Discussion Papers In Economics
And Business

Bias-Corrected Realized Variance under Dependent
Microstructure Noise

Kosuke Oya

Discussion Paper 09-39

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANBias-Corrected Realized Variance under Dependent
Microstructure Noise

Kosuke Oya

Discussion Paper 09-39
December 2009

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Bias-Corrected Realized Variance under Dependent
Microstructure Noise ∗Kosuke Oya † ‡Abstract
The aim of this study is to develop a bias-correction method for realized variance (RV) estimation,
where the equilibrium price process is contaminated with market microstructure noise, such asbid-ask bounces and price changes discreteness. Though RV constitutes the simplest estimator ofdaily integrated variance, it remains strongly biased and many estimators proposed in previous
studies require prior knowledge about the dependence structure of microstructure noise to ensure
unbiasedness and consistency. The dependence structure is unknown however and it needs to be
estimated. A bias-correction method based on statistical inference from the general noise dependence structure is thus proposed. The results of Monte Carlo simulation indicate that the new
approach is robust with respect to changes in the dependence of microstructure noise.

JEL Classifications: C01; C13; C51
Keywords: Realized variance; Dependent microstructure noise; Two-time scales∗

The author is grateful to Naoto Kunitomo, Michael McAleer and Nabil Maghrebi for helpfulcomments. This work is partially supported by the Japanese Ministry of Education, Science, Sports
and Culture, Grant-in-Aid for Scientiﬁc Research (A), 2009, under reference number 10153313 andJST, CREST.
† Graduate School of Economics, Osaka University, Toyonaka, Osaka, Japan. oya@econ.osakau.ac.jp‡ Japan Science and Technology Agency, CREST, Toyonaka , Osaka, Japan

1

Introduction

The estimation of the daily integrated variance of returns on ﬁnancial assets is importantfor derivatives pricing and risk management purposes. While realized variance (RV) constitutes a simple but useful estimator of daily integrated variance (IV), it remains also astrongly biased estimator, where the equilibrium price process is contaminated with market
microstructure noise. This microstructure noise can be induced by various market frictions
such as bid-ask bounces and the discreteness of price changes, inter alia. There are three
approaches to cope with noise contamination, including (i) use of returns on the appropriate
interval length based on optimal sampling frequency proposed by Bandi and Russell [2], (ii)subsampling and bias correction proposed by Zhang et al. [8] and (iii) kernel estimation following Barndorﬀ-Nielsen, et al. [4]. McAleer and Medeiros [6] provide an extensive review of
the recent literature on RV estimation. It is the time-dependent noise structure that ensuresthe unbiasedness and consistency of IV estimators. The estimations proposed in previous
studies ultimately require prior knowledge about this noise dependency, which needs to berather estimated. The present study addresses these estimation issues and uses the consistent cross-covariance and autocovariance estimators of microstructure noise, and the tests
statistics developed by Ubukata and Oya [7] to identify the noise dependence structure. The
selection procedure of time scales based on Aı̈t-Sahalia et al.[1]’s Two Scales RV (TSRV) is
also provided under the general conditions of dependent noise. An alternative bias-corrected
estimator of IV can be also proposed using the autocovariance of microstructure noise. The
remainder of this paper is organized as follows. The price process and market microstructure noise are presented in section 2. Section 3 discusses the realized variance and related
estimators are given. Section 4 provides a brief review of the autocovariance estimator of
microstructure noise proposed by Ubukata and Oya [7] followed by a discussion of of theselection two-scales TSRV estimator and alternative bias-corrected RV estimator. Section
5 presents the results of Monte Carlo simulation for the ﬁnite sample properties of the
proposed selection procedure and the new bias-corrected estimator under the general noise
dependence. Section 6 concludes the paper.

2

Price Process and Microstructure Noise

It is assumed that the equilibrium asset price follows a continuous semi-martingale process
dP ∗ (t) = µ(t)dt + σ(t)dW (t) where P ∗ (t) is the logarithmic equilibrium continuously compounded intra-daily price, W (t) is a standard Brownian motion, and both µ(t) and σ(t) > 0
are bounded measurable functions. The diﬀusion term σ(t) can be estimated according to–1–

