measuring transaction costs in the absence of …...measuring transaction costs in the absence of...
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Finance and Economics Discussion SeriesDivisions of Research & Statistics and Monetary Affairs
Federal Reserve Board, Washington, D.C.
Measuring Transaction Costs in the Absence of Timestamps
Filip Zikes
2017-045
Please cite this paper as:Zikes, Filip (2017). “Measuring Transaction Costs in the Absence of Timestamps,” Financeand Economics Discussion Series 2017-045. Washington: Board of Governors of the FederalReserve System, https://doi.org/10.17016/FEDS.2017.045.
NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminarymaterials circulated to stimulate discussion and critical comment. The analysis and conclusions set forthare those of the authors and do not indicate concurrence by other members of the research staff or theBoard of Governors. References in publications to the Finance and Economics Discussion Series (other thanacknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.
Measuring Transaction Costs in theAbsence of Timestamps
Filip Zikes∗
Federal Reserve Board
April 6, 2017
Abstract
This paper develops measures of transaction costs in the absence of trans-action timestamps and information about who initiates transactions, whichare data limitations that often arise in studies of over-the-counter markets.I propose new measures of the effective spread and study the performanceof all estimators analytically, in simulations, and present an empirical il-lustration with small-cap stocks for the 2005–2014 period. My theoretical,simulation, and empirical results provide new insights into measuring trans-action costs and may help guide future empirical work.
JEL Classification: C14, C15, G20Keywords: transaction costs, effective spread, simulated method of mo-ments, time-varying estimation
∗Board of Governors of the Federal Reserve System, Division of Financial Stability, 20thStreet and Constitution Avenue N.W., Washington, D.C. 20551, United States. Phone: +1-202-475-6617. Email: [email protected]. I am grateful to Evangelos Benos, Dobrislav Dobrev,Erik Hjalmarsson, Ivan Ivanov, John Schindler, Michalis Vasios, and Yang-Ho Park for valuablecomments, and to Eric Parolin and Margaret Yellen for excellent research assistance. The resultsreported in the paper were generated using programs written SAS and in the Ox language ofDoornik (2007); the programs are available on request. The views expressed in this paper arethe sole responsibility of the author and should not be interpreted as representing the views ofthe Federal Reserve Board or any other person associated with the Federal Reserve System.
1
1 Introduction
This paper develops measures of transaction costs that do not require observ-
ing the intraday transaction times or knowing who initiates trades (buyers vs.
sellers). A recent and growing literature on large and previously opaque over-the-
counter (OTC) markets employs transaction data that suffer from these limita-
tions. Examples include studies of the credit default swap market (Chen et al.,
2011; Benos, Wetherilt, and Zikes, 2013; Biswas, Nikolova, and Stahel, 2014), the
interest rate swap market (Chen et al., 2012; Benos, Payne, and Vasios, 2016),
the U.K. sovereign bond market (Benos and Zikes, 2016), and the U.S. corporate
bond market (Bessembinder et al., 2006). But to the best of my knowledge, the
literature has not yet formally tackled the problem of estimating transaction costs
when timestamps and trade direction are missing. The contribution of this paper
is to fill this gap.
I propose three consistent estimators of the effective spread and study their
sampling properties. The first one develops the idea of Benos and Zikes (2016),
who suggest inferring the effective spread from the dispersion of the transaction
prices from (1) some benchmark or reference price (e.g., end-of-day composite
quote) and (2) the average transaction price. The estimator is available in closed
form, which allows me to establish its finite-sample properties analytically and
compare them to some well-known (infeasible) measures, explicitly quantifying
the loss of information due to the missing timestamps, trade direction, or both.
The other two measures I propose are also moment-based and combine the ideas
of Corwin and Schultz (2012) and Benos and Zikes (2016). The first one is based
on the daily range, which is the difference between the daily high and low prices,
together with the sample variance of the transaction prices. The advantage of this
measure is that it is based solely on transaction prices and does not require a daily
benchmark or reference price, which may be difficult to obtain for some illiquid
assets. At the same time, it utilizes all available data (transaction prices), unlike
2
the Corwin and Schultz (2012) measure, which only uses the daily high and low
prices. If a benchmark price is available, however, it is, of course, optimal to use all
three moment conditions—the two dispersion metrics and the daily range—and this
is how I construct my third estimator. Because I do not assume that the number of
transactions is large, I have to resort to the simulated method of moments (SMM).
Despite having to approximate the expected range by simulation, the estimator
turns out to be computationally cheap and easy to implement in practice. I provide
simple yet accurate small-sample approximations to the unknown moment that
significantly speed up computations.
To summarize my theoretical results, I find that the absence of timestamps or
trade direction lead to a reduced convergence rate of the effective spread estima-
tors. While in the case of full information the effective spead can be estimated
n-consistently as the number of intraday transactions (n) increases, when times-
tamps or trade direction are missing, only√n-consistency can be achieved, and
when both are missing, the effective spread cannot be estimated consistently from
intraday data alone–averaging over an increasing number of days (T ) is necessary.
