Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji 073500106000000044

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Properties of Realized Variance Under Alternative
Sampling Schemes
Roel C. A Oomen
To cite this article: Roel C. A Oomen (2006) Properties of Realized Variance Under
Alternative Sampling Schemes, Journal of Business & Economic Statistics, 24:2, 219-237, DOI:
10.1198/073500106000000044
To link to this article: http://dx.doi.org/10.1198/073500106000000044

Published online: 01 Jan 2012.

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Properties of Realized Variance Under
Alternative Sampling Schemes
Roel C. A. O OMEN

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Department of Finance, Warwick Business School, The University of Warwick, Coventry CV4 7AL, U.K.
(roel.oomen@wbs.ac.uk )
This article investigates the statistical properties of the realized variance estimator in the presence of market microstructure noise. Different from the existing literature, the analysis relies on a pure jump process
for high-frequency security prices and explicitly distinguishes among alternative sampling schemes, including calendar time sampling, business time sampling, and transaction time sampling. The main finding
in this article is that transaction time sampling is generally superior to the common practice of calendar
time sampling in that it leads to a lower mean squared error (MSE) of the realized variance. The benefits
of sampling in transaction time are particularly pronounced when the trade intensity pattern is volatile.

Based on IBM transaction data over the period 2000–2004, the empirical analysis finds an average optimal
sampling frequency of about 3 minutes with a steady downward trend and significant day-to-day variation
related to market liquidity and a consistent reduction in MSE of the realized variance due to sampling in
transaction time that is about 5% on average but can be as high as 40% on days with irregular trading.
KEY WORDS: High-frequency data; Market microstructure noise; Optimal sampling; Pure jump
process.

1. INTRODUCTION
The recent trend toward the model-free measurement of asset
return volatility has been spurred by an increase in the availability of high-frequency data and the development of rigorous foundations for the realized variance estimator, defined as
the sum of squared intraperiod returns (see, e.g., Andersen,
Bollerslev, Diebold, and Labys 2003; Barndorff-Nielsen and
Shephard 2004; Meddahi 2002). Although the theory suggests
that the integrated variance can be estimated arbitrarily accurately by summing up squared returns at sufficiently high frequency, the validity of this result crucially relies on the price
process conforming to a semimartingale, thereby ruling out various market microstructure effects that are frequently encountered in high-frequency data. This apparent conflict between
the theory and practice of model-free volatility measurement
is what motivates this work. Based on a pure jump process
for high-frequency security prices, a framework is provided in
which to analyze the statistical properties of the realized variance in the presence of market microstructure noise. Closedform expressions for the bias and mean squared error (MSE) of
the realized variance are derived as functions of the model parameters and the sampling frequency. The principal contribution

of this work is that the analysis explicitly distinguishes among
different sampling schemes, which, to the best of our knowledge, has not yet been considered in the literature. Importantly,
both the theoretical and the empirical results suggest that the
MSE of the realized variance can be reduced by sampling returns on a transaction time scale as opposed to the common
practice of sampling in calendar time.
Inspired by the work of Press (1967, 1968), we use a
compound Poisson process to model the asset price as the accumulation of a finite number of jumps, each of which can
be interpreted as a transaction return with the Poisson process
counting the number of transactions. To capture the impact of
market microstructure noise, we allow for a flexible MA(q) dependence structure on the price increments. Further, we leave
the Poisson intensity process unspecified but point out that in

practice both stochastic and deterministic variation in the intensity process may be needed to capture trade duration and
return volatility dependence plus diurnal patterns in market activity. Despite these alterations, the model remains analytically
tractable in that the characteristic function of the price process
can be derived in closed form after conditioning on the integrated intensity process.
The move toward a semiparametric pure jump process with
discontinuous sample paths of finite variation marks a significant departure from the popular domain of diffusion-based
modeling. For instance, in the present framework the realized
variance is an inconsistent estimator of the ( jump analog of )

integrated variance. Also, in the presence of market microstructure noise, the bias of the realized variance does not tend to
infinity when the sampling frequency approaches 0, as is generally the case for a diffusive price process (see, e.g., Aït-Sahalia,
Mykland, and Zhang 2005; Bandi and Russell 2004a). Yet it
is important not to overstate these seemingly fundamental differences, because both the pure jump process and the diffusive
process can give rise to a number of similar results and intuition
regarding the statistical properties of the realized variance. The
main motivation for using the jump model here is that it provides a convenient framework in which to analyze the statistical properties of the realized variance for different sampling
schemes. Furthermore, the model has intuitive appeal and is
in line with a recent literature that emphasizes the important
role that pure jump processes play in the modeling of financial time series and pricing of derivatives (see, e.g., Cont and
Tankov 2003 and references therein).
On the theoretical side, this article makes three contributions.
First, it provides a flexible and analytically tractable framework
in which to investigate the statistical properties of the realized variance in the presence of market microstructure noise.
The optimal sampling frequency that leads to a minimum MSE

219

© 2006 American Statistical Association
Journal of Business & Economic Statistics

April 2006, Vol. 24, No. 2
DOI 10.1198/073500106000000044

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220

of the realized variance is readily available. In line with intuition, it is found that the main determinants of the optimal
sampling frequency are the number of trades and the level of
market microstructure noise, that is, quantities that vary over
time but are easily measured in practice. Second, the analysis
explicitly distinguishes among different sampling schemes, including transaction time sampling, business time sampling, and
calendar time sampling. The results here provide new insights
into the impact of a particular choice of sampling scheme on
the properties of the realized variance. In many cases, a superior (and inferior) sampling scheme can be identified. Third,
the modeling framework incorporates a general MA(q) dependence structure for the price increments, thereby allowing for
the flexible modeling of various market microstructure effects.
Further, on the empirical side, the article provides an illustration
of how the model can be used in practice to determine the optimal sampling frequency and gauge the efficiency of a particular
sampling scheme.

