k( Hh−1)+1FŴhF + Ŵ−h,

H = d0 n

1/2

,

ŴhF =

n
X
j =1

x j x j −h .

Here the Ŵh and ŴhF are not divided by the sample size. This means that the end conditions, the observations at the start and end of the sample, can have influential effects on Ŵh . With ŴhF we removed
this effect by starting the sum not at h + 1 but at 1. However, an implication of this is that the resulting
estimator is not guaranteed to be positive semi-definite whatever the choice of the weight function.
The alternative K F (X ) has the advantage that it converges at a n 1/4 rate and is close to the parametricefficiency bound. It has the disadvantage that it can go negative, while we see in the next subsection that
it is more sensitive to serial dependence in the noise and endogenous noise than K (X ). The requirementthat K (X ) be positive results in the bias-variance trade-off and reduces the best rate of convergence fromn 1/4 to n 1/5. The resembles the effects seen in the literature on density estimation with kernel functions.
R
The property, u 2 k(u)du = 0, reduces the order of the asymptotic bias, but kernel functions that satisfy
R 2
u k(u)du = 0 can result in negative density estimate, see Silverman (1986, sections 3.3 and 3.6).

There are three reasons that K F (X ) can go negative.9 The most obvious is the use of a kernel function
R∞
that does not satisfy, −∞ k(x) exp(i xλ)dx ≥ 0 for all λ ∈ R, such as the Tukey-Hanning kernel or the

cubic kernel, k(x) = 1 − 3x 2 + 2x 3 . The flat-top kernels give unit weight to γ1 and γ−1 , which can mean
K F (X ) may be negative. This can be verified by rewriting the estimator as a quadratic form estimator,

x ′ M x, where M is a symmetric band matrix M = band(1, 1, k( H1 ), k( H2 ), . . . , ). The determinant of the2upper-left matrix is given by − k( H1 ) − 1 , so that k( H1 ) = 1 is needed to avoid negative eigenvalues.

Repeating this argument leads to k( Hh ) = 1 for all h, which violates the condition that k( Hh ) → 0, as
h → ∞. Finally, the third reason that the flat-top kernel could produce a negative estimate was due toPthe construction of realized autocovariances, γh = nj =1 x j x j −h . This requires the use of “out-of-period”

intraday returns, such as x1−H . This formulation was chosen because it makes E{K (U )} = 0 whenU ∈ WN . However, since x−H only appears once in this estimator, with the term x1 x1−H , it is evident

9 The flat-top kernel is only rarely negative with modern data. However, if [Y ] is very small and the ω2 very large, which wesaw on slow days on the NYSE when the tick size was $1/8, then it can happen quite often when the flat-top realised kernel isused. We are grateful to Kevin Sheppard for pointing out these negative days.

27

that a sufficiently large value of x1−H (positive or negative, depending on the sign of x1 ) will cause theestimator to be negative. We have overcome the last obstacle by jittering the end-points, which makes theuse of “out-of-period” redundant. They can be dropped at the expense of a O(m −1 ) bias.
Table 8: Relative efficiency of the realised kernel K (X )
ω2 = 0.001

normalised bias2n
K BF (X)U ∈ WN
250
0.0
1,000
0.0
4,000
0.0
16,0000.0
U j = ǫ j + 0.5ǫ j −1
250
1.5
1,000
22.1
4,000
175.7
16,000898.5

normalised variancenormalised mse

K PF (X)K (X)K BF (X)K PF (X)K (X)K BF (X)K PF (X)

K (X)

0.0
0.0
0.0
0.0

0.8
2.5
3.1
4.6

16.2
11.7
10.4
10.5

16.3
12.1
10.4
9.5

18.0
16.9
19.0
20.8

16.2
11.7
10.4
10.5

16.3
12.1
10.4
9.5

18.8
19.4
22.1
25.4

1.2
7.3
18.5
41.0

0.6
2.2
3.2
4.4

15.3
11.0
9.3
9.0

15.7
11.9
10.2
9.4

17.6
16.9
19.0
20.9

16.9
33.0
185.0
907.6

16.9
19.2
28.8
50.4

18.2
19.1
22.2
25.4

96.9
588.0
1,490.4
3,326.8

3.9
6.1
5.0
5.5

27.5
44.8
73.1
88.6

24.2
20.4
13.9
10.9

18.3
16.9
19.3
20.8

150.2
1,813.9
14,268.2
72,886.2

121.1
608.3
1,504.4
3,337.7

22.2
23.0
24.3
26.3

U j = −0.5U j −1 + ǫ j
250
39.1
30.9
1,000
1,261.0
74.9
4,000
7,751.7
141.1
16,000 40,973.1
253.8

1.3
3.3
3.5
4.8

18.9
35.9
40.8
52.0

18.1
13.2
10.8
9.7

17.9
16.8
18.8
20.9

58.0
1,296.9
7,792.5
41,025.2

49.0
88.1
151.9
263.5

19.2
20.0
22.4
25.7

U j = 0.5U j −1 + ǫ j
250
0.5
1,000
9.6
4,000
96.0
16,000
505.8

0.3
1.5
2.7
4.2

14.8
9.8
8.5
8.5

15.3
10.8
9.7
9.2

17.7
16.6
19.1
21.1

15.3
19.4
104.4
514.3

15.7
17.1
49.2
150.7

18.0
18.2
21.8
25.3

U j = ǫ j − 0.5ǫ j −1
250
122.7
1,000
1,769.1
4,000 14,195.1
16,000 72,797.6

0.4
6.3
39.6
141.5

Relative efficiency of the realised kernel K (X ) and the flat-top realised kernel, K F (X ). Results for fivedifferent types of noise are presented. In the MA(1) and AR(1) designs, the variance of ǫ was scaled suchthat V ar(U ) = ω2 . The squared bias, variance, and MSE have been scaled by n 1/2 /ω. In the special casewith Gaussian white noise the asymptotic lower bound for the normalized MSE is 8.00 (the normalized
MSE for K PF (X ) converges to 8.54 as n → ∞ in this special case).
6.1.2