the integrated variance over a ﬁxed interval [0, T ]∗∗IV = ⟨P , P ⟩T =∫ T

σ 2 (t)dt(1)0using the observed logarithmic asset price of the asset for t ∈ [0, T ]. The market closing timeis denoted as T . We assume that the drift term µ(t) equals to zero since the trend term of
the price process is likely to be small during the trading hours on a given day. Suppose thatthe asset price can be observed at the discrete time points t0 = 0 < t1 < t2 < · · · < tn = T ,where ti represents the i-th transaction time. The length of the i-th interval is deﬁned as∆ti = ti − ti−1 . It is noted that ∆ti = T /n only under the restrictive conditions of regularsampling, and that ∆ti ̸= ∆tj for i ̸= j for non-regular sampling.
In order to examine the impact of market microstructure noise, it is also assumed that theobserved price process P (t) consists of the equilibrium continuously compounded intra-daily
price process P ∗ (t), which is unobservable, and the noise process η(t)P (t) = P ∗ (t) + η(t).

(2)

The market microstructure noise η(t) is also assumed to represent a serially dependent random variable. This is rather a plausible assumption given the behavior of noise determinants
such as bid-ask bounces, order-ﬂow clustering and other market imperfections. Thus, thefollowing set of assumptions about the microstructure noise can be made.
Assumption 1 (Market microstructure noise) Suppose (a) {η(t)} is a sequence of random variables with zero mean, (b) the noise process is covariance stationary with autocovariance function, which has a finite dependence structure in the sense that:γη (ℓ) = E[η(t)η(t − ℓ)] = 0, f or all | ℓ |> mwhere m is a finite positive integer, (c) there exists some positive number β > 1 that satisfies
4β
E η(t)η(s) < ∞ for all t, s and (d) the noise process is independent of the equilibrium
price process.

With respect to assumption (d), it is noted that as the number of observations increases,
the eﬀect of dependence is dominated by noise variation, even when the noise terms arecorrelated with equilibrium prices. Hansen and Lunde [5] suggest that the independence
assumption (d) does not signiﬁcantly aﬀect the analysis of asset prices with high-frequency
trading.
–2–

3

Realized Variance and Related Estimators

3.1 Realized Variance

The most widely used estimator of the integrated variance deﬁned in (1) is the realized
variance, which is deﬁned as the sum of squared returns given by
RV =

n
∑
i=1

ri2 =

n
∑

(P (ti ) − P (ti−1 ))2 ,

(3)

i=1

where the i-th transaction price is P (ti ) and the i-th intraday log return is deﬁned asri = P (ti ) − P (ti−1 ).
Given the time intervals I i = (ti−1 , ti ] for all i, it is possible to the expectation of RV
conditional on the stochastic arrival times as EI [ · ]. Thus, the conditional expectation of(3) can be written as
EI [RV ] =n∑

EI [(ri∗ + η(ti ) − η(ti−1 ))2 ] =

i=1

n
∑

EI [ ri∗ 2 ] + 2nγη (0) − 2

i=1

n
∑

γη (∆ti )

(4)

i=1

where ri∗ = P ∗ (ti ) − P ∗ (ti−1 ). It is straightforward to demonstrate that the variance γη (0)∑and the sum of autocovariances ni=1 γη (∆ti ) can introduce a bias in the estimation of RV
according to (3). For the sake of simplicity, it is assumed hereafter that the sampling scheme
is regular and that ∆ti = 1, i = 1, . . . , n. Under such conditions, the total bias is represented
by 2n(γη (0) − γη (1)).

3.2 Two-Scales Realized VarianceZhang et al. [8] proposed the Two Scales Realized Variance (TSRV) which is unbiased
when the microstructure noise is independent. Denote the original grid of observation(j)times as G = {t0 , t1 , . . . , tn }. G is partitioned into K nonoverlapping subgrids, GK , j =
(j)
(j)
(ℓ)
1, . . . , K, such that G = ∪K
j=1 GK , where GK ∩ GK = ϕ for j ̸= ℓ. Given the assumption of regular sampling scheme, the j-th nonoverlapping subgrid can be represented as(j)
GK = {tj−1 , tj−1+K , tj−1+2K , . . . , tj−1+nj K } for j = 1, . . . , K where nj is the integer making(j)(j)
tj−1+nj K the last element in the subgrid GK . Then the realized variance for the subgrid GK
can be written as
n
(j)

RVK =

j
∑

(P (t(j−1)+iK ) − P (t(j−1)+(i−1)K ))2 .

i=1

–3–

(5)

Let RV (all) be the realized variance for the full grid G. Then TSRV by Zhang et al. [8] canbe represented as
RVK = (1/K)