The principal finding that emerges from both the theoretical and the empirical analysis, is that transaction time sampling
is generally superior to the common practice of calendar time
sampling in that it leads to a reduction in MSE of the realized variance. Intuitively, transaction time sampling (or business time sampling for that matter) gives rise to a return series
that is effectively “devolatized” through appropriate deformation of the calendar time scale. It is precisely this feature of the
sampling scheme that leads to an improvement in efficiency of
the variance estimator. Further, it turns out that the magnitude
of this efficiency gain increases with an increase in the variability of trade intensity, suggesting that the benefits of sampling
in transaction or business time are most apparent on days with
irregular trading patterns. Only in exceptional circumstances,
where either the sampling frequency is extremely high and far
beyond its “optimal” level or market microstructure noise is
unrealistically dominant may calendar time sampling lead to
better results in terms of MSE. The empirical analysis confirms this. Based on IBM transaction data for January 2000–
December 2004, we estimate the model parameters and trade
intensity process and use this to determine the optimal sampling
frequency and MSE reduction of the realized variance when the
price process is sampled in transaction time as opposed to calendar time. We find that for each day in the sample, transaction
time sampling leads to the lowest MSE of the realized variance.
The median MSE reduction is a modest 5%, but can be as high
as 40% on days with dramatic swings in market activity. Further, the average optimal sampling frequency is about 3 minutes

in calendar time and 50 trades in transaction time but declines
steadily over time (from 5 minutes or 60 trades in 2000 to about
1.5 minutes or 30 trades in 2004). Simulations indicate that the
measurement error in the model parameters does not lead to
noteworthy biases in the results and that much of the day-to-day
variation in optimal sampling frequency is statistically significant.
The observation that market microstructure noise affects
the statistical properties of the realized variance is not original to this research. Andersen et al. (2000) were among the
first to document evidence on the relationship among sampling
frequency, market microstructure noise, and the bias of the
realized variance measure. Now a growing body of literature

Journal of Business & Economic Statistics, April 2006

emphasizes the crucial role of sampling of the price process
and the filtering of market microstructure noise. The articles
most closely related to the current one are those by Bandi and
Russell (2004a,b), Hansen and Lunde (2006), Zhang, Mykland,
and Aït-Sahalia (2005), and Aït-Sahalia et al. (2005), who built
on the work by Andersen et al. (2003) and Barndorff-Nielsen

and Shephard (2004) similarly to how this article builds on the
work of Press (1967). Because many of the results and intuition
are qualitatively comparable, they can be seen to complement
each other because they are derived under different assumptions on the price process. What differentiates this article is the
analysis of alternative sampling schemes. Note that the modeling framework proposed here was recently extended by Oomen
(2005) to study the properties of the first-order bias-corrected
realized variance measure of Zhou (1996) and by Griffin and
Oomen (2005) to analyze the difference between transaction
time sampling and tick time sampling.
The remainder of the article is structured as follows. Section 2 introduces the extended compound Poisson process as a
model for high-frequency security prices and informally shows
how it relates to the more familiar diffusion type models used
in this literature. We discuss how the model can account for
market microstructure noise and derive the joint characteristic
function of returns, which forms the basis for the analysis of the
realized variance. Section 3 presents a theoretical discussion of
the properties of the realized variance for alternative sampling
schemes in terms of bias and MSE. Section 4 briefly explores
the relation of the results derived in this article to those obtained
in a diffusion setting, and Section 5 discusses some possible extensions of the model. Section 6 outlines the estimation of the

model and presents the empirical results for IBM transaction
data. Section 7 concludes.
2. A PURE JUMP PROCESS FOR
HIGH–FREQUENCY SECURITY PRICES
In this article we model the price process as a pure jump
process. Although such processes have a long tradition in
the statistics literature (see, e.g., Andersen, Borgan, Gill, and
Keiding 1993; Karlin and Taylor 1981 and references therein),
they have received only moderate attention in finance following their introduction by Press (1967). Yet a number of recent
(and less recent) works provide compelling evidence that these
type of processes can be applied successfully to many issues in
finance, ranging from the modeling of low-frequency returns
(e.g., Ball and Torous 1983; Carr, Geman, Madan, and Yor
2002; Maheu and McCurdy 2004) and high-frequency transaction data (e.g., Bowsher 2002; Large 2005; Rogers and Zane
1998; Rydberg and Shephard 2003) to the pricing of derivatives
(e.g., Carr and Wu 2004; Cox and Ross 1976; Geman, Madan,
and Yor 2001; Mürmann 2001). In addition, for the purpose of
this article, a pure jump process appears to be ideally suited
for addressing the issues involved. Specifically, we consider
an extension of the compound Poisson process (CPP) of Press

(1967, 1968), which allows for time-varying jump arrival intensity and serially correlated increments to capture the effects
of market microstructure contaminations. The main reason to
adopt this model is that it provides a convenient framework in
which to analyze the properties of the price process, and hence

Oomen: Realized Variance

221

the realized variance, under alternative sampling schemes. Before presenting the detailed model, we provide an informal discussion of the basic CPP and its relation to the more familiar
stochastic volatility (SV) process to further motivate the choice
of model and provide the appropriate context.
Following Andersen et al. (2003), consider the following representation of the logarithmic price process P:

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P(t) = A(t) + C(t) + D(t),

t ∈ [0, T],

(1)

where A is a finite-variation predictable mean component and
the local martingales C and D are continuous sample path and
compensated pure jump processes. It is well known that the
mere existence of the decomposition in (1) implies that P belongs to the class of special semimartingales (see, e.g., Protter
2005). In what follows, we concentrate on the unit time interval (i.e., T = 1), and, because the focus is on the martingale
components, set A = 0 for simplicity of notation. In the finance
literature, C is commonly specified as an SV process,

t
C(t) = C(0) +
σ (u) dW(u),
(2)
0

where W is a Brownian motion and the instantaneous volatility
σ (t) is a positive càdlàg process. In line with Barndorff-Nielsen
and Shephard (2004), here it is assumed that σ is independent
of W, although this can be generalized, as shown by Jacod and
Protter (1998) and, more recently, Bandi and Russell (2004a)
and Barndorff-Nielsen, Graversen, Jacod, and Shephard (2006).
Specifications such as these have been the primary focus of
the literature on the realized variance, that is, continuous
semimartingales with jump component D = 0 (see, e.g., AïtSahalia et al. 2005; Andersen et al. 2003; Bandi and Russell
2004a,b; Barndorff-Nielsen and Shephard 2004; BarndorffNielsen, Hansen, Lunde, and Shephard 2004; Hansen and
Lunde 2006; Meddahi 2002; Zhang et al. 2005). In this article we take the opposite route and concentrate on the pure
jump component while eliminating the continuous part, that is,
C = 0. In particular, the starting point here is a CPP in the spirit
of Press (1967), which takes the form
D(t) = D(0) +

M(t)


εj ,

(3)

j=1

where ε ∼ N (0, σε2 ), dM(t) ∼ Poisson(λ(t)), and the instantaneous jump intensity λ(t) is a positive càdlàg process independent of ε. Comparing the two alternative specifications
in (2) and (3) immediately reveals a fundamental difference—
namely, C is a purely continuous diffuse process with sample
paths of infinite variation, whereas D is a purely discontinuous jump process with sample paths of finite variation. Further,
conditional on the volatility path up to time t, the characteristic
function of C can be expressed as


 

1
φC (ξ |t) = Eσ exp iξ(C(t) − C(0)) = exp − ξ 2 σ∗2 (t) ,
2

t
where σ∗2 (t) ≡ 0 σ 2 (u) du. Similarly, conditional on the jump
intensity path up to time t, the characteristic function of D is
given as
 

φD (ξ |t) = Eλ exp iξ(D(t) − D(0))





1 2 2
= exp (t) exp − ξ σε − 1 ,
2

t
where (t) = 0 λ(u) du. Hence, increments of C are conditionally Gaussian, whereas increments of D are conditionally
mixed Gaussian. Still, despite these apparent differences between the characteristics of the diffusive SV model and the pure
jump CPP model, there exists an intimate link between the two.
In particular, it can be seen that lim φD (ξ |t) = φC (ξ |t), where
the limit is taken such that (t) → ∞ and σε → 0 while keeping (t)σε2 = σ∗2 (t) < ∞ constant. Intuitively, when increasing
the (integrated) jump intensity and correspondingly decreasing
the innovation variance so as to keep the quadratic variation
of the process constant over a fixed time interval, the resulting
sample path is made up of an increasing number of progressively smaller jumps and is, in the limit, indistinguishable from
a diffusive sample path. In other words, C can be viewed as a
limiting case of D. (For a more formal discussion of this feature
of the model, see Oomen 2005.) A couple of points are worth
highlighting at this stage. First, the assumed homoscedasticity of ε is consistent with the specification of W, which also
has homoscedastic increments. Any deterministic or stochastic
variation in return volatility can be captured by the specification of λ(t) for the CPP model in an analogous fashion to the
specification of σ (t) for the SV model. Second, the equivalence between the diffusion process and the limiting case of
the pure jump process does not rely on the assumed normality
of ε, but holds for any iid random variable with finite variance
(due to Donsker’s theorem; see, e.g., Karatzas and Shreve 1991,
chap. 2.4). Finally, note that the celebrated Lévy–Khintchine
theorem states that any Lévy process can be constructed as the
convolution of a Brownian motion and a compound Poisson
process with a jump distribution as characterized by the Lévy
measure (see, e.g., Applebaum 2004; Protter 2005). This powerful insight highlights the fundamental importance of the CPP
and thereby provides a further motivation for its use here.
In this article we take the CPP of Press as a starting point and
extend it to allow for time-varying jump intensity and serially
correlated innovations. In particular, the observed logarithmic
asset price P is specified as
P(t) = P(0) +

M(t)

j=1

(εj + ηj ),

where ηj = ρ1 νj + ρ2 νj−1 + · · · + ρq νj−q+1 ,
N (µε , σε2 ),

(4)

where νj = νj − νj−1 , εj ∼ iid
νj ∼ iid N (0, σν2 )
with σν2 = γ σε2 , M(t) is a Poisson process with instantaneous
intensity λ(t) independent of ε and ν, and ρ1 = 1 for identification purposes. The model in (4) is referred to as the CPP–MA(q)
model when γ > 0 and as the CPP–MA(0) model when γ = 0.
Note that CPP–MA(0) corresponds to the model without noise,
CPP–MA(1) corresponds to iid noise, and CPP–MA(q), q > 1,
corresponds to autocorrelated noise.
Because the focus in this article is on the analysis of financial transaction data, we view the CPP–MA model as a model
for transaction prices and interpret λ(t) as the instantaneous arrival frequency of trades with the process M(t) counting the
number of trades that have occurred up to time t. Consequently,
the price process can be analyzed in “physical” or “calendar”
time t, giving rise to {P(t), t ∈ [0, 1]} with returns defined as
R(t|τ ) = P(t) − P(t − τ ), or “transaction” time M(t), giving rise
to {p(k) ≡ P(inf M −1 (k)), k ∈ {0, 1, . . . , M(1)}} with returns