Efficiency

An important question is how inefficient is K (X ) in practice compared to the flat-top realised kernel,
2
K F (X )? The answer is quite a bit when U ∈ WN . Table 8 gives E n 1/4 {K (X ) − [Y ]} /ω and
28



2
E n 1/4 K F (X ) − [Y ] /ω, the mean square normalised by the rate of convergence of K PF (X ) (whichis the flat-top realised kernel using the Parzen weight function. An implication is that the scaled MSE for
the K (X ) and K BF will increase without bound as n → ∞ because these estimators converge at a rate thatis slower than n 1/4). The results are given in the case of Brownian motion observed with different typesof noise. Results for two flat-tops are given, the Bartlett (K BF (X )) and Parzen (K PF (X )) weight functions.
Similar types of results hold for other weight functions.
Consider first the case with Gaussian U ∈ WN with variance of ω2 . The results show that thevariance of K (X ) is much bigger than its squared bias. For small n there is not much difference between
the three estimators, but by the time n = 4, 096 (which is realistic for our applications) the flat-top

K F (X ) has roughly half the MSE of K (X ) in the univariate case. Hence in ideal circumstances K F (X )has advantages over K (X ), but we are attracted to the positivity and robustness of K (X ).
The robustness advantage of K (X ) can be seen using four simulation designs where U j is modelled asa dependent process. We consider the moving average specification, U j = ǫ j − θǫ j −1, with θ = ±0.5 andthe autoregressive specification, U j = ϕU j −1 +ǫ j , with ϕ = ±0.5, where ǫ j is Gaussian white noise. The
bandwidth for all estimators were to be “optimal” under U ∈ WN , which is the default in the literature, soP
H BF = 2.28ω4/3 n 2/3 , H PF = 4.77ωn 1/2 , and H P = 3.51ω4/5 n 3/5 where ω2 = ∞h=−∞ cov(U j , U j −h ). The
results show the robustness of K (X ) and the strong asymptotic bias of K PF and K BF under the non-white
noise assumption. The specifications, θ = 0.5 and ϕ = −0.5 induce a negative first-order autocorrelation
while θ = −0.5 and ϕ = 0.5 induce positive autocorrelation. Negative first-order autocorrelation can be
the product of bid-ask bounce effects, this is particularly the case if sampling only occurs when the pricechanges. Positive first-order autocorrelation would, for example, be relevant for the noise in bid prices
because variation in the bid-ask spread would induce such dependence.

6.2 Preaveraging without bias correction
6.2.1

Estimating multivariate QV

In independent and concurrent work Vetter (2008, p. 29 and Section 3.2.4) has studied a univariate
suboptimal preaveraging estimator of [Y ] whose bias is sufficiently small that the estimator does notneed to be explicitly bias corrected to be consistent (the bias corrected version can be negative). Its rateof convergence does not achieve the optimal n −1/4 rate. Hence his suboptimal preaveraging estimator
has some similarities to our non-negative realised kernel. Implicit in his work is that his non-correctedpreaveraging estimator is non-negative. However, this is not remarked upon explicitly nor developed into
the multivariate case where non-synchronously spaced data is crucial.
Here we outline what a simple multivariate uncorrected preaveraging estimator based on refresh timePn−H′
−1/2 P H
hwould look like. We define it as V̂ =j =1 x j x j , where x j = (ψ2 H )h=1 g H x j +h , ψ2 =R1 2
0 g (u)du. Here g(u), u ∈ [0, 1] is a non-negative, continuously differentiable weight function, with29

the properties that g(0) = g(1) = 0 and ψ2 > 0. Now if we set H = θn 3/5 , then the univariate result in
Vetter (2008) would suggest that V̂ converges at rate n −1/5, like the univariate version of our multivariate

realised kernel. There is no simple guidance, even in the univariate case, as to how to choose θ.
In the univariate bias corrected form, Jacod et al. (2007) show that V̂ is asymptotically equivalent to
R1using a K F (X ) with k(x) = ψ2−1 x g(u)g(u − x)du and H ∝ n 1/2 . It is clear the same result will holdfor the relationship between V̂ and K (X ) in the multivariate case when H = θn 3/5 . A natural choice of
R1
g is g(x) = (1 − x) ∧ x, which delivers 0 g 2 (u)du = 1/12 and a k function which is the Parzen weightfunction. Hence one might investigate using θ = c0 as in our paper, to drive the choice of H for V̂ whenapplied to refresh time based high frequency returns.
Kinnebrock & Podolskij (2008a) have defined a bias corrected preaveraging estimator of the multivariate [Y ] with H = θn 1/2 , for which they derive limit theory. To define their high frequency returns they
use the Refresh Time idea — taken from an early draft of this paper. Their estimator has the disadvantagethat it it is not guaranteed to be positive semi-definite.
6.2.2

Estimating integrated quarticity

In order to construct feasible confidence intervals for our realised quantities (see Barndorff-Nielsen &Shephard (2002)) we have to estimate 9. Our approach is based on the no-noise Barndorff-Nielsen &Shephard (2004) bipower type estimator applied to suboptimal preaveraged data taking H = θn 3/5 . This

is not an optimal estimator, it will converge at rate n 1/5, but it will be positive semidefinite. The proposed
n

o
P
−1
(positive semi-definite) estimator of vec (9) is Q̂ = n n−Hc j c′j − 12 c j c′j +H + c j +H c j , wherej =1
c j = vec(x̄ j x̄ ′j ). That the elements of Q̂ is consistent using this choice of bandwidth is implicit in thethesis of Vetter (2008, p. 29 and Section 3.2.4).