K
∑

(j)

RVK − (n̄/n)RV (all)

(6)

j=1

∑

where n̄ = K
j=1 nj /K = (n − K + 1)/K. The ﬁrst term of (6) represents the average of(j)(j)
RVK estimators for the subgrid GK , j = 1, . . . , K, which is a biased estimator of IV . In
case of independent noise, the second term of (6) represents the bias-correction term since
the bias in the ﬁrst term is 2n̄γη (0) and RV (all) /(2n) is a consistent estimator of γη (0). The
reﬁnement to correct for the ﬁnite sample bias in (6) following Zhang et al. [8] is conduciveto the following expression(adj)
RVK = (1 − n̄/n)−1 RVK .
(7)
Although these two-scales realized variances are unbiased and consistent estimators of IV
under the independent noise assumption, these desirable features are not guaranteed when
the noise terms are not independently distributed.

3.3 Extended Two-Scales Realized VarianceAı̈t-Sahalia et al. [1] extended the TSRV to allow for dependent market microstructure noise.
The ﬁrst term of (6) can be rewritten in the form of the average lag K realized varianceunder regular sampling intervals.
(avg)

RVK

= (1/K)K∑(j)

RVK = (1/K)

j=1

n−K
∑

(P (ti+K ) − P (ti ))2 .

(8)

i=0

It is also possible to use an alternative lag J instead of lag K, (1 ≤ J < K ≤ n) in (8) and
express using two diﬀerent lags J and K, the extended TSRV as follows.
(avg)

RVJ,K = RVK(avg)− (n̄K /n̄J )RVJ

(9)

where n̄J = (n − J + 1)/J, n̄K = (n − K + 1)/K, 1 ≤ J < K ≤ n and K = o(n).
The main diﬀerence between TSRV expressed in (6) and extended TSRV estimators is
captured by the second term of (9) which represents a bias-correction term. It is easy toidentify the bias term in RVJ,K as
2n̄K (γη (ti+J − ti ) − γη (ti+K − ti )) = 2n̄K (γη (J) − γη (K)).

(10)

This bias consists of the autocovariances of microstructure noise γη (J) and γη (K). It should–4–be noted that these autocovariances become negligible when the selected lags J and K arelarge enough. Based on Assumption 1 that the microstructure noise process is m-dependent,
it is possible to select J such that J = m + 1 and K = O(n2/3 ) as in Aı̈t-Sahalia et al. [1].
The ﬁnite-sample correction of the extended TSRV can be thus expressed as
RVJ,K = (1 − n̄K /n̄J )−1 RVJ,K .
(adj)

4

(11)

Lags (J, K) Selection and Alternative Estimator

The extended TSRV is an appropriate estimator of IV when the assumption of independent
noise assumption is not valid. As discussed in the previous section however, the important
issue remains as to how the optimal lags J and K should be selected. Although Aı̈t-Sahalia
et al. [1] argue that the extended TSRV is robust with respect to lags (J, K) selection, thefollowing section 5.2 demonstrates that this estimation may still be signiﬁcantly sensitive
to lag J selection.
The present section discusses the new methodology that allows for the selection of theappropriate lag J, based on the testing procedure proposed in Ubukata and Oya [7]. Analternative IV estimator which utilizes a diﬀerent bias-correction method from the extended
TSRV is also brieﬂy introduced.

4.1 Microstructure Noise Autocovariance Estimation

Ubukata and Oya [7] proposed an unbiased and consistent estimator of the microstructurenoise autocovariance γη (ℓ), and derived its asymptotic properties. The test statistic of thenull hypothesis γη (ℓ) = 0 is also applied to examine the signiﬁcance of the microstructurenoise dependence. Suppose that the threshold value of microstructure noise dependence as
m, is such that γη (m + 1) = 0 and γη (m) ̸= 0. The threshold value m can be determinedthrough the test statistic deﬁned in Ubukata and Oya [7, section 3.2].
To obtain an unbiased estimator of γη (ℓ), it is possible to construct the product of returns(±)
Zℓ,ij for all i, j, such that ℓ = tj−1 − ti using the selected threshold value m(±)(−) (+)

Zℓ,ij = ri rj(+)

where tj(−)(+)

= (P (ti ) − P (ti−1 ))(P (tj ) − P (tj−1 ))(+)

is the ﬁrst transaction time, which follows tj subject to tj
–5–

(12)
(−)

