Estimation and Testing for Dependence of Market
Microstructure Noise
Masato Ubukata and Kosuke Oya
Graduate School of Economics, Osaka University
October 17, 2008

Abstract
Market microstructure noise, which is induced by market frictions such as the bid-ask
bounce and the discreteness of the price change, generates deviations of observed price fromfrictionless equilibrium price. Estimation of the noise dependence is of direct economic interest because it sheds light on market microstructure effects such that the bid-ask bounce andthe clustering of order flow are related to negative and positive autocorrelations of the noises,
respectively. This paper proposes a test statistic for the dependence and cross and auto covariance estimators of the bivariate noise processes, and derives their asymptotic distributions.
The asymptotic distributions provide another test statistics for statistical significance for the
cross and auto covariances. Monte Carlo simulation shows the covariance estimators and test
statistics provide good performance in a finite sample. An empirical illustration confirms the
proposed statistics and estimators capture various dependence patterns in market microstructure noise.

Keywords: test statistic; market microstructure noise; time dependence; nonsynchronous observations; high-frequency data.

JEL classification: C12; D49

1

1

Introduction

Market microstructure noise is induced by some market frictions: the bid-ask bounce, the discreteness of price change and asymmetric information of traders, etc. A study about each of thesemarket frictions have been pioneered in market microstructure literature. Roll (1984) derives a
simple measure of estimating the bid-ask spread based on negative autocovariance of observed
return. Other covariance spread model is established by Stoll (1989) and George, Kaul and Nimalendran (1991). A related literature includes inferences about the decomposition of the bid-ask
spread using an indicator function driven by the direction of trade as in Glosten and Harris (1988),
Huang and Stoll (1997) and Madhavan, Richardson and Roomans (1997). Harris (1990) studies therounding effects by the discreteness of transaction price. The asymmetric information of traderscauses adverse selection components within bid-ask spread in both market maker system and limit
order market (see, e.g., Glosten and Milgrom 1985, Easley and O’Hara 1987, Copeland and Galai
1983, Foucault 1999 and Glosten 1994).
A recent microstructure noise analysis has provided a way to measure how much impact thenoise causes to transaction price because it deviates from frictionless equilibrium price by microstructure effects. Bandi and Russell (2006) propose an estimator to evaluate full-information
cost which is a difference between the transaction price and the unobserved price reflecting all
private and public information about the asset. Medahhi (2002) and Hansen and Lunde (2006)
study the variance of noise and correlation between the noise and an instanteneous volatility of
semimartingale price process which appear in the market with no frictions.
Many empirical analyses exhibit autocorrelations and read-lag relationship of cross-correlationin observed intraday returns and these correlations could be induced by the dependence of microstructure noises. This paper studies a degree of the dependence of the microstructure noise.
The main contributions and results of the study are the followings: we propose a test statistic
for the dependence and cross and auto covariance estimators of bivariate noise processes and derive their asymptotic distributions. Furthermore, we provide test statistics for significance of thevariance and covariance of the noise. We confirm that the proposed cross and auto covariance2

estimators and test statistics have a good performance in finite samples through Monte Carlo simulation. As an empirical illustration, the statistics are applied to high-frequency asset prices on
the Osaka Securities Exchange and we find various dependence patterns in market microstructurenoise.
The measurement of the noise dependence enables bias correction for the integrated varianceand covariance estimation, which is important for option pricing, the measurement of value-atrisk, and portfolio evaluation. We must be careful to the noise which contaminates high-frequency
prices. Realized variance and covariance are not necessarily the best approaches.
In the literature on integrated variance estimation with noise, Zhou (1996) proposes a kernelbased estimator, and Zhang, Mykland and Aı̈t-Sahalia (2005) suggest two scales of realized variance as a linear combination of realized variances at two frequencies, with Zhang (2006) extending
this estimator to multiple scales. Although these studies are conducted under an i.i.d. noise assumption, market microstructure noise possibly has time dependence. Under the dependent noise
assumption, Aı̈t-Sahalia, Mykland and Zhang (2006) modify the two- and multiple-scale realized variances, and Hansen and Lunde (2005, 2006) and Barndorff-Nielsen, Hansen, Lunde andShephard (2008) develop the kernel-based estimator. Bandi and Russell (2008) show the optimal
frequency based on the minimization of the mean squared error. For integrated covariance estimation with dependent noise, Voev and Lunde (2007) show that Hayashi and Yoshida’s (2005)
cumulative covariance estimator is biased in the presence of cross-correlated noises and propose
modified cumulative covariance estimators based on kernel and subsampling methods. However,
it is important that the adequacy of these estimators such as their unbiasedness, consistency andefficiency depend on the dependence structure of the noise process. Therefore, we should knowwhether the market microstructure noise is time dependent and how the degrees of the dependence
are for the estimation of integrated variance and covariance.
Voev and Lunde (2007) propose a test statistic for the cross-sectional dependence of the noises
in order to determine the kernel bandwidth of their estimator. The main differences from Voev andLunde (2007) and our study are a choice of the intervals used for the cross-covariance estimationof the noises and an evaluation of a variance of the cross-covariance estimator of the noises. First,
3

Voev and Lunde’s (2007) cross-covariance estimator is based on subsampled interval whose lengthis large enough in order to guarantee the unbiasedness of the estimator. If the interval is too wide,
the cross-covariance estimator may have a large variance. To take care of it, we propose a crosscovariance estimator based on the interval with proper length where the dependence of the noise
disappears. The length of the interval is determined by a testing procedure proposed in this paper.
Second, although Voev and Lunde (2007) show the unbiasedness of the kernel-based cumulative
covariance estimator under dependent noise, their t-statistic uses an approximated variance of thecross-covariance estimator under i.i.d. noise. As suggested in their paper, it is natural that thisapproximation leads to t-statistics that are somewhat larger than they should be. On the other
hand, the test statistic derived in this paper does not require the i.i.d. noise approximation by using
subsampling methods.
The paper itself proceeds as follows. In section 2, the transaction price model and properties ofmarket microstructure noise are presented under a framework of high-frequency financial analysis.
We propose the test statistic for the cross-sectional dependence of noises in section 3. We provide
the cross and auto covariance estimators of the bivariate noise processes, their asymptotic distributions, and the test statistics for their significance in section 4. Section 5 includes a simulation
experiment and an empirical illustration. We conclude the paper with an appendix that provides
proofs for the several lemmas and theorems.

2

Price process and market microstructure noise

We assume logarithmic equilibrium price processes of two assets, {P1∗ (t)} and {P2∗ (t)}, which
follow two-dimensional Itô process without drift on a probability space (Ω, F, P):
dPl∗ (t) = σl (t)dWl (t), l = 1, 2, t ∈ [0, T ],d⟨W1 , W2 ⟩t = ρ∗ (t)dt,

4ρ∗ (t) ∈ (−1, 1),(1)where W1 and W2 are standard Brownian motions. We assume that the initial value of the pricePl∗ (0) is a constant and σl (t) > 0 is a bounded progressively measurable function. {P1∗ (t)} and{P2∗ (t)} correspond to the logarithmic prices which appear in the market without trading imperfections, frictions and informational effects.
We make some notation to represent intraday returns and an irregularly nonsynchronous trading. r1,i := P1 (ti ) − P1 (ti−1 ) is the i-th observed intraday return of an asset 1, and r2,j :=P2 (sj ) − P2 (sj−1 ) is the j-th one of an asset 2, where ti and sj are the end times of the i-th andj-th intervals. The different notations of the transaction times for the two assets are due to thenonsynchronous trading. A simple way to model high-frequency transaction price is to use hidden semimartingale processes, as named by Mykland and Zhang (2005). In this framework, thelogarithmic transaction price Pl is observed with market microstructure noise as follows:
P1 (ti ) = P1∗ (ti ) + η(ti ), P2 (sj ) = P2∗ (sj ) + δ(sj ),(2)

where Pl∗ (t) is the logarithmic equilibrium price described in (1). η(ti ) and δ(sj ) are the market
microstructure noises in asset 1 and 2 which have many sources, including the presence of bid ask
spreads, the discreteness of prices change and the difference in trade sizes, etc. We assume themarket microstructure noises have the following properties.
Assumption 1. Market microstructure noise.
Let a vector of market microstructure noise of asset 1 and 2 be u(t) = (η(t) δ(t))′ .
(1a) {u(t)} is a sequence of random variables with zero mean.
(1b) The bivariate noise processes are covariance stationary with autocovariance function, which
has finite dependence in the sense that: γη (ℓ) γηδ (ℓ) Γ(ℓ) = E[u(t)u′ (t − ℓ)] =  = 0, f or all | ℓ |> m.
γδη (ℓ) γδ (ℓ)
m is a finite positive number. Γ(ℓ) is a covariance matrix with finite elements.
5

¯
¯4β
(1c) There exists some positive number β > 1 that satisfies E¯u(t)u′ (s)¯ < ∞ for all t, s.
(1d) The noise process is independent of the equilibrium price process. Pl∗u(t), l = 1, 2.

To avoid complication of the subscript for (1b), γηδ (ℓ) and γδη (ℓ) are rewritten as γ(ℓ) and γ(−ℓ)
because of γηδ (ℓ) = E[η(t)δ(t − ℓ)] and γδη (ℓ) = E[η(t − ℓ)δ(t)] = E[η(t)δ(t + ℓ)]. The autoand cross correlation coefficients of the two noises are defined as ρη (ℓ), ρδ (ℓ) and ρ(ℓ). For (1d),even if Pl∗ and u(t) are correlated, the dependence between the noises generally dominates thedependence between the equilibrium price and noise as the number of high-frequency observations increases. Furthermore, Hansen and Lunde (2006) suggest that the independence assumption
between the equilibrium price and noise does not statistically damage the analysis of asset prices
with high trading intensities.
Our interest of the study is to estimate the covariance matrix in Assumption (1b) and characterize the asymptotic and finite sample properties of the auto and cross covariance estimators ofbivariate noise processes. A test statistic for the cross-sectional dependence of the noises using
the subsampling method is introduced in section 3. We provide the auto and cross covariance estimators of the bivariate noise processes, their asymptotic distributions, and test statistics for theirsignificance in section 4.

3

Cross-sectional dependence of noises

Microstructure noise as well as equilibrium price are unobservable. We introduce a simple identification approach to measure cross-sectional dependence of the noises and propose a test statistic
to detect a distance where the dependence of the noises disappears.

3.1 How to measure cross-sectional dependence of noises
First we define expectation and variance conditional on the stochastic arrival times. Denote theconditional expectation and variance given intervals I i := (ti−1 , ti ] and J j := (sj−1 , sj ] for all i, j
as EIJ [ · ] and VIJ [ · ]. We cannot identify the covariation of the equilibrium price processes that6

have martingale properties and the covariance of the market microstructure noises in the pair ofoverlapping intervals because the covariation of the equilibrium price processes is not zero. On theother hand, in the pair of the nonoverlapping intervals {I i ∩ J j = ∅}, we have:

EIJ

[∑

]
[∑
]
eη,i eδ,j 1{I i ∩J j =∅} ,r1,i r2,j 1{I i ∩J j ]=∅} = EIJ

(3)

i,j

i,j

where eη,i := η(ti ) − η(ti−1 ) and eδ,j := δ(sj ) − δ(sj−1 ). The product of returns on the nonoverlapping intervals is used to identify the covariance of the noises as in Voev and Lunde (2007).
For the nonoverlapping intervals {I i ∩ J j = ∅} and ti−1 − sj > 0, the distance between
the intervals is defined as ℓ = ti−1 − sj . In the case of sj−1 − ti > 0, the difference of thenonoverlapping intervals is denoted by ℓ = −(sj−1 − ti ). In Figure 1, the top panel (a) and thelower panel (b) illustrate the former and latter cases, respectively. The nonoverlapping adjacentintervals such that ti−1 − sj = 0 or sj−1 − ti = 0 are used in the case of ℓ = 0. In what follows, weconsider the case of ℓ > 0 because, for the other cases, we have only to replace the corresponding
definition of ℓ. It is noted that ℓ = ti−1 − sj > 0 implies {I i ∩ J j = ∅}. We define the product ofreturns on the i-th and j-th intervals satisfying ti−1 − sj = ℓ > 0 as follows:
Zℓ,ij = r1,i r2,j , for all, i, j, such that ti−1 − sj = ℓ.

(4)

The conditional expectation of Zℓ,ij is:
]
[
]
[
]
[
[
]
]
[
EIJ Zℓ,ij = EIJ η(ti )δ(sj ) − EIJ η(ti )δ(sj−1 ) − EIJ η(ti−1 )δ(sj ) + EIJ η(ti−1 )δ(sj−1 )= γ(ℓ + ∆ti ) − γ(ℓ + ∆ti + ∆sj ) − γ(ℓ) + γ(ℓ + ∆sj ),(5)where ∆ti := ti − ti−1 and ∆sj := sj − sj−1 . For all ℓ taking more than a large enough L such][that γ(L) = 0, we obtain γ(ℓ) = 0 and EIJ Zℓ,ij = 0 from (1b) in Assumption 1. Now supposes∗ := mins {s | γ(L − s) ̸= 0, s ≥ 0}. This implies γ(L) = γ(L − 1) = γ(L − 2) = · · · =
[
]
[
]
[
]
γ(L − s∗ + 1) = 0 and γ(L − s∗ ) ̸= 0, and EIJ ZL,ij = EIJ ZL−1,ij = EIJ ZL−2,ij = · · · =[][]EIJ ZL−s∗ +1,ij = 0 and EIJ ZL−s∗ ,ij ̸= 0 because of γ(L − s∗ ) ̸= 0. Denote ℓ∗ = L − s∗ .
7

Then we have γ(ℓ∗ + 1) = 0 and γ(ℓ∗ ) ̸= 0 and conclude that the distance ℓ∗ is the threshold[]value of dependence of the noises. We note that whether EIJ Zℓ,ij = 0 does not necessarily imply
whether γ(ℓ) = 0. This is because the sum of all cross-covariances in (5) possibly takes a valueof zero, even where γ(ℓ) ̸= 0. To avoid this situation, we apply the method of determination forℓ∗ described above. Thus, a test statistic for determining the threshold value can be constructed byusing a sample mean of Zℓ,ij , which satisfies the nonoverlapping intervals with the distance ℓ.