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222

Journal of Business & Economic Statistics, April 2006

defined as r(k|h) ≡ p(k) − p(k − h). As is clear from (4), it is
possible to decompose the price process into a latent martingale
M(t)
component j=1 εj that can be viewed as tracking the evolution of the efficient price process plus a market microstrucM(t)
ture noise component j=1 ηj that induces serial correlation
in observed returns through its MA(q) dependence structure.
Accordingly, the object of econometric interest in the realized
variance calculations that follow is
= (1)σε2 , that is, the
integrated variance of the efficient price process. Because the
parameter γ measures the level of market microstructure noise
relative to the uncontaminated martingale component, it is referred to as the “noise ratio.” Note that although the pure jump
behavior of the CPP is consistent with the discrete nature of
transaction prices, the efficient price process may still evolve
continuously in time. Yet only when a transaction takes place
is the efficient price reflected in the observed price, subject to
market microstructure noise.
The model’s ability to capture market microstructure-induced
serial correlation in returns is most easily illustrated using the
baseline CPP–MA(1) specification, where
R(t|τ ) =

M(t)


j=M(t−τ )

εj + νM(t) − νM(t−τ )−1 .

Hence returns exhibit a negative first-order autocorrelation that
is consistent with the well-documented presence of a bid–
ask spread (e.g., Niederhoffer and Osborne 1966; Roll 1984).
The impact of other and possibly more complicated market
microstructure effects can be captured by allowing for a higherorder MA structure in (4). Note that the assumed MA dependence structure and market microstructure interpretation of
the model is in line with those of Bandi and Russell (2004a),
Hansen and Lunde (2006), and Zhang et al. (2005), the only difference being in the assumed type of underlying price process
(i.e., pure jump versus continuous semimartingale). Interaction
between the noise and the efficient price process, as advocated
by Hansen and Lunde (2006), can be incorporated in the current framework by simply allowing for a correlation between
ε and ν (see Sec. 5 for more discussion).
As pointed out earlier, the CPP model can account for SV
through the specification of the jump intensity process in just
the same way that the SV model does through the specification
of the variance process. Moreover, because λ(t) represents the
trade intensity here, the model can also capture diurnal patterns
in market activity and serial dependence in trade durations.
For instance, a simple two-factor intensity process with one
slowly mean-reverting factor and one quickly mean-reverting
factor can generate serial dependence in both return volatility
at low frequency and trade durations at high frequency similar
to that of an SV model (Taylor 1986) and stochastic conditional
duration (SCD) model (Bauwens and Veredas 2003). The specification of the intensity process is not the focus of this article,
however (see Bauwens and Hautsch 2004; Oomen 2003 for further discussion). Instead, all of the calculations that follow are
performed conditional on the realization of λ(t). This is in line
with the diffusion-based literature in this area, in which typically all calculations are done conditional on the latent variance
process.

Before stating the main theorem that characterizes the distribution of returns, we introduce some further notation that
is used throughout the remainder of the article. Given two
nonoverlapping sampling intervals [ti − τi , ti ] and [tj − τj , tj ]
for tj − τj ≥ ti , define
λj = (tj ) − (tj − τj )

and
(5)

λi,j = (tj − τj ) − (ti ),
so that λj measures the integrated intensity over the sampling
interval associated with R(tj |τj ), whereas λi,j measures the integrated intensity between the sampling intervals associated with
R(ti |τi ) and R(tj |τj ). Keep in mind that although the dependence is suppressed for notational convenience, these quantities
of integrated intensity are always associated with a particular
sampling interval; that is, λj (λi,j ) is associated with tj and τj
(and ti ).
Theorem 1. For the CPP–MA(q) price process given in (4),
the joint characteristic function of nonoverlapping transaction
time returns r(k1 |h1 ) and r(k2 |h2 ) for m = k2 − h2 − k1 ≥ 0 is
given by


φTT (ξ |k, h, q) ≡ E exp{iξ1 r(k1 |h1 ) + iξ2 r(k2 |h2 )}

= exp iµε (ξ1 h1 + ξ2 h2 )
−

1 2
ξ
q (h1 ) + ξ22
q (h2 )
2 1



+ 2ξ1 ξ2 q (h1 , h2 , m) ,

(6)

where
q (·) and q (·) are given in (A.1) and (A.2) in Appendix A. Conditional on the intensity process λ, the joint
characteristic function of nonoverlapping calendar time returns
R(t1 |τ1 ) and R(t2 |τ2 ) for t2 − τ2 − t1 ≥ 0 is given by


φCT (ξ |t, τ, q) ≡ Eλ exp{iξ1 R(t1 |τ1 ) + iξ2 R(t2 |τ2 )}
= e−λ2 q (ξ1 ) + e−λ1 q (ξ2 )


