Estimating the Quadratic Covariation Matrix for
Asynchronously observed high frequency stock returns
corrupted by Additive Measurement Error1
Sujin Park

Seok Young Hong

GSA Capital Partners
Statistical Laboratory
LondonUniversity of Cambridge

Oliver Linton
Faculty of Economics
University of Cambridge

Abstract
This paper studies the estimation problem of the covariance matrices of asset returns
in the presence of microstructure noise and asynchronicity between the observations
across diﬀerent assets. Motivated by Malliavin and Mancino (2002, 2009) we propose
a new Fourier domain based estimator of multivariate ex-post volatility, which we callthe Fourier Realized Kernel (FRK). An advantage of this approach is that no explicit
time alignment is required unlike the time domain based methods widely adopted in theexisting literature. We derive the large sample properties and establish asymptotic normality of our estimator under some general conditions that allow for both temporal andcross-sectional correlations in the measurement error process. Our results can be viewed
as Frequency domain extension of the asymptotic theories for the multivariate realized
kernel estimator of Barndorﬀ-Nielsen et al. (2011). We show in extensive simulations
that our method outperforms the time domain estimators when two assets with diﬀerent
liquidity are traded asynchronously.
Keywords: Quadratic covariation, Market microstructure noise, Asynchronous observations, Fourier
Realized Kernel

1Introduction

Over the past two decades there have been many advances in the theory and application ofvolatility measurement from high frequency data. The ex-post measure of volatility called
the quadratic variation has been the focus of much attention. The theory has been developed in a series of papers including but not limited to: Andersen, Bollerslev, Dieboldand Labys (2001), Barndorﬀ-Nielsen and Shephard (2002, 2004), and Mykland and Zhang(2006). This literature has recently been extended to account for what is called the microstructure noise, namely the measurement error that distorts the underlying eﬃcient pricediﬀusion: Zhang, Mykland and Aı̈t-Sahalia (2005), Zhang (2006), Kalnina and Linton (2008),
Barndorﬀ-Nielsen, Hansen, Lunde and Shephard (2008, 2011), Aı̈t-Sahalia, Mykland andZhang (2011), and Bibinger, Hautsch, Malec and Reiss (2014). Jacod, Li, Mykland, Podolskij and Vetter (2009) introduced the pre-averaging method, which involves first averaging
the observed prices over a moderate number of time points to reduce the measurement error.
In the multivariate case an additional issue arises in the estimation procedure; since transactions most likely occur at diﬀerent time points for diﬀerent assets the observations exhibit
asynchronicity. Hayashi and Yoshida (2005) proposed an estimator of the integrated covariance that do not require synchronization. However, their estimator is inconsistent under thepresence of microstructure noise.
1
We thank Maria Elvira Mancino, Per Mykland, Victor Todorov and two anonymous referees for many
helpful comments and suggestions. Financial support from the European Research Council (ERC) is gratefully
acknowledged.

1

In a seminal paper Malliavin and Mancino (2009) proposed a Fourier domain approachthat does not require data alignment, and established consistency and asymptotic mixed
normality of their estimator under a general setting (but without deriving the limiting distribution under the presence of measurement error), extending their earlier work Malliavin
and Mancino (2002). Since the first version of this paper was written, Mancino and Sanfelici(2008) have shown that their univariate estimator is consistent in the presence of measurement error; they also provide mean squared error expansions for their uniform weighting
estimator. Furthermore, Curato, Mancino and Sanfelici (2014) derived the bias expression
due to microstructure noise, and showed consistency of their estimator. Time domain estimators addressing both asynchronicity and microstructure noise have been proposed by Zhang(2011), Barndorﬀ-Nielsen, Hansen, Lunde and Shephard (2011), and Aı̈t-Sahalia, Fan and
Xiu (2010). The estimators are consistent with convergence rates of O(n1/6 ), O(n1/5 ) andO(n1/4 ), respectively. The first two papers require aligning the data, although consistencyof their estimator is robust to the alignment. However, the hidden cost of data alignment
and non-synchronicity for these estimators is that the sample size n that appears in theconvergence rate is the sample size of the aligned data. Also, the drawback of Zhang (2011)
and Aı̈t-Sahalia et al. (2010) is that the estimator cannot be generalized to dimensions
higher than two unless the covariance matrix is estimated element-wise, which in turn does
not guarantee the estimated covariance matrix to be positive definite. See Park and Linton(2011) for a more detailed survey.
The goal of this paper is to propose an estimator of a general multivariate volatility
measure that is robust both to the microstructure noise and asynchronous data timing. The
method is based on Fourier domain techniques which have been widely used in discrete timeseries, and is broadly similar to that of Malliavin and Mancino (2009) although we allow for
more general kernel weighting in the Fourier domain. An advantage of this approach is thatit does not require an explicit time alignment. The by-product of the Fourier domain based
estimator is that we have a consistent estimator of the instantaneous co-volatility even under
the presence of quite general dependent microstructure noise. We provide a central limit
theorem for our estimator under some general conditions, and also discuss the bandwidth
choice issue based on the asymptotic mean squared error expressions. Our results allow for
the unbalanced case where one series may have many more observations than another, whichis common in practice since stocks vary considerably in terms of their trading intensities.
In Section 2 we give a setup of the model and assumptions regarding the sampling scheme.
In Section 3, we propose a Fourier domain based estimator of the integrated covariance.
Section 4 studies the large sample asymptotics of the proposed estimator and derives its
limiting distribution under the presence of microstructure noise. The Fourier method isfurther extended to estimate the instantaneous covariance matrix of diﬀusion process. We
carry out extensive simulations and empirical analysis, and report the results in Section 5.
As regards the notations, we take a ∧ b and a ∨ b to mean the smaller and bigger valueof the scalars a and b, respectively. The operator ∆(·) acting on a time series process ora function thereof is the diﬀerencing operator; for example, ∆tj = tj − tj−1 and ∆g(tj ) =g(tj ) − g(tj−1 ). We denote by =⇒stably stable convergence in law, see Aldous and Eagleson(1978) or Jacod (1997) for rigorous definition. Also, an ≃ bn means an and bn are the sameorder of magnitude. The matrix norm ∥.∥ = ∥.∥2 is understood as the Frobenius norm unlessspecified otherwise. Throughout this paper we reserve the following square bracket notation[·, ·] to denote the quadratic variation of a stochastic process. Lastly, by stationarity we mean
stationarity in wide sense rather than strict stationarity, unless stated specifically otherwise.

2

2

Model and assumptions

2.1

Eﬃcient Price and Parameter of Interest

The following standard assumption on the eﬃcient price process provides the general framework that will be used throughout this paper.
Assumption 1. The eﬃcient price process
follows a Brownian
semimartingale: For a p × 1()|
vector of logarithmic prices P (t) = P1 (t), . . . , Pp (t) defined on some filtered probability
space (Ω, F, (Ft )t≥0 , P), we have∫ t
P (t) =∫ tµ(u)du +0

σ(u)dW (u),0where µ(u) = [µ1 (u), . . . , µp (u)]| is a vector of predictable locally bounded drifts, σ(u) is asymmetricp × p matrix of locally bounded càdlàg process with finite integrated quarticity:∫t| ⊗ σ(u)σ(u)| du < ∞ a.s., and W (u) is a p × 1 vector of Brownian motion thatσ(u)σ(u)0
is independent of the volatility process.
The assumptions of locally bounded drift and of diﬀusion coeﬃcient are needed to apply
Girsanov’s theorem to remove the drift term in the theoretical derivation. Consider a discrete
time grid 0 = t0 < · · · < tn = T where T is fixed, and denote by P (ti ) the logarithmic priceobserved at ti ; i = 1, ..., n. The quadratic covariation matrix of P over a time interval [0, t],
for some t ≤ T is defined by
∑ {
}{
}|
P (ti ) − P (ti−1 ) P (ti ) − P (ti−1 ) .
[P, P ]t := plim(1)n→∞i;ti ≤t

The limit here is finite and well-defined with probability one, and is independent of the choiceof the time grid if P is a semimartingale. Furthermore, under Assumption 1 one can showthat (1) is almost surely equal to the integrated covariance matrix:∫ t[P, P ]t =σ(u)σ(u)| du,(2)0

where σ(u)σ(u)| =: Σ(u) is the instantaneous covariance matrix. We write [P, P ] := [P, P ]T ,∫T
and its j-th diagonal element [Pj , Pj ] = 0 Σj,j (u)du, the integrated variance of j-th asset.
We note that the integrated covariance is related to the covariance matrix of prices by
[∫ t
(∫ t
)| ] ∫ t
cov{P (t)} = E
σ(u)dW (u)σ(u)dW (u)=E [σ(u)σ(u)| ] du = E[P, P ]t .
000

A natural estimator of the quadratic covariation matrix is the realized covariance, the finite
sum given in the right hand side of (1), which is consistent by construction. BarndorﬀNielsen and Shephard (2002) showed in the univariate framework that the realized variance√is an unbiased and n−consistent estimator of the integrated variance under Assumption 1.
From a practical viewpoint, two important issues arise in estimating the integrated covariance matrix (2). First, in the multivariate setting prices of diﬀerent assets may be observed
at diﬀerent times, leading to the issue of asynchronicity in observations. Second, observed
prices are distorted by some noise due to market microstructure eﬀects, and do not satisfy
Assumption 1. The objective of this paper is to propose an estimation theory that is robust
to these two problems.

3

2.2

Sampling scheme

In this subsection we describe the main assumptions we make on the observation times. We
allow for both unequal spacing and asynchronicity in random observation times. However,since they are assumed to be strictly exogenous (see Assumption 2 below), we shall workwith the conditional distributions given the observation times; all statements below should
hence be interpreted as stochastic boundedness/convergence.
Assumption 2. The time span is fixed and is scaled to vary between [0, 2π]. The logarithmic
prices are observed at discrete time points: 0 = t0,ℓ < · · · < tnℓ ,ℓ = 2π for ℓ = 1, . . . , p,
where nℓ is the total number of observations for the ℓ-th asset. The discrete time points
are allowed to be stochastic and are assumed to be independent of the price and volatility
process. For asymptotics, we let the smallest number of sample sizes amongst all assets
n := minℓ (nℓ ) → ∞. For a, b, ℓ ∈ {1, . . . , p}:()
1) The discrete time points satisfy sup1≤i≤nℓ ti,ℓ − ti−1,ℓ = s∗ /nℓ = O(n−1ℓ ), for some∗finite constant s > 0.
2) Denote the intervals Ii,a = [ti−1,a , ti,a ) and Ij,b := [tj−1,b , tj,b ). The degree of nonsynchronicity satisfies()
1
sup |ti,a − tj,b |1{Ii,a ∩Ij,b ̸=∅} = O.
na ∧ nbi,jGiven any set of bivariate time points {ti,a , tj,b } with na < nb , we assume that)({}na ∨ nb
sup # tj,b ∈ [ti−1,a , ti,a )|1{Ii,a ∩Ij,b ̸=∅} = O.
na ∧ nb0≤j≤nb3) Define the empirical quadratic covariation process of time: for all a, b, c, d ∈ {1, . . . , p},∑(n)Qaabb (t) = (na ∧ nb )∆ti,a ∆tj,b 1{Ii,a ∩Ij,b ̸=∅}i,j:ti,a ,tj,b <t(n)Qabab (t) = (na ∧ nb )∑(ti,a ∧ tj,b − ti−1,a ∨ tj−1,b )

i,j,ℓ:ti,a ,tj,b ,tl,b <t× (ti,a ∧ tl,b − ti−1,a ∨ tl−1,b )1{Ii,a ∩Ij,b ̸=∅} 1{Ii,a ∩Iℓ,b ̸=∅} , for na < nb .
(n)The empirical quadratic covariation satisfies Qabcd (t) −→ Qabcd (t) as na ∧ nb → ∞,where the limit Qabcd (t) is continuously diﬀerentiable in t.
The array Qabcd (t) will appear in the asymptotic variance of our estimator for the integrated variance. In the univariate case, the expression in Assumption 2-3) specializes to∑(n)Qaa (t) = na i:ti,a <t (∆ti,a )2 . As regards Assumption 2-2) we note that {1{Ii,a ∩Ij,b ̸=∅} = 1}
if and only if {uij := (ti,a ∧ tj,b ) > (ti−1,a ∨ tj−1,b ) =: li,j }. Lastly, we see that the assumption
does not restrict the ratio of sample sizes of diﬀerent assets to be bounded away from zeroor infinity (see also Assumption 4). In summary, both unequal spacing and asynchronicity
in observation are allowed in a suﬃciently general way.
We introduce some notations that will be used in the sequel. Define{}{ (ab) }
t
∪
t
;
i
=
1,
.
.
.
,
n
,
j
=
1,
.
.
.
,
n
,
(3)
Tl
(ab) :=
i,a
a
j,b
b
1≤l≤N
T

(ab)

where NT (≤ na + nb ) is the total number of data points for the union of time stamps.
Denote the average interval size for asset ℓ by ∆tℓ := 2π/nℓ . As for two diﬀerent assets say aand b that are being compared, the average interval size of the more liquid asset is denoted
f ab := 2π/(na ∨ nb ).
by ∆t
4

3

Estimation

3.1

The Fourier Kernel Estimator

Motivated by the disadvantages of data alignment methods widely discussed in the literature,
see for example Aı̈t-Sahalia and Jacod (2014) for relevant discussions, we propose to use a
Fourier domain approach which does not require data synchronization. Our framework
follows the line of approach of Malliavin and Mancino (2002, 2009), where a nonparametricmethod based on Fourier analysis of returns was discussed. Frequency domain techniques arewidely used in discrete time series analysis; an important application of such an approachis the estimation problem of the long-run variance of a stationary time series (which is
proportional to the spectral density at zero frequency). We draw a natural link between
those classical theories and the estimation of the quadratic covariation of a continuous timeprocesses.
√
The Fourier basis {gt (q) := eiqt / 2π; q ∈ Z} where i2 = −1 constitutes an orthonormal
basis of L2 ([0, 2π]):{∫ 2π
1
1 if k = j
gt (k)gt (j)dt =0 otherwise,
2π 0
where gt (q) refers to the complex conjugate of gt (q).
In view of completeness of the Fourier basis, we can express the continuous time signal
Σ(t) ∈ L2 ([0, 2π]) as a linear combination of the Fourier basis with coeﬃcients denoted byF(Σ)(q):∞1 ∑F(Σ)(q)eiqt ,(4)
Σ(t) =
2π q=−∞
whereF(Σ)(q) =∫ 2πe−iqt Σ(t)dt;

q ∈ Z.