6.3 The case with [0, T ]Throughout the paper we have discussed estimating QV over a unit interval, now we extend this to the interval [0, T ]. Technically this is trivial, it is just a time-change argument. The results
q R are that the QV target
RTRT
T
is 0 6(u)du, while 9 = T 0 {6(u) ⊗ 6(u)} ◦ T (u)du. Finally, ξii = ii / T 0 6ii2 (u) ◦ T (u)du.

6.4 Finite sample improvements

The realised kernel is non-negative so we can use log-transform to improve its finite sample performance.
In particular

!2 

Z 1



κ
κ
Ls
n 1/5 log (K (X )) − log,4 R1.
σ 2 (u)du→ MN R 12
 σ 2 (u)du000 σ (u)duWhen the data is regularly spaced and the volatility is constant then κσ −2 = (ω/σ )2/5 k ′′ (0)which is slightly less dependent on σ 2 than the non-transformed version.
30

1/5

k•0,0

2/5

,

6.5 Subtlety of end effects
We have introduced jittering to eliminate end-effects. The larger is m the smaller is the end-effects,
however increasing m has the drawback that is reduces the sample size, n, that can be used to computethe realised autocovariances. Given N observations, the sample size available after jittering is n = N −2(m − 1), so extensive jittering will increase the variance of the estimator. In this subsection we study
this trade-off.
We focus on the univariate case where U ∈ WN . The mean square error caused by end-effects is

simply the squared bias plus the variance of U0U0′ + Un Un′ , which is given by 4m −2 ω4 + 4m −2 ω4 =8ω4 m −2 , as shown in Appendix A, see the proof of Lemma A.4. The asymptotic variance (abstracting
2/5  0,0
4/5 −2/5
from end-effects) is 5κ 2 n −2/5 = 5 k ′′ (0)ω2k• IQn. So the trade-off between contributions
from end-effects and asymptotic variance is given by
g N,ω2 ,IQ (m) = m −2 8ω4 + 5 k ′′ (0)ω2

2/5  0,0
4/5k• IQ
(N − m)−2/5 .

This function is plotted in Figure 4 for the case where N = 1, 000 and IQ = 1 and ω2 = 0.0025 and0.001. The optimal value of m ranges from 1 to 2. The effect of increasing n on optimal m can be seenfrom Figure 4, where the optimal value of m has increased a little from Figure 4 as n has increased to
5, 000. However, the optimal amount of jittering is still rather modest.

6.6 Finite lag refresh time
In this paper we roughly synchronise our return data using the concept of Refresh Time. Refresh Timeguarantees that our returns are not stale by more than one lag in Refresh Time.

Our proofs need asomewhat less tight condition, that returns are not stale by more than a finite number of lags. Thissuggests it may be possible to find a different way of synchronising data which throws information away
less readily than Refresh Time. We leave this problem to further research.

6.7 Jumps
In this paper we have assumed that Y is a pure BSM. The analysis could be extended to the situation
where Y is a pure BSM plus a finite activity jump process. The analysis in Barndorff-Nielsen et al.
(2008a, section 5.6) suggests that the realised kernel is consistent for the quadratic variation, [Y ], at thesame rate of convergence as before, but with a different asymptotic distribution.

7 Conclusions
In this paper we have proposed the multivariate realised kernel, which is a non-normalised HAC typeestimator applied to high frequency financial returns, as an estimator of the ex-post variation of asset

31

RMSE with Jittering (N=5,000)

RMSE with Jittering (N=1,000)

0.16530
0.16525
0.16520
0.16515
0.165100.16505
0.16500
0.16495
0.164900.164852

ω = 0.0025

1

2

3

4

5

6

7

8

0.11960
0.11958
0.11956ω2 = 0.0025

0.119540.11952
0.11950
0.119480.11946

9 10

1

2

3

0.13765
0.13760
0.13755
0.13750
0.137450.137400.13735
0.137300.137250.13720

ω2 = 0.001

1

2

3

4

5

6

7

5

6

7

8

9 10

8

9 10

Level of jittering (m)RMSE with Jittering (N=5,000)

RMSE with Jittering (N=1,000)

Level of jittering (m)

4

8

9 10

Level of jittering (m)0.099480.09947
0.09947
0.09947
0.09946
0.09946
0.099450.099450.099440.099440.09943ω2 = 0.001

1

2

3

4

5

6

7

Level of jittering (m)Figure 4: Sensitivity to the the choice of m. The Figure shows the RMSE as a function of m for the sample
sizes N = 1, 000 and N = 5, 000, and ω2 = 0.001 and ω2 = 0.0025.
prices in the presence of noise and non-synchronous trading. The choice of kernel weight function isimportant here — for example the Bartlett weight function yields inconsistent estimation in our case.
Our analysis is based on three innovations: (i) we used a weight function which delivers biased
kernels, however allowing us to use positive semi-definite estimators, (ii) we coordinate the collectionof data through the idea of refresh time, (iii) we show the estimator is robust to the remaining stalenessin the data. Using this setup we are able to show consistency and asymptotic mixed Gaussianity of ourestimator.
Our simulation study indicates our estimation procedure is close to being unbiased for covariances
under realistic situations.