3.2

Test statistic for cross-sectional dependence of noises

From Assumption (1b), the dependence of the two noises disappears when the noises are sufficiently separated. In this subsection, we propose a test statistic to detect the threshold value as
in the previous subsection. For the construction of the test statistic, we define a sequence thatℓarranges Zℓ,ij in ascending order of index i as {Zℓ,k }N
k=1 . Nℓ is the total number of the productsof returns on the nonoverlapping intervals with the distance ℓ. We define the k-th pair of the selected intervals as Ak and Bk . Then Zℓ,k is defined as a product of returns on nonoverlapping
intervals Ak and Bk . Figure 2 illustrates each pair of intervals (Ak , Bk ), (Ak+1 , Bk+1 ) and (Ak+2 ,∑ ℓ
Bk+2 ) for k = 1. We define the sample mean of Zℓ,k ; that is Z̄ℓ,Nℓ := N1ℓ N
k=1 Zℓ,k , as the esti(
)
[
] 1/2
mator of EIJ [Zℓ,k ]. Let fℓ,Nℓ := Z̄ℓ,Nℓ − EIJ Z̄ℓ,Nℓ Nℓ be the theoretical standardization forZ̄ℓ,Nℓ . We make the following assumption for the asymptotic variance of fℓ,Nℓ which is given by
[(
)2 ]
2
limNℓ →∞ EIJ fℓ,Nℓ= σℓ,f:[]∑ ′ +n2
Assumption 2. VIJ n−1/2 kk=k
→ σℓ,f, uniformly in any k ′ , as n → ∞. This means that′ +1 Zℓ,k¯¯ [ −1/2 ∑k′ +nN
]
2 ¯
ℓ
−
σ
Z
for any sequence {nNℓ } that tends to infinity with Nℓ , supk′ ¯VIJ nNℓ′ℓ,kℓ,f → 0k=k +1
as Nℓ → ∞.
ℓ
Next we consider the asymptotic normality of the estimator of EIJ [Zℓ,k ]. It is noted that {Zℓ,k }N
k=1

is a sequence of dependent and heterogeneously distributed random scalars because the variancedepends on the length of the irregularly observed interval and {Zℓ,k } is serially correlated. We
obtain the following lemma for the asymptotic normality of the estimator of EIJ [Zℓ,k ].

8

Lemma 1. Suppose Assumptions 1 and 2 hold. As Nℓ goes to infinity, we have:fℓ,Nℓ a→ N (0, 1).
σℓ,f

(6)

The proof is given in the Appendix.
2
is unknown, we can construct a consistent estimator
Although the asymptotic variance σℓ,f2of σℓ,fby applying a subsampling method first proposed by Carlstein (1986). Carlstein (1986)

considers variance estimation for a general statistic without specifying the dependence in a stationary sequence. Fukuchi (1999) and Politis, Romano and Wolf (1999) extend their results toheteroskedastic observations.
ℓ
We define the h-th subseries using {Zℓ,k }N
k=1 as follows:()[]{Zℓh } := Zℓ,hMℓ +1 , Zℓ,hMℓ +2 , · · · , Zℓ,(h+1)Mℓ , 0 ≤ h ≤ Kℓ − 1, Kℓ = Nℓ /Mℓ ,where [ · ] denotes the integer part of a real number, Mℓ is a number of observations within eachsubseries {Zℓh } and Kℓ is a total number of subseries. The superscript h of {Zℓh } represents thatthe subseries take the sample Zℓ,k with k = hMℓ + 1, · · · , (h + 1)Mℓ . The variance estimator isgiven by:
Mℓ
2
σ̂ℓ,f
=
Kℓ

K
ℓ −1 (
∑

h
Z̄ℓ,M
−
ℓ

h=0

Kℓ −1
)2
1 ∑
h
Z̄
,
Kℓ h=0 ℓ,Mℓ(7)hwhere Z̄ℓ,M
is a sample mean of subseries {Zℓh }. We have the following lemma for the varianceℓ2estimator σ̂ℓ,f.

Lemma 2. Suppose Assumptions 1 and 2 hold. Let Mℓ be s.t. Mℓ → ∞ and Mℓ /Nℓ → 0 as
Nℓ → ∞. Then we have:22→L2 σℓ,fσ̂ℓ,f

as Nℓ → ∞.

The proof is described in the Appendix.
9(8)

The purpose of this subsection is to construct a test statistic to detect the threshold value of thedependence of the noises. The test statistic is derived from the results of Lemmas 1 and 2. Let thenull hypothesis be EIJ [Zℓ,k ] = 0 for all k, for given ℓ. The alternative hypothesis consists of all
possible deviations from the null. Then we have the following theorem for the test statistic.
Theorem 1. Suppose Assumptions 1 and 2 hold. As Nℓ goes to infinity, we have:√τ (ℓ) :=

Nℓ Z̄ℓ,Nℓ a→ N (0, 1)
σ̂ℓ,f

(9)

under the null hypothesis. τ (ℓ) diverges under the alternative.
The asymptotic normality of the test statistic follows directly from Lemmas 1 and 2.
The large numbers of Mℓ and Kℓ are available for the variance estimation in (7) because thehigh-frequency transaction data yield a large number of Nℓ . However, it is difficult to determine2
the optimal numbers of Mℓ and Kℓ that minimize the mean squared error of σ̂ℓ,fbecause we do

not know the covariance structure of the noises. For fixed Nℓ , an increase in Mℓ (i.e. a decrease2
in Kℓ ) reduces the bias but increases the variance of σ̂ℓ,f. It is known that the optimal asymptotic
1/3

rate of Mℓ is proportional to Nℓ
1/3

formula is Mℓ = cNℓ

for the subsampling variance estimation; that is, the asymptotic

where c depends on the noise process. We investigate the influence of the
1/3

numbers of Mℓ and Kℓ selected under Mℓ = cNℓwith some c through Monte Carlo simulationin section 5.
Next, we summarize the procedure to identify the threshold value where the cross-covarianceof noises becomes zero. We first test the null hypothesis EIJ [ZL,k ] = 0 for all k with a large value
L using the test statistic (9). The null hypothesis EIJ [ZL,k ] = 0 would not be rejected because
the cross-covariance between sufficiently separated noises is zero from (1b) of Assumption 1. If
EIJ [ZL,k ] = 0 is not rejected as expected, we then test whether EIJ [ZL−1,k ] is zero. If EIJ [ZL−1,k ] = 0
is not rejected, we proceed to judge the statistical significance of EIJ [ZL−2,k ]. We continue to test
sequentially until the null is rejected. Finally we regard the distance where the null is rejected thefirst time as ℓ∗ = maxℓ {|τ (ℓ)| > c.v.} where c.v. is the critical value of the test statistic (9).

10

4Consistent covariance estimators of noises

Once the threshold value ℓ∗ as described in the section 3 has been selected, we find that the crosscovariance of bivariate noise processes γ(ℓ) is zero when ℓ ≥ ℓ∗ . However, γ(ℓ) is still unknown
for ℓ < ℓ∗ . In subsection 4.1, we derive the cross-covariance estimator of bivariate noise processes
and its asymptotic distribution, and propose a test statistic for statistical significance of the crosscovariance. The related statistics for autocovariance of univariate noise process are established insubsection 4.2.

4.1

Cross-covariance estimator of noises

We consider the estimation of the cross-covariance γ(ℓ) in this subsection. For construction ofan unbiased estimator, we only have to remedy the nonoverlapping intervals so that all crosscovariances in (5) except γ(ℓ) become zero by using the threshold value determined through thetest statistic (9). We use m+ and m− for the threshold values in the cases of ℓ > 0 and ℓ < 0instead of ℓ∗ . Suppose the bivariate noise processes have finite cross-sectional dependence in the(+)

sense that γ(ℓ) = 0 for ℓ > m+ > 0 and for −ℓ > m− > 0. When ℓ is positive, we define ti(+)

the first transaction time of asset 1, which follows ti subject to ti

as

(+)

− sj > m+ and sj−1 as the last(+)

transaction time of asset 2, which is followed by sj−1 subject to ti−1 −sj−1 > m+ . As ℓ is negative,(−)

we define sj

as the first transaction time of asset 2, which follows sj subject to sj −ti > m− and(−)

ti−1 as the last transaction times of asset 1, which is followed by ti−1 subject to sj−1 − ti−1 > m− .
( (+) )(+)(−)(+)
The returns on the intervals (ti−1 , ti ] and (ti−1 , ti ] are denoted by r1,i := P1 ti− P1 (ti−1 )( (−) )(−)(−)(+)
and r1,i := P1 (ti ) − P1 ti−1 . For asset 2, the returns on the intervals (sj−1 , sj ] and (sj−1 , sj ]( (−) )
( (+) )(−)(+)
are denoted by r2,j := P2 sj− P2 (sj−1 ) and r2,j := P2 (sj ) − P2 sj−1 , respectively. Then
(−)

(−)

(±)

Zℓ,ij , which modifies Zℓ,ij in (4), is defined as follows:

(±)

Zℓ,ij


(+) (+)

r1,i r2,j 1{ti−1 −sj =ℓ}
if ℓ > 0(+) (+)(−) (−)=
r1,i r2,j 1{ti−1 −sj =0} + r1,i r2,j 1{sj−1 −ti =0} if ℓ = 0 r(−) r(−) 1{s −t =−ℓ}
if ℓ < 0
2,j
1,i
j−1
i

11

(10)

The top panel (a), the middle panel (b) and the lower panel (c) in Figure 3 illustrate each pair ofintervals with ℓ > 0, ℓ = 0 and ℓ < 0, respectively. For all ℓ > 0, cross-covariances where thedistances between the noises are further than m+ are zero. The nonoverlapping intervals described(±)

in (b) and (c) are given by the same idea as in (a). The conditional expectation of Zℓ,ij where[ (±) ](±)
the indicator function takes one is EIJ Zℓ,ij = −γ(ℓ). We select Zℓ,ij for all i, j such that the(±)

indicator function takes a value of one and define a sequence that arranges the selected Zℓ,ij in(±)ℓ
ascending order of index i as {Zℓ,k }N
k=1 . Then the cross-covariance estimator is given by:

ℓ
1 ∑
(±)
γ̂(ℓ) = −Zℓ,k .
Nℓ k=1

N(11)

The following theorem states the asymptotic normality of the cross-covariance estimator γ̂(ℓ).
Theorem 2. Suppose Assumptions 1 and 2 hold. Then we have:a

1/2Nℓ (γ̂(ℓ) − γ(ℓ)) → N (0, ωℓ2 ),where ωℓ2 = limNℓ →∞ EIJ

[{

(12)

}2 ]
1/2
Nℓ (γ̂(ℓ) − γ(ℓ)) .

The proof is described in the Appendix. Then the subsampling variance estimator is given by:

ω̂ℓ2

(±),h

Kℓ −1 (
Kℓ −1
)2
Mℓ ∑
1 ∑
(±),h
(±),h
=
Z̄ℓ,Mℓ −Z̄ℓ,Mℓ ,Kℓ h=0Kℓ h=0(±),hwhere Z̄ℓ,Mℓ is a sample mean of subseries {Zℓ
(±),h

{Zℓ

(13)

} and the h-th subseries is:

( (±)
)
[
]
(±)
(±)
} := Zℓ,hMℓ +1 , Zℓ,hMℓ +2 , · · · , Zℓ,(h+1)Mℓ , 0 ≤ h ≤ Kℓ − 1, Kℓ = Nℓ /Mℓ .

The asymptotic distribution of γ̂(ℓ) established in Theorem 2 and the subsampling variance estimator provide the test statistic for the null hypothesis γ(ℓ) = 0 and the alternative γ(ℓ) ̸= 0.
Corollary 1. The test for cross-covariance of market microstructure noises.

12

As Nℓ goes to infinity s.t. Mℓ → ∞ and Mℓ /Nℓ → 0, we have:√∗τ (ℓ) :=

Nℓ γ̂(ℓ) a→ N (0, 1)
ω̂ℓ

(14)

under the null hypothesis. τ ∗ (ℓ) diverges under the alternative.
It is noted for the integrated covariance estimation that the bias of cumulative covariance estimatorby Hayashi and Yoshida (2005) is virtually zero when the test statistic (14) does not reject thenull γ(ℓ) = 0 for all ℓ. Even if the cumulative covariance estimator is unbiased, the noise has
a strong impact on its variance. An estimator applying subsampling method to the cumulative
covariance estimator by Hayashi and Yoshida (2005) is preferable for such case. It is the same as
the subsampling estimator by Voev and Lunde (2007).

4.2 Autocovariance estimator of noise
In the previous subsection, we have proposed the cross-covariance estimator of the noises and havederived the test statistic for the significance of the cross-covariance. This framework is applicablefor estimation of the autocovariance of a univariate noise process. We briefly describe a consistentautocovariance estimator of the noise in this subsection. We start with the construction of thetest statistic to find the threshold value of the dependence of univariate noise process. Define theproduct of the returns on the i-th and j-th intervals for asset 1 satisfying ℓ = tj−1 − ti ≥ 0 as
follows:

Z1,ℓ,ij = r1,i r1,j ,for all i, j, such that tj−1 − ti = ℓ ≥ 0.

(15)

N

1,ℓ
Let {Z1,ℓ,k }k=1
be a sequence that arranges Z1,ℓ,ij in ascending order of index i. N1,ℓ is the total

number of a sequence {Z1,ℓ,k }. The null hypothesis is EIJ [Z1,ℓ,k ] = 0 for all k for given ℓ, which
implies γη (ℓ) = 0, and the alternative hypothesis consists of all possible deviations from the null.