+ q (λ1,2 ) + e−λ1,2

× q (ξ1 )q (ξ2 ) + q (ξ1 , ξ2 ),

(7)

where q (ξz ) = exp(ξz2 c0 + λz (exp(cz ) − 1))q (λz exp(cz )),
q (x) = 1 − Ŵ(q, x)/ Ŵ(q), and cz and q (ξ1 , ξ2 ) are given by
(A.4) and (A.5) in Appendix A.
For the proof see Appendix A.
Theorem 1 completely characterizes the distribution of returns in transaction time and calendar time. In particular, the
characteristic function in (6) can be used to derive moments or
cumulants of transaction time returns, whereas the conditional
characteristic function (CCF) in (7) allows for the straightforward derivation of conditional moments or cumulants of returns
in calendar time. It is emphasized that the calculations in calendar time are conditional on the latent intensity process. But
in Section 6 we demonstrate that for realistic sample sizes, accurate estimates of λ(t) can be obtained so that the derived formulas are operational in practice. Furthermore, it is noted that
1 (ξ1 , ξ2 ) = 0, so that the CCF takes an especially simple form
for the baseline CPP–MA(1) model.

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Oomen: Realized Variance

To conclude this section, we highlight some important differences between the statistical properties of returns on the two
time scales. First, it is clear from the model specification in (4)
and the characteristic function in (7) that returns in transaction time are normal and the distribution of returns in calendar
time is mixed normal, allowing for fat tails. For instance, for
the CPP–MA(0) model with constant jump intensity, the excess
kurtosis of R(t|τ ) is equal to 3(τ )−1 , which can be substantial
over short time intervals, that is, at high frequency. Another notable difference is that returns in transaction time have an MA
dependence structure, whereas returns in calendar time have an
ARMA dependence structure. For example, as can be seen from
the moment expressions given in Appendix B, returns in calendar time have an ARMA(1, 1) dependence structure for the
CPP–MA(1) model. Finally, it is noteworthy that Theorem 1 is
particularly useful for calculating return moments of CPP models with high-order MA dependence structure, because then the
moment expressions are complicated and lengthy. However, directly deriving moments without using the CCF can provide
some additional insight into the mechanics of the model. As an
illustration, consider again the CPP–MA(1) model and note that
E[r(k|h)4 ] = E[(εk−h+1:k + νk − νk−h )4 ]

= E[(εk−h+1:k )4 ] + E[(νk − νk−h )4 ]
+ 6E[(εk−h+1:k )2 (νk − νk−h )2 ]

= 3σε4 (h + 2γ )2 ,

where we use the notation xa:b = bi=a xi . Based on the foregoing transaction return moment, the corresponding moment in
calendar time can be obtained as


Eλ [R(ti |τi )4 ] = Eλ EN r(M(ti )|Ni )4


= Eλ 3σε4 (Ni + 2γ )2 1Ni ≥1



= 3σε4 (λ2i + λi ) + 4γ λi + γ (1 − e−λi ) ,

where Ni = M(ti ) − M(ti − τi ). The foregoing derivation highlights the relationship between moments on the two time
scales—namely, return moments in calendar time can be
viewed as a probability-weighted average of the corresponding
transaction return moments.
3. REALIZED VARIANCE IN CALENDAR, BUSINESS,
AND TRANSACTION TIME
In this section we use the foregoing CPP–MA(q) model to
investigate the statistical properties of the realized variance
(i.e., the sum of squared intraperiod returns) as an estimator
of the integrated variance
≡ (1)σε2 . The analysis is carried out along two dimensions, namely by varying the sampling
frequency and varying the sampling scheme. The impact of a
change in sampling frequency on the statistical properties of
the realized variance is quite intuitive; an increase in sampling
frequency will lead to a reduction in the variance of the estimator due to the increase in the number of observations and an
increase in the bias of the estimator due to the amplification of
market microstructure noise-induced return serial correlation. It
is this tension between the bias and the variance of the estimator
that motivates the search for an “optimal” sampling frequency,

223

that is, the frequency at which the MSE is minimized. The contribution of this article on this front is the derivation of a closed
form expression for the MSE of the realized variance as a function of the sampling frequency which allows for the identification of the optimal sampling frequency. The results given herein
are closely related to those of Aït-Sahalia et al. (2005), Bandi
and Russell (2004a), and Hansen and Lunde (2006), who obtained similar expressions within their frameworks. The second
issue (i.e., the impact of a change in sampling scheme on the
statistical properties of the realized variance) is far less obvious, and to the best of our knowledge this article is the first to
provide a comprehensive analysis. As an illustration, consider
a trading day on which 7,800 transactions occurred between
9.30 AM and 4.00 PM. Sampling the price process in calendar
time at regular intervals of, say, 5 minutes will yield 78 return
observations. An alternative sampling scheme records the price
process each time that a multiple of 100 transactions is executed. Although this transaction time sampling scheme leads to
the same number of return observations, the properties of the
resulting realized variance measure turn out to be fundamentally different. Crucially, we can show that in many cases, sampling in transaction time as opposed to calendar time results in
a lower MSE of the realized variance.
To formalize ideas and facilitate discussion, we define three
different sampling schemes: namely calendar time sampling
(CTS), business time sampling (BTS), and transaction time
sampling (TTS):
• Calendar time sampling. CTSN samples the sequence of
c
−1
prices {P(tic )}N
i=0 , where ti = iN .
• Business time sampling. BTSN samples the sequence of
b
−1
prices {P(tib )}N
i=0 , where ti =  (iλ) and λ = (1)/N.
• Transaction time sampling. TTSN samples the sequence
tr
−1
of prices {P(titr )}N
i=0 , where ti = inf M (ih) and h =
M(1)/N is integer-valued (or, equivalently, {p(titr )}N
i=0 ,
tr
where ti = ih).
Based on this, the realized variance can be defined as
RV sN =

N

i=1

s
R(tis |tis − ti−1
)2

for s ∈ {c, b, tr}.

