Discussion Papers In Economics
And Business

A Test for Dependence and Covariance Estimator ofMarket Microstructure Noise
Masato Ubukata
Kosuke Oya

Discussion Paper 07-03-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Test for Dependence and Covariance Estimator ofMarket Microstructure Noise
Masato Ubukata
Kosuke Oya

Discussion Paper 07-03-Rev.2
March 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Test for Dependence and Covariance Estimator ofMarket Microstructure Noise ∗Masato Ubukata†
Kosuke Oya‡

Abstract
There are many approaches for estimating an integrated variance and covariance in thepresence of market microstructure noise. It is important to know a dependence of noise toconstruct the integrated variance and covariance estimators. We study a time dependence ofbivariate noise processes in this paper. We propose a test statistic for the dependence of thenoises and an autocovariance estimator of the noises and derive its asymptotic distribution.
The asymptotic distribution of the autocovariance estimator provides us to another test statistic
which is for significance of the autocovariances and for detection whether the noise exists ornot. We obtain good performances of the test statistics and autocovariance estimator of thenoises in a finite sample through Monte Carlo simulation. In empirical illustration, we confirm
that the proposed statistics and estimators capture various dependence patterns of the market
microstructure noises.

Keywords: test statistic; market microstructure noise; time-dependence; nonsynchronous observations; high frequency data.
JEL classification: C12; D49∗

Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Special Purposes, 90254135, 2007 and for
Scientific Research (A), 18203013, 2007.
†
Graduate School of Economics, Osaka University, 560-0043, Machikaneyama, Toyonaka, Osaka, Japan. E-Mail:
ubukata@econ.osaka-u.ac.jp‡
Graduate School of Economics, Osaka University, 560-0043, Machikaneyama, Toyonaka, Osaka, Japan. E-Mail:
oya@econ.osaka-u.ac.jp

1

Introduction

Estimating covariance matrix of two diffusion processes are important for option pricing, the measurement of value-at-risk, and portfolio evaluation. High frequency financial prices have been used
for estimation of integrated variance and covariance. However, we must be careful to the high frequency prices being contaminated with some noise. This noise, known as market microstructurenoise, has many sources, including the presence of bid-ask spread and the discreteness of theprices. The realized volatility and covariance are not necessarily the best approach to estimate
integrated variance and covariance in the presence of noise. For the literature on the integrated
variance estimation with noise, Zhou (1996) proposes the kernel-based estimator. Zhang, Mykland and Aı̈t-Sahalia (2005) propose two scales realized volatility which is the linear combinationof realized volatilities at two frequencies. Zhang (2006) extend their estimator to multiple scales.
Bandi and Russell (2006) show the optimal frequency based on the minimization of mean squared
error. Although these studies are conducted under i.i.d. noise assumption, the market microstructure noise possibly has a time dependence. Under the dependent noise assumption, Aı̈t-Sahalia,
Mykland and Zhang (2006) modify the two scales and multiple scales realized volatilities. Hansen
and Lunde (2005, 2006) and Barndorff-Nielsen, Hansen, Lunde and Shephard (2006) develop
the kernel-based estimator. For the integrated covariance estimation with the dependent noise,
Voev and Lunde (2007) show that Hayashi and Yoshida’s (2005) cumulative covariance estimatoris biased in the presence of cross-correlated noises and propose modified cumulative covarianceestimators based on kernel and subsampling methods.
However, we need to know a characteristic of the noises to guarantee the adequacy of theseestimators described above because the unbiasedness, consistency and efficiency of them depend
on some assumptions for the noises. Whether the market microstructure noise is a time dependent
or not and the amount of its dependence could be one of interesting objects on high frequency
financial analysis. In this paper, we propose a test statistic to detect the dependence of bivariate
market microstructure noise processes and an autocovariance estimator of the noises. Voev andLunde (2007) propose a test statistic for the cross-sectional dependence of the noises in order1

to determine kernel bandwidth for their estimator. The main difference from Voev and Lunde’s(2007) t-statistic is an evaluation of the variance of the cross-covariance estimator of the noises.
Althogh Voev and Lunde (2007) show the unbiasedness of the kernel-based cumulative covarianceestimator under the dependent noises, their t-statistic uses an approximated variance of the crosscovariance estimator under the i.i.d. noises and its approximation requires to estimate an integrated
variance and a variance of the noise in advance. As denoted in their paper, it is natural that theirassumption leads to somewhat larger t-statistic than it should be. On the other hand, we show thatthe test statistic can be constructed without such i.i.d. noise approximation by using subsampling
method and provide an autocovariance estimator of the bivariate noise processes and its asymptotic
distribution. Furthermore, we propose a test for significance of the autocovariance, including
a test for variance of the noise. We confirm good performances of the proposed test statistic
and autocovariance estimator in finite sample through Monte Carlo simulations. For an empiricalillustration, we apply these statistics to high frequency asset prices in Osaka Securities Exchange
and find that the market microstructure noises in some assets are significantly correlated.
The paper itself proceeds as follows. In section 2 we describe problem settings and backgroundon high frequency financial analysis. We propose the test statistic for cross-sectional dependenceof the noises in section 3. In section 4 we provide the autocovariance estimator of the bivariate
noise processes, its asymptotic distribution and the test statistic for its significance. Section 5
includes a simulation experiment and an empirical illustration. In section 6 we conclude the paperwith appendix that provide proofs about several lemmas and theorems.

2

Problem settings and backgroundConsider the logarithmic price processes of two assets, {P1∗ (t)} and {P2∗ (t)} which follow the
one-dimensional Itô process with no 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,

2ρ∗ (t) ∈ (−1, 1),(1)where W1 and W2 are standard Brownian motions. The initial value of the price Pl∗ (0) is a constant.
σl (t) > 0 of Pl∗ (t) is a bounded measurable function. The integrated variance and covariation of
P1∗ and P2∗ within time horizon [0, T ] is given by∫ T∫ T
σl2 (u)du,IVl,T =ICT =0

σ1 (u)σ2 (u)ρ∗ (u)du,l = 1, 2.

(2)0

We make some notation to represent intraday returns and irregularly nonsynchronous trading system. 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. Let ti and sj be the end times of the i-th andj-th intervals of the asset 1 and 2. The different notations for the transaction times in two assets are
due to the nonsynchronous trading.
Before showing the test statistic for the dependence and autocovariance estimator of noises, wefirst review Hayashi and Yoshida’s (2005) cumulative covariance (CC) estimator for the nonsynchronously observed process. Figure 1 illustrates a case of {(ti−1 , ti ] ∩ (sj−1 , sj ] ̸= ∅}. Then theCC estimator is given by

CC =

∑

r1,i r2,j 1{(ti−1 ,ti ]∩(sj−1 ,sj ]̸=∅} .

(3)

i,j

The product of r1,i and r2,j contributes to the estimation of the integrated covariance only if intervals (ti−1 , ti ] and (sj−1 , sj ] are overlapping. The indicator function in (3) enables the CC estimator
to use a raw data without any data interpolations. When the observed returns r1,i and r2,j are∗∗= P2∗ (sj ) − P2∗ (sj−1 ), the CC= P1∗ (ti ) − P1∗ (ti−1 ) and r2,jreturns from true price process r1,iestimator is unbiased. Its consistency and asymptotic normality are proved in Hayashi and Yoshida(2005, 2006).
However, the integrated volatility and covariance estimators with finer sampling do not converge to the true values in many empirical studies because the transaction prices in high frequency
data usually is contaminated with market microstructure noise. It is related to imperfections in the
trading system established in the market microstructure literature. A simple way to model high3

frequency transaction data is to use hidden semi-martingale processes, as named by Mykland andZhang (2005). In this framework the logarithmic price Pl is observed with market microstructurenoises,P1 (ti ) = P1∗ (ti ) + η(ti ), P2 (sj ) = P2∗ (sj ) + δ(sj ),(4)where Pl∗ (t) is the true logarithmic price described in (1) which appears in the market with no trading imperfections, frictions, or informational effects. η(ti ) and δ(sj ) are the market microstructurenoises in asset 1 and 2, respectively. We assume the properties of the market microstructure noises
as follows.
Assumption 1. The properties of the 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 means.
(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 decreasing function of | ℓ |.
¯¯4β(1c) There exists some positive number β > 1 satisfies E¯u(t)u′ (s)¯ < ∞ for all t, s.
(1d) The noise process is independent with true price process. Pl∗u(t), l = 1, 2.