13

The test statistic for this hypothesis is given by:√τη (ℓ) :=N1,ℓ Z̄1,ℓ,N1,ℓ,
σ̂1,ℓ,f

(16)

[(
(
)2 ]
2
2
σ̂1,ℓ,f
is a subsampling estimator of σ1,ℓ,f= limN1,ℓ →∞ EIJ f1,ℓ,N1,ℓ , where f1,ℓ,N1,ℓ := Z̄1,ℓ,N1,ℓ −[]) 1/2EIJ Z̄1,ℓ,N1,ℓ N1,ℓ and Z̄1,ℓ,N1,ℓ is a sample mean of {Z1,ℓ,k }.
The subsampling variance estimator is denoted by:
M1,ℓ
2
σ̂1,ℓ
=
K1,ℓ

K1,ℓ−1 (

∑

)2
1 ∑ h
Z̄1,ℓ,M1,ℓ ,K1,ℓ h=0
K1,ℓ−1hZ̄1,ℓ,M
−
1,ℓ

h=0

(17)

h
h
where Z̄1,ℓ,M
is sample mean of the h-th subseries {Z1,ℓ
}:
1,ℓ

(
)
[
]
h
{Z1,ℓ
} := Z1,ℓ,hM1,ℓ +1 , Z1,ℓ,hM1,ℓ +2 , · · · , Z1,ℓ,(h+1)M1,ℓ , 0 ≤ h ≤ K1,ℓ−1 , K1,ℓ = N1,ℓ /M1,ℓ .
aUnder the null we have τη (ℓ) → N (0, 1) as N1,ℓ goes to infinity s.t. M1,ℓ → ∞ and M1,ℓ /N1,ℓ → 0.
We define the threshold value of the finite dependence of noise for asset 1 as m1 in the sense thatthe autocovariance function γη (ℓ) for all ℓ > m1 is zero. The test statistic (16) enables us to
identify the threshold value m1 .
(±)

To derive the autocovariance estimator of the noise, we construct Z1,ℓ,ij for all i, j satisfying
tj−1 − ti = ℓ ≥ 0 using the selected threshold value m1 as follows:()( (−) ))( ( (+) )(±)(−) (+)
Z1,ℓ,ij = r1,i r1,j = P (ti ) − P ti−1
P tj
− P (tj−1 ) ,for all i, j, such that tj−1 − ti = ℓ ≥ 0,(+)

where tj(18)(+)

is the first transaction time, which follows tj subject to tj(−)− ti > m1 , and ti−1 is(−)

the last transaction time, which is followed by ti−1 subject to tj−1 − ti−1 > m1 . Then we have
(±)

(±)

EIJ [Z1,ℓ,ij ] = −γη (ℓ) for all ℓ. We define a sequence that arranges Z1,ℓ,ij in ascending order of
(±)

N

1,ℓ
. The autocovariance estimator of the noise and its asymptotic distribution
index i as {Z1,ℓ,k }k=1

14

are given by:N1,ℓ1 ∑ (±)γ̂η (ℓ) = −Z ,N1,ℓ k=1 1,ℓ,k

2= limN1,ℓ →∞ EIJ
where ωη,ℓ

[{

) a (
)
1/2 (
2
N1,ℓ γ̂η (ℓ) − γη (ℓ) → N 0, ωη,ℓ
,

(19)

}2 ]
1/2
2
N1,ℓ (γ̂η (ℓ) − γη (ℓ)) . The subsampling variance estimator of ωη,ℓis given by:
M1,ℓ
2
ω̂η,ℓ
=
K1,ℓ

K1,ℓ−1 (∑

1 ∑ (±),h )2Z̄
,K1,ℓ h=0 1,ℓ,M1,ℓ
K1,ℓ−1(±),hZ̄1,ℓ,M1,ℓ −

h=0

(±),h

(20)

(±),h

where Z̄1,ℓ,Mℓ is a sample means of the h-th subseries {Z1,ℓ }:
)
[
]
( (±)
(±),h
(±)
(±)
{Z1,ℓ } := Z1,ℓ,hM1,ℓ +1 , Z1,ℓ,hM1,ℓ +2 , · · · , Z1,ℓ,(h+1)M1,ℓ , 0 ≤ h ≤ K1,ℓ−1 , K1,ℓ = N1,ℓ /M1,ℓ .
The test statistic to find whether the autocovariance of the noise is zero is defined as follows.
Corollary 2. The test for autocovariance of market microstructure noise.
• Case of ℓ = 0; that is, test for the variance of the noise.
Let the null hypothesis and the alternative be γη (0) = 0 and γη (0) > 0. As N1,0 goes toinfinity s.t. M1,0 → ∞ and M1,0 /N1,0 → 0, the one-sided test statistic for the variance ofthe noise is asymptotically distributed as chi-square with one degree of freedom:τη∗ (0) =( √N

1,0 γ̂η (0)

ω̂η,0

)2

a

→ χ2 (1)(21)

under the null hypothesis. τη∗ (0) diverges under the alternative.
• Case of ℓ > 0.
Let the null hypothesis and the alternative be γη (ℓ) = 0 and γη (ℓ) ̸= 0. As N1,ℓ goes toinfinity s.t. M1,ℓ → ∞ and M1,ℓ /N1,ℓ → 0, the test statistic for the significance of the15

autocovariance of the noise is:√
N1,ℓ γ̂η (ℓ) aτη∗ (ℓ) :=→ N (0, 1)
ω̂η,ℓ

(22)

under the null hypothesis. τη∗ (ℓ) diverges under the alternative.

5 Monte Carlo simulation and empirical illustration
In this section, we investigate finite sample properties of auto and cross covariance estimatorsand the test statistics proposed in this paper through Monte Carlo simulation, and apply them tothe high-frequency transaction prices of several stocks on the Osaka Securities Exchange as anempirical illustration.

5.1 Finite sample properties of covariance estimators of noises
We employ the data generation process introduced in Voev and Lunde (2007). The equilibrium
price processes P1∗ and P2∗ follow the stochastic differential equations:
]
[√
(A)
(B)
dPl∗ (t) = σl (t) 1 − λ2l dWl (t) + λl dWl (t) ,(B)

dσl2 (t) = κl (θl − σl2 (t)) dt + ωl σl2 (t) dWl(·)

where Wl

(23)(t), l = 1, 2,

is a standard Brownian motion and σl2 (t) follows the generalized autoregressive con(A)ditional heteroskedasticity (GARCH) diffusion process. W1(A)

and W2are correlated; that is,d⟨W1 , W2 ⟩t = ρ∗ (t)dt. We use the anti-Fisher transformation to generate stochastic correla(A)(A)

tion ρ∗ (t):ρ∗ (t) =exp(2x(t)) − 1,
exp(2x(t)) + 1

dx(t) = κ3 (θ3 − x(t))dt + ω3 x(t)dW (t),16where x(t) follows the GARCH diffusion process. We take (λ1 , λ2 ) = (0.5, 0.5), (κ1 , κ2 , κ3 ) =(0.006, 0.037, 0.051), (θ1 , θ2 , θ3 ) = (2.719, 2.527, 0.200) and (ω1 , ω2 , ω3 ) = (0.0382, 0.216, 0.130).
The parameters κl , θl and ωl , for l = 1, 2 are obtained by Drost and Werker’s (1996) relationshipbetween the continuous time and discrete time parameters which are the daily GARCH(1,1) estimates based on Intel Corporation and Microsoft Corporation returns. The trading time per day is
set as 6.5 hours like in the NYSE and NASDAQ.

5.1.1

Bivariate moving average noise

In this simulation, we consider regular and synchronous sampling. First we employ bivariate
moving average noise processes which are truncated at some finite lag. The noise η(t) and δ(t) are
generated by the following bivariate MA(2) model every a transaction time:
η(t) = ϵη (t) − 0.5 ϵη (t − 1) + 0.4 ϵη (t − 2) − 0.1 ϵδ (t − 1) + 0.15 ϵδ (t − 2),
δ(t) = ϵδ (t) − 0.4 ϵη (t − 1) + 0.15 ϵη (t − 2) − 0.1 ϵδ (t − 1) + 0.3 ϵδ (t − 2).

To set variance of the noise we use a noise-to-signal ratio defined as the variance of the noise
divided by the integrated variance. Hansen and Lunde (2006) report that noise-to-signal ratios takea range from 0.0002 to 0.006 using transaction data for 30 stocks on the NYSE and NASDAQ. We
then set the variance of noise such that the noise-to-signal ratios of assets 1 and 2 average 0.005
and 0.002, respectively.
The transaction time interval is set to be 10, 15 and 30 seconds, respectively. Two transaction
price series, which are the sum of the equilibrium price and noise, are sampled synchronously at
equidistant times. In the regular and synchronous sampling, ℓ = 1 represents one tick time whose
length is equal to each sampling interval (10, 15 and 30 seconds). On each sampling interval, thesample size of Nℓ (N2,ℓ ) for ℓ > 0 per day is about 2300, 1550 and 770, respectively. On the other
hand, the irregular and non-synchronous sampling is applied in section 5.1.2.
We use the test statistic (9) and (16) to find the threshold value where the cross and auto covariances of the noises become zero, and estimate the cross and auto covariances of the bivariate noise17

processes by (11) and (19). We compare the performance of the following covariance estimators.
• Cross and auto covariance estimators constructed by the sample mean of Zℓ,k and Z2,ℓ,k ; that
∑ ℓ
∑N2,ℓ
1
is, γ̃(ℓ) := − N1ℓ N
k=1 Z2,ℓ,k . For example, the conditional
k=1 Zℓ,k and γ̃δ (ℓ) := − N2,ℓexpectation of Zℓ,k is equal to γ(ℓ + ∆ti ) − γ(ℓ + ∆ti + ∆sj ) − γ(ℓ) + γ(ℓ + ∆sj ) as
represented in (5). Therefore, we expect the cross-covariance estimator γ̃(ℓ) with ℓ less than
the threshold value is biased and the autocovariance estimator γ̃δ (ℓ) is also biased by thesame reason.
• The proposed cross and auto covariance estimators γ̂(ℓ) and γ̂δ (ℓ) in (11) and (19). Theseestimators are unbiased if the threshold values are correctly selected. For the variance es1/3

timation in the test statistic (9) and (16), we use the asymptotic formula Mℓ = cNℓ

and

1/3

M2,ℓ = cN2,ℓ with some c to determine Mℓ (the number of observations within each subseries) and Kℓ (the total number of subseries). Denote the cross and auto covariance estimators as γ̂(ℓ)[c] and γ̂(ℓ)[c] . We set c = 4, 2, 1 and 0.5 to investigate the influence of Mℓ andKℓ . The corresponding numbers of Mℓ (M2,ℓ ) and Kℓ (K2,ℓ ) are: (c, Mℓ , Kℓ ) =(4, 52, 44),(2, 26, 88), (1, 13, 176), (0.5, 7, 330) with sample size = 2300, (c, Mℓ , Kℓ ) =(4, 46, 33),
(2, 23, 67), (1, 12, 129), (0.5, 6, 258) with sample size = 1550 and (c, Mℓ , Kℓ ) =(4, 36, 21),(2, 18, 42), (1, 9, 85), (0.5, 5, 154) with sample size = 770, respectively.
• The Voev and Lunde’s (2007) cross-covariance estimator:
S1 ∑
S2
1 ∑
γ̈(ℓ) =
γ̈(ℓ)b ,S1 S2 b =1 b =1
12

where S1 and S2 are numbers of subgrids for asset 1 and 2. Voev and Lunde’s (2007) crosscovariance estimator γ̈(ℓ) is based on subsampling method. They do not conduct a selectionof the threshold value before the cross-covariance estimation. Instead of the selection, S1and S2 are taken as the values where length of subsampled interval is large enough in order
to eliminate a bias caused by three covariances γ(ℓ+∆ti ), γ(ℓ+∆ti +∆sj ) and γ(∆sj ). For
the bl -th subgrid (bl = 1, . . . , Sl , for l = 1, 2), define the subgrid returns r1,ib := P1 (tib ) −
18

Nb

Nb

1,ℓ
P1 (tib −1 ) with {tib }ib =1
and r2,j b := P2 (sj b ) − P2 (sj b −1 ) with {sj b }j b2,ℓ. The shorthand=1

subscript b is used in place of b1 and b2 . Then γ̈(ℓ)b is denoted by:

b
b
∑N1,ℓ
∑N2,ℓ

1

−
r br b1
for ℓ > 0,bb
j =1 1,i 2,j {tib −1 −sj b =ℓ} Nℓ,b i =1bb∑N1,ℓ ∑N2,ℓγ̈(ℓ)b =for ℓ = 0,
− N1ℓ,b ib =1
r br b1
j b =1 1,i 2,j {tib −1 −sj b =0}∪{sj b −1 −tib =0}


b
b


 − 1 ∑Nb 1,ℓ ∑Nb2,ℓ r1,ib r2,j b 1{s −t =−ℓ}
for ℓ < 0,
Nℓ,b
i =1
j =1
j b −1
ib

(24)

with

b
b
∑N1,ℓ
∑N2,ℓ


for ℓ > 0,
 ib =1 j b =1 1{tib −1 −sjb =ℓ}


b
b
∑N1,ℓ
∑N2,ℓ
Nℓ,b =−sj b =0}∪{sj b −1 −tib =0} for ℓ = 0,
b =1
b =1 1{t b
i
j
i −1


b
b


 ∑Nb 1,ℓ ∑Nb2,ℓ 1{s −t =−ℓ}
for ℓ < 0.
i =1
j =1
j b −1ib(25)

On the other hand, we propose a cross-covariance estimator based on the interval with proper
length where the dependence of the noise disappears. The length of the interval is determinedthrough the test statistic (9) and (16) proposed in this paper.
√
We also introduce Voev and Lunde’s (2007) test statistic τ ∗ (ℓ)V L = γ̈(ℓ)/ V[γ̈(ℓ)] to test
the significance of cross-covariance of the noises. Their t-statistic uses an approximatedvariance of the cross-covariance estimator under i.i.d. noise. The approximated variance isgiven by:V[γ̈(ℓ)] ≈







1
(S1 S2 )21(S1 S2 )2∑S1 S21
2 2
b=1 Nℓ,b (vx + 5ση σδ )for ℓ = 0,1
2 2
b=1 Nℓ,b (vx + 4ση σδ )for ℓ ̸= 0,∑S1 S2where vx := σx2 sec,1 σx2 sec,2 + 2ση2 σx2 sec,1 + 2σδ2 σx2 sec,2 ,σx2 sec,l =R1