− ti > m, and ti−1 is(−)

the last transaction time, which is followed by ti−1 subject to tj−1 − ti−1 > m. Thus, for a(±)
given ℓ = tj−1 − ti , expected product of returns can be expressed as E[Zℓ,ij ] = −γη (ℓ) for all
(±)
(±)
ℓ
i, j. Let {Zℓ,k }N
k=1 be the sequence that arranges Zℓ,ij satisfying ℓ = tj−1 − ti in ascendingorder of index i. Nℓ represents the number of observations in the sequence. It is then possible
to construct the unbiased autocovariance estimator of the microstructure noise using the(±)ℓ
sample mean of {Zℓ,k }N
k=1 .

Autocovariance Estimator (Ubukata and Oya [7]): The autocovariance estimator of
the microstructure noise and its asymptotic distribution are given as
Nℓ
1 ∑
a
(±)
1/2
γ̂η (ℓ) = −Zℓ,k , Nℓ (γ̂η (ℓ) − γη (ℓ)) → N (0, ωℓ2 )Nℓ k=1(13)

where ωℓ2 = limNℓ →∞ Nℓ E[(γ̂η (ℓ) − γη (ℓ))2 ].
The test statistic to examine the signiﬁcance of γη (ℓ) is also discussed in Ubukata and Oya[7, corollary 2]. Let the null hypothesis and the alternative be represented as γη (ℓ) = 0 andγη (ℓ) ̸= 0, respectively. The test statistic of the noise autocovariance can be expressed asτη∗ (ℓ) =√Nℓ γ̂η (ℓ)/ω̂ℓ .

(14)This test statistic is asymptotically distributed as standard normal under the appropriate
conditions, which are not further discussed herein for the sake of brevity. Reference can bemade to Ubukata and Oya [7] for more details including the explicit formulation of ω̂ℓ .

4.2 Lags (J, K) Selection

To select the appropriate lag J, it is required ﬁrst to examine whether γη (1) = 0 through thetest statistic τη∗ (ℓ). When the null is rejected, there is a need to verify whether γη (2) = 0,
and this test γη (ℓ) = 0 is reiterated until the null cannot be rejected. The distance ℓ forwhich the null hypothesis γη (ℓ) = 0 cannot be rejected for the ﬁrst time is denoted as Jˆ forthe extended TSRV.
The optimal choice of lag K is clearly provided in Zhang et al. [8] under the i.i.d. noise
assumption. However, the important issue of optimal lag selection remains under less restrictive conditions of dependent microstructure noise. It is possible to devise a simple approachto address this issue of optimal lag selection, where the original grid of observation times
denoted by G = {t0 , t1 , . . . , tn } as deﬁned in previous section, is supplemented by a new–6–

subgrid GJˆ = {t0 , tJˆ, t2Jˆ, . . . , t[n/J]ˆ } where Jˆ is the selected lag, representing the thresholdvalue of noise dependence based on the test statistic τη∗ (ℓ). It is reasonable to suppose thatˆ[n/J]

the microstructure noise { η(tiJˆ) }i=0 is an uncorrelated random sequence. The optimallag K can thus be obtained by applying the methodology proposed in Zhang et al. [8] toˆ[n/J]
the sequence of the observed transaction price { P (tiJˆ) }i=0 . The extended TSRV withˆ K̂) and the bias-adjusted extended TSRV are denoted hereafter as RV ˆ
selected lags (J,
J,K̂
(adj)
and RVJ,ˆ K̂ , respectively.

4.3 Alternative Bias-Corrected Estimator(bc)

In order to correct the bias of RV, an alternative IV estimator RVK
using autocovariance deﬁned in (13) as
(bc)

RVK

(avg)

= RVK

can also be constructed− 2n̄K γ̂η (0).

(15)

(bc)

The unbiasedness and consistency of RVK can be immediately established from the unbiasedness and consistency of the autocovariance estimator (13) and the result given in Zhang
et al. [8].