Not surprisingly the estimators of correlations is downward biased due tothe sampling variance of our estimators of variance. The empirical results based on our new estimator
are striking, providing much sharper estimates of dependence amongst assets than has previously been
available.
Multivariate realised kernels have potentially many areas of application, improving our ability toestimate covariances — allowing high frequency data to significantly improve our predictive models as
32

well as better understand the pricing and management of risk in financial markets.

References
Andersen, T. G., Bollerslev, T. & Diebold, F. X. (2008), Parametric and nonparametric measurement of volatility,
in Y. Aı̈t-Sahalia & L. P. Hansen, eds, ‘Handbook of Financial Econometrics’, North Holland, Amsterdam.
Forthcoming.
Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2000), ‘Great realizations’, Risk 13, 105–108.
Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2001), ‘The distribution of exchange rate volatility’,
Journal of the American Statistical Association 96, 42–55. Correction published in 2003, volume 98, page
501.
Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2003), ‘Modeling and forecasting realized volatility’,
Econometrica 71, 579–625.
Andersen, T. G., Bollerslev, T. & Meddahi, N. (2006), Market microstructure noise and realized volatility forecasting. Unpublished paper: Department of Economics, Duke University.
Andrews, D. W. K. (1991), ‘Heteroskedasticity and autocorrelation consistent covariance matrix estimation’,
Econometrica 59, 817–858.
Bandi, F. M. & Russell, J. R. (2005), Realized covariation, realized beta and microstructure noise. Unpublishedpaper, Graduate School of Business, University of Chicago.
Bandi, F. M. & Russell, J. R. (2006a), Market microstructure noise, integrated variance estimators, and the limitations of asymptotic approximations: A solution. Unpublished paper, Graduate School of Business, Universityof Chicago.
Bandi, F. M. & Russell, J. R. (2006b), ‘Seperating microstructure noise from volatility’, Journal of Financial
Economics 79, 655–692.
Bandi, F. M. & Russell, J. R. (2008), ‘Microstructure noise, realized variance, and optimal sampling’, Review of
Economic Studies 75, 339–369.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. & Shephard, N. (2008a), ‘Designing realised kernels to measure
the ex-post variation of equity prices in the presence of noise’, Econometrica . Forthcoming.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. & Shephard, N. (2008b), ‘Realised kernels in practice’, forthcoming in Econometrics Journal .
Barndorff-Nielsen, O. E. & Shephard, N. (2002), ‘Econometric analysis of realised volatility and its use in estimating stochastic volatility models’, Journal of the Royal Statistical Society, Series B 64, 253–280.
Barndorff-Nielsen, O. E. & Shephard, N. (2004), ‘Econometric analysis of realised covariation: high frequency
covariance, regression and correlation in financial economics’, Econometrica 72, 885–925.
Barndorff-Nielsen, O. E. & Shephard, N. (2007), Variation, jumps and high frequency data in financial econometrics, in R. Blundell, T. Persson & W. K. Newey, eds, ‘Advances in Economics and Econometrics. Theory
and Applications, Ninth World Congress’, Econometric Society Monographs, Cambridge University Press,
pp. 328–372.
Bollerslev, T., Tauchen, G. & Zhou, H. (2008), Expected stock returns and variance risk premia. Unpublished paper:
Department of Economics, Duke University.

33

Brownless, C. T. & Gallo, G. M. (2006), ‘Financial econometric analysis at ultra-high frequency: Data handlingconcerns’, Computational Statistics & Data Analysis 51, 2232–2245.
Campbell, J. Y., Lo, A. W. & MacKinlay, A. C. (1997), The Econometrics of Financial Markets, Princeton University Press.
de Pooter, M., Martens, M. & van Dijk, D. (2008), ‘Predicting the daily covariance matrix for s&p 100 stocks using
intraday data — but which frequency to use?’, Econometric Reviews 27, 199–229.
Dovonon, P., Goncalves, S. & Meddahi, N. (2007), Bootstrapping realized multivariate volatility measures. Unpublished paper: Tanaka Business School, Imperial College, London.
Drechsler, I. & Yaron, A. (2008), What’s vol got to do with it. Unpublished paper: The Wharton School, Universityof Pennsylvania.
Embrechts, P., Kluppelberg, C. & Mikosch, T. (1997), Modelling Extremal Events for Insurance and Finance,
Springer, Berlin.
Engle, R. F. & Gallo, J. P. (2006), ‘A multiple indicator model for volatility using intra daily data’, Journal of
Econometrics 131, 3–27.
Engle, R. F. & Kroner, K. F. (1995), ‘Multivariate simultaneous generalized ARCH’, Econometric Theory 11, 122–150.
Epps, T. W. (1979), ‘Comovements in stock prices in the very short run’, Journal of the American Statistical
Association 74, 291–296.
Falkenberry, T. N. (2001), High frequency data filtering, Technical report, Tick Data.
Fisher, L. (1966), ‘Some new stock-market indexes’, Journal of Business 39, 191–225.
Fleming, J., Kirby, C. & Ostdiek, B. (2003), ‘The economic values of volatility timing using ‘realized’ volatility’,
Journal of Financial Economics 67, 473–509.
Gallant, A. R. (1987), Nonlinear Statistical Models, John Wiley, New York.
Ghysels, E., Harvey, A. C. & Renault, E. (1996), Stochastic volatility, in C. R. Rao & G. S. Maddala, eds, ‘Statistical
Methods in Finance’, North-Holland, Amsterdam, pp. 119–191.
Glasserman, P. (2004), Monte Carlo Methods in Financial Engineering, Springer-Verlag New York, Inc.
Goncalves, S. & Meddahi, N. (2008), ‘Bootstrapping realized volatility’, Econometrica . Forthcoming.
Griffin, J. E. & Oomen, R. C. A. (2006), Covariance measurement in the presence of non-synchronous trading andmarket microstructure noise. Unpublished paper: Department of Statistics, University of Warwick.
Guillaume, D. M., Dacorogna, M. M., Dave, R. R., Müller, U. A., Olsen, R. B. & Pictet, O. V. (1997), ‘From thebird’s eye view to the microscope: a survey of new stylized facts of the intra-daily foreign exchange markets’,
Finance and Stochastics 2, 95–130.
Hansen, P. R. & Lunde, A. (2005), ‘A realized variance for the whole day based on intermittent high-frequency
data’, Journal of Financial Econometrics 3, 525–544.
Hansen, P. R. & Lunde, A. (2006), ‘Realized variance and market microstructure noise (with discussion)’, Journalof Business and Economic Statistics 24, 127–218.
Harris, F., McInish, T., Shoesmith, G. & Wood, R. (1995), ‘Cointegration, error correction and price discovery on
informationally-linked security markets’, Journal of Financial and Quantitative Analysis 30, 563–581.