To avoid complication of the subscript for (1b) we rewrite γηδ (ℓ) and γδη (ℓ) as γ(ℓ) and γ(−ℓ)
because γηδ (ℓ) = E[η(t)δ(t − ℓ)] and γδη (ℓ) = E[η(t − ℓ)δ(t)] = E[η(t)δ(t + ℓ)]. We define
the autocorrelation coefficients and the cross-correlation coefficient of two noises as ρη (ℓ), ρδ (ℓ)
and ρ(ℓ). For (1d), even if Pl∗ and u(t) are correlated, the dependence of the noises generally

4dominates the dependence between the true price and noise as the number of high frequency observations increases. Furthermore, Hansen and Lunde (2006) suggest that the independence assumption between the true price and noise does not cause the damage statistically for analysis ofasset prices in more trading intensities. We denote conditional expectation and variance given intervals I i := (ti−1 , ti ] and J j := (sj−1 , sj ] for all i, j as EIJ [ · ] and VIJ [ · ]. Then the conditional
expectation of the CC estimator is

EIJ

[∑

]
r1,i r2,j 1{I i ∩J j ]̸=∅} = EIJ

[∑

i,j

∗ ∗
r1,i
r2,j 1{I i ∩J j ̸=∅}

i,j

]
+ EIJ

[∑

]
eη,i eδ,j 1{I i ∩J j ̸=∅} ,i,j∫T
where eη,i := η(ti )−η(ti−1 ) and eδ,j := δ(sj )−δ(sj−1 ). The first term converges to 0 σ1 (u)σ2 (u)ρ∗ (u)du as shown in Hayashi and Yoshida (2005). The second term representing the bias of theCC estimator is further decomposed as

EIJ

[∑

]
eη,i eδ,j 1{I i ∩J j ̸=∅}

i,j

=

∑(

)
γ(ti − sj ) − γ(ti − sj−1 ) − γ(ti−1 − sj ) + γ(ti−1 − sj−1 ) 1{I i ∩J j ̸=∅} .

i,j

It is obvious that the CC estimator is biased when the market microstructure noises are crosscorrelated. Voev and Lunde (2007) propose the bias-corrected CC estimator based on kernels.
They show that it is unbiased when cross-sectional dependence of the noises is correctly specified. For their estimator, we need to identify the cross-sectional dependence of the noises by using
t-statistic proposed in Voev and Lunde (2007). In their t-statistic, they use an approximated variance of cross-covariance estimator which is derived under the i.i.d. and no cross-correlated noises
assumptions. Therefore, the variance of cross-covariance estimator is undervalued and it leads
to larger t-statistic than it should be. Further their t-statistic includes unknown parameters suchthat the integrated variance and the variance of the noises we have to estimate. To get rid of the
difficulties, we propose alternative test statistic for the cross-sectional dependence of the noises using subsampling method in section 3 and we provide the autocovariance estimator of the bivariate
noise processes, its asymptotic distribution and a test statistic for its significance in section 4.
5

3

The test statistic for cross-sectional dependence of noises

In this section, we propose a test statistic for cross-sectional dependence of noises. Although wecannot identify the covariation of the true price processes which have martingale property and thecovariance of the market microstructure noises in the pair of the overlapping intervals, we have

EIJ

[∑

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

i,j

(5)

i,j

because of the covariation of the true price processes being zero in the pair of the nonoverlapping
intervals {I i ∩ J j = ∅}. We use the product of returns on the nonoverlapping intervals to identify
the covariance of the noises if it exists, as well with Voev and Lunde (2007).

3.1 Details of the test statistic
From Assumption (1b), the cross-covariance of two noises disappears when the noises are separated enough. We propose a test statistic to detect the distance where the cross-covariances of
the noises become zero in this subsection. For the nonoverlapping intervals {I i ∩ J j = ∅} andti−1 −sj > 0, the distance between the intervals is defined as ℓ = ti−1 −sj . In case of sj−1 −ti > 0,
the difference of the nonoverlapping intervals is denoted by ℓ = −(sj−1 − ti ). The top panel (a)and the lower panel (b) in Figure 2 illustrate the former and the latter cases, respectively. The
nonoverlapping adjacent intervals such that ti−1 − sj = 0 or sj−1 − ti = 0 are used in the case ofℓ = 0. In what follows we consider the case of ℓ > 0 because we can construct the test statistic
for the other cases by replacing the definition of ℓ. We define the product of returns on the i-th andj-th intervals using the indicator function which takes one if {I i ∩ J j = ∅} and ℓ = ti−1 − sj > 0
as follows,

Zℓ,ij = r1,i r2,j 1{{I i ∩J j =∅} ∩ {ti−1 −sj =ℓ}} , for all i, j.

(6)

We take the conditional expectation of Zℓ,ij for any i, j such as the indicator function being one.
For any i, j satisfying {I i ∩ J j = ∅} and ti−1 − sj = ℓ, which is a nonoverlapping interval with6

distance ℓ,
[
]
[
]
[
]
[
]
[
]
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 )(7)

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 suppose s∗ := 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. It is[]
obvious that the source of EIJ ZL−s∗ ,ij ̸= 0 is γ(L − s∗ ) ̸= 0. Changing the point of view, we set a[][]large enough L such that EIJ ZL,ij = 0 and find the distance ℓ∗ such that ℓ∗ = maxℓ {EIJ Zℓ,ij ̸=0, ℓ ≤ L}. Then we conclude that γ(ℓ∗ + 1) = 0 and γ(ℓ∗ ) ̸= 0 and that the threshold value[]of the dependence becomes the distance ℓ∗ . It is noted that whether EIJ Zℓ,ij = 0 or not is notnecessarily implies whether γ(ℓ) = 0 or not. This is because the sum of all cross-covariances in(7) incidentally takes zero even in case of γ(ℓ) ̸= 0. To avoid such situation we apply the methodof determination for ℓ∗ described above. The test statistic is constructed by using a sample mean ofZℓ,ij which satisfies nonoverlapping intervals with the distance ℓ to determine the threshold value.
ℓ
For the construction of the test statistic, we define a sequence {Zℓ,k }N
k=1 as follows. First weℓselect Zℓ,ij which satisfies {I i ∩ J j = ∅} and ℓ = ti−1 − sj . {Zℓ,k }N
k=1 is a sequence which arranges

the selected Zℓ,ij in ascending order of index i. Nℓ is the total number of the products of 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 3 illustrates each pair of intervals (Ak , Bk ), (Ak+1 , Bk+1 ) and (Ak+2 , Bk+2 ) for k = 1. We∑ ℓdefine Z̄ℓ,Nℓ := N1ℓ N
k=1 Zℓ,k as a sample mean of Zℓ,k and make the following assumption forZℓ,k .
][∑ ′ +n2→ σℓ,f, uniformly in any k ′ , as n → ∞. This means:
Assumption 2. VIJ n−1/2 kk=k
′ +1 Zℓ,k¯¯ [ −1/2 ∑k′ +nN]
2 ¯ℓ
For any sequence {nNℓ } that tends to infinity with Nℓ , supk′ ¯VIJ nNℓk=k′ +1 Zℓ,k − σℓ,f → 0
as Nℓ → ∞.
7

This assumption states that the conditional variance of a standardized sample mean of {Zℓ,k′ +1 ,. . . , Zℓ,k′ +n } is close to some limiting value as a sample size n goes to infinity. Let fℓ,Nℓ :=([]) 1/2Z̄ℓ,Nℓ −EIJ Z̄ℓ,Nℓ Nℓ be the theoretical standardization for Z̄ℓ,Nℓ . Then the asymptotic variance
[(
)2 ]
2
ℓ
of fℓ,Nℓ is given by limNℓ →∞ EIJ fℓ,Nℓ= σℓ,f. It is noted that {Zℓ,k }N
k=1 is a sequence ofdependent and heterogeneously distributed random scalars because the variance depends on thelength of the irregularly observed interval and {Zℓ,k } is serially correlated. We obtain the following
lemma for the asymptotic normality.
Lemma 1. Suppose Assumptions 1 and 2 hold. As Nℓ goes to infinity, we havefℓ,Nℓ a→ N (0, 1).
σℓ,f

(8)

2
The proof is given in Appendix. Next we consider the estimation of σℓ,f
. We construct a consistent2
estimator of σℓ,fby applying a subsampling method which is first proposed by Carlstein (1986).