5Monte Carlo Simulation

5.1 Simulation Design

A series of Monte Carlo simulations is performed in order to examine the impact of lag (J, K)
selection on the extended TSRV. This simulation exercise allows also for the assessment of
the estimator (15) properties relative to the extended TSRV under dependent microstructurenoise. The return-generating process is deﬁned exactly as in Zhang et al. [8].
dP ∗ (t) = (0.05 − ν(t)/2)dt + σ(t)dB(t)dν(t) = 5(0.04 − ν(t))dt + 0.5ν(t)1/2 dW (t), ν(t) = σ 2 (t)The correlation between the two Brownian motions B and W is set to -0.5. A total of
10000 sample paths of the process by the Euler scheme are generated at time intervals∆t = 1 second. It is noted that T = 1 day and a day, which consists of 6.5 hours of open
trading, or the equivalent of 23400 seconds. The price levels are observed discretely under–7–

the microstructure noise. In the simulation, the time interval between observations is set tobe 5 seconds, i.e., t0 = 0, t1 = 5, . . . , t4680 = 23400. Since the focus is made on the case of
dependent microstructure noise, diﬀerent dependent patterns are considered, including thefollowing autoregressive model and moving average model for the noise process
AR(1) : η(ti ) = ρ η(ti−1 ) + ε(ti ),MA(3) : η(ti ) = ε(ti ) +3∑

θs ε(ti−s )s=1where ρ = -0.8, -0.4, 0.0, 0.4, 0.8 for AR(1), (θ1 , θ2 , θ3 ) = (-0.6, 0, 0), (0.6, 0, 0) for MA(1),(0.6, 0.5, 0) for MA(2) and (0.6, 0.5, 0.4) for MA(3). The variance of microstructure noiseE[η(t)2 ] should be carefully selected because the eﬀect of the microstructure noise may be
negligible only when E[η(t)2 ] is very small. Hansen and Lunde [5] reported that the Noiseto-Signal Ratio (NSR) deﬁned as E[η(t)2 ]/IV for a sample of stocks listed on the NYSE and
NASDAQ markets ranges from 0.0002 to 0.006. Consistent with on these previous empiricalﬁndings, the NSR for simulated paths E[η(t)2 ] is set to 0.004, which is equal to the averageof the above NSR reference values. The observed price is also given as P (ti ) = P ∗ (ti ) + η(ti ).
Under these conditions, it is possible in the following subsections, to examine the inﬂuenceof lag J selection on the extended TSRV estimators, with K = 50, 100, 200 and assess
the statistical properties of selected lag J using the test statistic τη∗ (ℓ). Furthermore, it isimportant to compare the extended TSRV estimator with selected lags (J, K), to the biasadjusted TSRV extended estimator as well as (15) proposed in this study. These distinct
estimators can be obtained for each simulated sample path.

5.2 Influence of Lag Selection(adj)

The integrated variance is estimated using RVJ,K and RVJ,K with J = 1, . . . , 40 and K=
50, 100, 200. The relative bias of these estimators represented by the sample means of(estimate − IV )/IV is reported in the ﬁrst and third columns of Figure 1. The sample rootmean-squared-errors (RMSE) of estimate/IV is also provided in the second and fourthcolumns. The ﬁrst and third rows of Figure 1 show the bias and RMSE for RVJ,K . Similarly,(adj)
the second and fourth rows refer to the bias and RMSE for RVJ,K . The horizontal axis isprovided for J=1, . . . , 40. The models used for the noise process are AR(1) with ρ = −0.8
(adj)
and 0.8, i.i.d. and MA(1) with θ=−0.6. It is clear that the bias of RVJ,K becomes negligible
after J exceeds the threshold value of noise dependence while the bias of RVJ,K grows as(adj)
J increases. Although the performance of the bias-adjusted estimator RVJ,K is relatively–8–better, the variance seems to be sensitive to this bias adjustment process. This eﬀect isclearly captured by the reported RMSE values.
(adj)

It is important to appropriately select the lag J for both RVJ,K and RVJ,K estimators
because their RMSEs can strongly depend on the lag J selection even under more favorable
conditions of independent microstructure noise. Judging also from the second and fourthcolumns of Figure 1, it appears that lag K selection does not exert signiﬁcant inﬂuence on
RMSE provided that lag J is properly selected.

Figure 1 :around here

5.3 Performance of Lag Selection Procedure

The selection of lag J is preformed by testing the null hypothesis γη (ℓ) = 0 for ℓ > 0 as
described in section 4.2. Again, it is possible to examine the behavior of the test statisticτη∗ (ℓ) in association with lag J selection. Figure 2 shows the empirical distribution of selectedlag J denoted as Jˆ for the representative cases. The mode of Jˆ is found to equal 8 and 2
for AR(1) noise dependence with ρ = -0.8 and -0.4, respectively. On the other hand, mode
estimates amount to just 1, 2, 3 and 4, for i.i.d., MA(1), MA(2) and MA(3), respectively.
Thus, it is possible to use the test statistic τη∗ (ℓ) as an essential criterion in the selection of lag
J. The results of lag K selection according to the procedure described in the previous section
are not reported here, but the evidence suggests that variance of the empirical distributionof K̂ rises as the noise dependence increases.