Thus, accurate estimates can only be obtained from weeks or months worth of
transaction data.
In practice, transaction costs may vary at a higher frequency, however. Is it pos-
sible, then, to estimate transaction cost that vary, say, every day when timestamps
and trade direction are missing? The answer is yes, provided the transaction costs
vary sufficiently smoothly. Employing the recent advances in time-varying estima-
tion by Giraitis, Kapetanios, and Yates (2013), I propose kernel-based estimators
of smoothly-varying effective spread and establish their asymptotic properties. Do-
ing so allows me to uncover smooth changes in transaction costs over time without
relying on an increasing number of intraday observations.
To corroborate the theoretical findings and to study how the various estimators
perform in small samples, I run Monte Carlo simulations. I also provide an em-
3
pirical illustration with small-cap stocks listed on the New York Stock Exchange
(NYSE) using data from the Trade and Quote Database (TAQ). To summarize, the
findings of these exercises show that the loss of information due to missing times-
tamps is large. The measures I propose in this paper deliver accurate estimates
of transaction costs in situations where the transactions costs are high relative to
the volatility of the efficient price. They may be, therefore, suitable for relatively
illiquid, infrequently traded assets that exhibit relatively low fundamental volatil-
ity, such as some corporate and municipal bonds. They should not be applied,
however, to highly liquid assets, such as listed equities, which trade with a tight
spread and tend to be quite volatile. Fortunately, for these assets, high-quality
time-stamped transaction data are typically available.
In OTC markets, timestamps are often inaccurate or outright missing for vari-
ous reasons. In the credit default swap and interest rate data mentioned previously,
transaction times are simply not reported, and the trade reporting time does not
necessarily correspond to the actual trade time, making it impossible to chronolog-
ically order transactions. In the U.K. sovereign bond market data used by Benos
and Zikes (2016), the timestamps are not accurate in the sense that two parties
to the same transaction report widely different transaction times. More generally,
though, the trading protocol in OTC markets often involves negotiation that may
stretch over a period of time, and so the exact timing of the trade may be ambigu-
ous; consider, for example, the “workup” protocol recently studied in Duffie and
Zhu (forthcoming).
Trade direction cannot be easily inferred because trades cannot be aligned
with intaday quotes when timestamps are missing, making it impossible to use
trade-signing algorithms such as that of Lee and Ready (1991). Researchers often
assume that clients initiate trades with dealers, motivated by the fact that dealers
are the main liquidity providers in these markets (Bessembinder, Maxwell, and
Venkataraman, 2006; Edwards, Harris, and Piwowar, 2007). However, as recently
4
shown by Choi and Huh (2017) for the U.S. corporate bond market, dealers often
initiate trades with clients as well, implying potenitally serious missclassification
issues associated with this identification method. Moreover, in some markets,
interdealer trades account for more than two-thirds of all transactions (Benos et
al., 2013) and so the vast majority of transactions cannot be signed in this way.
Thus, standard methods cannot be used to measure transaction costs in these large
and important financial markets.
Apart from the literature on measuring transaction costs (see Harris, 2015,
and the references therein), my paper is also related to the recent literature on
measuring volatility using high-frequency data starting with Andersen and Boller-
slev (1998) and Barndorff-Nielsen and Shephard (2002); see Aıt-Sahalia and Jacod
(2014) for a recent textbook treatment. Some of my estimators employ the range—
that is, the difference between intraday high and low prices—and here I draw on
the ideas of Christensen and Podolskij (2007) and Christensen, Podolskij, and Vet-
ter (2009). Although the data-generating process I assume is very similar to many
papers in this literature, the problem studied in my paper is different in three
important ways. First, my goal is to estimate the effective spread. Thus, what the
realized volatility literature (e.g. Aıt-Sahalia, Mykland, and Zhang, 2005; Zhang,
Mykland, and Aıt-Sahalia, 2005, Hansen and Lunde, 2006; Christensen, Podolskij,
and Vetter, 2009) treats as microstructure noise is precisely my object of interest,
and what that literature is interested in estimating—the variation of the efficient
price—is a source of noise in my framework. Second, in-fill asymptotics do not
always apply—that is, increasing the number of intraday observations (n) does
not generally improve the precision of the effective spread estimator. I have to rely
on an increasing number of days (T ) and employ large-T asymptotics or double
asymptotics (both n → ∞ and T → ∞). Finally, my estimation framework is
model based, unlike the estimation methods in the realized volatility literature
that operate in a model-free environment.
5
The rest of the paper is organized as follows. In Section 2, I set out my theoreti-
cal framework. In Section 3, I propose a closed-form measure of the effective spread
that does not require either timestamps or trade direction and study its properties
analytically, explicitly quantifying the loss of information associated with miss-
ing timestamps, trade direction, or both. In Section 4, I introduce range-based
estimators of the effective spread and propose simple computational methods to
implement the estimator in practice. Section 5 reports Monte Carlo simulations.