Although Carlstein (1986) considers a 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 to heteroskedastic observations.
ℓ
We define the subseries of {Zℓ,k }N
k=1 as follows.

(
)
[
]
hMℓ
{Zℓ,N
} := Zℓ,hMℓ +1 , Zℓ,hMℓ +2 , · · · , Zℓ,(h+1)Mℓ , 0 ≤ h ≤ Kℓ − 1, Kℓ = Nℓ /Mℓ ,ℓwhere [ · ] denotes the integer part of a real number, Mℓ is a number of element in subseries
hMℓ
{Zℓ,M
} and Kℓ is a total number of subseries. We assume that Mℓ → ∞ and Mℓ /Nℓ → 0 asℓNℓ → ∞. The variance estimator is given by
Mℓ
2
=
σ̂ℓ,f
Kℓ

K
ℓ −1 (
∑

hMℓ
Z̄ℓ,M
−
ℓ

h=0

Kℓ −1
)2
1 ∑
hMℓ
Z̄ℓ,M
,
ℓ
Kℓ h=0

(9)

hMℓ
hMℓ
where Z̄ℓ,M
is a sample mean of subseries {Zℓ,M}. We have the following lemma for the varianceℓℓ2
estimator σ̂ℓ,f.

8

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

as Nℓ → ∞.

(10)

The proof is described in Appendix.
Our purpose in this section is to construct a test statistic to detect the distance where crosscovariance of the noises becomes zero. The test statistic is derived from the results of Lemmas 1
and 2. Let the null 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

(11)

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 (9) because thehigh frequency transaction data yields a large number of Nℓ . However, it is difficult to determine2
the optimal numbers of Mℓ and Kℓ which minimizes mean squared error of σ̂ℓ,fsince we do notknow the covariance structure of the noises. The increase of Mℓ (i.e. the decrease of Kℓ ) in fixed2
Nℓ reduces a bias but increases a variance of σ̂ℓ,f. It is known that the optimal asymptotic rate of
1/3

Mℓ is proportional to Nℓ
1/3

is Mℓ = cNℓ

for the subsampling variance estimation, that is, the asymptotic formulawhere c depends on underlying process. We investigate an influence by numbers of
1/3

Mℓ and Kℓ selected under Mℓ = cNℓwith some c through Monte Carlo simulations in section 5.

Next we summarize a procedure to identify the distance where the cross-covariance of noises
becomes zero. First we test the null hypothesis EIJ [ZL,k ] = 0 for all k with a large value L
using the test statistic (11). The null hypothesis EIJ [ZL,k ] = 0 would not be rejected because9the cross-covariance between sufficiently separated noises is zero from (1b) of Assumption 1. If
EIJ [ZL,k ] = 0 is not rejected as expected, secondly we will test whether EIJ [ZL−1,k ] is zero or not.
If EIJ [ZL−1,k ] = 0 is not reject, we proceed to judge the statistical significance for EIJ [ZL−2,k ]. We
continue to test sequentially until the null being rejected. Finally we regard the distance where thenull is rejected the first time as ℓ∗ = maxℓ {|τ (ℓ)| > c.v.} where c.v. is a critical value of the test
statistic (11).

4 Consistent covariance estimators of noises
4.1

Cross-covariance estimator of noises

We consider an estimation of cross-covariance γ(ℓ) in this subsection. It is obvious that the sample
mean of Zℓ,k does not give an unbiased estimator of γ(ℓ) for any ℓ being less than a threshold value.
For construction of unbiased estimator, we have only to remedy the nonoverlapping intervals so as
to all cross-covariances in (7) except γ(ℓ) become zero by using the threshold value determinedthrough the test statistic (11). 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(+)

as the first transaction time of asset 1 which follows ti subject to ti− 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
(−)

to sj

as the first transaction time of asset 2 which follows sj subject− 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 the 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 (6) is defined as
(−)

(±)

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 (−) (−)
if ℓ < 0r1,i r2,j 1{sj−1 −ti =−ℓ}
10

(12)

The top panel (a), the middle panel (b) and the lower panel (c) in Figure 4 illustrate each pairof intervals with ℓ > 0, ℓ = 0 and ℓ < 0, respectively. For all ℓ > 0, cross-covariances where
the distances between the noises are further than m+ take zero. The nonoverlapping intervals(±)

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

(±)

ℓ
a sequence which arranges the selected Zℓ,ij in ascending order of index i as {Zℓ,k }N
k=1 . Then themodified cross-covariance estimator is given by
ℓ
1 ∑
(±)
γ̂(ℓ) = −Z .
Nℓ k=1 ℓ,k

N

(13)

We obtain the followings theorem for the modified cross-covariance estimator γ̂(ℓ).
Theorem 2. Suppose Assumptions 1 and 2 hold. Then we havea

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

[{

(14)

}2 ]
1/2
Nℓ (γ̂(ℓ) − γ(ℓ)) .

The proof is described in Appendix. The asymptotic distribution of γ̂(ℓ) established in Theorem 2
provides us 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.
Let a subsampling estimator of ωℓ2 be ω̂ℓ2 . As Nℓ goes to infinity, we have√τ ∗ (ℓ) :=Nℓ γ̂(ℓ) a→ N (0, 1)
ω̂ℓ

(15)

under the null hypothesis. τ ∗ (ℓ) diverges under the alternative.
It is noted that the bias of CC estimator (3) is virtually zero when the test statistic (15) does not
reject the null γ(ℓ) = 0 for all ℓ. Even if the CC estimator is unbiased, the noise has a strong impact
on its variance. Therefore, we should use the Voev and Lunde’s (2007) subsampling cumulative
covariance estimator without kernel method.
11

4.2 Autocovariance estimator of noise
We propose the cross-covariance estimator of the noises and derive the test statistic for significanceof cross-covariance in the previous subsection. The framework is applicable for estimation of an
autocovariance of univariate noise process. We will briefly describe a consistent autocovarianceestimator of the noise in this subsection. The autocovariance estimator of the noise also requires
the distance where the autocovariance becomes virtually zero as the cross-covariance estimator of
the noises does. Then we start with the construction of the test statistic to identify its distance. wedefine the product of returns on the i-th and j-th intervals for asset 1 using the indicator function
which takes one if ℓ = tj−1 − ti ≥ 0 as follows,

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

(16)

N

1,ℓ
Let {Z1,ℓ,k }k=1
be a sequence which arranges Z1,ℓ,ij satisfying ℓ = tj−1 − ti ≥ 0 in ascendingorder of index i. N1,ℓ is a total number of a sequence {Z1,ℓ,k }. The null hypothesis is EIJ [Z1,ℓ,k ]= 0 for all k, for given ℓ and the alternative hypothesis consists of all possible deviations from thenull. The test statistic for this hypothesis is given by√τη (ℓ) :=N1,ℓ Z̄1,ℓ,N1,ℓ,
σ̂1,ℓ,f(17)

22
where Z̄1,ℓ,N1,ℓ is a sample mean of {Z1,ℓ,k }. σ̂1,ℓ,f
is a subsampling estimator of σ1,ℓ,f= limN1,ℓ →∞
[(
]
)2
(
[
]) 1/2a
EIJ f1,ℓ,N1,ℓ , where f1,ℓ,N1,ℓ := Z̄1,ℓ,N1,ℓ −EIJ Z̄1,ℓ,N1,ℓ N1,ℓ . Under the null we have τη (ℓ) →N (0, 1) as N1,ℓ goes to infinity. We define the threshold value of the finite dependence of noise forasset 1 as m1 in the sense that the autocovariance function γη (ℓ) for all ℓ > m1 is zero. The test
statistic (17) enables to identify the threshold value m1 .
(±)

To derive the autocovariance estimator of noise, we construct Z1,ℓ,ij for all i, j using identified
threshold value m1 as follows,()( (−) ))( ( (+) )(±)(−) (+)− P (tj−1 ) 1{tj−1 −ti =ℓ ≥0} , (18)P tj
Z1,ℓ,ij = r1,i r1,j 1{tj−1 −ti =ℓ ≥0} = P (ti ) − P ti−1

12

(+)

where tj(+)

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 select Z1,ℓ,ij such as the indicator function taking one and define
(±)

(±)