34

Hayashi, T. & Yoshida, N. (2005), ‘On covariance estimation of non-synchronously observed diffusion processes’,
Bernoulli 11, 359–379.
Jacod, J. (2007), Statistics and high frequency data. Unpublished paper.
Jacod, J. (2008), A new type of limit theorems for discretized processes. Unpublished paper: Laboratoire de
Probabilites, Universite Paris VI.
Jacod, J., Li, Y., Mykland, P. A., Podolskij, M. & Vetter, M. (2007), Microstructure noise in the continuous case:the pre-averaging approach. Unpublished paper: Department of Statistics, University of Chicago.
Jacod, J. & Protter, P. (1998), ‘Asymptotic error distributions for the Euler method for stochastic differential equations’, Annals of Probability 26, 267–307.
Jacod, J. & Shiryaev, A. N. (2003), Limit Theorems for Stochastic Processes, 2 edn, Berlin, Springer.
Kalnina, I. & Linton, O. (2008), ‘Estimating quadratic variation consistently in the presence of correlated measurement error’, Journal of Econometrics . Forthcoming.
Kinnebrock, S. & Podolskij, M. (2008a), An econometric analysis of modulated realised covariance, regression andcorrelation in noisy diffusion models. Unpublished paper: Oxford Financial Research Centre, University of
Oxford.
Kinnebrock, S. & Podolskij, M. (2008b), ‘A note on the central limit theorem for bipower variation of generalfunctions’, Stochastic Processes and Their Applications 118, 1056–1070.
Large, J. (2007), Accounting for the Epps effect: Realized covariation, cointegration and common factors. Unpublished paper: Oxford-Man Institute, University of Oxford.
Malliavin, P. & Mancino, M. E. (2002), ‘Fourier series method for measurement of multivariate volatilities’, Financeand Stochastics 6, 49–61.
Martens, M. (2003), Estimating unbiased and precise realized covariances. Unpublished paper: Department of
Finance, Erasmus School of Economics, Rotterdam.
Mykland, P. A. & Zhang, L. (2006), ‘ANOVA for diffusions and Ito processes’, Annals of Statistics 34, 1931–1963.
Mykland, P. A. & Zhang, L. (2008), Inference for continuous semimartingales observed at high frequency: A
general approach. Unpublished paper: Department of Statistics, University of Chicago.
Newey, W. K. & West, K. D. (1987), ‘A simple positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix’, Econometrica 55, 703–708.
Phillips, P. C. B. & Yu, J. (2008), Information loss in volatility measurement with flat price trading. Unpublishedpaper: Cowles Foundation for Research in Economics, Yale University.
Protter, P. (2004), Stochastic Integration and Differential Equations, Springer-Verlag, New York.
Renault, E. & Werker, B. (2008), Causality effects in return volatility measures with random times. Unpublishedpaper: Department of Economics, University of North Carolina.
Reno, R. (2003), ‘A closer look at the Epps effect’, International Journal of Theoretical and Applied Finance
6, 87–102.
Silverman, B. W. (1986), Density Estimation for Statistical and Data Analysis, Chapman & Hall, London.
Vetter, M. (2008), Estimation methods in noisy diffusion models. Unpublished Ph.D. thesis, Institute of Mathematics, Ruhr University Bochum.

35

Voev, V. & Lunde, A. (2007), ‘Integrated covariance estimation using high-frequency data in the presence of noise’,
Journal of Financial Econometrics 5, 68–104.
Zhang, L. (2005), Estimating covariation: Epps effect and microstructure noise. Unpublished paper, Department of
Finance, University of Illinois, Chicago.
Zhang, L. (2006), ‘Efficient estimation of stochastic volatility using noisy observations: a multi-scale approach’,Bernoulli 12, 1019–1043.
Zhang, L., Mykland, P. A. & Aı̈t-Sahalia, Y. (2005), ‘A tale of two time scales: determining integrated volatility
with noisy high-frequency data’, Journal of the American Statistical Association 100, 1394–1411.
Zhou, B. (1996), ‘High-frequency data and volatility in foreign-exchange rates’, Journal of Business and Economic
Statistics 14, 45–52.
Zhou, B. (1998), Parametric and nonparametric volatility measurement, in C. L. Dunis & B. Zhou, eds, ‘Nonlinear
Modelling of High Frequency Financial Time Series’, John Wiley Sons Ltd, New York, chapter 6, pp. 109–
123.