2
0 σ1 (s)ds

23400/x(26)

is the x-second

average integrated variance of each asset l. As described in Voev and Lunde (2007), the noise
variances ση2 and σδ2 are estimated by subsampling realized variance with one-minute returns
and dividing by twice the number of returns. And the integrated variance is estimated by arealized kernel-based estimator of Barndorff-Nielsen, Hansen, Lunde and Shephard (2008).
On the other hand, the test statistic derived in this paper is constructed by the subsampling19

variance estimator which does not require the i.i.d. noise approximation.
The proposed cross and auto covariance estimators (11) and (19) require the threshold valueof noise dependence in advance. First we investigate an accuracy of the threshold value selectedby the test statistic (9) and (16). The bivariate MA(2) noise process has the true threshold values
m+ , m− and m2 being 2 tick times (| ℓ |= 2). As long as the selected threshold value is not lessthan 2 ticks, the proposed covariance estimators γ̂(ℓ) and γ̂δ (ℓ) are unbiased. Figure 4 plots theprobability of miss selection that the estimated threshold values m̂+ , m̂− and m̂2 are less than 2ticks, where the number of repetitions is one thousand. Although the probability of miss selection
with c = 0.5 is the lowest of all across the sampling interval, the differences of the probability
between c = 0.5 and the others are small. And the probabilities of miss selection approach tozero as the time interval becomes shorter (the sample size increases). The influence caused by themiss selection of threshold values are reported in Tables 1 and 2. In case of the sample size =
770 (sampling interval = 30 seconds) in Table 1, we find that γ̂(ℓ)[c] and γ̂δ (ℓ)[c] with any c havea somewhat bias by the miss selection of the threshold value. However, the bias of the estimatorswith the higher trading intensity like 10 seconds becomes close to zero because the probabilityof the miss selection of the threshold value is quite low. In case of the low trading intensity, we
use the sample over multiple days (for example 3 days) to obtain enough sample size. The fourth
panel in Table 1 shows the biases of γ̂(ℓ)[c] and γ̂δ (ℓ)[c] using sample over 3 days with the samplinginterval being 30 seconds are smaller than those in case of sample over 1 day (the third panel inTable 1). The behaviors of the proposed cross and auto covariance estimators are much improvedby the use of sample over multiple days even in the low trading intensity.
γ̃(ℓ) and γ̃δ (ℓ) with ℓ = 0, 1 are biased as mentioned above. The bias of Voev and Lunde’s(2007) cross-covariance estimator γ̈(ℓ) is virtually zero for all cases because it is constructed by
the subsampling returns with the intervals being wide enough. However, RMSE of γ̈(ℓ) is larger
than γ̂(ℓ)[c] with any c and RMSE ratio of γ̈(ℓ) to γ̂(ℓ)[1] is more than 1 as summarized in Table 2.
The large RMSE of γ̈(ℓ) comes from a large standard deviation induced by the longer intervals forconstruction of γ̈(ℓ) relative to that for γ̂(ℓ)[c] .
The large standard deviation of the covariance estimator also makes an impact on behavior
20

of the cross and auto correlations of the noises. We compare the cross-correlations computed by√
the proposed variance and covariance estimators; that is ρ̂(ℓ) = γ̂(ℓ)[1] / γ̂η (0)[1] γ̂δ (0)[1] , and themethod introduced in Voev and Lunde (2007). Voev and Lunde’s (2007) cross-correlation is computed by ρ̈(ℓ) = γ̈(ℓ)/(σ̂η σ̂δ ) where σ̂η2 and σ̂δ2 are estimates of the noise variances by subsampling
realized variance with one-minute returns and dividing by twice the number of returns. Figures 5
and 6 show empirical distributions of the cross-correlations in cases of the true cross-correlations
ρ(0) = 0.87 and ρ(2) = 0.26. We find that the cross-correlation estimate ρ̂(ℓ) have a higher degreeof concentration around the true correlation value than that of ρ̈(ℓ) in all sampling intervals.
Table 3 summarizes the size and power of test statistics for significance of cross and auto covariance estimators as proposed in (14), (21), (22) and Voev and Lunde (2007). The size of the
proposed test statistic takes around the nominal size, but the size of Voev and Lunde’ (2007) test
statistic τ ∗ (ℓ)V L is large. Figure 7 shows asymptotic distribution (standard normal) and empirical√
distributions of γ̂(ℓ)[1] and γ̈(ℓ) which are standardized by ω̂ℓ and V̂ [γ̈(ℓ)]. Although the standardized empirical distribution of γ̂(ℓ)[1] is close to the standard normal distribution, that of γ̈(ℓ)
is heavily tailed. The proposed test statistic also has a larger power as the sample size increases.
1/3

Next we summarize the influence of Mℓ and Kℓ selected at fixed Nℓ under Mℓ = cNℓon thevariance estimation. Although the value of c should be determined in accordance with the dependence of the noises, we do not know about the dependence in advance. Even in this simulation
setting, it is not easy to select the optimal value of c. In general, when noise processes have more
dependence, the value of c should be large. The behavior of the covariance estimators and the test
statistics would be also influenced by the value of c. However, we find that the performance of the
proposed statistics with any c does not change so much as we have seen in Tables 1, 2 and 3.
So far, we have assumed that the equilibrium price follows an Itô process without drift as in(1). Now we consider the equilibrium price follows an Itô process with drift. The product of thenoises dominates that of the drift term as the length of the interval shrinks to zero. However, it is
not clear whether the influence of the drift is negligible because the estimators and test statistics
proposed in this paper are based on the expanded intervals in (10) and (18). In this simulation, we

21

add a drift term to the stochastic differential equations:dPl∗ (t)

[√
]
(A)
(B)
= µl dt + σl (t) 1 − λ2l dWl (t) + λl dWl (t) , l = 1, 2,(27)

where µl is a constant and (µ1 , µ2 ) is set as (0.1, 0.1), (0.5, 0.5) and (1, 1). We estimate γ̂(ℓ)[1] andγ̂δ (ℓ)[1] under the above settings. Table 4 represents the bias and RMSE ratio of them to γ̂(ℓ)[1] andγ̂δ (ℓ)[1] , which are estimated using the data generation process without drift as in (22). The biasesof the estimators are quite small and the RMSE ratios take values of around one, even in the caseof rapid trend acceleration where (µ1 , µ2 ) = (1, 1). We find that the cross and auto covarianceestimators of the noises are not substantially affected by the presence of the nonzero drift term.

5.1.2

Bivariate autoregressive noise

We have investigated the behaviors of covariance estimators of moving average noise processes
and the test statistics for finding the threshold values (9), (16) and the significance (14), (21), (22)under the regular and synchronous sampling. In the next experiment, we consider the irregular andnon-synchronous sampling where the average observed time interval for asset 1 and 2 are 10 and
5 seconds because this sampling scheme is more realistic for the actual trading system. And we
employ bivariate autoregressive noise processes whose autocorrelation decays gradually to zero,but is not truncated at some finite lag. Although Assumption (1b) is approximately valid, it isimportant to investigate the finite sample properties of the covariance estimators of autoregressive
noises. We generate the noises η(t) and δ(t) by the following bivariate AR(1) model every one
second,η(t) = 0.6 η(t − 1) + 0.2 δ(t − 1) + ϵη (t),δ(t) = 0.2 η(t − 1) + 0.6 δ(t − 1) + ϵδ (t).

After the summing of equilibrium price and noise at the same period, the observed price is randomly selected by Poisson sampling with mean durations for asset 1 and 2 being 10 and 5 seconds.

22

In the irregular and non-synchronous sampling, ℓ = 1 represents one second. With these simulation settings, we obtain, on average, 900 realizations of sample size Nℓ with ℓ = 0. On the other
hand, the average sample size of Nℓ with | ℓ |> 0 is 450. Then the corresponding numbers of Mℓand Kℓ with some c are equal to (c, Mℓ , Kℓ ) =(4, 30, 15), (2, 15, 30), (1, 8, 45) and (0.5, 4, 110).
Tables 5 and 6 summarize the bias and RMSE of the proposed cross and auto covarianceestimators γ̂(ℓ)[c] , γ̂δ (ℓ)[c] and the Voev and Lunde’s (2007) cross-covariance estimator γ̈(ℓ). For
the cross-covariance estimators, the biases of γ̂(ℓ)[c] and γ̈(ℓ) are virtually zero, and γ̂(ℓ)[c] has asmaller RMSE than γ̈(ℓ) for all ℓ. The smaller RMSE of γ̂(ℓ)[c] comes from the shorter intervals
for construction of γ̂(ℓ)[c] relative to that for γ̈(ℓ). For the autocovariance estimators, the biases areclose enough to zero and the RMSEs are the almost the same value for all c. 1
Figure 8 shows the empirical distributions of cross-correlation and autocorrelaiton estimates√ρ̂(ℓ) = γ̂(ℓ)[1] / γ̂η (0)[1] γ̂δ (0)[1] and ρ̂δ (ℓ) = γ̂δ (ℓ)[1] /γ̂δ (0)[1] . We find that the correlation estimates take around the true correlation. Figures 9 and 10 show the standardized distribution of thecross and auto covariance estimators γ̂(ℓ)[c] and γ̂δ (ℓ)[c] with each value of c. The average sample
size within each subseries Mℓ is 4 when c is equal to 0.5. A few standardized empirical distributions have heavy tail caused by small Mℓ and approach to the standard normal as c increases.
In the empirical application, we should select the value of c not to be Mℓ too small. However,most histograms could be approximated by the standard normal distribution and the influences by
the different values of c are very small among the proposed statistics. Therefore, these simulation
results show that the proposed covariance estimator and test statistics have a good performanceand are robust for selection of c even in case of autoregressive noises.

5.2

Empirical illustration

We have established the covariance estimators (11) and (19) of bivariate noise processes and thetest statistics (9) and (16) for finding the threshold values and the significance (14), (21), (22) insections 3 and 4. In this subsection we provide the details of implementing them in practice. We
1

It is noted that the RMSE of γ̂δ (ℓ) for ℓ = 1, 2 takes somewhat larger values than that of γ̂δ (ℓ) for the other ℓ.
This is caused by smaller N2,ℓ for ℓ = 1, 2; that is, a sample size for asset 2. In this experiment, N2,1 and N2,2 areabout 1/6 and 1/2 of N2,ℓ for the other ℓ because the average observed time interval of asset 2 is set as 5 seconds.

23

illustrate a scheme of whole procedure for the estimation and testing for dependence of microstructure noise as follows:• Univariate noise process– Threshold value step: The threshold value m1 of univariate noise dependence is statistically selected through the test statistic (16).
– Estimation step: Estimate the variance and the autocovariance γη (ℓ) of the noise withℓ ≥ 0 using (19).
– Test significance step: Test the null hypotheses γη (0) = 0 by (21) and γη (ℓ) = 0 by
(22).
• Bivariate noise processes– Threshold value step: The threshold values of cross-sectional noise dependence m+
and m− are statistically selected through the test statistic (9).
– Estimation step: Estimate the cross-covariance γ(ℓ) of the noises using (11).
– Test significance step: Test the null hypothesis γ(ℓ) = 0 by (14).
We apply these statistics to high-frequency transaction prices of four stocks on the Osaka Securities Exchange: OMRON Corporation (OC) and Murata Manufacturing Co., Ltd. (MM) frommanufacturing industry, Ono Pharmaceutical Co., Ltd. (OP) and Santen Pharmaceutical Co., Ltd.
(SP) from medical industry. The trading day on the Osaka Securities Exchange is divided into
morning (9:00-11:00 local time) and afternoon (12:30-15:10) sessions. As illustrated in Andersen,
Bollerslev and Cai (2000), transaction prices at the opening and closing are more volatile than any
other transaction prices. This is due to the call auction to determine the opening and closing prices.
In this empirical illustration, we use the transaction prices during from 9:05 to 15:05 to estimate
the cross and auto covariances of the noises.
For univariate noise process, we start with testing whether EIJ [Z1,L,k ] = 0 or EIJ [Z1,L,k ] ̸= 0 for
all k with a large value L through the test statistic (16) for the identification of m1 as described24

in section 4.2. When the total number of {Z1,ℓ,k } for each ℓ is not large enough for the inference,
we only have to use samples in multiple days to obtain a large sample size. In the case of high(d)frequency data for D days, we define nℓ as the total number of {Z1,ℓ,k } on the d-th day whered = {1, · · · , D}. We set D = 245 where the sample period is from October 2, 2006 to September
28, 2007. We obtain {Z1,ℓ,1 , · · · , Z1,ℓ,n(1) } on the first day and {Z1,ℓ,n(1) +1 , · · · , Z1,ℓ,n(1) +n(2) } onℓℓℓ
∑D (d)
the second day. The total number of {Z1,ℓ,k } in the sample period is N1,ℓ = d=1 nℓ . We set
L = 70 seconds. If the null EIJ [Z1,70,k ] = 0 is not rejected, we test the null EIJ [Z1,69,k ] = 0. We
continue to test sequentially until the null is rejected. Finally, we regard the distance where thenull is rejected the first time as the estimator of threshold value m1 .
Next we estimate the variance of noise γη (0) by (19) and test the null hypothesis γη (0) = 0 using (21). If the test statistic (21) rejects the null, we estimate the autocovariance of the noise γη (ℓ)
by the estimator (19) and check its significance through the test statistic (22). The significance levelis set at 0.05. Table 7 (a) shows the variance estimate of the noise, the test statistic (21) for its significance and the noise-to-signal ratio which is the ratio between the variance estimate of the noise
and the integrated variance estimate using a realized kernel-based estimator of Barndorff-Nielsen,
Hansen, Lunde and Shephard (2008). We confirm that the variance of market microstructure noisein each asset is significantly larger than zero. Table 7 (b) shows the test statistic (22) for the significance of the autocovariance. The dependence of the noise in OC, MM, OP and SP disappear
at around 60 seconds. Figure 11 plots four autocorrelation functions ρ̂η (ℓ) = γ̂η (ℓ)/γ̂η (0). We cansee significantly negative and positive autocorrelations of microstructure noise in OC, MM and SP.
The negative autocorrelations at small lags imply that there exists the opposite orders within small
period that buying (selling) at present follows selling (buying) at small lags. On the other hand,the positive autocorrelations at higher lags would be induced by the clustering of order flow which
occurs in case of buying or selling pressure. These features are confirmed in Bandi and Russell(2006).
For bivariate noise processes, we use the test statistic (9) for the identification of m+ and m− ,estimate the cross-covariance by (11), and judge the significance of the cross-covariance throughthe test statistic (14). The results are reported in Table 8. In both pairs between the same industries
25and between different industries, there exist cross-covariances significantly different from zero
within −33 ≤ ℓ ≤ 66 for OC-MM, −57 ≤ ℓ ≤ 47 for OC-OP, −62 ≤ ℓ ≤ 0 for OC-SP,−48 ≤ ℓ ≤ 68 for MM-OP, −34 ≤ ℓ ≤ 68 for MM-SP, and −59 ≤ ℓ ≤ 42 for OP-SP, respectively.
√
Figure 12 plots the six cross-correlation functions ρ̂(ℓ) = γ̂(ℓ)/ γ̂η (0)γ̂δ (0). These plots showthat the market microstructure noises among some assets display asymmetric and different crosssectional dependence patterns. Especially the asymmetric cross-correlation in OC and SP captures
their lead-lag relationship. Although we do not go into the details, we find that the proposed test
statistics and cross and auto covariance estimators of the bivariate noise processes provide valuable
insights for the analysis of market microstructure.