N

1,ℓ
a sequence which arranges the selected Z1,ℓ,ij in ascending order of index i as {Z1,ℓ,k }k=1
. The

autocovariance estimator of the noise and its asymptotic distribution are given byN1,ℓ1 ∑ (±)γ̂η (ℓ) = −Z ,N1,ℓ k=1 1,ℓ,k

2
where ωη,ℓ= limN1,ℓ →∞ EIJ

[{

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

1/2

N1,ℓ (γ̂η (ℓ) − γη (ℓ))

}2 ]

(19)

. Once the asymptotic normality of γ̂η (ℓ) is

established, a test statistic to detect whether the autocovariance of the noise is zero or not 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 ση2 = 0 and ση2 > 0, and a subsampling22
estimator of ωη,0be ω̂η,0. As N1,0 goes to infinity, the one-sided test statistic for the varianceof the noise isτη∗ (0) =( √N

1,0 γ̂η (0)

ω̂η,0

)2

a

→ χ(1)(20)

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, and a subsampling22
estimator of ωη,ℓbe ω̂η,ℓ. As N1,ℓ goes to infinity, the test statistic for the significance of the

autocovariance of the noise isτη∗ (ℓ) :=√N1,ℓ γ̂η (ℓ) a→ N (0, 1)ω̂η,ℓunder the null hypothesis. τη∗ (ℓ) diverges under the alternative.
13

(21)

5

Monte Carlo simulation and Empirical illustration

5.1 Finite sample properties of autocovariance estimator of noises
We examine the bias and root mean squared error (RMSE) of the autocovariance estimator (13)and (19) of bivariate noise processes using threshold values identified by the test statistic (11) and(17) through Monte Carlo simulation. We use data generation process introduced in Voev andLunde (2007). The true price processes P1∗ and P2∗ follows the stochastic differential models,dPl∗ (t) = σl (t)

[√

(A)

1 − λ2l dWl(B)(t) + λl dWl(B)dσl2 (t) = κl (θl − σl2 (t))dt + ωl σl2 (t)dWl

]
(t) ,

(22)

(t), l = 1, 2,

(·)

(A)

where Wl is a standard Brownian motion and σl (t) follows the GARCH diffusion process. W1(A)

and W2are correlated, that is, d⟨W1 , W2 ⟩t = ρ∗ (t)dt. To generate stochastic correlation
(A)

(A)

ρ∗ (t), it is represented by the anti-Fisher transformation,exp(2x(t)) − 1,
exp(2x(t)) + 1

ρ∗ (t) =dx(t) = κ3 (θ3 − x(t))dt + ω3 x(t)dW (t),where x(t) follows the GARCH diffusion process. We take (λ1 , λ2 ) = (0.5, 0.5), (κ1 , κ2 , κ3 ) =(0.3, 0.2, 0.1), (θ1 , θ2 , θ3 ) = (0.1, 0.1, 0.1) and (ω1 , ω2 , ω3 ) = (0.2, 0.3, 0.1). The integrated variance and covariance of P1∗ and P2∗ within the time horizon [0, T ] is given by∫ T∫ T
σl2 (u)du,IVl,T =

ICT =

0

√
σ1 (u)σ2 (u) (1 − λ21 )(1 − λ22 )ρ∗ (u)du,l = 1, 2.

0

The subsequent simulation settings are the followings.
1. We set trading time per day as 6.5 hours like NYSE and NASDAQ, and the minimum timeinterval as one second.
2. We generate the noise process which follows a bivariate AR(1) process. To set a variance14

of the noise we use a noise-to-signal ratio defined as the variance of the noise divided byintegrated variance. Hansen and Lunde (2006) report the noise-to-signal ratios take a range
from 0.0002 to 0.006 using transaction data for thirty equities in NYSE and NASDAQ. Then
we set the variance of noise such that noise-to-signal ratios of asset 1 and 2 averagely take
0.005, 0.002, respectively. The average observed time intervals of asset 1 and 2 are chosen
as 10 and 5 seconds from Poisson observation arrivals.
3. We identify the distance where the covariance of the noises becomes virtually zero throughthe test statistic (11) and (17), and estimate the autocovariance estimator of the bivariate
noise processes (13) and (19).
On the cross-covariance of the noises, we evaluate performances of the following cross-covarianceestimators.
• a cross-covariance estimator constructed by using Zℓ,k , that is, γ̃(ℓ) := − N1ℓ∑Nℓ

k=1 Zℓ,k .

• the cross-covariance estimator γ̂(ℓ) in (13). To determine Mℓ (number of element in subseries) and Kℓ (total number of subseries) on the variance estimation in the test statistic (11),1/3

we use the asymptotic formula Mℓ = cNℓ . We denote the cross-covariance estimator asγ̂(ℓ)[c] . We set c = 4, 2, 1 and 0.5 to investigate an influence by Mℓ and Kℓ .
• the cross-covariance estimator (13) through Voev and Lunde’s (2007) t-statistic: γ̂(ℓ)V L . We
use their t-statistic to identify the distance where cross-covariance becomes virtually zero.
Table 1 summarizes the sample bias and RMSE of the cross-covariance estimators γ̃(ℓ), γ̂(ℓ)[1]and γ̂(ℓ)V L , where the number of repetition is one thousand. The second and third columns represent true cross-covariance γ(ℓ) and correlation coefficient ρ(ℓ) with each value of ℓ. γ̃(ℓ) is morebiased as ℓ approaches to zero as expected. The biases of γ̂(ℓ)[c=1] and γ̂(ℓ)V L are virtually zeroand γ̂(ℓ)[1] has the smaller RMSE than γ̂(ℓ)V L for all ℓ. The smaller RMSE of γ̂(ℓ)[1] comes fromthe shorter intervals for construction of γ̂(ℓ)[1] relative to that for γ̂(ℓ)V L .
In the next simulation experiment, we investigate the influence of Mℓ and Kℓ selected at fixed
1/3

Nℓ under Mℓ = cNℓon the variance estimation in the test statistic (11). Although we should
15determine the value of c in accordance with the dependence of the noises, we do not know about
the dependence in advance. Even in our simulation settings, it is not easy to select the optimalvalue of c. In general, when the noise processes have more dependence, the value of c will belarge. We conduct simulations under several values of c such that c = 4, 2, 1 and 0.5. In thissimulation settings, we obtain averagely 450 and 900 realizations of Nℓ with ℓ > 0 and ℓ = 0,
respectively. Then the average numbers of Mℓ and Kℓ with some c is (c, Mℓ , Kℓ ) =(4, 30, 15),(2, 15, 30), (1, 8, 45) and (0.5, 4, 110) in case of ℓ > 0. Table 2 represents RMSE ratios of γ̂(ℓ)[4] ,γ̂(ℓ)[2] , γ̂(ℓ)[0.5] and γ̂(ℓ)V L to γ̂(ℓ)[1] . Figure 5 plots the RMSE ratios in Table 2. γ̂(ℓ)V L uniformly
takes RMSE about 1.3 times as larger as the cross-covariance estimators with various c. The results
show that the cross-covariance estimator is robust for selection of c.
Table 3 summarizes the bias and RMSE of the autocovariance estimator (19) of univariate noise
through the test statistic (17) in asset 2. Figure 6 plots the bias and RMSE ratios of γ̂δ (ℓ)[4] , γ̂δ (ℓ)[2]and γ̂δ (ℓ)[0.5] to γ̂δ (ℓ)[1] . Although we will see some bias as c takes a smaller value, these biases
are enough close to zero and RMSE is the almost same value for all c. It is noted that RMSE ofγδ (ℓ) for ℓ = 1, 2 takes somewhat larger values than that of γδ (ℓ) for the other ℓ as in Table 3. Thisis caused by smaller numbers of N2,ℓ for ℓ = 1, 2. In this experiment, N2,1 and N2,2 are about 1/6,
1/2 of N2,ℓ for the other ℓ because the average observed time interval of asset 2 is set as 5 seconds.
So far, we have assumed that the true price follows Itô process with no drift as in (1). Now we
consider the case with non-zero drift. The product of the noises dominates that of the drift termas the length of interval shrinks to zero. However, it is not clear about whether an influence of thedrift is negligible or not since the estimators and test statistics proposed in this paper are based on
the expanded intervals as in (12) and (18).
In this simulation, we add the drift term to the true price process,
[√
]
(A)
(B)
dPl∗ (t) = µl dt + σl (t) 1 − λ2l dWl (t) + λl dWl (t) , l = 1, 2,(23)

where µl is a constant and (µ1 , µ2 ) is set as (0.05, 0.05), (0.1, 0.1) and (0.5, 0.5). We estimate
γ̂(ℓ)[1] and γ̂δ (ℓ)[1] under the above settings. Table 4 represents the biases and RMSE ratios of16

these estimates to γ̂(ℓ)[1] and γ̂δ (ℓ)[1] which are estimated using the data generation process withzero drift as in (22). The biases of the estimators are quite small and RMSE ratios take around oneeven in case of (µ1 , µ2 ) = (0.5, 0.5) which represents a rapid trend acceleration. We find that the
autocovariance estimator of the noises are not much effected by the presence of the non-zero drift
term.