Figure 2 :around here

5.4 Comparison of IV Estimators

The analysis so far focused on important issues related to the construction of the set of three
IV estimators under the assumption of dependent microstructure noise. The objective of thissubsection is to examine however the statistical properties of these estimators. It is noted(adj)
again that RVJ,ˆ K̂ and RVJ,ˆ K̂ represent the extended TSRV and bias-adjusted extended–9–ˆ K̂), while RV (bc) is the estimator proposed in the previous
TSRV with the selected lags (J,K̂
section with the selected lag K̂. The empirical distributions of (estimate − IV )/IV for eachestimator are given in Figure 3. It is easy to see that the empirical distributions of RVJ,ˆ K̂(adj)ˆ K̂) are found to be skewed to the right. It is noted in particular thatand RVJ,ˆ K̂ with (J,(bc)

under strong noise dependence, the empirical distribution of the proposed estimator RVK̂
is rather closer to symmetry than alternative estimators.

Figure 3 :around here

Table 1 :around here

The bias and RMSE values associated with IV estimators are reported in Table 1. The(bc)
bias terms in RVK̂ are generally smaller than those obtained with respect to RVJ,ˆ K̂ and(adj)
RVJ,ˆ K̂ . These RMSE values are rather comparable across these estimators, except in caseof strong noise dependence case. These simulation results suggest that the extended TSRVˆ K̂), as well as the proposed estimatorand bias-adjusted extended TSRV with selected (J,(bc)
RVK̂ are indeed robust to the dependence of microstructure noise.

6

Concluding Remarks

This paper proposes a new approach to the problem of appropriate lag selection for the twoscales realized variance under dependent microstructure noise. An alternative bias-adjusted
estimator based on the variance of microstructure noise is also proposed, along the lines of
Ubukata and Oya [7]. The evidence from Monte Carlo simulation suggests that the proposedlag selection procedure is appropriate as the the proposed estimator is associated withrelatively smaller bias and RMSE values. The proposed procedure for lag selection and the
new IV estimator can thus be useful for empirical studies based on transactions price data.
It should be noted that kernel-type estimators may be considered to be more eﬃcient. But
as noted by Bandi and Russell [3], this asymptotic property is not necessarily satisﬁed in
the presence of large samples typical of empirical studies based on transactions data. The
comparative analysis is nevertheless important and therefore warranted. It falls beyond thescope of the present study, which introduces a new approach that may open interesting
avenues for future research.
– 10 –

References[1] Y. Aı̈t-Sahalia, P. Mykland, L. Zhang, Ultra high frequency volatility estimation with dependent
microstructure noise, NBER Working Paper, 11380.
[2] F. M. Bandi, J. R. Russell, Microstructure noise, realized variance, and optimal sampling,
Review of Economic Studies 75 (2) (2008) 339–369.
[3] F. M. Bandi, J. R. Russell, Market microstructure noise, integrated variance estimators, andthe accuracy of asymptotic approximations, Journal of Econometrics, forthcoming.
[4] O. E. Barndorﬀ-Nielsen, P. Hansen, A. Lunde, N. Shephard, Designing realized kernels tomeasure the ex-post variation of equity prices in the presence of noise, Econometrica 76 (2008)
1481–1536.
[5] P. R. Hansen, A. Lunde, Realized variance and market microstructure noise (with discussions),
Journal of Business and Economic Statistics 24 (2006) 127–218.
[6] M. McAleer, M. C. Medeiros, Realized volatility: A review, Econometric Reviews 27 (2008)
10–45.
[7] M. Ubukata, K. Oya, Estimation and testing for dependence in market microstructure noise,
Journal of Financial Econometrics 7 (2009) 106–151.
[8] L. Zhang, P. Mykland, Y. Aı̈t-Sahalia, A tale of two time scales: Determining integrated
volatility with noisy high frequency data, Journal of the American Statistical Association 100(2005) 1394–1411.