6 Concluding remarks
Market microstructure noise occurs in the market with trading imperfections, frictions and informational effects. In this paper, we propose unbiased and consistent estimators of cross andauto covariances of the noises and derive the asymptotic distributions of them. Through Monte
Carlo simulation, we find that the proposed estimator has small mean squared error. For the crosscovariance estimation, our estimator has smaller mean squared error than Voev and Lunde’s (2007).
The larger mean squared error of Voev and Lunde’s comes from a large standard deviation induced
by a too wide time interval for construction of their cross-covariance estimator. Our estimator isbased on the interval with proper length such that the dependence of the noise disappears. The
length is determined by testing procedure proposed in this paper. Furthermore, we also propose
the test statistics for the significance of the cross and auto covariances of the noises. We confirm
that our test statistics have good empirical size and power properties.
The empirical illustration confirms that the proposed statistics enable to capture various noise
dependence patterns in several assets. The statistical analysis of market microstructure noise would
provide some evidence on the influence of market regularity and the trading mechanism on asset
pricing in financial markets. For that reason, the proposed method will shed more light on market
microstructure analysis.

26

Appendix
The proof of Lemma 1.
Let the start and end times of intervals Ak and Bk be Ak , Bk and Ak , Bk ; that is, Ak = ( Ak , Ak ]and Bk = ( Bk , Bk ]. We consider the dependence between Zℓ,k and Zℓ,k+h for any h such that
Bk+h − Ak ≥ 0. It is obvious that Zℓ,k has finite dependence from (1b) and (1d) in Assumption
1. Although there are some central limit theorems for finite dependence, as in Hoeffding andRobbins (1948) and Serfling (1968), we apply the results given by Theorem 3.1 in Politis, Romano
and Wolf (1997) because our studies are applicable to more general dependence cases like themixing sequence. Assumption 2 implies that the conditional variance of a standardized sample
2
mean of {Zℓ,k′ +1 , . . . , Zℓ,k′ +n } for any k ′ approaches the limiting value σℓ,f
. The condition for thestrong mixing coefficient in Theorem 3.1 of Politis, Romano and Wolf (1997) is satisfied from (1b)in Assumption 1. Therefore, it suffices to show the following condition (C1) for the application of
their central limit theorem.
(C1) EIJ |Zℓ,k |2β < ∞, for some β > 1.
Let ∆η(Ak ) := η(Ak ) − η(Ak ) and ∆δ(Bk ) := δ(Bk ) − δ(Bk ) be the differences between thenoises on each interval Ak and Bk . Zℓ,k is decomposed as:

Zℓ,k

∫
∫
)(
)
= P1 (Ak ) − P1 (Ak ) P2 (Bk ) − P2 (Bk ) =σ1 (u)dW1 (u)σ2 (u)dW2 (u)
Ak
Bk
∫
∫
+
σ1 (u)dW1 (u)∆δ(Bk ) +σ2 (u)dW2 (u)∆η(Ak ) + ∆η(Ak )∆δ(Bk ).
(28)
(

Ak

Bk

We take σ1 (t), σ2 (t) < C where C is a constant because σ1 (t) and σ2 (t) are bounded. On eachinterval Ak and Bk :
¯∫
¯
EIJ ¯
Ak

¯∫
¯2β ¯ ¯2β¯¯σ1 (u)dW1 (u)¯ < ¯C ¯ EIJ ¯

¯∫
¯2β
¯
¯
dW1 (u)¯ < ∞ and EIJ ¯
Bk

Ak

27

¯2β
¯
σ2 (u)dW2 (u)¯ < ∞.

Because Ak and Bk are nonoverlapping, the high-order absolute moment of the first term in (28):
¯∫
¯2β
¯∫
¯2β ¯ ∫
¯2β
∫
¯
¯
¯
¯
¯
¯
EIJ ¯ Ak σ1 (u)dW1 (u) Bk σ2 (u)dW2 (u)¯ = EIJ ¯ Ak σ1 (u)dW1 (u)¯ EIJ ¯ Bk σ2 (u)dW2 (u)¯is bounded. From (1c) in Assumption 1 and Minkowski’s inequality:
¯
¯2β (
)2β
EIJ ¯∆δ(Bk )¯ ≤ ∥ δ(Bk ) ∥2β + ∥ δ(Bk ) ∥2β< ∞,¯2β ) 1( ¯
where ∥ δ(·) ∥2β = EIJ ¯δ(·)¯ 2β . From (1d) in Assumption 1, the high-order absolute momentof the second term in (28) has:
¯∫
¯
EIJ ¯

¯2β
¯∫
¯
¯
σ1 (u)dW1 (u)∆δ(Bk )¯ = EIJ ¯

Ak

¯2β ¯
¯2β
¯
¯
¯
σ1 (u)dW1 (u)¯ EIJ ¯∆δ(Bk )¯ < ∞.

Ak

¯∫
¯2β
¯
¯
For the third term of (28), EIJ ¯ Bk σ2 (u)dW2 (u)∆η(Ak )¯ < ∞. From (1c) in Assumption 1, thehigh-order absolute moment of the fourth term of (28) has:¯
¯2β
EIJ ¯∆η(Ak )∆δ(Bk )¯¯2β¯= EIJ ¯η( Ak )δ( Bk ) − η( Ak )δ( Bk ) − η( Ak )δ( Bk ) + η( Ak )δ( Bk )¯()2β≤ ∥ η( Ak )δ( Bk ) ∥2β + ∥ η( Ak )δ( Bk ) ∥2β + ∥ η( Ak )δ( Bk ) ∥2β + ∥ η( Ak )δ( Bk ) ∥2β< ∞.

Finally, we have
( ∫
¯2β
¯
∫
∫
¯
¯
≤ ∥ Ak σ1 (u)dW1 (u) Bk σ2 (u)dW2 (u) ∥2β + ∥ Ak σ1 (u)dW1 (u)∆δ(Bk ) ∥2β
EIJ Zℓ,k)2β∫< ∞.
+ ∥ Bk σ2 (u)dW2 (u)∆η(Ak ) ∥2β + ∥ ∆η(Ak )∆δ(Bk ) ∥2β(29)
Condition (C1) holds. We then obtain the asymptotic normality of fℓ,Nℓ from the central limit
result in Politis, Romano and Wolf (1997).
The proof of Lemma 2.
28

To show the consistency of the variance estimator in (7), we apply L2 -convergence of the subsampling estimator given by Lemma 4.6.1 in Politis, Romano and Wolf (1999). Because the strong
mixing condition holds from (1b) in Assumption 1, it suffices to show the following conditions(C2) and (C3) for the application of Lemma 4.6.1.
(C2) Kℓ−1][
1/2 hMℓ
2
V
M
Z̄
IJ
ℓ
ℓ,Mℓ → σℓ,f s.t. Mℓ → ∞ and Mℓ /Nℓ → 0 as Nℓ → ∞.
h=0

∑Kℓ −1(C3) (fℓ,Nℓ )4 is uniformly integrable.
[

1/2 hMℓ
Denote VIJ Mℓ Z̄ℓ,Mℓ]

2. Then we have:as σℓ,h

Kℓ −1
Kℓ −1
¯ 2
¯
¯ 2
¯
1 ∑
1 ∑
2
2
2 ¯
2 ¯
¯σℓ,h − σℓ,f
σℓ,h − σℓ,f ≤= sup ¯σℓ,h− σℓ,f→0
Kℓ h=0Kℓ h=00≤h≤Kℓ −1

s.t. Mℓ → ∞ as Nℓ → ∞ from Assumption 2. Thus, (C2) holds.
¯¯4β
For (C3), we can show EIJ ¯Zℓ,k ¯ < ∞ for some β > 1 from (1c) in Assumption 1 by∗
the similar argument as the proof of (C1). Let the centered Zℓ,k be Zℓ,k
. It is obvious that¯∑¯4β¯ ∗ ¯4β()¯ Nℓ ∗ ¯∗¯ < ∞.
EIJ ¯ k=1
Zℓ,k ¯ = O Nℓ2β because the sequence {Zℓ,k} is m-dependent with EIJ ¯Zℓ,k
¯
¯
¯
¯4β
¯ −1/2 ∑Nℓ ∗ ¯4βTherefore, the order of EIJ ¯fℓ,Nℓ ¯ = EIJ ¯NℓZ
becomes O(1). Then (C3) holds
k=1 ℓ,k ¯¯¯4βbecause EIJ ¯fℓ,Nℓ ¯ < ∞ implies that (fℓ,Nℓ )4 is uniformly integrable. Finally these results yield
a

2
2
σ̂ℓ,f
→ σℓ,fas Nℓ → ∞.

The proof of Theorem 2.
The conditional expectation of γ̂(ℓ) is:
ℓ
1 ∑
(±)
EIJ [Zℓ,k ] = γ(ℓ).
EIJ [γ̂(ℓ)] = −Nℓ k=1N(30)

For the conditional variance of γ̂(ℓ), we have:[
VIJ [γ̂(ℓ)] = VIJ

Nℓ
Nℓ
Nℓ N
ℓ −k
]
∑
[ (±) ]
[ (±) (±) ]
2 ∑
1 ∑
1 ∑
(±)
Zℓ,k = 2
VIJ Zℓ,k + 2
CovIJ Zℓ,k , Zℓ,k+j−
Nℓ k=1Nℓ k=1Nℓ k=1 j=1

Nℓ
Nℓ ∑
m̃k
( 1 )
}
{ [
{¯
[ (±) (±) ]¯¯}
1 ∑
2 ∑
¯
(±) ]
max VIJ Zℓ,kmax ¯CovIJ Zℓ,k , Zℓ,k+j ¯ = O, (31)≤ 2+ 2
Nℓ k=1 k
Nℓ k=1 j=1 k
Nℓ

29

(±)

(±)

where m̃k is defined as maxj {CovIJ [Zℓ,k , Zℓ,k+j ]̸= 0, 0 < j ≤ Nℓ − k}. Because of the finite(±)ℓ
dependence of {Zℓ,k+j }N
k=1 , m̃k is finite. This implies VIJ [γ̂(ℓ)] → 0 as Nℓ goes to infinity and the[{
1/2
consistency of γ̂(ℓ) holds. Let the asymptotic variance of γ̂(ℓ) be ωℓ2 = limNℓ →∞ EIJ Nℓ (γ̂(ℓ)−}2 ]γ(ℓ)) . We find that the asymptotic normality of γ̂(ℓ) can be proved by a similar argument to the

(±)

Nℓ
ℓ
proof of Lemma 1. The difference between {Zℓ,k }N
k=1 and {Zℓ,k }k=1 is the amount of dependence.
(±)

(±)

Nℓ
ℓ
A sequence of {Zℓ,k }N
k=1 has more dependence than {Zℓ,k }k=1 because Zℓ,k is constructed by the

product of returns on the nonoverlapping intervals where the length of each interval is longer than(±)

those of Ak and Bk . However, Zℓ,k has finite dependence from (1b) and (1d) in Assumption 1.

30

Acknowledgements
The authors are grateful to the associate editor and two anonymous referees for helpful comments.
Special thanks go to the associate editor for constructive suggestions on previous drafts of thearticle and we go through many revisions about section 3 and 4. We would like to thank Toshiaki
Watanabe and participants of the 2007 Japanese Economic Association Annual Spring Meetingfor valuable comments. We also thank Center for the Study of Finance and Insurance, Osaka
University and Osaka Securities Exchange for providing the high-frequency data used in this paper.

Funding
Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Special Purposes, 90254135,
2007, for Scientific Research (A), 18203013, 2007 and for Young Scientists (Start-up), 20830047,
2008.

31

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34

r1,i
z
}|
(
ti−1

(a)

z
(

r2,j
}|

sj−1

(b)
z
(
ti−1

r1,i
}|

{
]
ti

asset 1{]| {z }
sjℓasset 2{]

asset 1tiz
| {z }(−ℓ sj−1

r2,j
}|

{
]
sj

asset 2

Figure 1: Pairs of returns on nonoverlapping intervals with (a) ℓ > 0 and (b) ℓ < 0.

35

I i :! (ti 1, ti ] J j :! (s j 1, s j ]Intervals for asset 1 and 2
(

](

I
(

I

](

J

1

](

1

2

I

](

J

](

I

](

2

J

](

3

3

4

](

J

]

I
](

4

J

5

asset 15]Jasset 2

6

Z ,k with k ! 1,2,3
(

r2,1

B1

]

r1, 2

(

A1
(

Z ,k !1

]

r2,3

B2

]

r1, 4

(

(

r2,5

B3

Z ,k ! 2

]

A2
]

(

r1,5

A3

]

Z , k !3Figure 2: An example of each pair of the intervals (Ak , Bk ) for k = 1, 2, 3.

36

(a)
(+)

r1,i

z
}|
(
](
ti−1
ti
←
−
−
−
−
−
−
−
−
−
−
−
−
→
+
(+)
m
r 2,j
z
}|
{
(
](
]| {z }(+)
sj−1 sj−1
sj ℓ←−−−−−−+−−−−−−→
m

{
]
(+)
ti

asset 1

asset 2

(b)
(+)

r1,i

z
}|
(
](
ti−1
ti
←
−
−
−
−
−
−−−−−→
(+)
r 2,j
m+
z
}|
{
]
(
](
(+)
sj−1
sj
sj−1
←−−−−−−−−−−→
m+

{
]
(+)
ti

asset 1

asset 2

(−)

r1,i

z
}|
{
(
](
]
(−)
ti−1 titi−1←−−−−−−−−−−→
(−)
r2,j
m−
}|
z
(
](
sj−1 sj←−−−−−−−−−−→
m−asset 1

{
]
(−)
sj

asset 2

(c)
(−)

r1,i

z
(
(−)
ti−1

}|
{
](
]
ti−1
ti
←−−−−−−−−−−−−→
(−)
m−
r2,j
}|
z
](
| {z }(
sj−1 sj−ℓ←−−−−−−−−−−−−−→
masset 1{
asset 2](−)
sj

Figure 3: A pair of modified intervals in cases of (a) ℓ > 0, (b) ℓ = 0 and (c) ℓ < 0.