5.2

Empirical illustration

We have established the test statistics and the autocovariance estimator of noises in section 3 and 4.
In this subsection we apply these statistics to high-frequency transaction prices of three stocks in
Osaka Securities Exchange; OMRON Corporation (OC), Murata Manufacturing Co., Ltd. (MM)and Nintendo Co., Ltd. (NI). We investigate whether the noise exists or not, estimate autocovariances of the noises and judge their significance if the noise exists.
First we test whether the noise exists or not using the test statistic (20) which requires the variance estimator (19). To identify m1 , 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 (17). In this empirical illustration, we set
L = 60. When the total number of {Z1,ℓ,k } for each ℓ is not large enough for empirical analysis, we
use multiple days to obtain a large sample size. In case of D days high-frequency data, we define(d)nℓ as the total number of {Z1,ℓ,k } in d-th day where d = {1, · · · , D}. We set D = 83 for theillustration and the sample period is from March 1, 2007 until June 29, 2007 for 83 trading days.
On the first day we obtain {Z1,60,1 , · · · , Z1,60,n(1) } and {Z1,60,n(1) +1 , · · · , Z1,60,n(1) +n(2) } is sampled
60
60
60∑D (d)
on the second day. The total number of {Z1,60,k } in the sample period is N1,60 = d=1 n60 . If thenull EIJ [Z1,60,k ] = 0 is not rejected, we will test whether EIJ [Z1,59,k ] is zero or not. We continue
to test sequentially until the null being rejected. Finally we regard the distance where the null isrejected the first time as m1 . Then we estimate the variance of noise γη (0) by (19) and test the null
hypothesis ση2 = 0 using (20). If the test statistic (20) rejects the null, we estimate the autocovariance of the noise γη (ℓ) by the estimator (19) and check its significance through the test statistic(21).
Table 5 (a) shows the test statistic (20) for the variance of noise at 5% significance level where
17the critical value is 3.84. We confirm that market microstructure noise in each asset exists because
the variance of noise is significantly larger than zero. The variances of the noise in OC, MM
and NI are estimated as 1.369 × 10−6 , 1.296 × 10−7 and 2.209 × 10−7 . Table 5 (b) shows thetest statistic (21) for the significance of the autocovariance where the critical value is 1.96 at 5%
significance level. The dependence of the noise differs for each asset, that is, the noise in OC is
uncorrelated and the dependences in MM and NI disappear over 17 and 8 seconds. Figure 6 plots
three autocorrelation functions ρ̂η (ℓ) = γ̂η (ℓ)/γ̂η (0). We find that the microstructure noise in each
asset except OC is negatively autocorrelated.
Next we estimate a cross-covariance of the noises for two assets. We use the test statistic (11)and the cross-covariance estimator (13) and judge the significance of the cross-covariance throughthe test statistic (15), similarly as the univariate noise case. The results are reported in Table 6.
In OC-MM and MM-NI, the cross-covariances are significantly different from zero within −8 ≤ℓ ≤ 15 and −3 ≤ ℓ ≤ 7 at 5% significance level. Figure 7 plots three cross-correlation functions√
ρ̂(ℓ) = γ̂(ℓ)/ γ̂η (0)γ̂δ (0). The negative and asymmetric cross-correlations are found in OC-MM
and MM-NI. These empirical illustration shows that the market microstructure noise for each asset
a various dependence patterns. In this way, the proposed test statistics and autocovariance estimatorof bivariate noise processes gives us an insightful analysis of market microstructure noise.

6

Concluding Remarks

Numerous studies have proposed integrated variance and covariance estimators in the presenceof market microstructure noise because realized variance, realized covariance and cumulative covariance estimators deteriorate with the noise. It is important to know a dependence of noise toconstruct a proper integrated variance and covariance estimator suited for the noise properties. In
this paper we propose a test statistic to detect a distance where the autocovariance of bivariate
noise processes becomes virtually zero for a consistent autocovariance estimator of the noises.
Furthermore we show the asymptotic distribution of the autocovariance estimator and propose ananother test statistic for its significance. We find that the test statistic and autocovariance estimator18

have good performance through Monte Carlo simulations. The empirical illustration confirms thatthe proposed statistics enable to capture various dependence patterns of the noises in several assets. The statistical analysis for the market microstructure noise will give us some evidence about
the influences of the market regularity and trading mechanism to the asset pricing in the financial
market. The proposed method will shed light on the market microstructure.

Appendix
The proof of Lemma 1.
Let the start times 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 forany 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 theorem for finite dependence suchthat Hoeffding and Robbins (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 the more generaldependence cases such as the mixing sequence. Assumption 2 implies the conditional variance ofa standardized sample mean of {Zℓ,k′ +1 , . . . , Zℓ,k′ +n } for any k ′ approaches to the limiting value
2σℓ,f. The condition for the strong 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
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

∫
∫
)(
)
σ2 (u)dW2 (u)σ1 (u)dW1 (u)= P1 (Ak ) − P1 (Ak ) P2 (Bk ) − P2 (Bk ) =
Bk
Ak
∫
∫
+
σ1 (u)dW1 (u)∆δ(Bk ) +σ2 (u)dW2 (u)∆η(Ak ) + ∆η(Ak )∆δ(Bk ).
(24)
(

Ak

Bk

19

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 ¯

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

Ak

¯2β
¯∫
¯
¯
dW1 (u)¯ < ∞ and EIJ ¯

Ak

¯2β
¯
σ2 (u)dW2 (u)¯ < ∞.

Bk

Since Ak and Bk are nonoverlapping, the high-order absolute moment of the first term in (24)
¯∫
¯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 moment of
the second term in (24) 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 (24), EIJ ¯ Bk σ2 (u)dW2 (u)∆η(Ak )¯ < ∞. From (1c) in Assumption 1 thehigh-order absolute moment of the fourth term of (24) 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

20

+∥

∫

σ (u)dW2 (u)∆η(Ak ) ∥2β + ∥ ∆η(Ak )∆δ(Bk ) ∥2β
Bk 2

)2β

< ∞.

(25)

The condition (C1) holds. Then we obtain the asymptotic normality of fℓ,Nℓ from the central limit
result in Politis, Romano and Wolf (1997).
The proof of Lemma 2.
To show the consistency of the variance estimator in (9) we apply L2 -convergence of the subsampling estimator given by Lemma 4.6.1 in Politis, Romano and Wolf (1999). Because strong mixing
condition holds from (1b) in Assumption 1, we suffice to show the following conditions (C2) and(C3) for the application of Lemma 4.6.1.
(C2)

Kℓ−1

[
]
1/2 hMℓ2
h=0 VIJ Mℓ Z̄ℓ,Mℓ → σℓ,f s.t. Mℓ → ∞ and Mℓ /Nℓ → 0 as Nℓ → ∞.