Appendices
Under the assumptions given in this paper, our line of argument will be as follows.
• Assume the data is synchronized and then a time-change allows us to think of the data as being
regularly spaced. This is clear from the arguments in Barndorff-Nielsen et al. (2008a).
• Show the staleness left by the definition of refresh time has no impact on the asymptotic distributionof the equally spaced realised kernel. This is shown in Appendix B.
• Show the realised kernel is consistent and work out its limit theory for synchronized and equally
spaced data. This is shown in Appendix A.

Appendix A: Proofs
Proof of Theorem 1. We note that for all i, j , (i)  Y
K (Y (i) )K (Y (i) , U ( j ))K=
≥ 0,
U ( j)K (Y (i) , U ( j ))K (U ( j ))which means that by taking the determinant of this matrix and rearranging we see that K (Y (i) , U ( j ))2 ≤p
K (Y (i) )K (U ( j ) ). Consequently, provided K (Y ) → [Y ] ,
q

q

(i)
(
j
)
K (X ) − [Y ] = K (Y ) − [Y ] + O p
maxi K Y
max j K U
+ K (U )p= K (Y ) − [Y ] + O p
K (U ) .
From this, together with the results of Lemmas A.1 and A.5, the conclusions of the Theorem follow
directly. 
Proof of Proposition 1. This is a special case of Theorem 2. 

36

Proof of Proposition 2. The asymptotic variance of K (Y ) is given in Barndorff-Nielsen et al. (2008a)
and the results concerning K (U ) follow from Lemma A.5, given below. Proof of Proposition 3. The problem is simply to minimize the squared bias plus the contribution
R1
from the asymptotic variance with respect to c0 . Set IQ = 0 σ 4 (u)du. The first order conditions of
minc0 −c0−4 k ′′ (0)2 ω4 + c0 4k•0,0 IQ yield the optimal value for c0c0∗ = ′′ 2 4 1/5
k (0) ωk•0,0 IQ
1/5
with c∗ = k ′′ (0)2 /k•0,0.

= c∗ ξ 4/5 ,With H ∗ = c∗ ξ 4/5 n 3/5 the asymptotic bias is given by− ′′ 2 4 −2/5
k (0) ωk•0,0 IQ

k ′′ (0)ω2 n −1/5 = k ′′ (0)ω2

and the asymptotic variance is ′′ 2 4 1/5
k (0) ωk•0,0 IQ

4k•0,0 IQn −2/5 = 4 k ′′ (0)ω2



1/5  0,0
2/5 −1/5n,k• IQ

2/5  0,0
4/5 −2/5k• IQn.

A.1 Proof of Theorem 2.
We decompose the realised kernel into four terms that we will analyze separately.


Z 1
Z 1
K (X ) −6(u)du = K (Y ) −6(u)du + E {K (U )} + {K (U ) − E [K (U )]}00+ {K (Y, U ) + K (U, Y )} .

R1
We start by deriving the asymptotic properties of K (Y ) − 0 6(u)du. The flat-top does not play a rolein the asymptotic analysis of K (Y ), so the result for the univariate case follows from Barndorff-Nielsen
et al. (2008a). The multivariate result is the following.
Lemma A.1 K (Y ) = [Y ] + O p ( Hn ), and with H = c0 n 3/5 , then
n

1/5



K (Y ) −

Z 1
0




Ls
6(u)du → MN 0, 4c0 k•0,0 9 .

noR1Proof of Lemma A.1. First we note that, for any fixed a ∈ Rd the convergence of a ′ K (Y ) − 0 6(u)du a
follows directly by applying the univariate results in Barndorff-Nielsen et al. (2008a). Stable convergence for multivariate statistics, such as the realised autocovariances, Ŵh , are established in Kinnebrock& Podolskij (2008b), see also Jacod (2007). With H ∝ n γ , the consistency and stable convergence follow from Jacod (2008, theorems 2.1 and 2.2). What remains is to derive the asymptotic variance. Fora, b, c, d ∈ Rd the asymptotic covariance between a ′ K (Y )b and c′ K (Y )d is given as the plim of!  n−1
n−1
X
X
X
X
n
c0
k( |h| )a ′ yi yi−h b k( H|l|+1 )c′ y j y j −l d 
H h=−n+1 H +1 ijl=−n+1
37


X
X
n 

′
′
= c0 
k( Hh+1 )2 a ′ yi yi′ cb′ yi−h yi−hd+k( Hh+1 )2 a ′ yi yi′ db′ yi−h yi−hc + O p ( Hn )
H h=l
h=−l


i= j

p

Z ∞

i= j +h

Z 1

 ′
k(s)2
a 6(u)cb′ 6(u)d + a ′ 6(u)db′ 6(u)c duds
−∞
0


Z 1
′
′
ab′ +ba ′ ′0,0{6(u) ⊗ 6(u)} duvec( cd +dc= vec( 2 ) 4c0 k•
).
2

→ c0

0

Here we used

R∞

2
0,0
−∞ k(s) ds = 2k• and
a ′ 6cb′ 6d+a ′ 6db′ 6c = c′ 6ab′ 6d + d ′ 6ab′ 6c = tr 6ab′ 6dc′ + 6ab′ 6cd ′= vec(ab′ )′ (6 ⊗ 6) vec(dc′ + cd ′ ) = 12 vec(ab′ + ba ′ )′ (6 ⊗ 6) vec(dc′ + cd ′ ).