37

Prob( m !
!

!

&

"

!

!

%

!

$

&

"

!

'

(

)

'

(

*

'

(

+

'

(

,

%

"

!

!

2)

"

-

.

$

#

"

!

#

!

"

"

#

#

"

%

Sampling intervalProb( m
!

!

!

2)

"

&

"

!

!

%

!

$

&

"

!

%

'

(

)

'

(

*

'

(

+

'

(

,

"

!

!

-

.

$

#

"

!

!

#

"

"

#

#

"

%

Sampling intervalProb(m 2
!

!

&

"

!

!

&

%

"

!

!

%

$

(

)

'

(

*

'

(

+

'

(

,

-

.

$

#

"

!

!

'

"

!

!

2)

"

#

"

"

#

#

"

%

Sampling interval

Figure 4: Probability of the estimated threshold value being less than 2.

38

Sampling interval 10 seconds
frequency

frequency

250

250

200

200

150

150

100

100

50

50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Sampling interval 15 seconds
frequency

frequency

250

250

200

200

150

150

100

100

50

50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

Sampling interval 30 seconds
frequency

frequency

250

250

200

200

150

150

100

100

50

50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

Sampling interval 30 seconds, 3 days
frequency

frequency

250

250

200

200

150

150

100

100

50

50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

0.4

Figure 5: Empirical distributions of ρ̂(0) (left side) and ρ̈(0) (right side), where true crosscorrelation ρ(0) = 0.87.

39

Sampling interval 10 seconds
175

frequency

175

150

150

125

125

100

100

75

75

50

50

25

25

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

frequency

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.50

0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.50

0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

!0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Sampling interval 15 seconds
175

frequency

175

150

150

125

125

100

100

75

75

50

50

25

25

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

frequency

1.00

0.75

Sampling interval 30 seconds
175

frequency

175

150

150

125

125

100

100

75

75

50

50

25

25

1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

frequency

1.00

0.75

Sampling interval 30 seconds, 3 days
175

frequency

175

150

150

125

125

100

100

75

75

50

50

25

25

!1.00

!0.75

!0.50

!0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

frequency

!1.00

!0.75

!0.50

Figure 6: Empirical distributions of ρ̂(2) (left side) and ρ̈(2) (right side), where true crosscorrelation ρ(2) = 0.26.

40

0.45

density

0.45

0.40

0.40

0.35

0.35

0.30

0.30

0.25

0.25

0.20

0.20

0.15

0.15

0.10

0.10

0.05

0.05

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

−8

density

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

Figure 7: Standardized empirical distributions of γ̂(ℓ) (left side) and γ̈(ℓ) (right side), and thestandard normal distributions (solid line).

41

100

U (1) 0.68

frequency

100

90

90

80

80

70

70

60

60

50

50

40

40

30

30

20

20

10

10

−1.00

100

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

−1.00

UG (1) 0.86

frequency

250

U (6) 0.21

frequency

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

1.00

1.25

1.50

1.75

2.00

UG (6) 0.30

frequency

90
80

200

70
60

150

50
40

100

30
20

50

10

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

Figure 8: Empirical distributions of ρ̂(ℓ) and ρ̂δ (ℓ).
Note: In the simulation for autocovariance estimator of univariate noise process, the sample sizesof N2,1 and N2,2 are about 1/6 and 1/2 of N2,ℓ for the other ℓ because the average observed timeinterval of asset 2 is set as 5 seconds. Empirical distributions of ρ̂δ (ℓ) is from sample over 3 days
to obtain a large sample size N2,1 .

42J (7) 0.08, U (7) 0.24
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

J (3) 0.018, U (3) 0.58
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

J (0) 0.028, U (0) 0.9
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

J (9) 0.003, U (9) 0.11
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

J (30) 0.000, U (9) 0.00
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

Figure 9: Standardized empirical distributions of the cross-covariance estimator of noises (histogram) and the standard normal distributions (solid line).

43

J G (0) 0.022, UG (0) 1
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

J G (2) 0.015, UG (2) 0.71
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

J G (4) 0.010, UG (4) 0.46
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

J G (11) 0.002, UG (11) 0.10
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

J G (30) 0.000, UG (11) 0.00
c 4
0.45

c 2

density0.45

c 1

density

0.45

c 0.5

density

0.45

0.40

0.40

0.40

0.40

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.30

0.25

0.25

0.25

0.25

0.20

0.20

0.20

0.20

0.15

0.15

0.15

0.15

0.10

0.10

0.10

0.10

0.05

0.05

0.05

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

density

0.05

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

−6

−5

−4

−3

−2

−1

0

1

Figure 10: Standardized empirical distributions of the autocovariance estimator of noises (histogram) and the standard normal distributions (solid line).

44

UˆK (")

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Figure 11: Autocorrelation functions of the noise in OC, MM, OP and SP.

45

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㻠㻜

㻡㻜

㻢㻜

㻣㻜

"

㻙㻣㻜

㻙㻢㻜

㻙㻡㻜

㻙㻠㻜

㻙㻟㻜

㻙㻞㻜

㻙㻝㻜

㻜

㻢㻜

"

Figure 12: Cross-correlation functions of the noises among the four assets.

46

㻣㻜

47

0
0
0.0088-0.01800.0269-0.0151
0.008000000.0088-0.01800.0269-0.0151
0.008000000.0088-0.01800.0269-0.0151
0.0080
0
0

ℓ
-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

-0.0001
0.00010.0000-0.01770.0415-0.0160
0.0001-0.0001
0.0000-0.0001
0.00000.0001-0.0179
0.0417-0.0160-0.0001
0.0002-0.0002-0.0001
0.00010.0000-0.01770.0415-0.0159-0.0001
0.00010.0000γ̃(ℓ)-0.0001
0.00010.0000-0.01770.0415-0.0160
0.0001-0.0001
0.0000-0.0048-0.0044-0.00100.0005-0.0003

sample size: 770, sampling interval: 30 seconds-0.0002-0.0003-0.00020.00100.00100.0009-0.0011-0.0012-0.0011-0.0059-0.0059-0.0058-0.0062-0.0046-0.0064-0.0051-0.0063-0.0053-0.0040-0.0058-0.0046-0.0059-0.0014-0.0011-0.0015-0.0013-0.00150.00070.00040.00090.00030.0009-0.0002-0.0004-0.0003-0.0005-0.0004-0.0016-0.0013-0.0005
0.0002-0.0001-0.0025-0.0022-0.0009
0.0004-0.0003-0.00020.0003-0.0007-0.0010-0.0019-0.0020-0.0008
0.0002-0.0002-0.00030.0009-0.0012-0.0055-0.0058-0.0052-0.00140.0008-0.0004-0.00030.0006-0.0010-0.0024-0.0032-0.0032-0.00120.0008-0.0004

sample size: 1550, sampling interval: 15 seconds-0.0003-0.0003-0.00030.00060.00070.0007-0.0011-0.0012-0.0012-0.0028-0.0032-0.0031-0.0037-0.0028-0.0040-0.0029-0.0037-0.0034-0.0024-0.0037-0.0025-0.0034-0.0012-0.0009-0.0012-0.0010-0.00130.00090.00020.00080.00030.0009-0.0004-0.0004-0.0003-0.0003-0.0003-0.0016-0.0013-0.0005
0.0001-0.0002-0.0043-0.0039-0.0009
0.0005-0.0002-0.0022-0.0019-0.0009
0.0004-0.0003c = 0.5γ̂(ℓ)[0.5]γ̂δ (ℓ)[0.5]-0.0001
0.0002-0.0005-0.0007-0.0014-0.0011-0.0016-0.0009-0.0006-0.0004
0.00030.0002-0.0001-0.0001

sample size: 2300, sampling interval: 30 seconds in 3 days
-0.0002-0.0002-0.00020.00030.00040.0004-0.0007-0.0008-0.0007-0.0014-0.0015-0.00140.0290-0.0022-0.0015-0.0022-0.0015-0.0021-0.0121 -0.0022-0.0012-0.0021-0.0012-0.00210.0000-0.0008-0.0005-0.0009-0.0005-0.0009
0.00000.00030.00030.00040.00020.00040.0000-0.0002-0.0001-0.0002-0.0001-0.00030.0290-0.01210.00000.0001-0.0001

0.0291-0.0121-0.0001
0.0001-0.0001

0.0290-0.01210.00000.00000.0000γ̃δ (ℓ)sample size: 2300, sampling interval: 10 seconds
c=4
c=2
c=1
γ̂(ℓ)[4]γ̂δ (ℓ)[4]γ̂(ℓ)[2]γ̂δ (ℓ)[2]γ̂(ℓ)[1]γ̂δ (ℓ)[1]-0.0002-0.0002-0.00020.00030.00030.0003-0.0006-0.0006-0.0006-0.0012-0.0013-0.0010-0.0019-0.0014-0.0020-0.0013-0.0018-0.0013-0.0018-0.0011-0.0020-0.0010-0.0019-0.0010-0.0007-0.0004-0.0008-0.0004-0.0007-0.0004
0.00030.00020.00040.00020.00030.0002-0.0001-0.0001-0.0002-0.0001-0.0002-0.0001

0.00010.00020.00020.0002-0.0001-0.0004-0.0004-0.0004-0.0004-0.0004-0.0003-0.0005-0.0007-0.0005-0.0005-0.0006-0.0004-0.00070.0001-0.0001
0.00010.00000.0000-0.0001-0.0002-0.0002-0.0002γ̈(ℓ)-0.0008-0.0008-0.0007-0.0007-0.0004
0.00000.00010.00000.0001

Note: The market microstructure noises are generated by a bivariate MA(2) process. ℓ = 1 represents one tick whose length is equal to each samplinginterval (5, 10, 15 and 30 seconds). The second and third columns represent true cross and auto covariances γ(ℓ), γδ (ℓ).

0.0217-0.0115
0.0061000.0217-0.0115
0.0061000.0217-0.0115
0.00610
0

true covariancesγ(ℓ)γδ (ℓ)
00
0.0088-0.01800.02690.0217-0.0151 -0.0115
0.00800.00610
0
0
0

Table 1: Bias of cross and auto covariance estimators of bivariate MA noise processes.

48

0
0
0.0088-0.01800.0269-0.0151
0.008000000.0088-0.01800.0269-0.0151
0.008000000.0088-0.01800.0269-0.0151
0.0080
0
0

-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

0.0217-0.0115
0.0061000.0217-0.0115
0.0061000.0217-0.0115
0.0061000.0217-0.0115
0.0061000.00330.00330.00330.01800.04170.01630.00330.00330.00330.00570.00570.00570.01890.04220.01690.00560.00570.00580.00390.00390.00390.01820.04180.01640.00400.00400.00390.00310.00320.00320.01800.04160.01630.00320.00320.0032γ̃(ℓ)

0.02920.01240.00240.00230.00240.02950.01280.00410.00410.00410.02930.01240.00280.00280.00280.02910.01240.00230.00220.0022γ̃δ (ℓ)

0.00310.00340.00360.00740.00770.00790.00370.00350.00340.00730.00740.00750.01240.01180.01230.00700.00750.00700.00390.00420.00410.00640.00640.00690.00370.00390.0036

sample size: 1550, sampling interval: 15 seconds
0.00310.00310.00330.00330.00370.00360.00820.00800.00710.00830.00720.00820.00690.00620.00830.00620.00820.00600.00320.00360.00320.00370.00320.00330.00340.00310.00340.00310.00310.00320.00300.00330.0030sample size: 770, sampling interval: 30 seconds
0.00710.00710.00720.00730.00720.00730.01280.01280.01280.01200.01260.01200.01270.01150.01250.01120.01250.01130.00820.00690.00770.00660.00780.00850.00720.00800.00710.00810.00810.00680.00780.00670.0078sample size: 2300, sampling interval: 30 seconds in 3 days
0.00390.00380.00390.00400.00410.00410.00400.00400.00400.00680.00680.00670.00670.00670.00680.00650.00670.00660.00720.00570.00700.00560.00710.00560.00390.00410.00370.00380.00360.00390.00400.00420.00380.00400.00380.00400.00380.00420.00350.00390.00350.00390.00710.00730.00720.01290.01190.01270.00720.00770.00710.00310.00340.00350.00800.00810.00820.00360.00350.00330.00480.00400.00200.00190.00180.00200.00230.00250.00510.00560.00600.00250.00240.00210.00480.00400.00200.00190.00180.00480.00410.00210.00200.00190.00210.00230.00260.00540.00590.00610.00250.00240.00210.00670.00570.00400.00420.00410.01250.01110.00790.00820.00780.00680.00580.00320.00320.00300.00460.00390.00210.00190.0018c = 0.5γ̂(ℓ)[0.5]γ̂δ (ℓ)[0.5]0.00200.00220.00250.00460.00520.00550.00240.00230.00200.00200.00230.00250.00540.00570.00590.00250.00240.0021

sample size: 2300, sampling interval: 10 seconds
c=4
c=2
c=1
γ̂(ℓ)[4]γ̂δ (ℓ)[4]γ̂(ℓ)[2]γ̂δ (ℓ)[2]γ̂(ℓ)[1]γ̂δ (ℓ)[1]0.00950.00950.00950.00950.00720.00930.00910.00910.00920.01650.01630.01650.01620.01280.01670.01640.01680.01620.01650.01620.01640.01620.01260.01620.01590.01610.01600.01560.01580.01560.01580.01210.01520.01520.01520.0153γ̈(ℓ)

2.45212.32992.34481.42981.0714
1.31752.48182.38272.62642.31812.23042.25691.26081.07201.33402.49112.35092.41045.30574.88334.54732.0107
1.5406
1.98874.35864.78684.87777.6173
6.9754
6.16663.08502.14032.5456
6.11376.4281
7.4416γ̈(ℓ)γ̂(ℓ)[1]RMSE ratio

Note: The market microstructure noises are generated by a bivariate MA(2) processes. ℓ = 1 represents one tick time whose length is equal to eachsampling interval (5, 10, 15 and 30 seconds). The second and third columns represent true cross and auto covariances γ(ℓ), γδ (ℓ).