∑Kℓ −1(C3) (fℓ,Nℓ )4 is uniformly integrable.
[]
1/2 hMℓ2
Denote VIJ Mℓ Z̄ℓ,M
as σℓ,h. Then we have
ℓ
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 ∗ ¯4β
becomes O(1). Then (C3) holds
Therefore, the order of EIJ ¯fℓ,Nℓ ¯ = EIJ ¯Nℓ
k=1 Zℓ,k ¯¯4β¯because 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.
It is to see that the conditional expectation of γ̂(ℓ) is
ℓ
1 ∑
(±)
EIJ [Zℓ,k ] = γ(ℓ).
Nℓ k=1N

EIJ [γ̂(ℓ)] = −

21

(26)

For the conditional variance of γ̂(ℓ), we have
Nℓ
Nℓ
Nℓ N
ℓ −k
[
]
∑
[ (±) (±) ][ (±) ]
1 ∑
1 ∑
2 ∑
(±)
CovIJ Zℓ,k , Zℓ,k+j
VIJ [γ̂(ℓ)] = VIJ −Zℓ,k = 2
VIJ Zℓ,k + 2
Nℓ k=1Nℓ k=1Nℓ k=1 j=1
Nℓ
Nℓ ∑
m̃k
{¯
( 1 )
{ [
}
[ (±) (±) ]¯¯}
1 ∑
2 ∑
¯
(±) ]
max ¯CovIJ Zℓ,k , Zℓ,k+j ¯ = O≤ 2max VIJ Zℓ,k
+ 2
, (27)
Nℓ k=1 k
Nℓ k=1 j=1 k
Nℓ

(±)

(±)

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 the asymptotic normality of γ̂(ℓ) can be proved by the similar argument as the

(±)

Nℓ
ℓ
proof of Lemma 1. The difference point of {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 is not correlated with Zℓ,k+h for the distance of them being
large enough.

22

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 Meeting forvaluable comments.

ReferencesAı̈t-Sahalia, Y., P. A. Mykland, and L. Zhang (2006), ”Ultra High Frequency Volatility Estimation
with Dependent Microstructure Noise,” Working Paper, Princeton University.
Bandi, F. M. and J. R. Russell (2006), ”Separating microstructure noise from volatility,” Journalof Financial Economics, 79(3), 655-692.
Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde and N. Shephard (2006), ”Designing Realised
Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise,” Working
Paper, University of Oxford, Department of Economics.
Carlstein, E. (1986), ”The Use of Subseries Values for Estimating the Variance of a General Statistic from a Stationary Sequence,” The Annals of Statistics 14(3), 1171-1179.
Fukuchi, J. (1999), ”Subsampling and model selection in time series analysis,” Biometrika 86(3),591-604.
Hansen, P. R. and A. Lunde (2005), ”A realized variance for the whole day based on intermittent
high-frequency data,” Journal of Financial Econometrics 20(3), 525-554.
Hansen, P. R. and A. Lunde (2006), ”Realized Variance and Market Microstructure Noise.” Journalof Business and Economic Statistics 24(2), 127-218.
Hayashi, T. and N. Yoshida (2005), ”On Covariance Estimation of Nonsynchronously Observed
Diffusion Processes,” Bernoulli 11(2), 359-379.
Hayashi, T. and N. Yoshida (2006), ”Estimating Correlations with nonsynchronous observations
in Continuous Diffusion Models,” unpublished manuscript.
23

Hoeffding, W. and R. Herbert (1948), ”The central limit theorem for dependent random variables,”Duke Math. J. 15, 773-780.
Mykland, P. A. and L. Zhang (2005), ”Comment: A Selective Overview of Nonparametric Methods in Financial Econometrics,” Statistical Science 20(4), 347-350.
Politis, D. N., J. P. Romano, and M. Wolf (1997), ”Subsampling for heteroskedastic time series,”
Journal of Econometrics 81, 281-317.
Politis, D. N., J. P. Romano, and M. Wolf (1997), ”Subsampling,” Springer Series in Statistics.
Serfling, R. J. (1968), ”Contributions To Central Limit Theory For Dependent Variables,” The
Annals of Mathematical Statistics 39(4), 1158-1175.
Voev, V. and A. Lunde (2007), ”Integrated Covariance Estimation using High-frequency Data in
the Presence of Noise,” Journal of Financial Econometrics 5(1) 68-104.
Zhang, L. (2006), ”Efficient estimation of stochastic volatility using noisy observations: a multiscale approach,” Bernoulli 12(6), 1019-1043.
Zhang, L., P. A. Mykland, and Y. Aı̈t-Sahalia (2005), ”A tale of two time scales: Determining integrated volatility with noisy high frequency data,” Journal of the American Statistical Association
100, 1394-1411.
Zhou, B. (1996), ”High-Frequency Data and Volatility in Foreign-Exchange Rates,” Journal ofBusiness and Economic Statistics, 14, 45-52.

24

r1,i

z
(

}|

ti−1
r2,j

}|

z
(
sj−1

{
]
ti

asset 1{]sj

asset 2

Figure 1: A case of r1,i and r2,j be overlapping.

(a)

z
(
ti−1
z
(

r2,j
}|

sj−1

(b)

z
(

ti−1

r1,i
}|

r1,i
}|

{
]
ti

{
]| {z }
sjℓasset 1

asset 2{]

asset 1

ti
r2,j
z
}|
| {z }(−ℓ sj−1{]sjasset 2

Figure 2: Pairs of returns on nonoverlapping intervals with (a) ℓ > 0 and (b) ℓ < 0.

25

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

26

(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)
(−)

z
(
(−)
ti−1

r1,i

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

Figure 4: A pair of modified intervals in cases of (a) ℓ > 0, (b) ℓ = 0 and (c) ℓ < 0.

27

RMSE ratio ( )[1]2 ( )[ 4 ] ( )[ 0.5]

 ( )[ 2 ] ( )VL

1.6

1.2

0.8

20

0

20

Figure 5: RMSE ratio plots of γ̂(ℓ)[4] , γ̂(ℓ)[2] , γ̂(ℓ)[0.5] and γ̂(ℓ)V L standardized by RMSE of γ̂(ℓ)[1] .

28

! ( )[ 4 ]! ( )[ 2 ]

bias
0.001

! ( )[1]! ( )[ 0.5]

0

0.001
0

10

20
! ( )[ 4 ]! ( )[ 2 ]RMSE ratio
1.530! ( )[1]! ( )[ 0.5]

1

0.5
0

10

20

30

Figure 6: The bias and the RMSE ratio plots of γ̂δ (ℓ)[4] , γ̂δ (ℓ)[2] and γ̂δ (ℓ)[0.5] to γ̂δ (ℓ)[1] .

29

! ( )
!

"

!

"

OC

(

)

!

!

"

)

!

"

(

!

"

'

!

"

&

!

"

%

!

"

$

!

"

#

!

)

!

(

!

! ( )

'

!

"

!

"

!

"

!

"

!

"

(

!

"

'

!

"

&

!

"

%

!

"

$

!

"

#

!

&

!

%

!

$

!

MM

(

)

!

)

!

)

!

(

! ( )

!

!

"

!

"

!

"

!

"

!

"

(

!

"

'

!

"

&

!

"

%

!

"

$

!

"

#

'

!

&

!

%

!

$

!

NI

(

)

!

)

!

)

!

(

!

'

!

&

!

%

!

$

!

Figure 7: Autocorrelation functions of the noise in OC, MM and NI.

30

OC-MM

( )
!

"

!

"

%

&

!

!

"

&

!

"

%

!

"

$

!

"

#

!

#

!

%

!

!

%

!

#

!

!

'

'

( )
!

"

!

"

!

"

!

"

!

"

%

!

"

$

!

"

#

OC-NI

%

&

!

&

!

#

!

%

!

!

%

!

#

!

!

'

'

( )
!

"

!

"

!

"

!

"

!

"

%

!

"

$

!

"

#

MM-NI

%

&

!

&

!

#

!

%

!

!

'

%

!

#

!

!

'

Figure 8: Cross-correlation functions of the noises in OC-MM, OC-NI and MM-NI.