(5)0This is the continuous time Fourier transform of the instantaneous covariance matrix. Notethat (5) corresponds to the integrated covariance when q = 0; we will propose an estimator
for the general q case. The Fourier pair above suggests that once we estimate the Fourierbcoeﬃcient by a suitable estimator F(Σ)(q),the signal can be reconstructed via:
n
∑
iqt
b = 1
b
Σ(t)
F(Σ)(q)e.
2π q=−n

We now discuss the estimation procedure of (5) based on Fourier transform of the returnprocess. Define∫
2πF(dPℓ )(α) =e−iαt dPℓ (t);α ∈ Z, ℓ = 1, ..., p,(6)0where Pℓ (t) refers to observed price of the ℓ-th asset satisfying Assumption 1, and let
Fn (dPℓ )(α) =nℓ∑e−iαtj,ℓ ∆Pℓ (tj,ℓ ),

(7)

j=1

{
}|
whose vector version Fn (dP )(α) = Fn (dP1 )(α), . . . , Fn (dPp )(α) , α ∈ Z, can be definedto account for multiple assets altogether.
Now we consider a class of kernel called the spectral window KH : [−π/H, π/H] → R,
where KH (λ) ≃ H · K(Hλ) and K(·) is a function called the spectral window generator.
Detailed analysis of these weighting functions is given in the next subsection (Assumption
53′ , (13) and their relevant discussions); see also Parzen (1967) and Rao and Gabr (1984) for
earlier discussions. The estimator we propose for the integrated covariance (5) is given by
(
)
[
][
]|
∑
b
b i,j )(q) =F(Σ)(q)= F(ΣKH (λα ) Fn (dP )(α) Fn (dP )(q − α) ,(8)|α|≤m/2

where λα = 2πα/n and n := maxℓ=1,...,p nℓ . Let m = n/H where the bandwidth H =Hn → ∞ but not as fast as n = minℓ nℓ → ∞ so that m → ∞. We are smoothing over theinterval [−π/H, π/H], where H controls the width of the smoothing window. We shall refer
to our estimator as the Fourier Realized Kernel (FRK). Note that the estimator consideredin Malliavin and Mancino (2009) can be compared with ours with K(x) = (1/π)(1 − |x|/π).
The q = 0 case deserves a special attention. In this case Fn (dP1 )(α)Fn (dP2 )(−α) =:I12 (α) is the realized cross periodogram between assets 1 and 2, say. Then, the (1, 2)-thbentry of F(Σ)(0)is given by kernel smoothing the realized cross periodogram around zerofrequency:∑
b 12 )(0) =F(ΣKH (λα )I12 (α).
(9)|α|≤m/2

Positive definiteness of the estimators (9) is guaranteed provided that the spectral window
is non-negative.

3.2

Comparison with some Time domain estimators

For data that is synchronized at {τi }, its realized autocovariance function is defined as∑γ12 (h) =∆P1 (τi )∆P2 (τi−h ); h ∈ Z,
(10)
i

∑
∑
where i = h<i≤n for h ≥ 0, and i = 1≤i≤n+h for h < 0. In the aligned case therealized periodogram is closely related to the realized autocovariance. In particular when τiare equally spaced and synchronous, i.e. τi = τj + (i − j)2π/n, it can be easily shown thatthe realized ∑cross periodogram is the Fourier transform of the realized autocovariance; that
is, I12 (α) = |h|<n e−iαh2π/n γ12 (h).
Hayashi and Yoshida (2005) considered a covariation estimator defined as the realized
cross periodogram at zero frequency over the interval that overlaps:
∑

∑

HY =

n1 ∑
n2
∑

∆P1 (ti,1 )∆P2 (tj,2 )1{Ii,1 ∩Jj,2 ̸=∅} .

i=1 j=1

Then the estimator can be expressed in terms of the centered (i.e. frequency zero) realized
cross periodogram I1,2 (0). In particular, we have the following decomposition:
I1,2 (0) − [P1 , P2 ] =

n1 ∑
n2
∑

∆P1 (ti,1 )∆P2 (tj,2 ) −M1 = HY −∫ 2π
Σ12 (t)dt,
0

Σ12 (t)dt = M1 + M2 ,0

i=1 j=1

where

∫ 2π

M2 =

n1 ∑
n2
∑
i=1 j=1∆P1 (ti )∆P2 (sj )1{Ii,1 ∩Jj,2 =∅} .

√Hayashi and Yoshida (2008) showed that nM1 is asymptotically zero mean Gaussian (whendata is Poisson sampled), and M2 is mean zero with leading term of order Op (1). Hence,
provided there is no microstructure noise the Hayashi and Yoshida estimator is unbiased,
6

Figure 1: Examples of the lag and spectral windows satisfying Assumption 3 and 3′
Time domain

Frequency domain
50
BartlettHanning
GaussianParzen1
0

0.8

Magnitude (dB)

Amplitude

−50

0.6

−100

0.4
−150

0.2

0

−200

10

20

30
Samples

40

50

−250

60

0

0.1

0.2

0.3
0.4
0.5
0.6
0.7
Normalized Frequency (×π rad/sample)

0.8

0.9

√
and achieves n-consistency without requiring synchronization. The realized periodogramis also unbiased but is inconsistent due to the presence of M2 .
We next compare our estimator (9) with the (multivariate) realized kernel estimatore It is given by kernel smoothing
by Barndorﬀ-Nielsen et al. (2008, 2011), denoted by Σ.
the realized autocovariances of the data aligned on {τi }N
i=1 using the refresh time sampling
scheme. For example, its (1, 2)th entry is
e 12 :=
Σ

∑
|h|<n

(
k

h
H

)
γ12 (h) =NN ∑∑i=1 j=1(∆P1 (τi )∆P2 (τj )k

i−j
H

)
,

(11)

where k(·) is a smoothing window in time domain. To establish the link between the time
domain estimator (11) and our frequency domain estimator (9), we discuss the properties ofsmoothing windows; we introduce the lag window for continuous time denoted by k(x), x ∈ R
and compare it with the spectral kernel for continuous and bandlimited frequency. The lag
window k is assumed to satisfy the following conditions (introduced as Assumption K in
Barndorﬀ-Nielsen et al. (2011)). Note that the prime notation is taken to mean diﬀerentiation with respect to the argument.
Assumption 3. The lag window k(·) satisfies the following conditions:∫ ∞(i) k is 2twice con2′
tinuously diﬀerentiable; (ii) k(0) = 1, and k (0) = 0; (iii) ∥k∥ := −∞ |k(x)| dx <, ∞,
∫∞
∫∞
∫∞
2
∥k 2 ∥ := −∞ |k(x)|4 dx < ∞, ∥k ′ ∥2 := −∞ |k ′ (x)|2 dx < ∞, ∥k ′′ ∥2 := −∞ |k ′′ (x)|2 dx < ∞;∫∞(iv) −∞ k(x) exp(−iλx)dx ≥ 0, ∀λ ∈ [−π, π].
We define the spectral window generator K as a Fourier transform of the lag window k:∫∞∫π1−iλt dt
K(λ) = 2πk(t) = −π K(λ)eiλt dλ,(12)−∞ k(t)e
where λ denotes the angular frequency. Then it follows from simple algebra and Parseval’s
identity that Assumption 3 on the lag window k translates to the following conditions on the
spectral window generator.

7

Assumption
3′ . The∫spectral window generator K(·) satisfies
the following conditions:∫π∫ππ(i) −π K(λ)dλ = 1, −π λK(λ)dλ = 0; (ii) ∥K∥2 := −π |K(λ)|2 dλ < ∞, µ21 (K) :=∫π∫
22 (K) := π |λ2 K(λ)|2 dλ < ∞; (iii) K(λ) ≥ 0, ∀λ ∈ [−π, π].
|λK(λ)|
dλ
<
∞
and
µ
2
−π
−π
Figure 1 shows some examples of weighting functions k and K that satisfy Assumption 3
and 3′ . In this sequel, Assumption 3 is taken to mean both Assumptions 3 and 3′ .
As remarked by Barndorﬀ-Nielsen et al. (2011), condition (iv) in Assumption 3 (andequivalently (iii) in Assumption 3′ ) guarantees positive semi-definiteness of the estimators
defined in (11) and (9) via Bochner’s theorem. The realized periodogram is also positive
semi-definite and is Hermitian as long as the spectral window is non-negative.
Consider the discrete time and discrete frequency Fourier pair:( h ) iλ h( h ) ∑m/2−1
1 ∑
α ;
(13)
KH (λα ) = 2πk H
= α=−m/2 KH (λα )e−iλα h .
|h|≤H k H e
We refer to the weighting function KH defined in (13) as the spectral window. To avoid thealiasing problem we assume that the signal is zero for frequencies that falls outside of the
Nyquist critical frequency. As shown in Parzen (1967, page 130), the spectral window thatis used to smooth the realized periodogram is related to a spectral window generator via
KH (λ) = H

∞
∑

(
)
K H(λ + 2πj) ≃ HK(Hλ).

j=−∞

Now with λα = 2πα/(na ∨ nb ), α ∈ Z for some a, b = 1, ..., p, upon substituting h =(na ∨ nb ) · (ti − sj )/(2π) into the latter expression of (13) we see that∑

KH (λα )Iab (α) =|α|<m/2nb
na ∑∑∆Pa (ti )∆Pb (sj )=KH (λα )e−iα(ti −sj )|α|<m/2

i=1 j=1
nb
na ∑
∑

∑

∆Pa (ti )∆Pb (sj )kH (ti − sj ),(14)

i=1 j=1

f ab H)). Further, since an equally spaced and synchronized
where kH (ti −sj ) := k((ti −sj )/(∆ttime grid satisfies ti = sj + (i − j)2π/(na ∨ nb ), the following key statement trivially holds:
Remark 1. When trading times are synchronized and equally spaced, the Fourier realized
kernel at zero frequency (9) and the multivariate realized kernel (11) are identical.
We will show later that when the data is not synchronously observed, using all the data andimplementing (14) delivers a superior estimator.
It is of interest how our Fourier kernel estimator is related to other time domain estimators such as the multivariate two time scale estimator of Zhang (2011), and the modulated
realized covariance (multivariate pre-averaging estimator) of Christensen, Kinnebrock andPodolskij (2010). In the univariate setting, Jacod et al. (2009) showed that their preaveraging estimator, the univariate two time scale estimator of Zhang et al. (2005), andthe flat-top realized kernel of Barndorﬀ-Nielsen et al. (2008) can be written as a smoothed
realized autocovariances, and the diﬀerence between the estimators comes from the contribution of the end points. This result holds also for the multivariate versions of the threeestimators when observation points are synchronized. Our estimator can be expressed asa realized kernel only when sampling points are equally spaced and aligned. The relation
between the smoothed periodogram to estimate the spectrum and data tapering (i.e. Fouriertransforming the weighted return) is analogous to the relation between our estimator andthe pre-averaging estimator.
8

4

Asymptotic Properties

4.1

Bandwidth conditions

We introduce the rate conditions on the bandwidth we require for our asymptotic theories.
Our conditions allow the sample sizes of diﬀerent assets to be of diﬀerent order of magnitude.
As aforementioned, such a situation arises often in practice as some assets may be tradedmuch more frequently than others. As defined previously in Assumption 2, n is taken tomean the minimum amongst the individual sample sizes of the assets throughout, unlessstated otherwise.
Assumption 4. The bandwidth sequence H = Hn is of order H = O(nα ) where α ∈ (0, 1) sothat H → ∞ and n/H (=: m) → ∞ as n → ∞. Moreover, the ratio of the bigger to smaller
sample size between two assets satisfies (na ∨ nb )/(na ∧ nb ) = o(H) for any a, b ∈ {1, . . . , p}.
The first condition regulates the speed at which the bandwidth is allowed to increase;this is consistent with the usual setting in nonparametric literature. The next requirement(na ∨ nb )/(H(na ∧ nb )) = o(1) specifies the asymptotic behaviour of the ratio of the larger tosmaller sample sizes; this will be shown to control the bias from the leading term. We define
β (= maxa,b βab ) ≥ 1 to be the degree of liquidity parameter where
βab = limlog (na ∨ nb )n→∞ log (na ∧ nb )

.