000.0088-0.01800.0269-0.0151
0.0080
0
0

-4
-3
-2
-1
0
1
2
3
4

ℓ

true covariancesγ(ℓ)γδ (ℓ)

Table 2: RMSE of cross and auto covariance estimators of bivariate MA noise processes.

49

0.00000.00000.0088-0.01800.0269-0.0151
0.00800.00000.00000.00000.00000.0088-0.01800.0269-0.0151
0.00800.00000.0000

-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4
0.0217-0.0115
0.00610.00000.00000.0217-0.0115
0.00610.00000.0000

0.951
0.882
0.658
0.064
0.053

0.864
0.809
0.232
0.054
0.040

sample size: 770, sampling interval: 30 seconds
0.038
0.041
0.038
0.046
0.044
0.052
0.316
0.308
0.324
0.859
0.864
0.870
0.929
0.845
0.937
0.860
0.952
0.794
0.765
0.820
0.798
0.835
0.237
0.232
0.234
0.219
0.240
0.047
0.048
0.040
0.044
0.045
0.038
0.037
0.029
0.033
0.035

sample size: 2300, sampling interval: 30 seconds in 3 days
0.048
0.053
0.055
0.067
0.071
0.067
0.72
0.715
0.726
0.939
0.945
0.951
1.00
0.989
0.939
0.996
0.944
0.997
-0.53
0.913
0.851
0.923
0.876
0.926
0.638
0.635
0.648
0.657
0.643
0.28
0.00
0.067
0.060
0.060
0.062
0.060
0.039
0.058
0.035
0.055
0.040
0.00

1.00
-0.53
0.28
0.00
0.00

0.983
0.921
0.597
0.054
0.053

sample size: 1550, sampling interval: 15 seconds
0.037
0.036
0.030
0.052
0.042
0.050
0.637
0.624
0.646
0.973
0.981
0.983
1.000
0.973
1.000
0.974
1.000
0.944
0.909
0.952
0.930
0.961
0.54
0.552
0.554
0.557
0.565
0.046
0.046
0.042
0.044
0.051
0.051
0.047
0.040
0.049
0.046

sample size: 2300, sampling interval: 10 seconds
c=4
c=2
c=1
γ̂(ℓ)[4]γ̂δ (ℓ)[4]γ̂(ℓ)[2]γ̂δ (ℓ)[2]γ̂(ℓ)[1]γ̂δ (ℓ)[1]
0.045
0.042
0.045
0.055
0.058
0.058
0.843
0.838
0.854
1.000
1.000
1.000
1.000
0.998
1.000
0.998
1.000
0.999
0.991
0.990
0.992
0.987
0.993
0.990
0.77
0.775
0.756
0.793
0.779
0.801
0.061
0.058
0.058
0.047
0.056
0.053
0.047
0.050
0.042
0.047
0.048
0.051

0.067
0.080
0.753
0.956
0.997
0.941
0.677
0.074
0.046

0.059
0.071
0.363
0.886
0.960
0.845
0.286
0.064
0.042

0.054
0.076
0.697
0.987
1.000
0.970
0.590
0.071
0.058

0.955
0.887
0.658
0.075
0.068

0.880
0.808
0.275
0.068
0.048

0.988
0.931
0.613
0.082
0.062

c = 0.5γ̂(ℓ)[0.5]γ̂δ (ℓ)[0.5]
0.055
0.073
0.891
1.000
1.000
1.000
0.996
0.990
0.819
0.819
0.071
0.068
0.056
0.072

0.386
0.383
0.584
0.845
0.986
0.800
0.501
0.354
0.362

0.281
0.271
0.333
0.537
0.846
0.482
0.339
0.281
0.274

0.484
0.460
0.549
0.687
0.918
0.628
0.519
0.484
0.486

γ̈(ℓ)
0.560
0.577
0.580
0.724
0.932
0.698
0.596
0.546
0.526

Note: The market microstructure noises are generated by a bivariate MA(2) processes. ℓ = 1 represents one tick time whose length is equal to eachsampling interval (5, 10, 15 and 30 seconds). From second to fifth columns represent true cross and auto covariances γ(ℓ), γδ (ℓ) and true cross and auto
correlation coefficients ρ(ℓ), ρ(ℓ).

0.00
0.00
0.29
-0.58
0.87
-0.49
0.26
0.00
0.00

0.00
0.00
0.29
-0.58
0.87
-0.49
0.26
0.00
0.00

1.00
-0.53
0.28
0.00
0.00

-4
-3
-2
-1
0
1
2
3
4
0.0217-0.0115
0.00610.00000.00000.00000.00000.0088-0.01800.0269-0.0151
0.00800.00000.0000

ℓ
-4
-3
-2
-1
0
1
2
3
4

0.00
0.00
0.29
-0.58
0.87
-0.49
0.26
0.00
0.00

true covariances and correlations
γ(ℓ)
ρ(ℓ)
γδ (ℓ)ρδ (ℓ)
0.00000.00
0.00000.00
0.00880.29-0.0180 -0.58
0.02690.87
0.02171.00-0.0151-0.49-0.0115-0.53
0.00800.26
0.00610.28
0.00000.00
0.00000.00
0.00000.00
0.0000
0.00

Table 3: Size and power of test statistics for significance in (14), (21) and (22).

50-0.0333
0.0704-0.1163-0.3067-0.3686-0.3384-0.12530.0890-0.0349-0.0168
0.0931-0.1132-0.5770-0.6288-0.5895-0.1517
0.0917-0.0378-0.02440.0359-0.0719-0.1356-0.2098-0.2076-0.0913
0.0365-0.0269

ℓ
-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

-4
-3
-2
-1
0
1
2
3
4

-0.02450.0358-0.0720-0.1357-0.2100-0.2078-0.0915
0.0363-0.0271-0.0168
0.0931-0.1133-0.5770-0.6288-0.5894-0.1516
0.0918-0.0377-0.0333
0.0703-0.1163-0.3067-0.3686-0.3385-0.12530.0890-0.0349-0.02190.0891-0.1172-0.5848-0.6351-0.5918-0.15290.0901-0.0380-0.0292
0.0312-0.0767-0.1404-0.2167-0.2127-0.0979
0.0305-0.0326

1.00261.00211.00591.0063
1.0060sample size: 770, sampling interval: 30 seconds
1.0006-0.0179
1.00311.0007
0.09231.00401.0007-0.11361.00401.0001-0.5794
1.00051.00001.0002-0.6304-0.4805
1.00020.99990.9999-0.5898-0.4357
0.99940.99961.0011-0.1519-0.1016
0.99870.9997 1.0012
0.09140.04700.99860.99961.0011-0.0381-0.0293
0.9984

sample size: 2300, sampling interval: 30 seconds in 3 days
1.0003-0.0258
1.00241.00030.03461.00220.9999-0.0733
1.00001.0002-0.1370
1.0011-0.1577 1.0002
1.0005-0.2114-0.1591
1.00151.0031-0.1321 1.0001
1.0005-0.2087-0.1332
1.00041.0031-0.0490 1.0003
1.0003-0.0934-0.0509
1.00251.00140.02021.0004 1.0003
0.03540.01871.00091.0022-0.0067 1.0006
1.0005-0.0291-0.0078
1.00301.0035-0.4782-0.4339-0.0998
0.0488-0.0275-0.1638-0.1368-0.05450.0150-0.0114-0.4845-0.4382-0.10470.0442-0.0326

1.00701.00571.00201.00261.0062
1.0037
1.00791.00771.00901.00741.00861.00831.00161.00230.99900.99620.99700.9925

1.00711.00701.00501.00591.00811.00611.00541.01401.01481.01400.99930.9976
0.9981
0.99820.99720.99790.99750.99880.9984
0.9975-0.2496-0.2179-0.09120.0374-0.0275-0.2522-0.2199-0.0923
0.0365-0.0286-0.03430.0693-0.1173-0.3077-0.3705-0.3400-0.1259
0.0876-0.0357

sample size: 1550, sampling interval: 15 seconds
0.9999-0.03350.99950.99980.0701
0.99881.0001-0.1165
1.00050.9999-0.30700.99960.99990.9984-0.3690-0.2498
0.99981.00000.9981-0.3387-0.21800.99981.00000.9997-0.1255-0.09131.00030.9999 0.9996
0.08880.03720.99980.99980.9994-0.0351-0.02760.9992bias × 100-0.0156
0.0321-0.0594-0.1001-0.1798 -0.1280-0.1944 -0.1024-0.0675 -0.0439
0.03460.0179-0.0167 -0.0125

0.9994
0.99781.00130.99910.99980.99971.00081.00000.9985(1,1)
bias × 100
RMSE ratio-0.0161
1.00100.03121.0000-0.0597
0.9992-0.10120.9999-0.1809-0.1271
1.0006
0.9977-0.1951-0.1016
1.00020.9976-0.0682-0.0443
1.00030.99980.03400.01731.00031.0030-0.0174-0.01271.00141.0030sample size: 2300, sampling interval: 10 seconds(0.5,0.5)
RMSE ratio
bias × 100
RMSE ratio
1.0001-0.0157
1.00041.00010.03160.99931.0000-0.0593
0.99891.0000-0.1008
0.99981.00001.0001-0.1803-0.1270
1.0004
0.99711.00001.0001-0.1946-0.10171.0001
0.99641.00001.0000-0.0678-0.04361.00011.00051.0000 1.0002
0.03440.01751.00011.00021.00011.0002-0.0169-0.0124
1.00061.0010(0.1,0,1)

Note: The market microstructure noises are generated by a bivariate MA(2) processes. ℓ = 1 represents one tick time whose length is equal to eachsampling interval. In the simulation we use the stochastic differential equations with drift parameter (µ1 , µ2 ) = (0.1, 0.1), (0.5, 0.5) and (1, 1).

-0.1575-0.1320-0.04880.0203-0.0065-0.4794-0.4351-0.0997
0.0489-0.0272-0.2507-0.2190-0.09120.0372-0.0275

no driftbias × 100-0.0155
0.0321-0.0594-0.1001-0.1798-0.1280-0.1944-0.1024-0.0675-0.04390.03470.0179-0.0167 -0.0125

Table 4: The bias and RMSE ratio of γ̂(ℓ)[1] and γ̂δ (ℓ)[1] with drift parameter (µ1 , µ2 ) to γ̂(ℓ)[1] and γ̂δ (ℓ)[1] without drift.

51

c=4
γ̂(ℓ)[4] γ̂δ (ℓ)[4]0.00000.00010.00000.0000-0.00030.0000-0.0003-0.0001
0.0001-0.0001-0.00020.0000-0.0001-0.0001
0.0000 -0.0002-0.0002 0.0001-0.0001 -0.0003
0.0000 -0.0001-0.0002 -0.0001
0.0001 -0.0001-0.0001 -0.0001-0.0003 -0.0002-0.0003 0.0000
0.0000 -0.0002-0.0002 -0.0001-0.0003 0.0000
0.0001 0.0000
0.0002 0.0000
0.0001 0.0000c=2γ̂(ℓ)[2] γ̂δ (ℓ)[2]-0.0001
0.0001-0.0001-0.0001-0.0004-0.0001-0.0004-0.0003-0.0001-0.0002-0.0003-0.0003-0.0004-0.0002-0.0002 -0.0003-0.0003 0.0000-0.0002 -0.0005
0.0000 -0.0002-0.0002 -0.0002
0.0000 -0.0001-0.0002 -0.0002-0.0002 -0.0002-0.0002 0.0000-0.0002 -0.0002-0.0001 -0.0002-0.0002 -0.0001
0.0001 0.0000
0.0001 0.0000
0.0000 0.0000c=1γ̂(ℓ)[1] γ̂δ (ℓ)[1]-0.0001
0.0000-0.00030.0000-0.0004-0.0002-0.0004-0.0003-0.0002-0.0002-0.0003-0.0002-0.0005-0.0003-0.0003 -0.0003-0.0004 -0.0001-0.0002 -0.0004
0.0000 -0.0002-0.0002 -0.0002
0.0000 -0.0002-0.0002 -0.0002-0.0002 -0.0002-0.0002 0.0000-0.0002 -0.0002-0.0001 -0.0002-0.0001 0.0000
0.0002 0.0000
0.0002 0.0000
0.0000 0.0000

c = 0.5γ̂(ℓ)[0.5] γ̂δ (ℓ)[0.5]-0.0001
0.0000-0.0001-0.0001-0.0004-0.0003-0.0005-0.0004-0.0001-0.0002-0.0004-0.0003-0.0005-0.0003-0.0004 -0.0003-0.0004
0.0000-0.0002 -0.0006-0.0002 -0.0003-0.0003 -0.0003
0.0000-0.0002-0.0002 -0.0002-0.0002 -0.0003-0.0001
0.0000-0.0002 -0.0002-0.0002 -0.0002-0.00020.00000.00010.00000.00010.00000.00000.0000γ̈(ℓ)-0.0010-0.0008-0.0008-0.0005-0.0005-0.0011-0.0007-0.0007-0.0006-0.0004-0.0001-0.0010-0.0004-0.0007-0.0003-0.0005-0.0001-0.0005-0.0004
0.0002-0.0001-0.0005-0.0004-0.0005-0.0006-0.0002-0.0003-0.0004-0.0003

Note: The market microstructure noises are generated by a bivariate AR(1) processes. From second to fifth columns represent true cross and auto
covariances γ(ℓ), γδ (ℓ) and true cross and auto correlation coefficients ρ(ℓ), ρ(ℓ).