– 11 –

Table 1: Relative Bias and RMSE of estimators
Bias

RMSE
(adj)

RVK̂

(bc)

RVJ,ˆ K̂

RVJ,ˆ K̂

0.035

0.008

0.326

0.366

0.259

-0.0640.009-0.006

0.208

0.218

0.214

AR: ρ = 0.4

0.097

0.181

0.041

0.336

0.374

0.295

AR: ρ = 0.8

0.123

0.344

0.077

0.543

0.688

0.385

i.i.d.

-0.074-0.019-0.005

0.160

0.141

0.213

MA(1)

0.016

0.081

0.025

0.334

0.358

0.292

MA(2)

0.058

0.146

0.032

0.400

0.446

0.314

MA(3)

0.088

0.196

0.037

0.441

0.506

0.336

Noise Type

RVJ,ˆ K̂

RVJ,ˆ K̂

AR: ρ = −0.8

-0.131

AR: ρ = −0.4

– 12 –

(adj)

(bc)

RVK̂

200

20

30

40

0

10

20

30

40

adj)
RV(J,Ki.i.d.

30

40

0

10

20

30

40

0.8
0.4

40

0.8
0

10

20

30

40

RVJ,K MA: θ = −0.6

10

20

30

40

0

10

20

30

40

adj)
RV(J,K
MA: θ = −0.6

0

10

20

30

40

0

10

Fig. 1. Figure 1: Eﬀect of selection lag J with diﬀerent lag K values– 13 –

30

0.4

RMSE
40

0.4

Bias

0.0
20

30

adj)
RV(J,K
MA: θ = −0.6

0.4

RMSE

0.4
−0.8 −0.2

10

20

RMSE
0

0.8

adj)
RV(J,Ki.i.d.

0

10

0.4

Bias

0.4

40

20

adj)
RV(J,K
AR: ρ = 0.8

RVJ,K MA: θ = −0.6

0.8
30

10

0.0
0

0.0

RMSE

0.4
−0.8 −0.2

20

0

0.8

10

RVJ,K i.i.d.

J
10

40

0.4
0

RVJ,K i.i.d.

0

30

0.0

40

20

0.8

30

10

adj)
RV(J,K
AR: ρ = 0.8

Bias

0.4

RMSE
20

0.0
0

0.0
10

−0.8 −0.2

40

0.4

30

RMSE

0.4
20

adj)
RV(J,K
AR: ρ = −0.8

0.4
−0.8 −0.2
0

Bias

10

0.8

adj)
RV(J,K
AR: ρ = −0.8

Bias

Bias
0

0.4

40

RMSE

30

RVJ,K AR: ρ = 0.8

0.0

20

−0.8 −0.2

10

0.4
0.0

RMSE

0.4

Bias

−0.8 −0.2

J
0

Bias

RVJ,K AR: ρ = 0.8

0.8

RVJ,K AR: ρ = −0.8

−0.8 −0.2

100−0.8 −0.2

K=50

RVJ,K AR: ρ = −0.8

20

30

40

AR: ρ = −0.4

0

2

4

6

8 10
^
J

13

16

Selected Ratio
0.0
0.4
0.8

i.i.d.

Selected Ratio
0.0
0.4
0.8

Selected Ratio
0.0
0.4
0.8

AR: ρ = −0.8

0

2

4

6

13

16

2

4

6

2

4

6

8 10
^
J

13

16

8 10
^
J

13

16

13

16

Selected Ratio
0.0
0.4
0.8

MA(3)

Selected Ratio
0.0
0.4
0.8
0

0

MA(2)

Selected Ratio
0.0
0.4
0.8

MA(1)

8 10
^
J

0

2

4

6

8 10
^
J

13

16

0

2

4

6

8 10
^
J

Fig. 2. Empirical distribution of selected lag J using τη∗ (ℓ))^
RV^J(adj
AR: ρ = −0.8
,K

−1

0

1

2

3

4

2.0
0.0
−1

1

2

3

4

0

1

2

3

4

−1

0

1

2

3

4

2.0

RV(K^bc) AR: ρ = 0.8

0.0

1.0

2.0
1.0
0.0

0.0

1.0

2.0

0

)
^
RVJ^(adj
AR: ρ = 0.8,KRV^J,K^ AR: ρ = 0.8−1

RV(^Kbc) AR: ρ = −0.8

1.0

2.0
1.0
0.0

0.0

1.0

2.0

RVJ^,K^ AR: ρ = −0.8

−1

0

1

2

3

4

−1

0

Fig. 3. Empirical distribution of (estimate − IV )/IV

– 14 –

1

2

3

4