From the foregoing definitions, it is clear that all schemes sample the price process at equidistantly spaced points on their respective time scale, namely t for CTS, (t) for BTS, and M(t)
for TTS. The focus on equidistant sampling is largely motivated by what researchers do in practice, and also because there
is no obvious alternative. It is noted, however, that when the
trade intensity varies over time, equidistant sampling in calendar time corresponds to irregular sampling in business and
transaction time, and vice versa. Furthermore, to isolate the impact of sampling on the properties of the realized variance, we
mostly consider the case where the number of sampled returns
N is the same for each scheme. Hence what distinguishes one
scheme from the other is the information set that it generates by
the location (not the number) of its sampling points. Although
it is clear that the properties of the realized variance change
when the information set is changed, it is not clear a priori in
what way they change; this is precisely the issue that this article
addresses.

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224

In the realized variance literature, CTS is the most widely
used sampling scheme. For instance, one of the first article in
this area, by Andersen and Bollerslev (1998a), sampled the
price process at 5-minute intervals (see also Hsieh 1991, in
which 15-minute data were used in a related context). More recently, Andersen, Bollerslev, Diebold, and Ebens (2001) used
5-minute data, Barndorff-Nielsen and Shephard (2004) used
10-minute data, and Andersen et al. (2003) used 30-minute data
to compute the realized variance. Although in some cases the
sampling scheme (and frequency) is dictated by the data restrictions, in other cases transaction data is available and TTS
becomes a feasible and, as this article shows, appealing alternative.
The BTS scheme deforms the calendar time scale in such a
way that the resulting jump intensity on a business time scale
is constant and the price process is transformed into a homogenous CPP. Of course, if the jump intensity is constant to start
off with, BTS and CTS are equivalent. Because implementation of BTS requires the intensity process to construct the sampling points, it is an infeasible scheme when the intensity is
latent. Clearly, in practice a feasible BTS scheme can be based
on an estimate of the intensity process (i.e., 
λ(t) for t ∈ [0, 1]),
which, as discussed later, can be obtained with standard nonparametric smoothing methods using transaction times. The
ϑ -time scale introduced by Dacorogna, Müller, Nagler, Olsen,
and Pictet (1993) is similar to the business time scale considered here, with the main difference that ϑ -time removes seasonality of volatility (typically across days) through deformation
while business time removes any deterministic and stochastic
variation in trade intensity and thus volatility. Note that following the initial draft of this article, BTS was also considered by
Barndorff-Nielsen et al. (2004) in a diffusion setting.
The key distinction between BTS and TTS is that the former
scheme samples prices based on the expected number of transactions [i.e., (t)], whereas the latter scheme samples prices
based on the realized number of transactions [i.e., M(t)]. As
a result, even when the intensity process is deterministic and
known, TTS and BTS are not equivalent. Interestingly, however, because Eλ [M(t)] = (t), TTS can be viewed as a feasible
BTS scheme when the intensity process is latent. In the foregoing definition of TTS, h = M(1)/N must be integer valued. For
given M(1), this may severely restrict the values that N can take
on. In fact, when M(1) is a prime number, N can take on only
two values! In practice, a sensible solution is to round M(1)/N.
This will then cause all returns but one to be sampled equidistantly in transaction time and is thus likely to have a negligible
impact on the results, especially when M(1) and N are large, as
is often the case in practice. Finally, it is important to emphasize that the number of sampling points for TTS is bounded by
the number of transactions [i.e., N ≤ M(1)], whereas for CTS
and BTS, N can be set arbitrarily high.
3.1 Absence of Market Microstructure Noise
In the absence of market microstructure noise (i.e., γ = 0),
the realized variance is an unbiased estimator of the integrated
variance
regardless of the sampling scheme. Thus the MSE

Journal of Business & Economic Statistics, April 2006

of the realized variance is equal to its variance. Using the moment expressions in (B.1)–(B.4) in Appendix B (set γ = 0), the
MSE of the realized variance for CTS can be expressed as
Eλ [(RV cN −
)2 ]
= 3σε2
−
2 + 2σε4

N

i=1

λ2i + σε4

N 
N


(8)

λi λj ,

i=1 j=1

where λi = (iN −1 ) − ((i − 1)N −1 ). By definition, for BTS,
λi = λ, which leads to the simplified MSE expression
Eλ [(RV bN −
)2 ] = 2
2 /N + 3σε2
,

(9)

whereas for TTS, the MSE of the realized variance is equal to
2
2
Eλ [(RV tr
N −
) ] = 2
/N.

(10)

Comparing the MSE expressions for alternative sampling
schemes provides some interesting insights, for instance,
Eλ [(RV cN

2

−
)

] − Eλ [(RV bN

2

−
)

] = 2σε4

N


ϑi2 > 0,

i=1

where ϑi = λi − λ measures the difference between the integrated intensity over the ith sampling increment associated with
c , tc ]) and associated with BTS (i.e., [tb , tb ]).
CTSN (i.e., [ti−1
N
i
i−1 i
In words, the MSE reduction associated with BTS relative to
CTS increases with the variability of the intensity process. Furthermore, the difference in MSE of the realized variance for
BTS and TTS is equal to
2
4
Eλ [(RV bN −
)2 ] − Eλ [(RV tr
N −
) ] = 3σε (1) > 0.