(15)

The definition suggests that if we write na ∧ nb =: nab , then we have na ∨ nb = nβabab ; this inturn implies that 1 ≤ βab < 2 for all a, b ∈ {1, . . . , p} because otherwise (na ∨nb )/(H(na ∧nb ))would not converge, violating Assumption 4. As a slight abuse of notation, we will hereafter
omit the subscript of β whenever it is clear from the context which assets are being considered.

4.2

Assumption on the microstructure noise

Empirical evidence from the volatility signature plot suggests that the observed price deviatesfrom the semimartingale assumption. In fact, various studies document that the observedhigh frequency returns have infinite quadratic variation. The following assumption is proposed to account for this phenomena.
Assumption 5. The observed logarithmic price of the ℓ-th asset, denoted Xℓ (tj,ℓ ), is the sumof two components. The first component is a discretely observed continuous signal Pℓ (tj,ℓ )that satisfies Assumption 1, and the other one is a noise process with respect to the realizationof transaction time Uℓ (tj,ℓ ) that has an infinite quadratic variation: i.e. for each ℓ = 1, ..., pXℓ (tj,ℓ ) = Pℓ (tj,ℓ ) + Ul (tj,ℓ ).

(16)This type of additive microstructure noise model has been well-studied in the literature,
particualrly in the univariate context; see for recent examples, Mancini and Sanfelici (2008)
and Curato et al. (2014) where the noise Uℓ (tj,ℓ ) is assumed to be i.i.d. There has notbeen much empirical work that studied cross autocorrelation of the microstructure noise formultiple assets. Amongst the few includes Aı̈t-Sahalia et al. (2010) where the noise is set tobe i.i.d in time but uncorrelated across diﬀerent assets. Zhang (2011) assumed (covariance-)
stationarity and an exponential alpha mixing condition with respect to the observation time.
In this paper we shall impose the following dependence condition on the microstructurenoise:

9

Assumption 6. The stochastic process Uℓ (.) is stationary, mean zero, and is independent of
the eﬃcient price process Pℓ (.). Furthermore, the covariance function of the noise process
f ab ) satisfies
defined as E[Ua (ti,a )Ub (tj,b )] =: γ(|ti,a − tj,b |/∆t)(n∑a −1 nb −1∑|ti,a − tj,b |1γ−→ Γab (< ∞),f abna ∧ nb∆t(17)

i=1 j=1

where the finite limit Γab is the (a, b)-th element of the p×p positive semi-definite covariancematrix Γ. Also, the fourth moments satisfy |E(Ua (ti,a )Ub (tj,b )Uc (tr,c )Ud (tl,d ))| ≤ ρ(M ) < ∞,f gs ] and ρ(·) is a function such that for some δ > 0,
where M := supu,v,g,s [(tu,g − tv,s )/∆t∞∑ρ(ν)(1 + δ)ν < ∞.

(18)

ν=1

Our assumption allows both cross-sectional and temporal correlations in the measurement
error
process. Note that when the data is equally spaced and balanced, (17) simplifies to
1 ∑n ∑ni=1
j=1 |γ(i − j)| → Γab < ∞. This standard absolute summability condition for thenautocovariance function is often assumed in the time series literature, and is well-known tobe suﬃcient for ergodicity and necessary for (a certain class of) mixing under stationarity,
see Ibragimov and Linnik (1971). The second condition (18) implies that the (supremum of)
the fourth moment decays suﬃciently fast as the maximum cross-lag M = max |i−j|, |h−l|
increases.
4.2.1

Asymptotic Normality

We now introduce our main theoretical results. At the end, the limiting distribution of ourestimator (8) will be derived under the presence of the microstructure noise. Before we
proceed, we impose a condition on the end points. This condition is in line with what isassumed in the existing literature in order to ensure consistency of the estimator at the end
points, see Jacod et al. (2009) and Barndorﬀ-Nielsen et al. (2011, section 2.2).
Assumption 7. The observed prices at two end points, i.e. Xℓ (t0,ℓ ) and Xℓ (tnℓ ,ℓ ) for eachℓ = 1, ..., p are respectively the arithmetric average of ξ number of distinct observations on
the interval [t−1,ℓ , t0,ℓ ) and [tnℓ ,ℓ , tnℓ +1,ℓ ).
Since Assumption 5 implies ∆X(ti ) = ∆P (ti ) + ∆U (ti ), our estimator (8) can be decomposed as follows:b
F(Σ)(q)− F(Σ)(q)
][
]| }
{[
][
]| [
∑
=
KH (λα ) Fn (dP )(α) Fn (dP )(q − α) − F(dP )(α) F(dP )(q − α)|α|≤m/2+∑|α|≤m/2+∑|α|≤m/2

+

∑

[
][
]|
KH (λα ) F(dP )(α) F(dP )(q − α) − F(Σ)(q)[][]|KH (λα ) Fn (dU )(α) Fn (dU )(q − α)
{[
][
]| [
][
]| }
KH (λα ) Fn (dU )(α) Fn (dP )(q − α) + Fn (dP )(α) Fn (dU )(q − α)|α|≤m/2= (i) + (ii) + (iii) + (iv).

(19)
10

The first term (i) is the discretization error due to sampling the continuous time signal
at discrete points, and (ii) is the error due to kernel smoothing. Moreover, (iii) can beinterpreted as the contribution from the smoothed realized periodogram applied to a microstructure noise. Lastly, (iv) is the sum of cross terms between the eﬃcient price andthe noise, which is of a smaller order than (iii), see Theorem 1 of Barndorﬀ-Nielsen et al.
(2011). Below we derive the asymptotic order of the noise contribution (iii), and establish
asymptotic normality of (i) + (ii). All proofs are contained in the appendix.
Proposition 1. Suppose Assumptions 2-7 hold. Then for any a, b = 1, ..., p we have


 ∑
[
][
] ( s∗ )2 n ∨ nab ′′KH (λα ) Fn (dUa )(α) Fn (dUb )(q − α)≃E|k (0)|Γab
2


2π
H
|α|≤m/2

2
(
)
 ∑
[
][
] 
(na ∨ nb )2 2µ−3E
KH (λα ) Fn (dUa )(α) Fn (dUb )(q − α)
=O
H
,


na ∧ nb|α|≤m/2

where µ is some constant such that 0 < µ < 1, and s∗ and Γab are as defined in Assumption
2 and Assumption 6, respectively.
The proposition above implies that the leading bias term (iii) is of order Op ((na ∨nb )/H 2 );a necessary condition for this bias contribution from the microstructure noise to be negligible in large sample is α ∈ (1/2, 1). We now move on to the main term (i) + (ii) =:b P )(q) − F (Σ)(q), and derive its asymptotic bias and variance expressions.
F(Σ()
b P )(q)Proposition 2. Suppose Assumptions 1-4 hold. Then the bias and variance of F(Σab(a, b = 1, ..., p) denoted respectively by Bab and Vab are given by:
)2
(
∫ 2π
1 2 ′′na ∨ nb
A k (0)e−itq d |[Pa , Pb ]| (t)Bab =(na ∧ nb )H
2 ab
0
[∫
2π
H
2
Vab =
∥k∥
e−i2tq [Pa , Pa ]′ (t)[Pb , Pb ]′ (t)dQaabb (t)na ∧ nb
0
]
∫ 2π
−i2tq
′ 2
e
([Pa , Pb ] ) (t)dQabab (t) ,+0with Baa = 0 (for all a = 1, ..., p), where
Aab :=na ∧ nb
sup |ti − sj |1{Ii,a ∩Ij,b =∅} .
na ∧nb →∞
2π
i,j
lim

This result gives the asymptotic mean squared error of our estimator in the absence of
microstructure noise but with asnychronous trading. It may be compared with BarndorﬀNielsen et al. (2011) Theorem A.5. Malliavin and Mancino (2009) Theorem 4.1 gives a CLT
for their estimator in the absence of microstructure noise. Our result in Proposition 2 holds
under diﬀerent restrictions on the bandwidth sequence, and so is not directly comparable.2
We note that Aab can be thought of as a measure of the degree of non-synchronicity
between two assets a and b; Assumption 2 implies that Aab ∈ [0, ∞). In particular, when
two series in consideration are perfectly synchronized and balanced, then Aab = 0. Otherwise
it is a bounded constant i.e. O(1) by Assumption 2.2.
2Note that their estimator has a specific weighting, and models the asynchronicity in a slightly diﬀerent
way.

11

From Proposition 2 we see that the order of asymptotic variance is O(H/(na ∧ nb )). The
squared bias is given by O((na ∨ nb )2 /H 4 ) as this dominates that of the main term. The
optimal bandwidth H := O(nα ) is obtained by balancing those two quantities. Denote byn = (na ∧ nb ) = na say, and let nb = (na ∨ nb ) = O(nβ ). Then solving n2β−4α = nα−1 wehave
2β + 1
H = C0 nαopt , αopt =,(20)
5where C0 is some positive constant. In particular, when two sample sizes na and nb are of
the same order (i.e. β = 1), then the optimal order of the bandwidth is given by αopt = 3/5.
The rate of convergence of the estimator is therefore (na ∧nb )ϑ , where ϑ = (2−β)/5 ∈ (0, 1/5].
This suggests that the best rate (na ∧ nb )1/5 is achieved when the sample sizes are of thesame order, and the rate slows down as β ∈ [1, 2), the degree of relative liquidity between
assets in consideration increases.
Theorem 1. Suppose that Assumptions 1-7 hold. Then provided that the optimal bandwidth(20) is chosen, we have for each q ∈ Z
{
}
(
)
b
Dn vech F(Σ)(q)− F (Σ)(q) =⇒stably N B, V
as n → ∞, where2−β,{Dn∗ }a,b = (na ∧ nb )ϑ , ϑ =5()
B = (2π)−2 C0−2 s∗,2 |k ′′ (0)| · vech Γ ,V = (Vab ),
∫ 2π
{
}
2
−i2tq
′
′
′ 2
Vab = ∥k∥e
[Pa , Pa ] (t)[Pb , Pb ] (t)dQaabb (t) + ([Pa , Pb ] ) (t)dQabab (t) .
Dn =diag {vech(Dn∗ )} ,0

4.3

Estimation of the Instantaneous covariance matrix

Here we extend the Fourier method discussed above to estimate the instantaneous covariancematrix of the diﬀusion process under the presence of microstructure noise. We can construct
an estimator of the instantaneous covariation matrix by Fourier inverting (8):
1 ∑
b
b
Σ(t)
=
KH (λq ) exp(iqt)F(Σ)(q).
(21)
2π|q|≤m/2

Suppose that the modulus of continuity of Σ(t) denoted by C(h) is given byC(h) := sup ∥Σ(t) − Σ(s)∥2 .