31

Table 1: The bias and RMSE of cross-covariance estimators γ̃(ℓ), γ̂(ℓ)[1] and γ̂(ℓ)V L .
ℓ
γ(ℓ)
ρ(ℓ)
γ̃(ℓ)
γ̂(ℓ)[1]γ̂(ℓ)V L
bias
RMSE
bias
RMSE
bias
RMSE
-30 0.0000 0.0000 0.0000 0.0022 0.0000 0.0025 -0.0001 0.0035-25 0.0001 0.0000 0.0000 0.0022 0.0002 0.0025 0.0002 0.0036-20 0.0004 0.0100 -0.0001 0.0022 0.0000 0.0028 -0.0002 0.0038-15 0.0012 0.0400 -0.0003 0.0022 0.0000 0.0029 0.0002 0.0036-14 0.0015 0.0500 -0.0007 0.0024 0.0001 0.0026 0.0001 0.0037-13 0.0019 0.0600 -0.0009 0.0024 0.0001 0.0029 -0.0002 0.0037-12 0.0024 0.0700 -0.0011 0.0025 0.0000 0.0028 0.0002 0.0036-11 0.0029 0.0900 -0.0014 0.0027 -0.0001 0.0028 0.0000 0.0039-10 0.0037 0.1200 -0.0017 0.0028 -0.0001 0.0029 0.0001 0.0038-9 0.0046 0.1400 -0.0020 0.0030 0.0000 0.0029 0.0001 0.0040-8 0.0057 0.1800 -0.0026 0.0034 -0.0001 0.0028 0.0000 0.0038-7 0.0072 0.2300 -0.0032 0.0040 0.0001 0.0028 0.0000 0.0039-6 0.0090 0.2800 -0.0040 0.0045 0.0000 0.0029 0.0001 0.0040-5 0.0112 0.3500 -0.0051 0.0056 -0.0001 0.0030 0.0004 0.0040-4 0.0139 0.4400 -0.0063 0.0067 -0.0001 0.0030 0.0001 0.0038-3 0.0172 0.5400 -0.0079 0.0082 -0.0001 0.0029 0.0001 0.0038-2 0.0210 0.6600 -0.0098 0.0100 -0.0003 0.0031 0.0001 0.0039-1 0.0251 0.7900 -0.0122 0.0124 -0.0004 0.0032 -0.0001 0.0040
0 0.0286 0.9000 -0.0138 0.0139 -0.0004 0.0029 0.0000 0.0037
1 0.0222 0.7000 -0.0098 0.0101 -0.0003 0.0030 0.0001 0.0040
2 0.0175 0.5500 -0.0077 0.0081 -0.0001 0.0030 0.0001 0.0038
3 0.0139 0.4400 -0.0062 0.0066 -0.0001 0.0029 0.0002 0.0039
4 0.0111 0.3500 -0.0051 0.0055 -0.0002 0.0030 0.0001 0.0038
5 0.0089 0.2800 -0.0040 0.0046 -0.0001 0.0028 -0.0001 0.0040
6 0.0071 0.2200 -0.0032 0.0040 -0.0001 0.0029 0.0002 0.0038
7 0.0057 0.1800 -0.0024 0.0033 -0.0002 0.0029 -0.0001 0.0037
8 0.0045 0.1400 -0.0020 0.0030 -0.0001 0.0028 -0.0001 0.0039
9 0.0036 0.1100 -0.0017 0.0029 -0.0001 0.0029 0.0001 0.0037
10 0.0029 0.0900 -0.0014 0.0027 0.0000 0.0028 0.0001 0.0040
11 0.0023 0.0700 -0.0010 0.0024 0.0001 0.0028 -0.0004 0.0039
12 0.0018 0.0600 -0.0008 0.0023 -0.0001 0.0028 0.0000 0.0037
13 0.0015 0.0500 -0.0006 0.0023 0.0001 0.0028 0.0001 0.0036
14 0.0012 0.0400 -0.0005 0.0024 0.0001 0.0027 0.0001 0.0039
15 0.0010 0.0300 -0.0004 0.0022 0.0000 0.0028 0.0001 0.0037
20 0.0003 0.0100 0.0001 0.0022 0.0002 0.0027 0.0000 0.0037
25 0.0001 0.0000 -0.0002 0.0022 0.0002 0.0028 -0.0002 0.0038
30 0.0000 0.0000 0.0000 0.0022 0.0002 0.0025 -0.0001 0.0036
Note: The market microstructure noises are generated by a bivariate AR(1) process. γ(ℓ) and ρ(ℓ) are true
cross-covariance and cross-correlation. γ̃(ℓ) is the estimator constructed by using Zℓ,k . γ̂(ℓ)[1] is the cross(±)

covariance estimator (13) using Zℓ,k with c = 1 on the variance estimation in the test statistic (11). γ̂(ℓ)V L
is the estimator through Voev and Lunde’s (2007) t-statistic.

32

Table 2: The RMSE ratios of γ̂(ℓ)[4] , γ̂(ℓ)[2] , γ̂(ℓ)[0.5] and γ̂(ℓ)V L to γ̂(ℓ)[1] .
ℓ
-30
-25
-20
-15
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
20
25
30

γ̂(ℓ)[4]
1.023
1.065
0.990
1.037
1.061
1.006
1.026
0.988
0.999
0.979
1.017
1.005
1.002
0.998
0.993
0.986
0.993
0.955
0.941
0.999
0.979
1.001
0.961
1.043
1.015
0.974
1.018
0.995
0.990
0.995
0.996
1.015
1.000
1.032
1.035
1.008
1.049

γ̂(ℓ)[2]
1.066
1.023
1.046
1.061
1.037
1.015
1.016
0.997
1.005
0.995
1.001
1.000
0.983
1.010
0.988
0.969
1.001
0.977
0.993
1.003
0.991
1.008
0.988
1.033
1.031
0.978
0.981
0.983
1.026
1.007
0.994
1.000
0.977
0.964
1.054
1.075
1.078

γ̂(ℓ)[0.5]
0.976
0.989
0.961
0.965
1.046
1.020
1.015
1.021
1.014
0.993
1.007
1.015
0.992
0.999
1.015
1.001
0.999
1.004
1.010
1.031
0.997
1.008
0.966
1.024
1.006
0.993
1.004
0.995
1.001
0.982
0.964
0.984
0.994
0.970
0.991
0.967
0.965

γ̂(ℓ)V L
1.363
1.432
1.326
1.239
1.415
1.287
1.266
1.416
1.288
1.381
1.370
1.365
1.378
1.357
1.253
1.327
1.267
1.248
1.256
1.337
1.280
1.332
1.258
1.448
1.319
1.276
1.402
1.259
1.404
1.386
1.345
1.300
1.416
1.294
1.374
1.378
1.403

Note: γ̂(ℓ)[4] , γ̂(ℓ)[2] , γ̂(ℓ)[1] and γ̂(ℓ)[0.5] are the cross-covariance estimator (13) with c = 4, 2, 1, 0.5 on
the variance estimation in the test statistic (11). γ̂(ℓ)V L is the estimator through Voev and Lunde’s (2007)
t-statistic.

33

Table 3: The bias and RMSE of autocovariance estimators γ̂δ (ℓ)[4] , γ̂δ (ℓ)[2] , γ̂δ (ℓ)[1] and γ̂δ (ℓ)[0.5] .
ℓγδ (ℓ)ρδ (ℓ)

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
30

0.02530.02090.01700.01370.01100.00880.00710.00570.00450.00360.00290.00230.00190.00150.00120.00100.00080.00060.00050.00040.00030.00010.0000

1.00
0.83
0.67
0.54
0.44
0.35
0.28
0.22
0.18
0.14
0.11
0.09
0.07
0.06
0.05
0.04
0.03
0.02
0.02
0.02
0.01
0.00
0.00

γ̂δ (ℓ)[4]
bias
RMSE
-0.0001 0.0015
0.0001 0.0056-0.0001 0.0035
0.0000 0.0027
0.0000 0.0024
0.0000 0.0021
0.0000 0.0022-0.0001 0.0022
0.0001 0.0022
0.0000 0.0022
0.0000 0.0022
0.0000 0.0022-0.0001 0.0021
0.0000 0.0020
0.0000 0.0019-0.0001 0.0019-0.0001 0.0018-0.0001 0.0017
0.0000 0.0018-0.0001 0.0018
0.0000 0.0017
0.0000 0.0015
0.0000 0.0014γ̂δ (ℓ)[2]
bias
RMSE
-0.0002 0.0018-0.0001 0.0055-0.0003 0.0034-0.0001 0.0028-0.0001 0.0023-0.0001 0.0021
0.0000 0.0021-0.0001 0.0022
0.0000 0.0022-0.0001 0.0021
0.0000 0.0022
0.0000 0.0022
0.0000 0.0021
0.0000 0.0019
0.0000 0.0018
0.0000 0.0019
0.0000 0.0018
0.0000 0.0018
0.0000 0.0018-0.0001 0.0018
0.0001 0.0017
0.0000 0.0015
0.0000 0.0014γ̂δ (ℓ)[1]
bias
RMSE
-0.0004 0.0019-0.0003 0.0054-0.0003 0.0034-0.0002 0.0027-0.0002 0.0023-0.0002 0.0021-0.0001 0.0021-0.0002 0.0022-0.0001 0.0021-0.0001 0.0022-0.0001 0.0022-0.0001 0.0021-0.0001 0.0021
0.0000 0.0019-0.0001 0.0018-0.0001 0.0019-0.0001 0.0017-0.0001 0.0018
0.0000 0.0017-0.0001 0.0018-0.0001 0.0017-0.0001 0.0015
0.0000 0.0014γ̂δ (ℓ)[0.5]
bias
RMSE
-0.0004 0.0020-0.0004 0.0055-0.0004 0.0034-0.0002 0.0027-0.0003 0.0023-0.0002 0.0021-0.0001 0.0021-0.0002 0.0022-0.0001 0.0021-0.0001 0.0021-0.0001 0.0022-0.0001 0.0022
0.0000 0.0020-0.0001 0.0019-0.0001 0.0018-0.0001 0.0019-0.0001 0.0017-0.0001 0.0017
0.0000 0.0017-0.0001 0.0018-0.0001 0.0017-0.0001 0.0014
0.0000 0.0014