These results are summarized in the following proposition.
Proposition 1. For the price process in (4), given N, and in
the absence of market microstructure noise (γ = 0), TTSN leads
to the most efficient realized variance and CTSN leads to the
least efficient realized variance. Moreover, the efficiency gain
of TTSN (BTSN ) over BTSN (CTSN ) increases with an increase
in the innovation variance and the level (and variability) of trade
intensity.
3.2 Presence of Market Microstructure Noise
We now turn to the case where market microstructure noise is
present and start with the analysis of the CPP–MA(1) model. As
we make clear later, this model is already sufficiently flexible to
capture much of the market microstructure noise in actual data.
Relevant moments for the CPP–MA(1) are given in (B.1)–(B.4)
in Appendix B (set ρ2 = 0). Now that price innovations are serially correlated, the realized variance is a biased estimator of
the integrated variance. Under CTS, this bias can be expressed
as
Eλ [RV cN

−
] = 2γ σε2

N

(1 − e−λi ).
i=1

Under BTS and TTS, the bias expression simplifies to 2γ σε2 ×
N(1 − e−λ ) and 2γ σε2 N. Clearly, the negative first-order serial
correlation of returns leads to a positive bias in the realized variance for all sampling schemes considered here. Similar to the
foregoing MSE comparison, it is possible to rank the magnitude

Oomen: Realized Variance

225

of the bias associated with the different sampling schemes, that
is,
Eλ [RV cN − RV bN ] = 2γ σε2 e−λ

N

(1 − e−ϑi ) < 0
i=1

and
2 −λ
Eλ [RV bN − RV tr
< 0.
N ] = −2γ Nσε e

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These results are summarized in the following proposition.
Proposition 2. For the price process given in (4), given N,
and in the presence of first-order (q = 1, γ > 0) market microstructure noise, the bias of the realized variance is largest
for TTSN and smallest for CTSN . Moreover, the increase in bias
for TTSN (BTSN ) relative to BTSN (CTSN ) increases with an
increase in the noise variance (and the variability of trade intensity) but decreases with the level of trade intensity.
Intuitively, the reason that the bias is maximized for TTS is
that with the use of this scheme, the underlying MA-dependence structure is recovered, which leads to maximum serial
correlation in sampled returns. The implication of this is that the
bias can be minimized by adopting a sampling scheme that is
as “different” from TTS as possible; that is, set the length of the
sampling intervals proportional to the number of trades leading
one to sample (in) frequent when markets are (fast) slow. Even
though such a scheme can reduce the bias, it may well have
the undesirable side effect of increasing the variance of the estimator. It thus seems appropriate to consider the MSE of the
realized variance, which for TTS is equal to
2
Eλ [(RV tr
N −
) ]

= 2
2 /N + 8
γ σε2 + 4γ 2 σε4 (N 2 + 3N − 1).

(11)

Further, for BTS, the MSE is equal to


2
2
−λ
Eλ [(RV bN −
)2 ] = Eλ [(RV tr
N −
) ] +
σε 3 + 4γ e


− 4γ 2 σε4 4Ne−λ − e−(1)


+ 4γ 2 σε4 N(N − 1) e−2λ − 2e−λ , (12)

and finally, for CTS, the MSE can be expressed as
Eλ [(RV cN −
)2 ]

= Eλ [(RV bN −
)2 ] − 2
2 /N
− 4e−λ
γ σε2 + 12e−λ Nγ 2 σε4
+ 4γ 2 σε4

+2

N−1


N

 −λ
(e j − 1)(e−λi − 1)(e−λi,j + 2)

i=1 j=i+1


2 

− e−λ − 1 e(i−j+1)λ + 2

N

 4 2

σε λi − 2γ σε2 (3γ σε2 −
)e−λi
i=1

− 4γ σε4

N−1


N


(λj e−λi + λi e−λj ).

i=1 j=i+1

(13)

In the presence of market microstructure noise, the two main
questions that need to be addressed are determining the optimal
sampling frequency and the optimal sampling scheme. In the
absence of market microstructure noise, the answers are clear
cut—namely, it is optimal to sample as frequently as possible
in transaction time. However, as is clear from the foregoing expressions [(11) in particular], an increase in N does not necessarily reduce the MSE of the realized variance when γ > 0. In
fact, for a given sampling scheme, it is possible to determine the
“optimal” sampling frequency at which the MSE of the realized
variance is minimized, that is,
Ns∗ = arg min Eλ [(RV sN −
)2 ]
N

for s ∈ {c, b, tr}.

(14)

Intuitively, Ns∗ balances the trade-off between a reduction
in the variance of estimator and an increase in the market
microstructure-induced bias when increasing the sampling frequency. For BTS and TTS, the optimal sampling frequency
is obtained by solving the appropriate first-order condition,
whereas for CTS it is computed by numerically minimizing the
MSE for a given (estimated) realization of the intensity path.
Interestingly, when (1) ≫ γ , the optimal sampling frequency
for TTS can be accurately approximated as