(22)|t−s|≤h

The continuity assumption is met when each element of Σ(t) in Assumption 1 does not contain jumps, for example Σ(t) is a Brownian semimartingale.
Theorem 2. Suppose that the regularity conditions of Theorem 1 and (22) hold. Then, there
exists a sequence δ(n) → 0 such that
lim

sup

b − Σ(t)∥2 = 0.
∥Σ(t)n→∞ δ(n)≤t≤2π−δ(n)Practitioners often encounter a problem of running a regression between variables thatare asynchronously observed; for example, we might be interested in the eﬀect of returns
and order book information of one asset on another asset. Hannan (1975) and Robinson(1975) are the earlier literature on using frequency domain to solve such problems. Mykland
and Zhang (2006) discussed a general setup of the analysis of variance for continuous timeregression.
12

5

Numerical Study

5.1

Comparison of estimators of co-volatility

Recall that our estmimator (at zero frequency q = 0) can be written as:∑

KH (λα )I(λα ) :=αn∑∆P1 (ti )∆P2 (sj )KH (λα )e−i(ti −sj )α

α

i,j=1
n
∑

∆P1 (ti )∆P2 (sj )kH (ti − sj )

=(i)

i,j=1

∑

=(ii)

∑

(
k

|h|<n

h
H

)
γh ,

where (ii) holds only when the discretization points are synchronous and equally spaced.
The form of the estimator we propose to implement is (i). In the theory sections above,
we assumed no leverage eﬀect; in the simulation studies however, we relax this assumption
and see if our estimator is robust to the presence of the leverage. We consider two data
generating processes (DGP) for asset returns. For the first simulation, we consider thestochastic volatility model with a perfect leverage given in Barndorﬀ-Nielsen et al. (2011).
The volatility process is continuous whereas the instantaneous co-volatility is constant. For
assets indexed by j = 1, 2,√(23)
dPj (t) = 0.03dt − 0.3σj (t)dBj (t) + 1 − (0.3)2 σj (t)dW (t)σj (t) = exp{−5/16 + 1/8ϱj (t)} ;dϱj (t) = −1/40ϱj (t)dt + dBj (t),and ϱj (t) is initialized by ϱj (0) ∼ N (0, 20). The model implies that the covariance between
the returns are EdP1 (t)dP2 (t) = 0.91σ1 (t)σ2 (t)dt. There is a perfect statistical leverage sincea single Brownian motion Bj (t) which is present in the return equation, drives the volatility
process.
Figure 2: Simulated intraday instantaneous co-volatility and variance(b) continuous and jump component of variance(a) correlation
0.86

0.18

0.16

0.84

0.14
0.82
0.12
0.8
0.1
0.78
0.08
0.76
0.06

0.74

0.72

0.04

0

0.5

1

1.5

2

0.02

2.5
4

0

0.5

1

1.5

2

2.5
4

x 10

x 10

For the second simulation, the stochastic volatility is specified as a jump diﬀusion process
and the instantaneous co-volatility coeﬃcient follows the Cox-Ingersoll-Ross (CIR) process.
13

Figure 3: Simulated price and variance - per second observation(a) Prices series (100 day per second prices)(b) volatility

5.5

0.65
Asset2
Asset1

0.6

5
0.55
0.5
4.5
0.45
4

0.4
0.35

3.5
0.3
0.25
3
0.2
2.5

0

1

2

3

4

5

0

1

2

3

4

4

5
4

x 10

x 10

This is a modification of the DGP considered in Aı̈t-Sahalia et al. (2010) and BarndorﬀNielsen et al. (2004). For j = 1, 2, we havedPj (t) = σj (t)dWj (t)(24)

dσj2 (t) = κj {σ̄j2 − σj2 (t)} + aj σj (t)dBj (t) + σj (t− )Jj (t)dNj (t),where the jump size follows Jj (t) = exp{zj (t)} with zj (t) ∼ N (µj , sj ), and Nj (t) is a poisson process with intensity λj . The leverage eﬀect here is EdWj (t)dBj (t) := δj dt, and thecovariance between the Brownian motions that are present in the price equation is given by
EdB1 (t)dB2 (t) = ρt dt. The parameter values taken from Aı̈t-Sahalia et al. (2010). We let
ρ(t) = (e2x(t) − 1)/(e2x(t) + 1), where x(t) follows the CIR processdx(t) = 0.03(0.64 − x(t))dt + 0.118x(t)dBxt .
Figure 2(a) shows the time series plot of ρt , and (b) shows σ12 (t) decomposed into a continuous and a discontinuous components. Figure 3(a) shows the time series plot of Pj (t), j = 1, 2
and (b) shows σj2 (t), j = 1, 2. The DGP of the microstructure noise is formed with respect
to the transaction time. We consider a correlated AR(1) noise processes with smoothly
decaying cross autocovariances. This can be implemented by
Uj (ti,j ) = Ūj (ti,j ) + ε(ti,j );Ūj (ti,j ) = αŪj (ti−1,j ) + ϵj (ti,j ),(25)

where the idiosyncratic errors are independent Gaussian, i.e. ϵj (ti,j ) ∼ N ID(0, 1). The common disturbance that drives the correlation between two microstructure noise is simulatedbyεl = 0.5εl−1 + ξl , for {Tl }1≤l≤NT = {ti,1 ∪ tj,2 , i = 1, . . . , n1 , j = 1, . . . , n2 }, ξ ∼ N ID(0, 1).
Then we define {ε(ti,1 )}1≤i≤n1 as {εl }1≤l≤NT sampled at {Tl ∩ ti,1 } points. {ε(tj,2 )}1≤j≤n2 isdefined similarly.
√ The∑variance of the noise is set to be proportionate to the sample integratednj
2
quarticity; ζnj −1 i=1
σj4 (ti,j ), where ζ = {0, 0.0.001, 0.01} is the noise to signal ratio.
We simulate the one second data assuming 6.5 hours of daily trading time, yielding 23,400
daily data points over 100 Monte Carlo sample. The simulation is designed to assess the14

Figure 4: Covariation signature plot for simulated series - bias induced by data alignment
for diﬀerent sampling rates
Refresh time bias

Fixed clock bias

0.1

0.1
0

0.05
−0.1
0

−0.2
−0.3
Bias

Bias

−0.05
−0.4

−0.1
−0.5
−0.6

(1.5,2)−0.15(5,10)
−0.7
−0.2
−0.8
−0.25

0

5

10

15
20
per transaction

25

−0.9

30

0

10

20

30
40
per x*10 second

50

60

impact of the asynchronicity on the estimator: We Poisson sample the data at the rate of{(3/2, 30), (20, 30)}, where sampling at rate (a, b) means on average observation is made at
every a and b seconds. To create a balanced sample for the rate (3/2, 30), for the first asset,
we sample on average at 1.5 second for the first half of the sample and at 30 second for thelast half of the sample. For the second asset, we do this in reverse order. Then we have two
assets that have the same number of transactions each day but traded very asynchronously.
Finally, we examine the properties of the estimators in higher dimensions. We consider
a simple setting where the log prices are given by P (t) = AB(t); P (t) is the 10 × 1 vector of
prices, B(t) is the 3 × 1 independent Brownian motion and A is a factor loading matrix. Thisis poisson sampled at rate {2, 2, 4, 4, 8, 8, 10, 10, 30, 30} and masked by i.i.d gaussian noise.
Table 2 and 3 at the end of the paper report the results for estimating the 2-dimensional
covariation matrix, where the first asset is more often traded then the second asset. Table 4reports the results for a higher dimension.
5.1.1

Realized Covariation: bias induced by data synchronization

Table 1 reports the finite sample properties of the realized covariance (RC). The eﬃcient pricefollows Brownian semimartingale, given in (23). The transaction time follows a homogenous
poisson process, and the microstructure noise are correlated AR(1) processes given in (25).
Asynchronous data is aligned using the 5 minute fixed clock time and the refresh time. The
negative bias when there is no noise is consistent with the result of Proposition 2.
When microstructure noise is present, the variance estimate has a large positive bias. The
sparse sampling (5 minute aligned data) is able to reduce such bias. However, the covarianceestimate has a negative bias induced by the Epps eﬀect which dominates the positive bias
induced by the microstructure noise. The degree at which Epps eﬀect dominates the noise
eﬀect depends on the degree of non-synchronicity. Figure 4 shows the covariation signatureplot for the simulated series when the price is observed without the noise. It shows that given
varying degrees of non-synchronicity (rate at which assets is traded), the higher frequency
we align the data (moving leftwards in x axis) the more bias it induces in estimating the
integrated covariance.
15

Figure 5: Time stamp of two assets traded at opposite liquidity
1

0.9

0.8

0.7

0.6
t
i
sj

0.5

0.4

0.3

0.2

0.1

0

5.1.2

0

1000

2000

3000

4000

5000

6000

7000

Balanced Sample example

We elaborate further on the aforementioned sampling procedure: We first simulate theequally spaced data per one second for two assets. For the first asset, we sample on average at 1.5 second for the first half of the sample and at 30 second for the last half of thesample. For the second asset, this is done in reverse order - sample at 30 second for the firsthalf and 1.5 second for later part. See Figure 5. Then we have two assets that have the same
number of transactions each day but traded very asynchronously. This is like a case where
two assets have opposite liquidity profile over a day.
The sample size is 607, 774 over one hundred days and the refresh time aligned data reduces to a size around 750 per day. The large reduction in the sample size of the aligned data
is due to severe non-synchronicity by simulation design. We compare the realized kernel andthe proposed method over the range of bandwidths, H = {1, 5, 10, 20, 50, 100, . . . , 750, 800}.
The Figure 6 shows that the proposed estimator is less sensitive to the choice of bandwidth- especially for large H. With large H, we can reduce the bias for the oﬀ-diagonal element
more than we can do for the realized kernel. Our estimator is less sensitive to the choice of
bandwidth for large values of H.
5.1.3

Unbalanced Sample example

We carried out the same exercise as above but with the unbalanced sample sizes. We Poisson
sample the data at rate {(3/2, 30), (3/2, 2), (20, 30)}. For example, sampling rate (3/2, 2)means that we sample the first asset on average per 1.5 second and the second asset per 2
second. The first rate is to examine the eﬀect of diﬀerent liquidity and diﬀerent sample sizes.
The second and third rates are to examine the eﬀect of sparse and intense sampling of asset
prices of similar liquidity.
Figure 6 shows the results for sampling rate (3/2, 30). The proposed estimator has aless bias and is less sensitive to the choice of the bandwidth for large values of H. Whenthe noise is not present, the proposed method estimates the variance of more liquid asset
more precisely. When the noise is present, the bandwidth should be large for the proposed
estimator to perform better. The conclusion is similar for sampling rate (20, 30) as shownin Figure 8. The improvement of using the Fourier domain estimator is most evident whenestimating the variance of more liquid asset when two sample sizes are very diﬀerent. The
16

proposed covariation estimator performs better under large bandwidth. For sampling rate
(3/2, 2) in Figure 9, the diﬀerence of two estimator is less pronounced.
Each of these figures also show the accuracy of estimating the scalar function of thecovariation matrix.
√ We examined the maximum eigenvalues and the variance of portfoliowith weight [0.5, 0.75]. Under the realistic noise to signal ratio and when two assets are ofdiﬀerent liquidity, the proposed method delivers superior estimates. Regardless of sampling
scheme, the proposed method does better in estimating these quantities when the eﬀect of
microstructure noise is not too dominant.
5.1.4

Overall Comparison and Higher Dimension Case

Table 2 and 3 shows that the proposed estimator has the best bias profile. With carefully
chosen bandwidth we can achieve the best root MSE under the presence of noise. When nonoise is present, the Hayashi and Yoshida estimator performs well. The refresh time aligned
method often performs better in estimating the integrated variance of the less traded asset;(2,2) element. This is because it eﬀectively aligns on the time stamp of less traded asset. As
shown in the analysis of asymptotic bias, when there is no noise and the number of refreshtime sample is smaller in size, then the realized kernel underperforms in terms of bias. The
proposed estimator overall estimates the oﬀ-diagonal elements better. We observe also thatthe realized covariance estimator aligned on sparesly sampled data often performs well this is because there are two opposing eﬀects in terms of bias: negative bias from the Epps
eﬀect and positive bias from the microstructure noise. The advantage of our estimator ismost clear when we estimate the covariance matrix of higher dimension as shown in Table
4. We estimate 10 dimensional integrated covariance matrix and compare the maximum of
eigenvalues and variance of the equally weighted portfolio. Under no presence of noise therefresh time based method has a large bias. We calculate the optimal bandwidth as given in
Theorem 1 for each element of covariance matrix and take the minimum, maximum of theseand average them. We note that our estimator seems to yield a large variance, however it
performs best under the optimal bandwidth.

5.2

Empirical Application

In this section we apply the Fourier Realized Kernel to high frequency data. We analyzed
five stocks of diﬀerent liquidity - Microsoft (MSFT), Dell (Dell), JP Morgan (JPM) andless frequently traded Caterpillar Inc (CAT) and Banco de Chile (BCH) from WRDS TAQ
database. The period of analysis is for 20 days during 05-30 March 2007. The order ofleast to most liquid stocks is BCH, CAT, DELL, JPM, and MSFT, whose average daily
sample sizes are given by {48, 7526, 8337, 10337, 11451}, respectively. We may calculate theoptimal bandwidth for individual asset by equalizing the squared bias and the variance term
given in Section 4. Let nℓ = nβℓ where n is a minimum of all sample sizes, then it is given
4/5 1+2βℓ
by Hℓ = {η|k ′′ (0)|/∥k∥}2/5 ζℓ n 5 , where η =√(s∗ /2π)2 . Note that ζℓ2 is the squared
noise to signal ratio for each asset given by Γℓℓ / IQℓℓ , where Γℓℓ is the limit defined inAssumption 6 and IQ refers to the integrated quarticity. We may estimate the variance ofthe microstructure noise by the Realized Variance applied on the tick data divided by 2n,i.e. E(U 2 ) ≃ RV /2n. See Zhang et al. (2005). The square root of the integrated quarticity
is estimated by the Realized Variance applied on the sparsely sampled data e.g. 10 minutes.
We applied the maximum, minimum and average of the above individual bandwidths. Figure
10 and Figure 11 compare the realized covariance and the proposed method in estimating thedaily covariation matrix. Since the first asset is least traded, the refresh time is eﬀectivelyaligned on the trading time of the first asset. In estimating the integrated variance, theproposed method lies between the RV using pairwise refresh time (which will be dominated17

by the microstructure noise) and the RV using all refresh time (which is more sparsely
sampled, therefore less aﬀected by the noise). The most interesting case is the performancein estimating the covariation for assets of diﬀerent liquidity - i.e. (1, 4) and (1, 5)-th elementof the estimator in our case. The daily Realized Covariance take values closer to zero due tothe Epps eﬀect whereas the proposed estimator clearly gives us non-trivial estimates.