In Section 6, I propose kernel-based time-varying estimation of the effective spread.
In Section 7, I present an empirical application to small-cap equities, and Section
8 concludes. Proofs are collected in the Appendix.
2 Framework
The effective spread is defined as two times the difference between the actual
transaction price (P ) and the prevailing mid-quote or some proxy for the true
value of the asset (efficient price) (M) at the time of the transaction. It can be
expressed in absolute terms—that is, 2|P − M |—or in relative terms—that is,
2|P −M |/M or 2| log(P )− log(M)|. Like the bid-offer spread, the effective spread
measures round trip transaction costs, but it is based on actual transaction price
rather than on quoted prices. The effective spread can also be seen as a measure
of the price impact of a trade, and because the price impact and transaction costs
tend to vary inversely with liquidity, it is frequently used as a measure of liquidity
(Foucault, Pagano, and Roell, 2013).
My theoretical framework is essentially borrowed from Roll (1984) and it can
be easily cast in continuous time as in Christensen, Podolskij and Vetter (2009).
Suppose we have a sample of T days and divide each day into n subintervals
of equal length. I assume that a transaction arrives at the beginning of each
of these subintervals and that the associated logarithmic transaction prices—pi,t;
6
i = 1, ..., n; t = 1, ..., T—are related to the logarithmic efficient price, mi,t, by
pi,t = mi,t +s
2qi,t, (1)
where s is the proportional effective spread and qi,t is a binary variable indicating
whether the i-th transaction on day t is buyer initiated (qi,t = 1) or seller initiated
(qi,t = −1). I initially assume that the efficient price is observable at the end of the
day—that is, at the end of the last subinterval n. I later relax this assumption and
propose estimators that do not require observing m at all. Following Roll (1984)
and Benos and Zikes (2016), I assume that the logarithmic efficient price m follows
a random walk with independently and identically distributed (iid) increments:
mi+1,t = mi,t + εi+1,t, (2)
where E(εi,t) = 0 and E(ε2i,t) = σ2/n. Thus, the daily integrated variance of the
efficient price equals σ2 for any n and t. Finally, I assume that qi,t is uncorrelated
with mj,s for all i, j, t, s and that qi,t is serially uncorrelated with E(qi,t) = 12—that
is, there is the same number of buyer- and seller-initiated trades on average. I
make no assumptions on the overnight return of the efficient price, m0,t −mn,t−1.
3 Baseline estimator
Inspired by Jankowitsch, Nashikkar, and Subrahmanyam (2011), Benos and Zikes
(2016) rely on the dispersion of transaction prices around some benchmark price,
but they recognize that the dispersion metric is affected by the intraday volatility
of the benchmark price in a nontrivial way. They suggest using two dispersion
metrics,
d2t =1
n
n∑i=1
(pi,t −m0,t)2, d2t =
1
n− 1
n∑i=1
(pi,t − pt)2, (3)
7
and show that under the assumptions stated in the previous section, the two met-
rics satisfy
E(d2t ) =s2
4+σ2
2
(n+ 1
n
), E(d2t ) =
s2
4+σ2
6
(n+ 1
n
). (4)
Solving for s2, censoring at zero, and taking the square root yield the relative
effective spread measure:
ESt =
√max{2(3d2t − d2t ), 0}. (5)
My baseline estimator develops the idea of Benos and Zikes (2016). I define
s2t = 2(3d2t − d2t ), where d2t and d2t are given in equation (3) and start by deriving
the variance of s2t , as it will be invoked repeatedly in the paper.
Proposition 1 Provided that the fourth moment of ε1,1 exists,
Var(s2t ) =9s4
2n(n− 1)+
2s2σ2(2n2 + 3n+ 1)
n2(n− 1)+
2(2nσ4 + σ4 + 2κ)(2n3 + 7n2 + 7n+ 2)
15n3(n− 1),
(6)
where κ = E(ε41,1)− 3σ4 is the excess kurtosis of ε1,1.
Equation (6) implies that although s2 is an unbiased estimator of s2, it is not
consistent as the number of intraday transactions increases because Var(s2t ) =
815σ4 + O(n−1) as n → ∞. This result is due to the fact that we are averaging
random walks in levels (prices) as opposed to first differences (returns), which
cannot be constructed due to missing timestamps.