A.1.1 Results concerning K (U )We derive the asymptotic properties of K (U ) under the assumption that U ∈ CS. The following definitions lead to a useful representation of K (U ). For h = 0, 1, . . . we define
Vh =

n−1
X

j =h+1

U j U ′j −h + U j −h U ′j ,′Z h = U0Uh′ + Uh U0′ + (Un Un−h
+ Un−h Un′ ).

andLemma A.2 The realised autocovariances of U can be written as
Ŵ0 (U ) = V0 − V1 + 12 Z 0 − Z 1

(A.1)

′

Ŵh (U ) + Ŵh (U ) = −Vh−1 + 2Vh − Vh+1 + Z h − Z h+1 ,(A.2)

Proof. The first expression, (A.1), follows fromŴ0 (U ) =
−

n
X
j =1

n−1
X
j =2

′

(U j − U j −1)(U j − U j −1 ) =

n−1
X
j =1

(U j U ′j + U j U ′j ) + Un Un′ + U0U0′′(U j U ′j −1 + U j −1U ′j ) − (Un Un−1
+ Un−1 Un′ + U0U1′ + U1U0′ ),and (A.2) is proven similarly. 
Lemma A.3 The realised kernel for U has the exact representation:
n−1
X

 h+1
K (U ) = k(0) − k( H1+1 ) V0 −k( H +1 ) − 2k( Hh+1 ) + k( Hh−1
) Vh
+1

(A.3)

h=1

+ 12 Z 0 −n−1Xk( Hh+1 ) − k( Hh−1
) Zh .
+1h=1

Proof. Follows from the definition of K (U ) and Lemma A.2. 
Now we prove the result concerning the end-effects. We note that U0 and Un are absent from Vl , for
all l = 0, 1, . . . , so end-effects can only have an impact on K (U ) through Z h , h = 0, 1, . . . .
38

Lemma A.4 E (Z 0 ) = 4m −1

Pm−1

h=−m+1Pm−|h|h and E(Z h ) = 2m −1 m+h−1( j + ′j ).
j =hm

Proof of Lemma A.4. Recall that U0 = m −1
E(U0U0′ ) = m −2

m−1
X m−1
X
j =0 i=0

Pm−1

j =0 U (t j ). So that E(U0 ) = 0 andE{U (t j )U (ti )′ } = m −1

m−1
X

m−|h|
h ,
m

h=−m+1and similar for E(Un Un′ ). So the first result follows from E (Z 0 ) = 2{E(U0U0′ ) + E(Un Un′ )}. Next, for
h > 0,
E(U0Uh′ ) = m −1

m−1
X
j =0

E{U (t j )U (tm−1+h )′ } = m −1

m−1
X
j =0

− j −h = m −1

m−1
X
j =0

′j +h ,P′and similarly we find E(Un Un−h
) = m −1 m−1
j =0  j +h , and the second and last result follows. It is a simple matter to compute the bias of the realised kernels caused by the noise.
Lemma A.5 For large n, H and m we have.
E{K (U )} = −k ′′ (0)nn + O(m −1 ) + o( 2 )2HHand the asymptotic variance of K (U ) is o(n H −3+ǫ ) + O(H −1 m −1 ) for any ǫ > 0.
Remark. So with H ∝ n 3/5 and m → ∞, we note that Var{K (U )} = o(n −3/5 ). This implies thatn 1/5 [K (U ) − E{K (U )}] = o p (1), so K (U ) only contributes to the bias term in the asymptotic distribution, not to the asymptotic variance when H ∝ n 3/5.
Proof. First we prove the result concerning the bias. We have,k(0) − k

1
H +1



′′

k (ε H ),= − 2(H+1)2

for some 0 ≤ ε H ≤ H1+1 ,since k ′ (0) = 0. Now we define a0 = −k ′′ (ε H ) and ah = H 2 {−k( |h|+1) + 2k( H|h|) − k( |h|−1)}, then
H +1
+1
H +1
n−1
X

 h+1
1
k(0) − k( H +1 ) E(V0) −k( H +1 ) − 2k( Hh+1 ) + k( Hh−1
) E(Vh )
+1
h=1

=H

−2

= H −2n−1
X

h=−n+1X√|h|≤ Hah (n − 1 − h)E(U j U ′j −h ) = H −2

ah (n − 1 − h)h + H −2

X√|h|> Hn−1
X

h=−n+1ah (n − 1 − h)hah (n − 1 − h)h .

By the continuity of k ′′ (x) it follows thatsup√|h|≤ HH2
ah (n − 1 − h) + k ′′ (0) → 0,n

as H, n → ∞so that the first term
X
n
n
H −2
ah (n − 1 − h)h = −k ′′ (0) 2  + o( 2 ).
H
H√|h|≤ H

39

with H/n = o(1),For the second term
X

H −2

√
|h|> Hah (n − 1 − h)h ≤

n X
n X
|a
|
|
|

≤
sup
h
h
h
√
H2
H2
√
√
|h|≥ H|h|> HH 2 ah ,|h|> H

which vanishes exponentially fast. For the Z -terms we have by Lemma A.4 that E(Z 0 ) = O(m −1 ), and
n−1
n−1
m−1
X
X
X


−1
h
h−1
h
h−1
E
k( H +1 ) − k( H +1 ) Z h =k( H +1 ) − k( H +1 ) 2m
( j +h + ′j +h )
h=1

j =1

h=1

≤

n−1
X
h=1

∞
X

−1
h
h−1
k( H +1 ) − k( H +1 ) 2m
| j +h + ′j +h |j =1

≃ m −1 2

Z

∞
X
 ′ h
k ( H +1 ) + O(H −1 )j | j + ′j | = O(m −1 ).
j =1Next, we turn to the asymptotic variance of K (U ). Consider for simplicity the univariate case. First
we choose some η > 0. We see from expression (A.3) that the contribution from terms involving Vh ,h = 0, 1, . . . has a total variance
X X
X
var(H −2ah
2Ui Ui−h ) ≤ 4H −4ah al E(Ui Ui−h U j U j −l ) + 4n 4 sup ̺ M ,
h