5.3

Concluding Remarks

In this section we have shown that the Fourier domain estimator performs well even under
extreme asynchronicity and the presence of microstructure noise. It may be possible to
make further improvements, given the literature on discrete time estimation of the spectral
density. For example, Xiao and Linton (2002) and Hirukawa (2006) developed multiplicative
bias reduction methods that can improve the performance under stronger conditions without
sacrificing positive semi-definiteness.

6

Proofs

We shall derive the results conditionally on the volatility matrix and the discretization time
points, both of which are hence treated as deterministic throughout. Note that C > 0 is
taken to mean some generic constant that may take diﬀerent values in diﬀerent places. In
most cases, proofs are done under the most general framework; that is, when time stamps
are asynchronous and sample sizes are unbalanced. The transaction time of the first andsecond assets is denoted either by {ti,1 , tj,2 } or {ti , sj }.
Lemma 1. Suppose P (t) defined on some filtered probability space (Σ, F, (Ft )t≥0 , P) satisfies
Assumption 1 with µ(t) = 0. Let f (t, s; q) be a bounded, measurable and square-integrable
function. Then, it follows that for any a, b, c, d = 1, . . . , p,
]
[∫
∫
∫
∫
2π

2π

2π

2π

f (t, s; q)dPa (s)dPb (t),E00∫ 2π ∫ 2π=
0

f (t, s; q ′ )dPc (s)dPd (t)

(26)

0

0

f (t, s; q)f (t, s; q ′ ) d[Pa , Pc ](s)d[Pb , Pd ](t)0∫ 2π ∫ 2π+
0

f (t, s; q)f (s, t; q ′ ) d[Pa , Pd ](s)d[Pb , Pc ](t),0where the double stochastic integral is Wiener-Itô sense.
Proof of Lemma 1. From the definition of the double Wiener-Itô integral and standard
properties of symmetrization operation, it is straightforward to see that∫ 2π ∫ 2π
f (t, s; q)dPa (s)dPb (t)00∫ 2π ∫ t∫ 2π ∫ t=f (t, s; q)dPa (s)dPb (t) +f (s, t; q)dPb (s)dPa (t).
0

0

0

0

The integrand here is measurable with respect to Ft , and two terms above are martingale.
Consequently, (26) can be expressed as∫ 2π ∫ t][ ∫ 2π ∫ tf (t, s; q)dPa (s)dPb (t) + 0 0 f (s, t; q)dPb (s)dPa (t),∫ 02π ∫ 0t
∫
∫
.
2π t
′
′
0
0 f (t, s; q )dPc (s)dPd (t) + 0
0 f (s, t; q )dPd (s)dPc (t)18Then one of the cross terms is given by][ ∫ 2π ∫ t∫ 2π ∫ t′
E
f (t, s; q)dPa (s)dPb (t),f (s, t; q )dPd (s)dPc (t)0000) (∫ t)∫ 2π (∫ t′=E
f (t, s; q)dPa (s)f (s, t; q )dPd (s) d[Pb , Pc ](t).
000) (∫ t)∫ 2π (∫ t=
E
f (t, s; q)dPa (s)f (s, t; q ′ )dPd (s) d[Pb , Pc ](t).
000∫ 2π ∫ t=f (t, s; q)f (s, t; q ′ ) d[Pa , Pd ](s)d[Pb , Pc ](t).
0

(27)

(28)

0

Interchange of the expectation operator and the integral is justified by Fubini’s Theorem.
Note that d[Pb , Pc ](t) = [Pb , Pc ]′ (t)dt, where the prime denotes the time derivative. The
same arguments apply to the remaining terms, completing the proof.

Lemma 2. Consider the following kernel-weighted oﬀ-diagonal step functions:∑fn (t, s; q) =e−iqtj kH (ti − tj )1[ti−1 ,ti [ (t)1[tj−1 ,tj [ (s)

(29)

i̸=j

gn (t, s; q) =

n
n ∑
∑

e−iqtj kH (ti − tj )1[ti−1 ,ti [ (t)1[tj−1 ,tj [ (s) ,i=1 j=1

where {ti ; i = 1, (. . . , n} are the discretization
points that satisfy Assumption 2, and the kernel)kH (ti − tj ) =: k (ti − tj )/(H∆t) where k (·) is a lag window satisfying Assumption 3. Then
∫ 2π ∫ 2π {}fn2 (t, s; q) + fn (t, s; q)fn (s, t; q) d[P, P ](t)d[P, P ](s)
0
0
∫ 2π
2
−→ 2∥k∥e−i2tq ([P, P ]′ (t))2 dQ(t),

n
H

(30)

0

where Q(t) is the limit of the empirical quadratic covariation Q(n) (t) = n∑

2
i:ti <t (∆ti ) .

Proof of Lemma 2. We first show that fn can be replaced by gn in the integrand with anerror of order o(1). Given a single asset observed at discretizing points {ti } we have
∫ 2π ∫ 2π0[]
gn (t, s; q) − fn (t, s; q) dsdt
0
∫ 2π ∫ 2π ∑
n ∑
n
∑
=
−
· e−itj q kH (ti − tj ) · 1[ti−1 ,ti [ (t)1[tj−1 ,tj [ (s)dsdt
00
i=1 j=1=∫ 2π ∫ 2π { ∑
n
0

(

≤C

0

j=1

sup ∆tj
1≤j≤n

e

)

−itj qi̸=j}1[tj−1 ,tj [ (t)1[tj−1 ,tj [ (s) dsdt

( )
1
=O
n

(31)

by Assumption 2.

19

Generalising this idea we can rewrite (30) as∫ 2π ∫ 2π {}
gn2 (t, s; q) + gn (t, s; q)gn (s, t; q) d[P, P ](t)d[P, P ](s) + o(1)
0
0
[
n−1
n−h
n ∑∑
2
=
(e−itj 2q + e−itj q e−itj+h q )kH(tj+h − tj )[P, P ]′ (tj+h )[P, P ]′ (tj )∆tj+h ∆tj
HnH

h=0 j=1

+

n−1
∑

n
∑

]
(e

−itj 2q

h=1 j=1+h

=

[ n−1
n ∑
H
+

(

k2

h=0
n−1
∑

k

(
2

h=1

=

[n−1
1 ∑
H
×n

k2

−itj 2q(e

j=1−→ 2∥k∥th /∆t
H

∫ 2π

2

e

2
)kH
(tj−h − tj )[P, P ]′ (tj−h )[P, P ]′ (tj )∆tj−h ∆tj+ o(1)) n−h∑(e−itj 2q + e−itj q e−itj+h q )[P, P ]′ (tj+h )[P, P ]′ (tj )∆tj+h ∆tj
j=1−th /∆t
H

(

h=0
n
∑

th /∆tH+e−itj q −itj−h q

) ∑
n

(e

+e

−itj q −itj−h q

e

′

′

)[P, P ] (tj−h )[P, P ] (tj )∆tj−h ∆tj + o(1)

j=1+h

)

+e

]
−itj 2q

+

n−1
∑

(
k2

h=1
−itj q −itj+h qe−th /∆t
H

(32)

]

)
+1−1

( )
1
)[P, P ] (tj+h )[P, P ] (tj )∆tj+h ∆tj + o(1) + O
n
′

′

(33)

e−it2q ([P, P ]′ (t))2 dQ(t),

(34)

0

It is straightforward to see that the error in approximating the kernel kH (ti − ti−h ) by kH (th )in (32) is negligible. Taylor expanding the lag window we haveti − ti−h − th ′kH (ti − ti−h ) − kH (th ) ≃
k
∆tH
where

(

th
∆tH

)

1
+
2

(

ti − ti−h − th
∆tH

)2
k

′′

(

th
∆tH

)
,

(
)
h
1
ti − ti−h − th ∑supl ∆tl=(∆tl+i−h − ∆tl )≤ 2h= Ch/H.
∆tH
∆tH
∆tH
l=1

Then the approximating error in (32) is bounded by{( ) ( )2 ( )}
∫
n
h ′ h
n ∑ ∑
h
h
1 ∞ 2 ′′′′C sup(∆tj−h ∆tj )k+k≃Cx k (x)dx,
H
H
H
H
H
H −∞
j,h
|h|<n j=1+h

by Assumption 2 and 3. Further, the approximation error from (32) to (33) is given by[ n−1 (]) n−1
∑ ( −th /∆t )n ∑ 2 th /∆t
2
k
+
k
H
H
H
h=0
h=1


h
∑
× 2  (e−itj 2q + e−itj q e−itj+h q )[P, P ]′ (tj+h )[P, P ]′ (tj )∆tj+h ∆tj 
j=1≤ C( sup ∆tj ) n
2

n−1
∑
|h|<n

h 2
k
H

(

h
H

)

1
≃ C
n

20

∫ ∞
−∞

xk 2 (x)dx.

( ) ∑nn ∑2 hLastly, (34) is justified by the fact that both H|h|<n kj=1 ∆tj+h ∆tj andH( h ) ∑nn ∑2
due to the presence of the kernel|h|<n kj=1 ∆tj ∆tj approach to the same limitHH∫∞ 2weights, and by Riemann approximation of the integral −∞ k (x)dx.
Lemma 3. Consider the following oﬀ-diagonal step functions
n1 ∑
n2
∑
e−isj q e−i(ti −sj )α 1[ti−1 ,ti [ (t)1[sj−1 ,sj [ (s)1{Ii,1 ∩Ij,2 =∅} (t, s)
fn (t, s; q) =gn (t, s; q) =i=1 j=1n2
n1 ∑∑e−isj q e−i(ti −sj )α 1[ti−1 ,ti [ (t)1[sj−1 ,sj [ (s),i=1 j=1

where the discretization points {ti , sj } satisfy Assumption 2. Then we have()
∫ 2π ∫ 2π∫ 2π ∫ 2π
1
fn (t, s; q)dsdt =gn (t, s; q)dsdt + O.
n1 ∧ n2
0
0
0
0

(35)

Proof of Lemma 3. The arguments closely follow those in (31). The only diﬀerence isthat here we consider two diﬀerent assets with discretizing points {ti , sj }.
The diﬀerence between the integrals in (35) is given by∫ 2π ∫ 2π ∑
n1 ∑
n2 [
]

1 − 1{Ii,1 ∩Ij,2 =∅} (t, s) e−iqsj e−iα(ti −sj ) 1[ti−1 ,ti [ (t)1[sj−1 ,sj [ (s) dsdt
0

0

=

0

≤

i=1 j=1∫ 2π ∫ 2π ∑
n1 ∑
n2
0

n1 ∑
n2
∑
i=1 j=1∗ 2

e−iqsj e−iα(ti −sj ) 1[ti−1 ,ti [ (t)1[sj−1 ,sj [ (s)1{Ii,1 ∩Ij,2 ̸=∅} (t, s)dsdt

i=1 j=1∆ti ∆sj 1{Ii,1 ∩Ij,2 ̸=∅} ≤ sup (∆ti ) sup (∆sj )1≤i≤n1(s )#{ti ∪ sj , 0 ≤ ti , sj ≤ 2π} = O
=
n1 n2

(

1≤j≤n21
n1 ∧ n2

)

n1 ∑
n2
∑

1{Ii,1 ∩Ij,2 ̸=∅}i=1 j=1,

because the total number of union of time two stamps is of order O(n1 ∨ n2 ).
Lemma 4. Suppose that P (t) is defined on some probability space (Σ, F, (Ft )t≥0 , P) andsatisfies Assumption 1, and denote by G = σ(P ) a sub-σ-field of F. Let Z be a standard
Gaussian variable on a suitable extension of the probability space, and V be the G-measurable
stochastic variance. Then for fn (·) given in Lemma 2,√ ∫ 2π ∫ 2π√nfn (t, s; q)dP1 (s)dP2 (t) =⇒stably VZ,
H 00provided that the conditions of Theorem 3.2 of Jacod (1997) are met.
Proof of Lemma 4. Given the discretized filtration Fi , i = maxj {tj ≤ t} and the discretized sequence adapted to Fi :∑√
χni = n/H∆P1 (ti )∆P2 (sj )kH (ti − sj )e−isj q ,j:sj <ti∫tn
we show stable convergence of Ztn :=maxi {ti ≤t} χi to Zt = 0 vs dWs , a Ft -conditional
Gaussian martingale. Then by Theorem 3.2 of Jacod (1997) we have Z n =⇒ Z stably. For
future reference we note that a suﬃcientcondition for the conditional Lindeberg condition∑in (4) is the Lyapunov condition i E({χni }2+ε |Fi−1 ) →p 0, for some ε > 0. We will show
this for ε = 2 in the proof of Theorem 1.
∑

21

6.1

Proof of Proposition 1

Proof. For the sake of simplicity of notation take a = 1 and b = 2, w.l.o.g.
Using (14) we may write∑[][]
KH (λα ) Fn (dU1 )(α) Fn (dU2 )(q − α)|α|≤m/2=∑

KH (λα )|α|≤m/2

=

n1 ∑
n2
∑

[n
1
∑

][ n
]
2
∑
e−iαti ∆U1 (ti )e−i(q−α)sj ∆U2 (sj )i=1[∆U1 (ti )∆U2 (sj )e−iqsj

=

∑

]

KH (λα )e−iα(ti −sj )|α|≤m/2

i=1 j=1
n2
n1 ∑
∑

j=1

∆U1 (ti )∆U2 (sj )e−iqsj kH (ti − sj ).