To derive a consistent estimator of s based on s2t , we need to average s2t over
an increasing number of days before censoring at zero and taking the square root
as in equation (5). The resulting estimator, which I denote by ES(1)T , is thus given
by
ES(1)T =
√√√√max
{1
T
T∑t=1
2(3d2t − d2t ), 0
}. (7)
8
Given the nonlinear nature of the estimator, E(ES(1)T ) and Var(ES
(1)T ) are not
available in closed form. I employ a Taylor series expansion of ES(1)T around s,
s > 0, together with equation (6), to establish the leading terms (as T →∞). The
leading term of the bias reads
limT→∞
E[T (ES(1)T − s)] = − 1
15
σ4
s3−(
1
3
σ4
s3+
1
15
κ
s3+
1
2
σ2
s
)1
n+O
(1
n2
), (8)
implying that ES(1)T tends to underestimate the true effective spread. The limiting
variance reads
limT→∞
Var[√T (ES
(1)T − s)] =
2
15
σ4
s2+
(2
3
σ4
s2+
2
15
κ
s2+ σ2
)1
n+O
(1
n2
). (9)
As expected, the absolute bias and variance decrease with the signal-to-noise ratio
(SNR) s/σ, so it is more difficult to estimate the effective spread when it is small
relative to the volatility of the efficient price. The absolute bias and variance of
ES(1)T also increase with excess kurtosis, but this only matters when the number
of transactions is small; the contribution of κ vanishes as n → ∞. The second
terms in the expansions also show that for sufficiently large n, the absolute bias
and variance of ES(1)T decrease with the number of transactions, as the coefficients
on the n−1 terms in equations (8) and (9) are always positive.
It follows from the assumptions stated in Section 2 and standard limit theorems
that as T → ∞, ES(1)T →p s and if s > 0 and κ < ∞,
√T (ES
(1)T − s) →d
N(0, ω2), where ω2 = 14s2
Var(s2). The limiting variance of ES(1)T has a particularly
simple form if we consider the asymptotics where both T, n → ∞, which may be
appropriate in situations where the number of daily transactions is large. Then,
from equation (9), we obtain√T (ES
(1)T − s)→d N(0, 2σ4
15s2). Feasible inference can
be obtained by replacing the unknown s and σ2 in the limiting variance with their
9
sample counterparts, ES(1)T and σ2
T , respectively, where
σ2T = max
{1
T
T∑t=1
3(d2t − d2t ), 0
}(10)
is a consistent estimator of σ2.
3.1 Comparison with estimators that use timestamps or
trade direction
In this section, I compare the baseline estimator with several well-known measures
of the effective spread that require timestamps or trade direction, or both. The
goal is to assess how serious the loss of information associated with these data
limitations is.1
3.1.1 Observable timestamps
Should timestamps be available, one would typically use the Roll (1984) estimator,
which is equal to minus 4 times the sample first-order autocovariance of intraday
returns:
γ2t = − 4
n− 2
n∑i=3
(pi,t − pi−1,t)(pi−1,t − pi−2,t). (11)
It is easy to show that in my setup the estimator is unbiased for s2, and its variance
reads
Var(γ2t ) =
(16σ4
n2+
16s2σ2
n+
2s4(n− 3)
(n− 2)+ 3s4
)1
n− 2. (12)
Clearly, Var(γ2t ) = 5s4
n+ O(n−2), so γ2t is a consistent estimator of s2 as n → ∞.
Similar to ES(1)T , γ2t can be averaged over T days and censored at zero to obtain a
1In the rest of this section, all analytical results are presented without proof to save space.The derivations follow similar steps as the derivation of equation (6) and can be obtained uponrequest.
10
nonnegative estimator of s:
RollT =
√√√√max
{1
T
T∑t=1
γ2t , 0
}. (13)
Unlike ES(1)T , RollT is consistent for s as n→∞ for any T .
But the Roll estimator is not the only√n-consistent measure of s. Christensen,
Podolskij, and Vetter (2009) propose an estimator based on realized volatility,
which can be tailored to my framework as follows:
ω2t =
2
n− 1
n−1∑i=1
(pi+1,t − pi,t)2. (14)
It is straightforward to show that
E(ω2t ) = s2 +
2σ2
nand Var(ω2
t ) =s4
n− 1+
8s2σ2
n(n− 1)+
4(κ− σ4)
n2(n− 1)(15)
where κ = E(ε41,1), which implies that ω2 is asymptotically unbiased and its variance
satisfies Var(ω2t ) = s4
n+ O(n−2). The limiting variance is five times smaller than
that of the Roll estimator γ2t , but the finite-sample bias can be large when σ2 is
large. Unlike s2 or γ2, the estimator ω2t is non negative by construction, and hence
there is no need for censoring when constructing a consistent estimator of s based
on T days worth of data:
RV allT =
√√√√ 1
T
T∑t=1
ω2t . (16)
Similar to RollT , and unlike ES(1)T , RV all
T is consistent for s as n→∞ regardless
of T .