ℓ
-30
-25
-20
-15
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
15
20
25
30

Table 5: Bias of cross and auto covariance estimators of bivariate AR noise processes.

true covariancesγ(ℓ)ρ(ℓ) γδ (ℓ) ρδ (ℓ)
0.0000 0.00
0.0001 0.00
0.0004 0.01
0.0013 0.04
0.0039 0.12
0.0048 0.16
0.0060 0.19
0.0075 0.24
0.0094 0.30
0.0117 0.38
0.0145 0.47
0.0179 0.58
0.0217 0.70
0.0255 0.83
0.0278 0.90 0.0217 1.00
0.0210 0.68 0.0186 0.86
0.0163 0.53 0.0154 0.71
0.0129 0.42 0.0125 0.58
0.0102 0.33 0.0101 0.46
0.0081 0.26 0.0081 0.37
0.0065 0.21 0.0065 0.30
0.0052 0.17 0.0052 0.24
0.0042 0.13 0.0042 0.19
0.0033 0.11 0.0033 0.15
0.0027 0.09 0.0027 0.12
0.0009 0.03 0.0009 0.04
0.0003 0.01 0.0003 0.01
0.0001 0.00 0.0001 0.00
0.0000 0.00 0.0000 0.00

52

1.00
0.86
0.71
0.58
0.46
0.37
0.30
0.24
0.19
0.15
0.12
0.04
0.01
0.00
0.00

0.00340.00360.00370.00380.00390.00390.00390.00390.00420.00400.00420.00420.00410.00440.00360.00430.00430.00400.00400.00410.00410.00410.00400.00390.00390.00380.00370.00350.00340.00180.00590.00370.00290.00260.00250.00230.00240.00240.00240.00240.00220.00210.00200.00190.00340.00350.00370.00380.00380.00390.00400.00400.00420.00390.00430.00410.00410.00460.00360.00440.00430.00410.00390.00410.00390.00410.00400.00380.00400.00370.00370.00340.00340.00190.00590.00380.00290.00270.00250.00240.00240.00240.00240.00240.00230.00200.00200.0018c=2γ̂(ℓ)[2] γ̂δ (ℓ)[2]0.00330.00350.00370.00380.00380.00390.00400.00390.00420.00400.00430.00420.00420.00450.00350.00430.00410.00400.00390.00410.00400.00410.00390.00390.00390.00360.00360.00330.00330.00180.00570.00360.00290.00260.00250.00230.00230.00240.00240.00230.00220.00200.00200.0018c=1γ̂(ℓ)[1] γ̂δ (ℓ)[1]0.00330.00340.00350.00380.00390.00380.00390.00400.00410.00400.00430.00410.00430.00460.00360.00440.00410.00410.00400.00400.00390.00410.00390.00390.00380.00360.00360.00330.00330.00190.00570.00370.00290.00260.00250.00230.00230.00240.00240.00230.00220.00200.00190.0018c = 0.5γ̂(ℓ)[0.5] γ̂δ (ℓ)[0.5]0.01850.01880.01850.01810.01840.01820.01850.01890.01830.01790.01900.01880.01870.01850.01350.01800.01820.01880.01840.01830.01820.01820.01760.01800.01830.01840.01800.01840.0174γ̈(ℓ)
5.60225.3962
5.05314.76654.81534.69984.67684.78584.37394.46784.37494.49234.47034.13873.80684.16034.41454.6493
4.69394.46724.55234.48154.51724.63494.74455.11605.00895.55345.2596γ̈(ℓ)γ̂(ℓ)[1]RMSE ratio

Note: The market microstructure noises are generated by a bivariate AR(1) processes. From second to fifth columns represent true cross and auto
covariances γ(ℓ), γδ (ℓ) and true cross and auto correlation coefficient ρ(ℓ), ρ(ℓ).

0.02170.01860.01540.01250.01010.00810.00650.00520.00420.00330.00270.00090.00030.00010.0000

-30
-25
-20
-15
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
15
20
25
30

0.00
0.00
0.01
0.04
0.12
0.16
0.19
0.24
0.30
0.38
0.47
0.58
0.70
0.83
0.90
0.68
0.53
0.42
0.33
0.26
0.21
0.17
0.13
0.11
0.09
0.03
0.01
0.00
0.00

0.00000.00010.00040.00130.00390.00480.00600.00750.00940.01170.01450.01790.02170.02550.02780.02100.01630.01290.01020.00810.00650.00520.00420.00330.00270.00090.00030.00010.0000ℓc=4γ̂(ℓ)[4] γ̂δ (ℓ)[4]Table 6: RMSE of cross and auto covariance estimators of bivariate AR noise processes.

true covariancesγ(ℓ)ρ(ℓ) γδ (ℓ) ρδ (ℓ)

Table 7: Autocovariance of univariate noise process.
(a) The variance estimates and the test statistics for the variance.
Industry

Manufacturers
OCMM

noise variance 1.17 × 10
test statistics
2240.84∗NSR
0.0049-6

MedicalsOP1.37 × 10
159.09∗0.0009-7SP

2.54 × 10
142.27∗0.0014-7

1.04 × 10-6
1251.67∗
0.0042(b) The test statistics for the autocovariance.
ℓ
1
2
3
4
5
6
7
8
9
10
20
22
24
30
40
43
50
52
54
55
60
61
62
63
70

OC
-3.22
(25132)-3.11 ∗ (15821)-2.71 ∗ (14909)-2.58 ∗ (14696)-1.92(12958)∗-2.04(12599)-2.00 ∗ (11752)-0.68(11599)-1.52(10923)-1.36(11018)-0.14(9123)
0.17(8870)
1.16(8726)
1.74(9672)
0.33(8284)-0.04(8148)-0.45(8211)
1.00(8101)
0.16(8093)-0.02(7985)
5.79 ∗ (9709)
1.71(8705)
0.82(8497)-0.36(8336)-0.45(7989)∗

MM-5.00(43753)-3.55 ∗ (31658)-1.52(29671)-3.66 ∗ (28661)-3.18 ∗ (27087)-0.99(27243)-4.02 ∗ (25902)-2.60 ∗ (25476)-1.63(24362)∗-2.88(23916)-1.54(21664)-2.10 ∗ (21306)-0.65(21264)-1.07(21414)0.23(20047)∗-2.15(19972)-0.50(19996)-1.10(19868)-1.83(19886)-0.54(19890)4.89 ∗ (21240)-0.59(20239)0.31(19754)-0.53(19989)-1.06(19547)∗OP-1.19(14838)-0.46(8542)
1.07(7227)-0.08(6415)-0.35(5680)
0.21(5567)-0.45(5130)-0.63(5125)
0.22(4855)
0.51(4879)
0.72(3643)
1.46(3432)
0.13(3476)
0.23(3730)
1.81(3072)∗
3.05(3077)
0.56(2983)
0.63(2879)-0.70(2927)-0.85(2903)-0.43(3521)
1.96 ∗ (3273)
0.68(3009)
0.55(2911)
1.24(2850)

SP-1.99(12163)-0.95(6936)
0.30(5813)
0.24(5677)-0.83(4855)-0.72(4786)-0.37(4385)
0.25(4390)-0.71(3956)-0.03(3960)
1.46(3223)
2.51 ∗ (3032)
2.60 ∗ (3048)
0.76(3581)
0.22(2690)
0.63(2749)
0.39(2775)-3.14 ∗ (2666)-2.00 ∗ (2798)-0.40(2582)∗
3.19(3545)-0.20(3018)
2.52 ∗ (2823)
2.81 ∗ (2843)
0.22(2598)∗

Note: In the top table (a), the test statistic for the variance of noise is given by (21). The critical value at
5% significance level is equal to 3.84. NSR is an abbreviation for noise-to-signal ratio which is the ratio
between the variance estimate of the noise and the integrated variance estimate using a realized kernel-based
estimator of Barndorff-Nielsen, Hansen, Lunde and Shephard (2008). The bottom table (b) shows the test
statistic (22) for the significance of the autocovariance of noise with ℓ > 0. The critical value of the test
statistic (22) is ±1.96 at 5% significance level. Superscript ∗ denotes significance at the 5% levels. The
numbers in parentheses represent a sample size N1,ℓ .

53

Table 8: Test statistics for the cross-covariance of bivariate noise processes.
ℓ
-70
-62
-59
-57
-55
-48
-34
-33
-30
-25
-17
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
15
20
30
42
43
47
64
66
68
70

OC-MM
-0.91
-0.50
1.01
0.19
-0.30
-1.60
0.93
-2.68 ∗
0.92
-0.95
-0.13
-1.71
-2.62 ∗
-1.85
-2.05 ∗
-3.19 ∗
-1.45
-2.72 ∗
-4.29 ∗
-1.90
-2.52 ∗
-2.32 ∗
-3.37 ∗
-2.64 ∗
-3.19 ∗
-2.99 ∗
-2.17 ∗
-1.78
-3.87 ∗
-3.54 ∗
-4.08 ∗
-1.87
-2.98 ∗
-2.83 ∗
-1.98 ∗
-0.48
-1.04
-1.55
-0.05
2.86 ∗-0.67
0.04(10615)(10801)(10987)(11071)(10966)(10905)(10832)(10826)(10959)(10917)(10862)(11441)(11197)(11214)(11295)(11457)(11411)(11358)(11577)(11671)(11942)(27263)(12401)(12046)(11628)(11473)(11454)(11371)(11308)(11486)(11307)(11247)(11265)(11053)(11014)(10801)(10696)(10989)(10916)(10848)(10927)(10733)

OC-OP
1.30
1.07
1.28
2.25 ∗
0.90
-1.26
-1.32
-0.79
-0.05
-1.25
-0.19
-1.16
-2.47 ∗
-0.87
-1.09
-1.35
-0.47
-0.19
-1.01
-1.25
-1.49
-0.21
-1.39
-1.94
-1.67
-2.03 ∗
-1.61
-2.08 ∗
-1.96
-2.27 ∗
-3.57 ∗
-1.13
-2.17 ∗
-2.97 ∗
-3.06 ∗
-0.68
-0.99
-2.25 ∗
-0.47
0.61
-0.72
-0.24

(4217)(4245)(4207)(4092)(4143)(4171)(4205)(4317)(4225)(4251)(4290)(4398)(4335)(4506)(4528)(4575)(4627)(4703)(4776)(4808)(5019)(12576)(5239)(4832)(4712)(4790)(4619)(4496)(4554)(4602)(4541)(4399)(4373)(4368)(4281)(4206)(4187)(4116)(4225)(4067)(4127)(4146)

OC-SP
0.74-2.72 ∗
1.13
-0.30
2.18 ∗
-0.57
-0.88
0.67
-1.06
-0.88
-0.29
-0.77
-1.12
-1.74
-0.03
-2.25 ∗
-2.53 ∗
-1.34
-1.13
-0.88
-0.53
3.58 ∗
-0.67
0.07
-0.68
-0.78
-1.37
-1.78
0.24
0.38
0.72
-0.97
-0.73
0.89
0.53
-0.02
-0.42
-0.29
-1.83
0.63
-0.39
-1.25

MM-OP

(3952)(4105)(4043)(3939)(3863)(3825)(3889)(3874)(3959)(4051)(4069)(4016)(4225)(4096)(4108)(4082)(4129)(4200)(4404)(4474)(4657)(11042)(4751)(4461)(4437)(4245)(4235)(4073)(4026)(4141)(4089)(4205)(4168)(4049)(4024)(3961)(3981)(3897)(3913)(3996)(4011)(3922)

-0.55
-1.06
-0.87
-0.72
-0.68
2.51 ∗
-0.96
-0.28
-0.44
0.80
-2.36 ∗
-1.74
-1.98 ∗
-2.44 ∗
-1.97 ∗
-2.86 ∗
-3.88 ∗
-2.90 ∗
-2.97 ∗
-2.96 ∗
-2.46 ∗
-4.51 ∗
-2.98 ∗
-3.56 ∗
-4.81 ∗
-4.39 ∗
-3.23 ∗
-3.30 ∗
-3.24 ∗
-3.16 ∗
-2.93 ∗
-2.32 ∗
-2.45 ∗
-1.35
-2.18 ∗
-2.16 ∗
-2.77 ∗
-1.93
-2.39 ∗
-1.86
-2.16 ∗
1.73

(6527)(6500)(6467)(6528)(6355)(6500)(6572)(6496)(6562)(6464)(6634)(6868)(6863)(7025)(6993)(7158)(7121)(7129)(7300)(7410)(7824)(17583)(7648)(7391)(7149)(7141)(7197)(7154)(7081)(6987)(6975)(6770)(6874)(6556)(6700)(6531)(6462)(6547)(6398)(6341)(6560)(6445)

MM-SP
-0.36
-0.56
0.80
-0.70
-0.80
0.74
-2.22 ∗
-1.06
-0.59
1.56
0.31
-0.72
-0.79
-0.51
-1.62
-1.82
-1.45
-2.90 ∗
-2.17 ∗
-2.96 ∗
-0.78
-0.81
-0.67
-1.54
-1.33
0.24
-2.05 ∗
-0.34
0.03
-2.14 ∗
-1.02
-1.75
-0.85
1.19
-0.44
1.91
2.28 ∗
0.11
-0.91
1.19
2.04 ∗-0.02(6114)(6124)(6006)(6161)(6107)(6283)(6213)(6166)(6306)(6191)(6283)(6431)(6549)(6474)(6556)(6564)(6512)(6659)(6800)(6739)(6948)(15319)(6822)(6787)(6681)(6637)(6659)(6568)(6547)(6554)(6233)(6362)(6347)(6300)(6251)(6247)(6156)(6064)(6073)(6108)(6092)(6140)

OP-SP
0.89
0.53
2.48 ∗
0.82
-0.52
-0.68
-0.07
0.60
1.47
-1.06
-0.38
-0.91
-1.49
-1.52
0.02
-0.57
-0.34
-2.23 ∗
-1.87
-2.53 ∗
-0.41
2.03 ∗
-1.54
-1.54
-1.32
-1.12
-1.64
-0.73
-1.62
-1.91
-1.42
-1.31
-3.34 ∗
-1.53
-0.30
4.18 ∗
0.44
-0.66
-0.65
-0.10
1.04
1.63

(2366)(2360)(2456)(2478)(2417)(2455)(2442)(2386)(2506)(2442)(2448)(2547)(2649)(2668)(2663)(2616)(2706)(2824)(2894)(2980)(3414)(9256)(3257)(2922)(2807)(2853)(2773)(2647)(2613)(2572)(2640)(2551)(2459)(2419)(2382)(2345)(2369)(2336)(2337)(2424)(2372)(2381)

Note: The test statistic for the significance of the cross-covariance of the bivariate noise processes is givenby (14). The critical value of the test statistics (14) is ±1.96 at 5% significance level. Superscript ∗ denotesignificance at the 5% levels. The numbers in parentheses represent a sample size Nℓ .

54