(36)

i=1 j=1

We first consider the terms that are not aﬀected by the end points (i.e. tn1 , t0 , sn2 and s0 ).
Expanding the diﬀerence operators we see that those terms are given byn∑
1 −1 n∑
2 −1

]
{
[
)
(
−iqsj
U1 (ti )U2 (sj ) e− kH ti+1 − sj + kH (ti − sj )i=1 j=1+e−iqsj+1[]}
kH (ti+1 − sj+1 ) − kH (ti − sj+1 ) .

(37)

By repeated use of Taylor expansion the terms in the curly bracket of (37) become
{
[
(
)
(
)]
ti − sj + ∆ti+1
ti − sj−isj q
e
−k
+k
H̄
H̄
[ (
)
(
) ]}
t
−
s
+
∆t
−
∆s
t
−
s
−
∆s
i
j
i+1
j+1
i
j
j+1
+ e−isj+1 q k
−k
H̄
H̄
}
{
(
)]
[
(
)]
[
t
−
s
t
−
s
−
∆s
∆t
∆t
i
j
i
j
j+1
i+1 ′
i+1 ′
= − e−isj qk+ e−isj+1 qk+ o(1)
H̄
H̄
H̄
H̄
{
[
(
)
(
)]
∆ti+1 ′ ti − sj − ∆sj+1−isj q ∆ti+1 ′ ti − sj
= −e
k
−
k
H̄
H̄
H̄
H̄
}
[
(
)]
(
)
−isj+1 q ∆ti+1 ′ ti − sj − ∆sj+1−isj q ∆ti+1 ′ ti − sj − ∆sj+1
+e
k
−e
k
+ o(1)
H̄
H̄
H̄
H̄
{
}
[
(
)]
[
(
)]
∆ti+1 ′ ti − sj
−isj q ∆ti+1 ∆sj+1 ′′ ti − sj
−isj q−i∆sj+1 q
= e
k
+e
(e
− 1)
k
+ o(1) ,
H̄ 2
H̄
H̄
H̄
f 12 . In this sequel ∆t
f is taken to mean ∆t
f 12 for the sake of simplicity. Note
where H̄ = H ∆t
1that under Assumptions 2 and 4 ∆ti+1 ∆sj+1 H̄ 2 = o(1) and ∆ti+1 H̄1 = O(H −1 ) = o(1).
When samples are equally spaced and balanced, the curely bracket terms can be simplified
further as follows:
{ }
(
)
(
)
(
)1 ′
i−j−1
−isj q 1 ′′ i − j−isj q −i∆sj+1 q· = −e
k
+e
e
−1
k
.
H2
H
H H
H

22

It now follows that (37) can be written as a sum of the following two terms (up to additionalterms of negligible order o(1)):[()]
n1 −1 n∑2 −1
1 ∑−isj q′′ ti − sj
A1 + A2 = 2e
U1 (ti )U2 (sj )k∆ti+1 ∆sj+1(38)
H̄ i=1 j=1
H̄
[
(
)]
n1 −1 n∑2 −1( −i∆s q)
1 ∑−isj q′ ti − sj
j+1
+
e
· e
− 1 U1 (ti )U2 (sj )k∆ti+1 .
H̄ i=1 j=1
H̄

(39)

We show that A2 is of negligible order of magnitude, and that A1 is the leading term that
dominates in large sample. The expectation of A2 is bounded above by:()
1 −1 n∑2 −1[] ′ ti − s j() n∑1−iq∆s(j+1)
sup 1 − e
E U1 (ti )U2 (sj ) · k
E(A2 ) ≤sup ∆ti+1
f
H̄
j
i
∆tH
i=1 j=1n1 ∨ n2 1= C
H n1 n2≤

C
H(n1 ∧ n2 )

{

[

∑
√
f
|ti −sj |/∆t≤
H

sup

√
f
|ti −sj |/∆t≤
H

(
+

=

sup

√
f
|ti −sj |/∆t>
H

γ

k

′

+

(

ti − sj
f
∆t

]

∑
√
f
|ti −sj |/∆t>Hti − sj
H̄

)

·

)
·

[
]
E U1 (ti )U2 (sj ) · k ′√f|ti −sj |/∆t≤
H

∑
√
f
H
|ti −sj |/∆t>{}C
O(1)O(n1 ∧ n2 ) + o(1) = O
H(n1 ∧ n2 )

(

∑

(

1
H

k

′

(

γ

ti − s jf∆tti − s jH̄(ti − sj
H̄

)

)

) }

)

= o(1),

where in the second and the last equality we used Taylor approximation and the fact thatk ′ (0) = 1 along with Assumption 6, respectively.
Now consider a set S := {i, j, r, l; (ti − tr )/∆t < H µ , (sj − sl )/∆s < H µ } where µ is
some positive constant between (0, 1). Note that maxi,r (ti − tr )/∆t = n, and recall H = nα ,α ∈ (0, 1). We then see that E(A22 ) is bounded above by


(
) (
)

[
]
−2
∑
ti − s jCHtr − slE U1 (ti )U2 (sj )U1 (tr )U2 (sl ) k ′
k′
(40)
2

(n1 ∧ n2 ) H̄H̄
i,j,r,l [] (t − s ) (t − s )
−2
∑
∑
CH
i
j
r
l
=
+
E U1 (ti )U2 (sj )U1 (tr )U2 (sl ) k ′k′(n1 ∧ n2 )2 H̄H̄ci,j,r,l∈S≤

i,j,r,l∈S∑
CH −2
sup EU1 (ti )U2 (sj )U1 (tr )U2 (sl ) ·2(n1 ∧ n2 ) i,j,r,l∈S∑

i,j |h|,|v|<H µ

+

k′

(

ti − sj
H̄

)

k′

(

ti−h − sj−v
H̄

CH −2
n2 n2 sup EU1 (ti )U2 (si−h )U1 (tr )U2 (sr−v ) = B1 + B2 .
(n1 ∧ n2 )2 1 2 i,j,r,l∈S c

When the data is balanced and equally spaced, B1 simplifies to[Csup
EU1 (ti )U2 (si−h )U1 (tr )U2 (sr−v )(n1 ∧ n2 )2 H 2 |i−r|<H µ ,|h−v|<H µ
(
) (
) ]
∑
′ ti − si−h
′ tr − sr−v
k
.
×
k
H̄
H̄
µ
µ
|i−r|<H ,|h−v|<H

23

(41)

)

In particular, when the sample size of two assets is equal, it holds that (ti − si−h )/H̄ =h/H + o(1) under Assumption 2. Hence the squared bracket term in (41) is given by( ) ( )( ) ( )
∑
∑
∑
h
h
v
′
′ v
µ
′
k
k
= 2H n
k
k′
H
H
H
H
µ
µ
|h−v|<H µ|i−r|<H |h−v|<H


( ) (
)
( ) (
)
n−l
n
 ∑ ∑
∑
∑
h−l
h+l
h
h
= 2H µ n
k′
k′
+
k′
k′


H
H
H
H
µ
µ
≤ 4H 2µ n0≤l<H h=1+ln∑ { ( h )}2
′

k

H

h=1

0<l<H h=1.

In the unbalanced case, the Riemann approxmination givesn∑
1 −1 n∑2 −1 {k′(i=1 j=1

ti − sj
H̄

)}2

(

)

= n1 ∧ n2 H

∫ ∞−∞{k ′ (x)}2 dx + o(1),−trand the order of #{0 ≤ i, r ≤ n1 ; ti∆t
< H µ } is the same as when the data is equally spaced
under Assumption 2; therefore, it follows that[
( 2µ−1 )
∫ ∞(
)2 ]
CH −2
H
2µ+1
′
= o(1).
B1 =×
4ρ(0)(n
∧
n
)H
k
(x)
dx
=
O
1
2
(n1 ∧ n2 )2n
1 ∧ n2−∞

Also, we have B2 = CH −2 (n1 ∧ n2 )−2 n21 n22 · sup|τ |>H µ ρ(τ ) which vanishes in large sample asa consequence of (17) in Assumption 6.
We next move on to A1 . Its expectation is given by
(
)
[
]
∑
−2
−isj q ′′ ti − sjH̄E U1 (ti )U2 (sj ) ek∆ti+1 ∆sj+1H̄√f|ti −sj |/∆t≤ H
(
)
[
]
∑
−2
−isj q ′′ ti − sj
+ H̄E U1 (ti )U2 (sj ) ek∆ti+1 ∆sj+1 = D1 + D2 ,H̄√f|ti −sj |/∆t>H

where D2 is bounded above byH̄−2sup(∆ti+1 ) sup(∆sj+1 )
i

(
≤ C

n1 ∨ n2
H2

j

)

sup

√
f
|ti −sj |/∆t>
H

(

sup √

f
|ti −sj |/∆t>Hγ|ti − sj |f∆tand moreover(n1 ∨ n2 )2 s∗ s∗D1 ≤(2π)2 H 2 n1 n2∑√f|ti −sj |/∆t≤H( ∗ )2sn1 ∨ n2=k ′′ (0)22πH (n1 ∧ n2 )EU1 (ti )U2 (sj )

)

·H

∫ ∞

∑
√

k

′′

f
H
|ti −sj |/∆t>(ti − s jH̄k ′′ (x) dx = o(1),−∞[] (t − s )ijE U1 (ti )U2 (sj ) k ′′e−isj qH̄∑√f|ti −sj |/∆t≤
H[]
E U1 (ti )U2 (sj ) e−isj q + o(1)()( ∗ )2n1 ∨ n2 ′′n1 ∨ n2sk (0) Γ12 + o(1) = O
≤
2π
H2
H2
by Assumption 6.
24)

The expectation of the square of (38) is bounded above by


(
)2
n∑
1 −1 n∑2 −1 [
]ti − s j
n1 ∨ n2CE
U1 (ti )U2 (sj ) k ′′2
H (n1 ∧ n2 )H̄i=1 j=1()2∫ ∞( ′′ )2
n1 ∨ n2
2µ+1
=C
ρ(0)
·
4(n
∧
n
)H
k (x) dx + o(1)12H 2 (n1 ∧ n2 )−∞()(n1 ∨ n2 )2 2µ−3=OH.
n1 ∧ n2
)2

(

(42)

All other terms involving the end points are of smaller order by similar arguments given in
Barndorﬀ-Nielsen et al. (2011) along with Assumption 7. The proof is now complete.

6.2

Proof of Proposition 2 and Theorem 1

Proof. The proof consists of three parts; we show stable convergence of the diagonal and oﬀdiagonal terms of the centered estimator in the first and second parts, respectively, followed
by a brief justification for joint convergence via the Cramér-Wold device.
Without loss of generality consider the first element of the centred estimator (without
noise); that is, the term involving the price process of the first asset:
[
]
(
(
))
b P )(q) − F (ΣP )(q)b P )(q) − F(ΣP )(q)E1 = F(Σ= vech F(Σ
11
1
[
][
]
∑
=
KH (λα ) Fn (dP1 )(α) Fn (dP1 )(q − α) − F (Σ11 )(q).
|α|≤m/2

For the sake of simplicity we drop the subscript denoting asset; for example P (tj ) and n are
hereafter taken to mean P1 (tj,1 ) and n1 , respectively. In view of the relationship between
the spectral window K and the lag window k (13), we have the following ‘bias-variance’
decomposition:
] ∫
[ n
][ n
2π
∑
∑
∑
e−i(q−α)tk ∆P (tk ) −e−iqt d[P, P ](t)E1 =KH (λα )e−iαtj ∆P (tj )|α|≤m/2

=

[
n
n ∑
∑
∑
j=1 k=1

=

n ∑
n
∑

j=1

]

KH (λα ) e=∆P (tj )∆P (tk )e

−iqtk

−

∫ 2π

e−iqt d[P, P ](t)0|α|≤ m2kH (tj − tk )∆P (tj )∆P (tk )e

−iqtk

−

∫ 2π
0

j=1 k=1[ n∑(−iα(tj −tk )

0

k=1

)2
∆P (tj ) e−iqtj −

]