11
3.1.2 Observable trade direction
As suggested by Warga (1991) and Schultz (2001), when the trade direction is
observable one can simply regress the difference between the transaction price
and some benchmark price on the trade indicator. Here I continue assuming that
the benchmark price equals the end-of-day mid-quote and suggest running the
following OLS regression:
pi,t −m0,t = β(1)t qi,t + u
(1)i,t . (17)
If the daily benchmark prices are not available, one can use the average transaction
price instead and run the regression
pi,t − pt = β(2)t qi,t + u
(2)i,t . (18)
If model (1) is the data-generating process, the regression innovations are given by
u(1)i,t = mi,t −m0,t and u
(2)i,t = mi,t − mt − (s/2)qt, respectively. It is easy to show
that the ordinary least squares (OLS) estimators of β(1)t and β
(2)t in regressions (17)
and (18) satisfy, under my assumptions:
E(2β(1)t ) = s, Var(2β
(1)t ) =
2σ2(n+ 1)
n2, (19)
E(2β(2)t ) = s− s
n, Var(2β
(2)t ) =
2σ2(n+ 1)(n− 1)
3n3+ 2s2
(1
n2− 1
n3
).(20)
Replacing mi,t with m0,t or pt therefore does not render the regression-based esti-
mator inconsistent as it did for d2t and d2t in Section 3. The OLS estimators 2β(i)t ,
i = 1, 2 will converge in probability to s at rate√n as n→∞ as did the Roll and
RV-based measures in Section 3.1.
Similar to the effective spread measures in the previous section, none of these
OLS estimators are guaranteed to be non-negative. Thus, I censor them at zero
12
and denote the regression-based estimators of the effective spread s for day t by
RS(i)t = max{2β(i)
t , 0}, i = 1, 2. When estimating the effective spread using the full
sample of T days with n transactions each, one simply runs the regressions (17)
and (18) using all nT observations. I denote these estimators by RS(i)T , i = 1, 2.
Note that in practice, n does not have to be the same for all days in the sample; I
only make this assumption here to simplify derivations.
3.1.3 Observable timestamps and trade direction
Finally, I investigate the loss of efficiency associated with both missing timestamps
and trade direction. If timestamps were available, one could run the regression of
p on q in first differences:
∆pi,t = β(3)t ∆qi,t + u
(3)i,t . (21)
The gain in efficiency compared with the regressions in levels is simply due to
the fact that u(3)i,t = εi,t, which has much smaller variance than either u
(1)i,t or
u(2)i,t and is serially uncorrelated. A complication with the standard OLS estima-
tor β(3) in regression (21) is that it is not always well defined. We have β(3) =∑ni=2 ∆pi,t∆qi,t/
∑ni=2(∆qi,t)
2 and it is not difficult to show that P(∑n
i=2(∆qi,t)2 =
0) = (1/2)n−1. But because 1n−1
∑ni=2(∆qi,t)
2 converges in probability to 2 as
n → ∞, an asymptotically equivalent, well-defined estimator can be obtained by
simply setting∑n
i=2(∆qi,t)2 equal to 2(n− 1) in β
(3)t whenever
∑ni=2(∆qi,t)
2 = 0—
that is, I define
β(3)t =
∑ni=2 ∆pi,t∆qi,t
1{∑n
i=2(∆qi,t)2 = 0}2(n− 1) +
∑ni=2(∆qi,t)
2. (22)
Clearly, E(2β(3)t ) = s, so 2β
(3)t is an unbiased estimator of s. It is difficult to
derive the exact variance of β(3)t , but it is easy to show that the limiting variance
satisfies limn→∞ n2Var(2β
(3)t ) = 2σ2 and that 2β
(3)t is a n-consistent estimator of s.
13
Thus, observing both timestamps and trade direction at the same time improves
the convergence rate further: recall that the RMSE of the estimators RollT , RV allT ,
RS(i)T , i = 1, 2 only decays at rate n1/2. In small samples, β
(3)t can be negative with
positive probability, so I define RS(3)t = max{2β(3)
t , 0} as an estimator of s for day
t and RS(3)T = max{2β(3)
T , 0}, where β(3)T is obtained by running regression (21)
using all nT observations (T days with n transactions each).
3.1.4 Summary
The analytical comparison reveals that the absence of timestamps and/or trade
direction reduces the convergence rates of the effective spread estimators. In the
full-information case, one can achieve n-consistency, while in the absence of either
timestamps or trade direction, only√n-consistency is possible. In the absence of
timestamps, the limiting RMSE only depends on s (equations (12) and (15)), while
in the case of missing trade direction, it is solely driven by σ2 (equations (19)).
Finally, when both are missing, consistency cannot be achieved by increasing the
number of intraday observations and averaging over an increasing number of days
is necessary. The limiting variance of the effective spread estimator depends on
the ratio of σ2 and s, see equation (9).
Now, in practice this means that the relative performance of the various esti-
mators depends on the parameter configuration and the number of intraday ob-
servations. To illustrate that, I plot in Figure 1, the RMSE as a function of n on
a log-log scale. Clearly, when the signal-to-noise ratio σ2/s is high, the absence
of time stamps and trade direction lead to significant deterioration in RMSE for
any n. But when σ2/s is small (left panel), there exists a wide range for n where
the infeasible estimators do not really improve much upon the estimator that does
not require either timestamps or trade direction. This is a useful result because
in practice it is precisely illiquid, infrequently traded asses for which these data
limitations occur.