M≥H η

|i− j |<H η|h−l|<H ηiAssumption 1 guarantees that the last term vanishes at exponential rate. For the first term we have
X
X
4H −4
ah al E(Ui Ui−h U j U j −l ) ≤ 4H −4 2H η nE(Ui4 )ah al|i− j |<H η|h−l|<H η|h−l|<H η

≤ 8H

−3+η

nE(Ui4)2H η

Z

′′



k (s) ds + o(1) = O(n H −3+2η )E(Ui4 ).
2

The other terms involving Z h , h ≥ 0 have the form
#
" n−1
n
X
X
 ′ h
2
−2
h
h−1
var
k( H +1 ) − k( H +1 ) U0Uh ≤ H
k ( H +1 ) + O(H −1 ) E(U02Uh2 ) = O(H −1 m −1 ).
h=1

i=1



A.1.2 Results concerning K (Y, U ) and K (U, Y )Lemma A.6 K (Y, U ) = O p (H −1/2) so that K (Y, U ) = o p (n −1/5 ) when H ∝ n 3/5 .
Proof. For simplicity we prove the result for the case d = 1. We haveXX
 X
K (Y, U ) =k( Hh+1 )y j U j −h − U j −h−1 ≃k( Hh+1 ){Wh − Wh−1 },
h

j

h

Pn

Pn−1

−1 ′ |h|where Wh =j =h+1 y j U j −h , so that K (Y, U ) ≃h=−n+1 H k ( H +1 )Wh . Write EY (·) = E(·| {Y }).
Then


n
n
X
X
X
X
EY 
yi Ui−h y j U j −l  =yi y j EY (Ui−h Ui−l ) +yi y j EY (Ui−h U j −l )i=h+1 j =l+1

i6 = j

i= j

40

=
The first term converges to h−l
n−1
X

H

k ( H|h|
)Wh
+1

−1 ′

h=−n+1≃ H −2
≃H

−2

X

h,l
|h−l|<H γ

0 σ

n−1
X

2

i= j

X

q+l−hq6 =0

Hk ( H|l|+1 )Wl ≃ H −2−1 ′l=−n+1X

yi y j l−h +Xyi yi−q .

i(u)du in probability while the second term vanishes. So thatk ′ ( H|h|)k ′ ( H|l|+1 )h−l
+1

X
2
)
k ′ ( H|h|
+1
h

R1

X

h−l

l
|h−l|<H γ

Z 1

X
h,l

k ( H|h|)k ′ ( H|l|+1 )h−l
+1
′

Z 1

σ 2 (u)du0

σ 2 (u)du

0

Z 1
0

2

σ (u)du ≃ H

−1



Z 1

2

σ (u)du

0

Z ∞

−∞

{k ′ (u)}2 du.

Thus K (Y, U ) = O p (H −1/2). With H = n 3/5 we have n 1/5 K (Y, U ) = O p (n −1/10) such that this term
does not contribute to the asymptotic distribution when H 3/5. Appendix B: Errors induced by stale prices
The stale prices induce a particular form of noise with an endogenous component. The assumptions
that we made about the noise were formulated for prices sampled with the Refresh Time. It may be
more natural to formulate assumptions for the noise that is tied to actual observation times rather than
the artificial refresh times. Here we show that the limit distribution for K (X ) is the same under bothassumptions.
The price indexed by time τ j is, in fact, the price recorded at time t (i)j ≤ τ j , for i = 1, . . . , d. With(i)
Refresh Time we have τ j ≥ t j > τ j −1 so that(i)(i) (i)(i)(i) (i)(i)
X (i) (τ j ) = Y (i) (t (i)j ) + U (t j ) = Y (τ j ) + U (t j ) − {Y (τ j ) − Y (t j )}.
|{z}Ũ (i) (τ j )(i)This shows that if the dependence in {U } was specific to U (i) (t (i)j ), rather than U (τ j ), then the actual(i)(i)(i) (i)measurement error, Ũ (τ j ) = X (τ j ) − Y (t j ), has an endogenous component given by Z (i)=j(i)(i) (i)Y (τ j ) − Y (t j ). The implication is that K (X ) = K (Y + U + Z ), when we define the noise using the(i) (i)observation times U (i)j = U (t j ) for i = 1, . . . , d.

no′(d) (d)Theorem B.1 Suppose that H ∝ n 3/5 and that Assumption 1 holds for U j = U (1)(t (1)
),
.
.
.
,
U
(t
)
.
j
j
Then K (Y + U + Z ) − K (Y + U ) = o p (n 1/5).

The implication is that the asymptotic distribution is unaffected by Refresh Time.
Proof. We prove the result by showing that K (Z ), K (Y, Z ), K (U, Z ) are all o p (n −1/5). First we note that
Z j , j = 1, . . . , n are increments in Y, computed over non-overlapping intervals. So {Z j } is effectively aheteroskedastic independent process, where E(Z 2j ) is bounded by a term that is of orderδn = supjZ τjτ j −1σ 2 (s)ds = O(n −1 ).
41

So we can apply our analysis of K (U ), and analogous to Lemma A.5 we find that
E{K (Z )} =nδn = O(n −6/5 ),H2and the asymptotic variance of K (Z ) is o(n H −2 )δn2 = o(n −11/5). Using the same argument as in the proofof Theorem 1, we have K (Y, Z )2 ≤ K (Y )K (Z ) and K (U, Z )2 ≤ K (Y )K (Z ), which proves that both
K (Y, Z ) and K (U, Z ) are o p (n −1/5 ), as was needed. 

42