∫ 2π
e

−iqt

d[P, P ](t) +0j=1

e−iqt d[P, P ](t)
[

∑

]
−iqtk

kH (tj − tk )∆P (tj )∆P (tk )ej̸=k=: M1 + M2 .
√We first show that n/HM1 converges in probability to zero.
Since Itô’s lemma gives
∫ tj
∫ tj
[
]2
∆P (tj ) e−iqtj = 2{P (t) − P (tj−1 )} · e−iqtj dP (t) +e−iqtj d[P, P ](t),
tj−1

tj−1

25

we can further decompose M1 into a martingale M11 and a predictable finite variation component A, where
M11 = 2A=n ∫ tj∑{P (t) − P (tj−1 )}e−iqtj dP (t) = Op (n−1/2 ),j=1 tj−1n ∫ tj∑(e−itq − e−itj q ) d[P, P ](t) = O(n−1 ).

j=1

tj−1

This is the Euler discretization error whose distribution
is given by√the Theorem 5.5 of Jacod√and Protter (1998). Therefore, it follows that n/HM1 = Op (1/ H) = op (1).
Now we show stable convergence of M2 . Using the oﬀ-diagonal step function fn (t, s; q)defined in Lemma 3:∑
fn (t, s; q) =e−iqtk kH (tj − tk )1[tj−1 ,tj [ (t)1[tk−1 ,tk [ (s),j̸=kwe writeM2 =∑kH (tj − tk )∆P (tj )∆P (tk )e−iqtk∫ 2π ∫ 2π=

fn (t, s; q) dP (s)dP (t).
0

j̸=k

0

Clearly, the expectation of M2 is zero. Furthermore, using Lemma 1 we see that∫ 2π ∫ 2π
E[M2 , M2 ] = 2
fn2 (t, s; q) d[P, P ](s)d[P, P ](t),00whose limiting behaviour is given by Lemma 2. Consequently the distribution of E1 follows
by Lemma 4, provided the conditional ∑Lindeberg condition holds. It suﬃces to verify the
Lyapunov condition: For some ε > 0, j E({χnj }2+ε |Fj−1 ) →p 0, as we shall do here forε = 2.
For later reference we note that the fourth moment of the return is given by([]4 )E ∆P (tj )
=E

( ∫

{

tj

2
(∫

tj−1
tj

= 4E

}
P (t) − P (tj−1 ) dP (t) +

)2

∫ tj
d[P, P ](t)
tj−1

{

)2
(∫
)2
tj
}
∆P (tj ) dP (t) + E
d[P, P ](t)

tj−1

tj−1

(∫
tj

∫ tj ∫ t= 2E

d[P, P ](s)d[P, P ](t) + E
tj−1

(∫

tj−1

tj

= 3E

)2

)2
d[P, P ](t)tj−1d[P, P ]t
tj−1

due to Itô’s lemma and the isometry property.
Now letting√]∑[ n {( −it q)}−itj qnk∆P (tj )∆P (tk )kH (tj − tk ) · e
+e
,
χj :=
H
k<j

26

we can construct an upper bound of E|χnj |4 for j = n:
)}4
th
E
2
∆P (tj )∆P (tj−h ) · k
H
∆tH
h=1
n
( )2 ∑ () ()] ( t )
h
4
4
4 n
= 2
E [∆P (tj )] E [∆P (tj−h )]
k4
H
∆tH

E|χnj |4 ≤

4

{ n
∑

( n )2(h=1+ 24 · 6= 2 ·342n( n )2 ∑

H

h,m=1

n
( n )2 ∑H+ 24 · 6 · 3
×E

(
) (
) (
)
2
2
E [∆P (tj )]4 E [∆P (tj−h )]2 E [∆P (tj−m )]2 kH(th )kH
(tm )
(∫

(∫
tj−m

)2

tj−h

′

[P, P ] (t)dt[P, P ] (t)dt

tj−1

[

H

)2 (∫

′

E

h=1

( n )2

tj

(∫

n
∑

k

tj−h−1tj

E)2[P, P ]′ (t)dt

(∫
tj−h

)

′

th
∆tH
)

)

[P, P ]′ (t)dt

E

tj−1

h,m=1

(
4

tj−h−1([P, P ] (t)dt k 2

tj−m−1

th
∆tH

)

(
k2

tm
∆tH

)]

{
}4
≤ 144 · n2 × sup [P, P ]′ (t) × sup (∆ti 4 )
t
1≤j≤n
[
(
)
(
)∑
(
)]
n
n
n
∑
th
2 ∑ 2
th
tm
4
−1 1
2
k
× H
+ 2
k
k
H
H
∆tH
∆tH
∆tH
m=1
h=1
h=1
(∫ ∞
)
(∫ ∞
)2
= n−2 H −1 C1
k 4 (x)dx + 2n−2 C2
k 2 (x)dx = O(n−2 ) = o(1),
0

(43)

0

∫∞
provided 0 k 4 (x)dx < ∞, where the last line is due to Assumption 2.1. Consequently the
distributional result in Lemma 4 follows.
We now give a result for the oﬀ-diagonal element of the estimator. When time stamps
are synchronous and sample sizes are balanced, the proof is same as the univariate case.
We will give a proof for the most general case; that is, when time stamps are asynchronousand sample sizes are unbalanced. We first show for the bivariate case and will extend theresult to general p × p dimension. Denote the transaction time of the first asset ti,1 = ti andthe second asset tj,2 = sj for the sake of simplicity. The centered estimator in (8) can be
decomposed into, E = M1 + M2 , where∫ 2π∑−isj qM1 =ekH (ti − sj )∆P1 (ti )∆P2 (sj )1{Ii,1 ∩Ij,2 ̸=∅} −e−iqt d[P1 , P2 ](t)
0

i,j

M2 =

∑

e−isj q kH (ti − sj )∆P1 (ti )∆P2 (sj )1{Ii,1 ∩Ij,2 =∅} .

i,j

We first derive the asymptotic bias. Let uij = ti ∧ sj and lij = ti−1 ∨ sj−1 . Then,
E(M1 ) = E(∑e−isj q−Ei,j∫ ui,jdP1 (t)

li,j

i,j

(∑

∫ ui,je−isj qli,j∫ ui,j∫ ui,jdP1 (t)

li,j

li,j

dP2 (s)1{Ii,1 ∩Ij,2 ̸=∅} −

∫ 2π
e
0

−iqt

)
d[P1 , P2 ](t))
dP2 (s){1 − kH (ti − sj )}1{Ii,1 ∩Ij,2 ̸=∅} .

27

By multivariate Itô calculus, conditionally on 1{Ii,1 ∩Ij,2 =∅} , E(M1 ) can be written as the
expectation of following sums
∫ ui,j∫ ui,j∑−isj q−isj qe{P1 (t) − P1 (li,j )}dP2 (t) + e{P2 (t) − P2 (li,j )}dP1 (t)
(44)
i,j

li,j

+

(e−isj q − e−itq )d[P1 , P2 ](t)∑ ∫ ui,j
li,j

i,j

−

∑

−isj qli,j∫ ui,je∫ ui,jdP1 (t)

li,j

i,j

(45)

dP2 (s){1 − kH (ti − sj )}.

(46)

li,j

Using the notation of (3), we see that the order of magnitude of the first term in (44) is∫ ui,j∑−isj qe{P1 (t) − P1 (li,j )}dP2 (t)li,j1≤i≤n1 ,1≤j≤n2=

NT ∫ Tl∑l=1{P1 (t) − P1 (Tl−1 )}dP2 (t) −

Tl−1
−1/2

= Op (NT∑(1 − e−isj q∫ ui,j)−1/2) + Op (n−12 NT

{P1 (t) − P1 (li,j )}dP2 (t)

li,j

i,j

).

The order of the magnitude for the second term in (44) is derived in a similar way. Thechange of discretization points to the union of the time points are without error and holds
analytically. As for the second term we discretize the deterministic function e−itq over the
−1
time
of∑
sj ; the order∫ sj of (45) is therefore Op (n2 ) since we can replace the summation∑ ∫stamp
ui,ji,j li,j by
1≤j≤n2 sj−1 . This term is zero for the integrated (co)variance estimator,q = 0. Now the asymptotic bias term conditional on the volatility path is given by∫ ui,j∫ ui,j∑Ee−isj qdP1 (t)dP2 (s){1 − kH (ti − sj )}1{Ii,1 ∩Ij,2 ̸=∅} 
i,j

li,j

∑

∫ ui,j

li,j

(
)2 }
t
−
s
1
i
j
1{Ii,1 ∩Ij,2 ̸=∅}≃e−isj qd[P1 , P2 ](t) − k ′′ (0)
f
2
∆tH
l
i,j
i,j
}2
(
)2 {
∫ ui,j∑n1 ∨ n2
1 n1 ∧ n2′′−isj q≤sup|ti − sj |1{Ii,1 ∩Ij,2 ̸=∅}k (0)d |[P1 , P2 ]| (t)e(n1 ∧ n2 )H
2
2π
i,j
li,j
i,j
)2
(
∫ 2π
n1 ∨ n2
1 2 ′′A k (0)e−itq d |[P1 , P2 ]| (t).
(47)=(n1 ∧ n2 )H20{

The first approximation holds by Taylor expansion of {k(0) − kH (ti − sj )}, Assumption 2.3−1/2
and k ′ (0) = 0. Then the order of the stochastic bias M1 is given by Op (NT ) + Op (n−1
2 )+
−1/2
n1 ∨n2n1 ∨n22
Op ({ (n1 ∧n2 )H } ) for the estimator at non-zero frequency and Op (NT ) + Op ({ (n1 ∧n2 )H }2 )for the integrated (co)variance estimator. In both cases, the leading order term for the bias
is the last term under the optimal bandwidth.
We next move on to M2 , and write∫ 2π ∫∫ 2π ∫M2 =fn (t, s; q)dP2 (s)dP1 (t) +fn (s, t; q)dP1 (s)dP2 (t),
0

s<t

0

s<t

where fn (t, s; q) is given in Lemma 3. Its expectation is zero, and moreover by Lemma 1 the28expectation of the quadratic variation E [M2 , M2 ] is given by∫ 2π ∫ 2π
E [M2 , M2 ] =fn2 (t, s; q) d[P2 , P2 ](s)d[P1 , P1 ](t)00∫ 2π ∫ 2π+
fn (t, s; q)fn (s, t; q) d[P2 , P1 ](s)d[P1 , P2 ](t).
0

(48)
(49)

0

Now using similar arguments of the proof of Lemma 2, we can show that∫∫n1 ∧ n2 2π 2π 2
fn (t, s; q)d[P1 , P1 ](t)d[P2 , P2 ](s)
H
0
0
∫ 2π
−→ ∥k∥2e−i2tq [P1 , P1 ]′ (t)[P2 , P2 ]′ (t)dQ1122 (t)0∫∫n1 ∧ n2 2π 2πfn (t, s; q)fn (s, t; q)d[P1 , P2 ](t)d[P2 , P1 ](s)
H
0
0
∫ 2π
2
−→ ∥k∥
e−i2tq ([P1 , P2 ]′ )2 (t)dQ1212 (t).

(50)

(51)

0

Hence the proof of Proposition 2 is complete in view of (47), (50) and (51).
Now similar arguments as in the univariate case yields supi E|χni |4 = O((n1 ∧ n2 )−2 ),
where
∑ √
m∆P1 (ti )∆P2 (sj )kH (ti − sj )e−isj q 1{Ii,1 ∩Ij,2 =∅} ;χni =j:sj <tiso stable convergence can be established using Lemma 4 and Proposition 1. Higer-dimensional
extension of the asymptotic results involves the use of the Cramér-Wold device; it is suﬃcient to show that any linear combination of the elements of the matrix estimator convergesbto the corresponding univariate Gaussian random variable. Denote by R(q) := F(Σ)(q)−⊤⊤F(Σ)(q) and consider the linear combinations a R(q)b and c R(q)d, where a, b, c, d aresome arbitrary constant vectors of conformable dimension. Note that a⊤ R(q)cb⊤ R(q)d =vech(ab⊤ )⊤ (R(q) ⊗ R(q))vech(dc⊤ ), and that its expectation depends on E{R(q) ⊗ R(q)}.
So the same arguments as above using lemmas in Section 6.1 complete the proof.