14
4 Range-based estimators
The baseline estimator is simple to compute, but it requires observing the bench-
mark price m0,t. When these prices or mid-quotes are not available, dt cannot be
calculated and we need an alternative moment condition. Inspired by Corwin and
Schultz (2012), I use the daily range:
r2t = (maxjpj,t −min
jpj,t)
2. (23)
The range has a long tradition in financial econometrics, dating back to Parkin-
son (1980), and has been widely used for estimating variance from intraday data
(Brandt and Diebold, 2006; Christensen and Podolskij, 2007; Christensen, Podol-
skij, and Vetter, 2009; Dobrev, 2007). It is clear that r2t is expected to depend on
both s and σ, as do d2t and d2t . Corwin and Schultz (2012) combine equation (23)
with a second moment condition based on the squared range over two consecutive
days, that is, (maxj pj,t:t+1−minj pj,t:t+1)2, where pj,t:t+1 denotes the j-th transac-
tion price in a two-day window starting on day t. To derive their estimator, Corwin
and Schultz (2012) make two strong assumptions that I do not want to make here:
continuous intraday trading and zero overnight returns.2 Moreover, relying solely
on the one-day and two-day range means throwing away a lot of data, so I de-
velop measures that use all available transaction data. My range-based estimators
therefore use only the daily range in equation (23) and work for any finite n.
I continue with the assumptions stated in Section 2 and additionally assume
that the innovations of the efficient price are normally distributed. The expectation
of the squared range can then be approximated by simulation for any finite n
and the SMM employed to consistently estimate s. I proceed as follows. Let
θ = (s, σ2)′ and let p∗s = (p∗1s, p∗2s, ..., p
∗ns)′ denote a random draw from model (1)
2Corwin and Schultz (2012) suggest a simple correction for non-zero overnight return, but thecorrection does not eliminate the overnight return problem completely and the estimator remainsgenerally biased and inconsistent.
15
given θ. Taking S independent draws, I approximate the expectation of r2 by
mS(θ, n) =1
S
S∑s=1
(maxjp∗js −min
jp∗js)
2. (24)
The SMM estimator is then obtained by
θT = arg minθ∈R++
g′TgT , (25)
where gT = 1T
∑Tt=1 gt, gt = (g1t, g2t)
′, g1t(θ, n) = d2t − E(d2t ), and g2t(θ, n) =
r2t −mS(θ, n). The objective function g′TgT must be minimized numerically under
the restrictions that s and σ are nonnegative. The range-based estimator of s,
which I denote by ES(2)T , is then given by ES
(2)T = θ1,T . It follows that if T/S → 0
as T →∞, ES(2)T
p→ s and the SMM estimator is asymptotically equivalent to the
generalized method of moments (GMM) (see chapter 2 in Gourieroux and Monfort,
1996), and the usual GMM inference applies.
My final estimator follows naturally from the previous two. If the benchmark
prices are observable, it is clearly desirable to use all three moment conditions at
the same time. Formally, define g3t(θ, n) = d2t − E(d2t ) and gt = (g1t, g2t, g3t)′,
where g1t and g2t are previously given. The over-identified SMM estimator of θ is
given by
θT = arg minθ∈R++
g′TWTgT (26)
for some positive definite matrix WT . I follow the standard two-stage approach,
whereby I first use WT = I to obtain a preliminary estimate θT and then use the
optimal WT (sample variance of gt evaluated at θT ) in the second stage to obtain
θT . My third estimator of s is then given by ES(3)T = θ1,T . Again, if T/S → 0 as
T →∞, ES(3)T
p→ s, and we obtain asymptotic equivalence with GMM.
16
4.1 Computational aspects
When the number of transactions is large, the previously described simulation-
based estimation may be slow. The minimization must be done numerically and the
evaluation of the objective function can be costly. Fortunately, simulation-based
or analytical approximations for E(r2t ) can be devised that significantly speed up
computations.
Observe that the price process in equation (2) can be approximated by a process
σW (t) + s2q(t), where W (t) is standard Brownian motion and q(t) is a continuous-
time process such that for any t, q(t) = 1 with a probability of 1/2, and q(t) = −1
otherwise. Now due to continuity of Brownian motion, the range of σW (t) + s2q(t)
equals the range of σW (u) plus s. Thus, for large n, we can approximate the range
of pi,t by the range of mi,t plus s:
E[(maxjpj,t −min
jpj,t)
2] ≈ E[(σ(maxjzj −min
jzj) + s)2], (27)
where zj, j = 0, ..., n, is a discretized Brownian motion on [0, 1]. All that has to be
simulated, then, is the expectation of the range and squared range of a discretized
Brownian motion. This simulation needs to be done only once, before the SMM
estimation begins, and not every time the objective function is evaluated. When
n is large, this approximation leads to significant gains in computational speed.