Note: The market microstructure noises are generated by a bivariate AR(1) process. γ̂δ (ℓ)[4] , γ̂δ (ℓ)[2] ,γ̂δ (ℓ)[1] and γ̂δ (ℓ)[0.5] are the estimator (19) with c = 4, 2, 1, 0.5 in asset 2.

34

Table 4: The results of autocovariance estimators. Case of true price process with non-zero drift.
(a) The bias and RMSE ratio of γ̂(ℓ)[1] with drift parameter (µ1 , µ2 ) to γ̂(ℓ)[1] without drift(µ1 , µ2 )ℓno drift(0.05, 0.05)(0.1, 0.1)
(0.5, 0.5)
bias×100 bias×100 RMSE ratio bias×100 RMSE ratio bias×100 RMSE ratio-30
0.00100.00101.00010.00101.00010.00121.0006-25
0.02090.02100.99990.02100.99990.02160.9993-20 -0.0017-0.00171.0000-0.0018
1.0001-0.00221.0004-150.00010.00010.99990.00010.99970.0004
0.9988-10 -0.0115-0.0115
1.0000-0.0116
1.0001-0.0116
1.0004-5-0.0059-0.00590.9999-0.0060
0.9998-0.0065
0.9988-4-0.0069-0.0069
1.0001-0.0069
1.0002-0.0073
1.0012-3-0.0120-0.0120
0.9999-0.0120
0.9999-0.01250.9995-2-0.0335-0.0335
1.0001-0.0335
1.0002-0.0337
1.0011-1-0.0353-0.0353
1.0000-0.0353
0.9999-0.03540.99980-0.0396-0.0397
1.0000-0.0397
1.0000-0.0397
1.0002
1-0.0297-0.0297
1.0000-0.0298
0.9999-0.0304
0.99972-0.0095-0.0095
0.9999-0.0095
0.9998-0.0099
0.99903-0.0076-0.0076
1.0000-0.0076
1.0001-0.0080
1.00044-0.0219-0.0219
1.0000-0.02190.9999-0.02190.99965-0.0053-0.0053
1.0000-0.00540.9999-0.0058
0.9996100.00250.00251.00000.00251.00000.00290.9999150.00090.00091.00000.00090.99990.00090.999720
0.02360.02361.00010.02361.00030.02391.001425
0.02280.02281.00000.02291.00000.02330.999930
0.01780.01781.00000.01790.99990.01840.9996(b) The bias and RMSE ratio of γ̂δ (ℓ)[1] with drift parameter (µ1 , µ2 ) to γ̂δ (ℓ)[1] without drift.
ℓ
0
1
2
3
4
5
10
15
20
25
30

(µ1 , µ2 )no drift(0.05, 0.05)(0.1, 0.1)
(0.5, 0.5)
bias×100 bias×100 RMSE ratio bias×100 RMSE ratio bias×100 RMSE ratio-0.0361-0.0361
1.0000-0.0361
1.0000-0.0365
0.9998-0.0256-0.0256
1.0000-0.0257
1.0001-0.02621.0004-0.0315-0.0315
1.0000-0.0315
1.0000-0.0318
1.0000-0.0176-0.01770.9998-0.01770.9997-0.0183
0.9984-0.0227-0.0227
1.0000-0.0227
1.0000-0.0230
1.0004-0.0155-0.0155
0.9999-0.0155
0.9998-0.0158
0.9989-0.0083-0.00830.9999-0.00830.9997-0.0085
0.9989-0.0076-0.0076
1.0000-0.0076
1.0001-0.0080
1.0005-0.0083-0.0083
1.0000-0.00830.9999-0.00850.9997-0.0073-0.0073
1.0001-0.0074
1.0001-0.0078
1.0007-0.0038-0.0038
1.0001-0.0038
1.0002-0.00391.0011

Note: In the simulation we use the stochastic differential model with non-zero drift in (23). We set (µ1 , µ2 )as (0.05, 0.05), (0.1, 0.1) and (0.5, 0.5).

35

Table 5: Autocovariance of univariate noise process.
(a) The variance estimates and the test statistics for the variance.
estimates
test statistics

OC
1.369 × 106
1411.64∗MM
1.296 × 107
185.70∗NI
2.209 × 107
3412.69∗(b) The test statistics for the autocovariance.
ℓ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
30
40
50
60

OC
-0.91
-0.95
-0.24
-0.44
-1.07
-1.04
-0.73
0.64
-0.95
-1.18
-0.49
0.17
-0.98
-0.46
1.32
0.40
-1.71
-1.01
-0.71
-0.66
0.66
0.41
-0.43
1.69

MM
-3.98∗-2.05∗-3.04∗-2.26∗-3.72∗-2.18∗-2.19∗-2.12∗-1.59-2.16∗-3.19∗-2.37∗-1.66-2.40∗-2.67∗-2.57∗-2.63∗
-0.97
-0.62
-0.38
-0.77
-0.54
-0.42
1.59

NI
-4.34∗-4.91∗-3.82∗-3.81∗-4.61∗-3.03∗-1.97∗-2.50∗
-0.18
-1.06
-0.91
0.03
0.96
0.03
0.95
-0.27
-1.07
-0.58
0.64
-0.62
1.28
-0.52
0.53
0.92

Note: In the top table (a), the test statistic for the variance of noise is given by (20). The critical value at 5%
significance level is 3.84. The bottom table (b) shows the test statistic (21) for the autocovariance of noisewith ℓ > 0. The critical value of the test statistic (21) is 1.96 at 5% significance level. Superscript ∗ denotes
significance at the 5% levels.

36

Table 6: Test statistics for a cross-covariance of bivariate noise processes.
ℓ OC-MM
-60
0.30
-50
-0.64
-40
0.51
-30
1.85
-20
-1.74
-15
-0.98
-10
-0.89
-9
-1.31
-8
-3.29∗-7-3.37∗-6-3.23∗-5-2.87∗-4-3.33∗-3-2.17∗-2-2.17∗-1-2.27∗
0-2.07∗
1-2.43∗
2-2.14∗
3-2.40∗
4
-1.81
5
-1.10
6
-1.05
7
-1.34
8
-2.83∗
9
-1.51
10
-1.40
15
-1.99∗
20
-1.61
30
-0.53
40
0.54
50
0.39
60
0.69

OC-NI MM-NI
0.51
0.67
1.15
0.91
0.05
-1.01
0.03
-1.29
-0.11
-0.95
-0.68
1.30
-1.35
-0.66
0.02
-0.96
-0.36
0.72
-0.34
0.56
-0.99
-1.49
-1.56
0.18
-0.42
0.37
0.31
-2.48∗
0.01-2.24∗
-0.83
-1.27
0.20
-3.41∗-0.63-4.66∗-0.86-4.02∗-0.15-2.25∗-1.66-3.94∗-0.71-4.35∗
0.06-2.33∗-0.77-2.48∗
-1.27
-1.51
-0.21
-1.03
-0.66
-0.82
1.25
0.40
1.25
-1.85
-0.17
0.74
1.79
-0.31
-0.86
-0.73
-1.84
-0.79

Note: The test statistic for the cross-covariance of the bivariate noise processes is given by (15). The criticalvalue of the test statistics (15) is 1.96 at 5% significance level. Superscript ∗ denote significance at the 5%
levels.

37