6.3

Proof of Theorem 2

By the triangle inequality we haveΣ(t) −

12π∑|q|≤m/2≤ Σ(t) −(+b
KH (λq ) exp(iqt)F(Σ)(q)

1
2π

2

1
2π

)2

∑

KH (λq ) exp(iqt)F(Σ)(q)|q|≤m/2∑

2(

KH (λq )KH (λℓ ) exp i(q + ℓ)t)b.
F(Σ)(q) − F(Σ)(q)2|q|,|ℓ|≤m/2

where m = n/H. Now Theorem 1 implies that (8) converges in probability to F(Σ)(q) for
each q. So if we assume the modulus of continuity of Σ(t) is available and is given by (22)then there exists a sequence δ(n) → 0 such that
( )
∑
4
b
Σ(t) −KH (λq ) exp(iqt)F(Σ)(q) ≤ Csup,m
δ(n)≤t≤2π−δ(n)|q|≤m/2

2

as required.
29

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31

Table 1: Realized CovarianceSampling
Equal(3/2,30)(3/2,2)(20,30)Sampling
Equal(3/2,30)(3/2,2)(20,30)Realized Covariance 5 min aligned
BIAS
rMSE
(2,2)
(1,2)
(1,1) (2,2)
(1,2)
0.00
(0.00)
0.17
0.17
0.16
0.03
0.01
0.17
0.18
0.16
0.26
0.13
0.31
0.34
0.23
0.01
(0.08)
0.17
0.17
0.17
0.03
(0.08)
0.17
0.18
0.17
0.26
(0.07)
0.32
0.33
0.18
0.01
(0.00)
0.17
0.17
0.16
0.03
0.01
0.18
0.18
0.16
0.26
0.09
0.34
0.35
0.21
0.01
(0.07)
0.17
0.17
0.17
(0.07)
0.18
0.02
0.17
0.17
0.22
(0.04)
0.31
0.30
0.19

NoiseSignal
0
0.001
0.01
0
0.001
0.01
0
0.001
0.01
0
0.001
0.01

(1,1)
(0.01)
0.02
0.24(0.00)
0.02
0.25(0.00)
0.03
0.27(0.00)
0.02
0.24

NoiseSignal
0
0.001
0.01
0
0.001
0.01
0
0.001
0.01
0
0.001
0.01

Realized Covariance Refresh Time aligned
BIAS
rMSE
(1,1) (2,2)(1,2)(1,1) (2,2)(1,2)(0.00)
0.00(0.00)
0.01
0.01
0.01
4.52
4.54
2.10
4.52
4.54
2.10
45.24 45.41 21.07 45.24 45.41 21.07
0.01
0.01
(0.02)
0.08
0.08
0.08
0.23
0.23
0.07
0.25
0.24
0.10
2.24
2.25
0.92
2.25
2.26
0.93
(0.00)
0.00(0.17)
0.02
0.02
0.17
1.78
1.80
0.53
1.78
1.80
0.53
17.83 17.98
6.77 17.84 17.99
6.78
0.01
0.00
(0.27)
0.09
0.07
0.28
0.16
0.15
(0.24)
0.18
0.17
0.25
1.55
1.55
0.08
1.56
1.56
0.15

32

33

0.01

0.001

0

0.01

0.001

0

0.01

0.001

0

NRS

0.6
17.2
3.1
18.2
26.1
34.0

(0.9)
17.4
1.4
17.9
23.5
31.7

(0.1)
17.1
2.2
17.2
24.5
31.6

(6.7)
17.5
(6.1)
18.0
(1.7)
20.4

(8.2)
18.0
(8.0)
18.5
(6.9)
20.3

(7.6)
17.1
(7.7)
17.2
(6.6)
17.9

1.3
17.8
3.7
18.8
26.0
34.3

0.9
17.8
3.8
18.7
27.3
35.2

0.9
16.8
3.3
17.6
25.5
33.0

Realized Cov
5 min(1,1) (1,2) (2,2)

0.8
9.3
4.9
11.0
41.1
43.6

0.4
6.6
9.5
11.7
95.7
96.8

1.0
7.6
10.3
12.9
96.4
97.6

(1,2)

HY

0.2
8.4
2.0
16.3
3.8
24.3

0.0
7.0
2.0
15.7
2.6
24.0

1.0
8.2
1.6
16.7
3.5
25.3

(23.7)
24.8
0.1
15.0
2.2
23.4

(1.9)
7.0
1.8
14.6
2.3
23.3

(1.3)
7.5
0.9
15.2
2.3
23.7

0.7
8.2
2.8
16.2
4.7
26.7

0.6
7.3
3.1
15.6
4.6
25.9

0.8
8.0
1.6
15.4
4.0
25.5

Realized Kernel
Refresh Time(1,1) (1,2) (2,2)

0.9
20.2
1.1
23.3
1.7
32.2

1.0
15.9
1.5
10.5
1.9
18.1

0.9
20.9
1.0
30.2
0.6
43.8

(0.4)
19.1
0.1
22.1
0.6
30.8

(3.1)
14.8
(12.5)
15.5
(1.9)
16.7

(0.6)
19.6
0.8
28.9
(1.2)
40.1

1.8
20.9
1.9
24.2
1.9
34.5

1.4
15.9
5.0
12.2
11.6
21.7

1.4
20.8
1.7
32.3
(1.5)
46.7

FRK
average H(1,1) (1,2)(2,2)

Table 2: 2 dimensional covariation matrix - continuous SV (·/100)

Realized Cov
Refresh Time(1,1) (1,2)(2,2)
Sampling: (3/2,30) Balanced
bias
1.0
(1.9)
0.8
rmse 8.0
7.6
8.1
bias
23.2
7.3
22.9
rmse 24.7 10.6
24.6
bias
225
92.3
225
rmse 22693.6
226
Sampling: (3/2,30) Unbalanced
bias
0.1
(2.6)
0.5
rmse 6.7
6.9
7.0
bias
23.5
6.6
21.3
rmse 24.7
9.6
22.6
bias
239
92.5
210
rmse 24093.9
211
Sampling: (20,30) Unbalanced
bias
0.1 (27.1)
0.7
rmse 8.3
28.0
8.2
bias
16.8 (23.1)
16.2
rmse 19.0
24.4
18.6
bias
162
10.4
155
rmse 164
16.5
156
0.9
20.2
0.8
26.9
1.0
35.8

1.1
22.2
1.0
15.2
1.6
23.9

0.9
20.9
0.9
38.2
0.6
43.8

(0.4)
19.1
0.4
25.9
(0.5)
34.1

(0.1)
21.0
(4.2)
14.4
0.3
22.9

(0.6)
19.6
(0.1)
35.5
(1.2)
40.1

1.8
20.9
1.5
28.7
0.2
39.4

1.3
23.1
2.9
15.3
4.9
25.8

1.4
20.8
0.1
41.2
(1.5)
46.7

FRK
opt H
(1,1) (1,2) (2,2)

34

0.01

0.001

0

0.01

0.001

0

0.01

0.001

0

NRS

(1.4)
16.8
1.4
17.1
24.9
32.2

(1.6)
16.9
0.8
17.6
23.1
32.1

(2.2)
17.2
0.5
16.8
24.2
30.8

(5.4)
21.7
(4.7)
21.6
0.1
22.3

(6.6)
22.0
(6.6)
22.4
(6.1)
24.2

(6.0)
21.9
(5.6)
22.0
(3.6)
22.6

(20.1)
26.3(17.6)
24.7
4.9
20.9

(21.2)
26.1(19.0)
24.7
3.4
20.8

(20.4)
25.4(17.2)
23.0
7.0
19.9

Realized Cov
5 min(1,1) (1,2) (2,2)

(0.0)
19.2
3.7
19.6
40.1
45.5

(0.1)
18.2
10.0
21.1
97.8
100.7

(0.3)
18.6
9.8
21.1
98.8
101.8

(1,2)

HY

0.1
6.8
1.6
14.5
3.0
22.7

(0.2)
7.1
0.7
14.0
1.4
22.1

(0.2)
7.1
1.5
13.4
2.6
19.8

(14.7)
20.4
(0.4)
21.2
1.2
25.8

(1.6)
17.9
0.5
20.7
1.0
24.7

(1.5)
18.3
1.3
22.0
2.1
24.5

(18.0)
19.6(17.9)
21.7(15.9)
24.1(18.8)
20.5(17.5)
21.5(15.7)
23.8(18.6)
20.3(16.9)
21.8(15.8)
25.3

Realized Kernel
Refresh Time(1,1) (1,2)
(2,2)

(0.0)
17.6
0.1
19.7
0.8
30.2

0.2
13.7
1.5
9.2
1.8
15.2

(0.1)
18.0
0.0
29.1
1.4
42.6

(0.9)
23.3
(0.4)
24.5
0.9
30.6

(2.5)
21.1
(8.3)
19.1
(1.6)
21.6

(0.7)
23.4
0.1
29.2
1.3
38.9

(18.9)
24.2(18.5)
25.1(17.8)
32.6(19.1)
23.0(14.6)
18.0
(8.6)
17.3

(18.8)
24.6(19.3)
31.9(18.0)
42.8

FRK
average H(1,1) (1,2) (2,2)

Table 3: 2 dimensional covariation matrix - jump diﬀusion SV (·/100)

Realized Cov
Refresh Time(1,1) (1,2)(2,2)
Sampling: (3/2,30) Balanced
bias (0.3) (2.0)(18.7)
rmse 6.7
18.1
20.3
bias
23.3
8.2
4.9
rmse 24.7 19.8
9.9
bias
233
96.7
211
rmse 234
99.6
212
Sampling: (3/2,30) Unbalanced
bias (0.3) (1.9)(18.8)
rmse 7.1
17.7
20.4
bias
24.0
8.0
2.6
rmse 25.5
19.7
9.3
bias
243
94.1
192
rmse 244
97.1
193
Sampling: (20,30) Unbalanced
bias (0.1) (16.7)(18.0)
rmse 6.8
21.5
19.5
bias
15.3 (13.2)(2.7)
rmse 17.1 19.1
8.9
bias
159
20.6
136
rmse 160
26.8
137
(0.2)
24.2
0.1
23.9
6.2
17.9

0.0
20.1
0.7
12.8
3.8
12.6

(0.1)
18.0
0.0
29.1
10.2
17.2

(1,1)

(0.1)
27.0
(0.0)
26.7
0.7
22.6

(0.5)
24.4
(3.2)
20.5
(3.3)
19.6

(0.7)
23.4
0.1
29.2
(0.7)
21.2

(18.9)
28.3(18.8)
28.0(10.5)
17.7(18.9)
25.3(17.5)
21.6
2.6
14.8

(18.8)
24.6(19.3)
31.9(10.0)
17.1

FRK
opt H
(1,2)
(2,2)

Table 4: Scalar function of 10 dimensional covariation matrix
max (eigenvalue)
portfolio
Noise to Signal Ratio=0
Bias rMSE
Bias(2.34)
2.75
1.76
RV refresh
RV fixed(0.85)
3.18
0.14
Realized Kernel(2.21)
2.65
1.66
Fourier RK(1.18)
2.17
0.26
Noise to Signal Ratio=0.001
7.22
7.50
27.20
RV refresh
RV fixed
0.40
3.28
4.14
Realized Kernel
0.38
3.00
1.67
Fourier RK minH
(0.28)
1.95
3.88
avgH
(0.47)
2.64
0.73
maxH
(0.52)
3.13(0.20)
Noise to Signal Ratio=0.01
127.24 127.81
256.02
RV refresh
RV fixed
15.42
16.54
40.37
Realized Kernel
1.29
4.82
3.23
Fourier RK minH
1.22
3.27
6.88
avgH
0.13
4.02
1.76
maxH
(0.03)
4.92
0.67

35

rMSE
2.74
4.09
2.67
2.51
27.47
6.31
4.16
4.87
3.67
3.99
257.05
41.99
6.86
8.55
5.58
5.96

36

a
Comparison of 2 × 2 covariance matrix and its scalar function estimated by Realized Kernel and Fourier Realized Kernel for range of bandwidths. Sample size of two
assets are unbalanced and time stamps are poisson sampled at rate = {3/2, 30}.

Figure 6: Simulation Result : Balanced, Sampled at = {3/2, 30}. aFigure 7: Simulation Result : Unbalanced, Sampled at = {3/2, 30}

37

Figure 8: Simulation Result : Unbalanced, Sampled at = {20, 30}

38

Figure 9: Simulation Result : Unbalanced, Sampled at = {3/2, 2}

39

40

(co)variance estimate

5

15

20

Day

15

20

−0.01

0

0.01

0.02

5

10

2,3

15

20

−0.02

0

0.02

0.04

−0.05

−0.04
10

0

−0.02

5

0.05

0

0.1

−0.1

0

0.02

−0.05

0.05

1,4

0

0.1

10

0.05

0.2

0.15

0.1

0.25

1,1−th element

0

5

5

5

10

2,4

10

1,5

10

1,2

15

15

15

20

20

20

Figure 10: Covariation matrix estimates : element (1, 1) to (2, 5)

−0.02

0

0.02

0.04

0.06

0.05

0.1

0.15

0.2

0.25

−0.06

−0.04

−0.02

0

0.02

5

5

5

10

2,5

10

2,2

10

1,3

15

15

15

RV, pair refresh

RV, all refresh

FRK, H avg

FRK, H max

FRK, H min

20

20

20

41

(co)variance estimate

0

0.1

0.2

0.3

0

0.1

0.2

0.3

5

5

10

4,4

10
Day

15

15

RV, pair refresh

RV, all refresh

FRK, H avg

FRK, H max

FRK, H min

3,3−th element

20

20

20

20

0.1

0.12

0.14

0.04
0.02

0.01
0

0.06

15

15

0.02

10

4,5

10

0

0.01

0.02

0.03

0.04

0.08

5

5

3,4

0.03

0.04

0.05

0.06

−5

0

5

10

15

x 10

−3

Figure 11: Covariation matrix estimates : element (3, 3) to (5, 5)

5

5

10

5,5

10

3,5

15

15

20

20