The approximation can be further improved by using the decomposition of
Christensen, Podolskij, and Vetter (2009), Lemma A.1, where the maximum of
the efficient price is only taken over buyer-initiated transactions and the minimum
over seller-intiated transactions when calculating the range of z in (27). Formally,
let bi, i = 0, ..., 1 be an iid binary process independent of z, where bi = 1 with a
probability of 1/2 and bi = −1 otherwise. Given a sample path of b and z, the range
of z is calculated over the set I = {(i, j)|bi = 1, bj = −1]}. The approximation
17
then becomes
E[(maxjpj,t −min
jpj,t)
2] ≈ E[(σ max(i,j)∈I
(zi − zj) + s)2]. (28)
As before, the range on the right-hand side of equation (28) needs to be simulated
only once and not every time the objective function is evaluated.
But the simulation can be avoided altogether because accurate analytical ap-
proximations for the range of discretized Brownian motion in equation (27) exist.
Using Lemma A.8 in Andersen, Dobrev, and Schaumburg (2013) together with
equation (27) leads to the approximation
E[(maxjpj,t −min
jpj,t)
2] ≈ (4 log 2)σ2 + 2
√8
πσs+ s2 +
ζ(1/2)√2π
(√8
πσ2 + σs
)4√n,
(29)
where ζ(1/2)/√
2π ≈ −0.5826.
To see how these approximations work, I plot expressions (27),(28), and (29)
together with the true value E[(maxj pj,t−minj pj,t)2] in Figure 2 for different values
of n. I find that all approximations are generally quite close to the true value for
n > 1000. Interestingly, there is virtually no difference between the analytical
approximations in equations (29) and 27); clearly, the first-order correction in
Andersen, Dobrev, and Schaumburg (2013) works very well, even for small n. But
both of these approximations are significantly upward biased when n is small.
Fortunately, the approximation in equation (28), based on the idea of Christensen,
Podolskij, and Vetter (2009), is significantly more accurate for all values of n and
is very close to the true value when n > 100. Thus, it seems that in practice one
should simulate E[(maxj pj,t−minj pj,t)2] when n is small—say, less than 100—and
use the approximation in equation (28) to speed up computations when n > 100.
For very large values of n, one can avoid simulations altogether and use equation
(29).
18
Another issue that obviously arises in practice is that the number of transac-
tions is not the same every day. This issue poses no problems for my estimators.
All one needs to do is to replace n in equations (3) and (29) with nt, where nt
denotes the number of transactions on day t. Similarly, in the SMM estimation,
one would simply simulate the squared range with the appropriate nt for each t.
5 Simulations
To assess the performance of my estimators ES(i)T , i = 1, 2, 3 in a controlled en-
vironment, I run a Monte Carlo experiment. I set the daily integrated volatility
of the efficient price (σ) to 35 basis points, which is approximately equal to the
daily volatility of the 10-year Treasury futures price, and let the efficient price in-
novations follow a normal distribution. I vary the true effective spread (s) between
5 and 50 basis points, the number of daily transactions (n) between 10 and 250,
and the number of days (T ) in the sample between 25 and 250. Recall that the
absolute values of s and σ are not that important—what matters are their relative
values. Each simulation is based on 10,000 Monte Carlo replications.
Table 1 reports the average effective spread obtained in the simulation together
with the associated RMSE. Starting with the results for the baseline estimator
ES(1)T , which are reported in the top two rows of each panel, I find that the bias
of the estimator can be either positive or negative in small samples depending on
the true effective spread. But as predicted by theory (see equation (8)), the bias
does become negative for sufficiently large T before eventually converging to zero
as T →∞. The RMSE of the estimator approaches zero at a rate that is broadly
in line with√T consistency.
Turning to the just-identified range-based estimator, ES(2)T , reported in rows 3
and 4 of each panel in Table 1, I find that the estimator exhibits a bias that can
be either positive or negative depending on n, T , and s, but both the bias and
19
the RMSE decline as T → ∞, as expected. Comparing the performance of ES(2)T
with the baseline estimator ES(1)T , I find that the two estimators can perform quite
differently. On the one hand, ES(1)T does well when s is large and T is small; for
example, when s = 50, n = 50, and T = 50, the RMSE of ES(1)T is around three
times smaller than that of ES(2)T . On the other hand, ES
(2)T works relatively well
when s is small and T is large; for example, when s = 5, n = 250, and T = 250,
the RMSE of ES(1)T is more than three times larger than that of ES
(2)T .
It is therefore not surprising that the over-identified estimator, ES(3)T , which
combines the moment conditions underlying ES(1)T and ES
(2)T using the optimal
weighting matrix, generally performs the best. The results reported in rows 5
and 6 of Table 1 show that the estimator is typically the most precise in terms of
RMSE, except when T is very small.