(1) 2/3

.
(15)
Ntr∗ =
2γ

This expression complements the rule of thumb proposed by
Bandi and Russell (2004a) for calendar time sampling in a diffusion setting (see Sec. 4 for further discussion) and confirms
the intuition that the optimal sampling frequency increases with
an increase in trade activity and a decrease in the level of market microstructure noise. The requirement that the expected or
average number of transactions far exceeds the noise ratio is
generally immaterial in practice. For instance, it is not uncommon for IBM to trade more than 5,000 times a day with a noise
ratio of about 2 (see Sec. 6). Moreover, because estimates of
(1) and γ are straightforward to obtain, the expression in (15)
provides a reliable and easily computed measure of the optimal
sampling frequency for TTS. The only caveat here is that for
small values of γ , it can occur that 
Ntr∗ > M(1). In practice,
this clearly poses no problem, and one simply sets the optimal
sampling frequency equal to the minimum of 
Ntr∗ and M(1).
Turning to the properties of the alternative sampling schemes,
we note that in the presence of market microstructure noise
there is no one sampling scheme that is superior. In fact,
whether CTS, BTS, or TTS leads to a lower MSE of the realized
variance depends on the specific model parameters and the evolution of the intensity process. To illustrate this, we compute the
MSE of the realized variance associated with each of the sampling schemes across a range of noise ratios γ and sampling frequencies N. The integrated intensity is fixed at (1) = 4,000,
whereas σε2 is set to ensure that
= (20%)2 annualized. Further, for the MSE calculations for CTS we consider two distinct intensity paths, one path with little variation beyond the
ubiquitous seasonal pattern and the other with large variation
[see Fig. 1(a)]. Both intensity paths are rescaled nonparametric
smoothing estimates based on IBM data for August 25, 2003
(dashed line) and June 7, 2000 (solid line) and thus represent
realistic patterns (for details, see Sec. 6). To facilitate presentation, we report the “BTS loss” and “CTS loss,” which are

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226

Journal of Business & Economic Statistics, April 2006
(a)

(b)

(c)

(d)

Figure 1. “Inefficiency” of BTS and CTS Across Sampling Frequency and Noise Ratio. (a) The nonparametric smoothing estimates of the trade
intensity for IBM on June 7, 2000 (
) and August 25, 2003 (
) rescaled to ensure that Λ(1) = 4,000. (b) The isocurves of the BTS loss across
√
a range of sampling frequencies converted to minutes (i.e., 6 .5N×60 = 390
N ) on the horizontal axis and the square root of the noise ratio (i.e., γ ) on
the vertical axis. (c) and (d) The corresponding isocurves for the CTS loss using the respective intensity paths of (a). The “+” indicate the optimal
sampling frequency for BTS, and while the “◦” indicate the optimal sampling frequency for TTS (b) and CTS [(c) and (d)].

defined as the percentage increase in MSE when moving from
TTS to BTS and from BTS to CTS, keeping N fixed. These
measures thus represent the cost in terms of MSE associated
with the use of a suboptimal sampling scheme.
The results, reported in Figures 1(b)–1(d), can be summarized as follows. Consistent with the foregoing discussion, in
the absence of market microstructure noise (i.e., γ = 0) TTS
outperforms BTS and BTS outperforms CTS for any given sampling frequency. With noise added to the process, these relationships break down, and both the BTS loss and CTS loss can turn
negative. This occurs when the sampling frequency and noise
ratio combination lies to the left of the solid line. Comparing
Figures 1(c) and 1(d) shows that, in line with Proposition 1, the
magnitude of the CTS loss is greater when the intensity process
is volatile. Additional simulations (not reported here) suggest
that an increase in the level of the intensity process causes the
solid line to shift leftward, thereby reducing the area in which
the CTS and BTS loss is negative. Ultimately, however, these
results say little about the performance of the various sampling

schemes in practice unless a sampling frequency is specified.
Therefore, to provide the appropriate perspective, we indicate
the optimal sampling frequency in the graph, because this is the
frequency most relevant for empirical applications. It is clear
that in the region around the optimal sampling frequency, TTS
outperforms BTS and BTS outperforms CTS, as was the case
in the absence of market microstructure noise. Moreover, it is
interesting to note that the optimal sampling frequencies for the
various schemes lie very close together, suggesting that the simple formula in (15) is more widely applicable to other schemes,
at least to a first approximation. All of the foregoing results are
robust to alternative choices of model parameters and intensity
path specifications.
An issue that remains open at this point is whether (and if so,
how) the foregoing results are expected to change when the order of the MA process is raised. In particular, what will happen
to the MSE and optimal sampling frequency in such a case, and
will the properties of the sampling schemes change? To answer
the first question, we investigate how the optimal sampling frequency and magnitude of the MSE change when moving from

Oomen: Realized Variance

227

an MA(1) specification to an MA(2). In doing so, we distinguish among two cases: change ρ2 while keeping the variance
of ν constant, and change ρ2 while keeping the variance of η
constant. The first case corresponds to adding noise, and we find
that this leads to an increase in MSE and a decrease of optimal
sampling frequency. The second case corresponds to altering
the structure of the noise dependence, in which case the results
can go either way. Both of these finding are quite intuitive and
are robust to the specific choice of parameter values and the
order of the MA process. To answer the second question, we
redo the foregoing analysis for the MA(2) case, and find that
these results are also qualitatively unchanged with higher-order
dependence.

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4. A SMALL JUMP TO A DIFFUSION?
We now provide an informal discussion of some important
differences between the properties of the realized variance derived earlier and those obtained in a diffusion setting. First, it
is easy to see from (9) that the realized variance is an inconsistent estimator of the integrated variance in the current framework. For BTS, the variance of the realized variance converges
to 3σε2
> 0 when N → ∞. Intuitively, because the price path
follows a pure jump process, there is a point beyond which increasing the sampling frequency does not generate any additional information; instead, increasing the sampling frequency
will just lead to the addition of more and more 0’s into the sampled return series, and hence the inconsistency. This is clearly
not the case for a diffusive price process that is of infinite variation and for which realized variance is c