6 Time-varying effective spread
In this section, I allow the effective spread to vary over time. As shown previously,
due to the lack of transaction timestamps and trade direction, I am not able to
estimate the effective spread consistently from data spanning a fixed time period
such as a day or week, even if the number of transactions within that period
increases without bound. Transaction costs do fluctuate over time, however, and
so it is worth exploring the conditions under which one can recover the path of
the time-varying effective spread using all available data. It turns out that this is
possible if the effective spread process is sufficiently smooth.
For any given period t (for example, a day), I assume that the effective spread
within the period is constant and equal to st. I follow the recent advances by
Giraitis, Kapetanios, and Yates (2013) and Giraitis et al. (2016) and adopt a
nonparametric approach whereby the law of motion of the parameters is left un-
specified, up to a class of processes, and the parameters are estimated by local
20
averaging. The processes I consider for st are bounded stochastic and/or deter-
ministic processes satisfying the smoothness condition:
supj:|j−t|<h
||st − sj||2 = Op(h/t) (30)
as t → ∞, h → ∞, and h = o(t). Examples of processes that belong to this
class include st = t−1xt, st = xtmaxj≤T |xj |
, and st = g(tn
), where {x} is a unit root
process with stationary increments and g is a smooth deterministic function on the
unit interval. These processes are bounded in probability and are smoother than
random walks.
To estimate st, I take some weights wj,t = wj,t/∑
j wj,t, where wj,t = K((j −
t)/H) for some kernel function K and bandwidth parameter H. I then define the
kernel estimator of the effective spread at time t as
ES(1)t,T =
√√√√max
{2
T∑j=1
wj,t(3d2j − d2j), 0
}. (31)
The kernel can be uniform, leading to simple rolling estimation with a window
of size H, or have unbounded support, such as the Gaussian kernel. For an un-
bounded kernel, I require that K(x) ≥ 0, x ∈ R, is a continuous bounded func-
tion with a bounded first derivative such that∫K(x)dx = 1. K(x) = O(e−cx
2),
∃c > 0, |K ′(x)| = O(|x|−2), x→∞. Several popular kernel functions satisfy these
conditions—for example, the Gaussian and quartic kernels. Similar to the time-
varying spread st, I assume that the volatility of the efficient price is constant
intraday but varies smoothly over days—that is, σi,t = σt for all i, and that σt is a
bounded stochastic and/or deterministic process satisfying the same smoothness
condition (30). This assumption, of course, contains constant volatility as a special
case.
The key to achieving consistency of the time-varying estimator ES(1)t,T is the
21
choice of H relative to T . Define H = H if the kernel has bounded support and
H = H log1/2H for an unbounded kernel. Then we have the following.
Proposition 2 For any t = [τT ], 0 < τ < 1, if H → ∞, H/T → 0 as T → ∞,
ES(1)t,T
p→ st.
The proposition shows that the bandwidth parameter has to grow with T but not
as fast as T . When choosing H, one faces the familiar tradeoff between bias and
variance: Smaller (larger) H produces less (more) biased and more (less) volatile
estimates. There is currently no data-driven method for choosing H, but taking
H =√T seems to work well in existing applications (Giraitis, Kapetanios, and
Yates, 2013; Giraitis et al. 2016).
The advantage of the ES(1)t,T estimator is that it is available in closed form and
easy to calculate. However, the simulation results reported in the previous section
show that the range-based estimators can perform better. It is therefore desirable
to adapt those estimators to the time-varying parameter setting as well. This
method turns out to be more involved because the expectation of the range is not
known in closed form. Some theoretical results are nonetheless possible to obtain
under the assumption that both n→∞ and T →∞.
To construct the kernel analog to the estimators ES(2)T and ES
(3)T , define
g(1)t,T =
T∑j=1
wjt
(d2j −
s2
4− σ2
2
), (32)
g(2)t,T =
T∑j=1
wjt
(d2j −
s2
4− σ2
6
), (33)
g(3)t,T =
T∑j=1
wjt
(r2j − 4 log 2σ2 − 2
√8
πsσ − s2
), (34)
and gt,T = (g(1)t,T , g
(2)t,T , g
(2)t,T )′. Then the kernel estimator of θt = (st, σ
2t )′ is given by
θt,T = arg minθ∈R++
g′t,TWt,Tgt,T (35)
22
for some positive definite matrix Wt,T and ES(3)t,T = θ
(1)t,T . When using only the
second and third moment conditions, we obtain ES(2)t,T . The following proposition
establishes the consistency of these estimators.
Proposition 3 For any t = [τT ], 0 < τ < 1, if n → ∞, H → ∞, H/T → 0 as
T →∞, ES(i)t,T
p→ st, i = 2 and 3.
7 Empirical illustration
Having explored the behavior of the various effective spread estimators analytically
and through simulations, I now turn to an empirical illustration. The purpose of
this exercise is to assess the performance of my measures against (1) the true effec-
tive spread one would simply calculate if m, q, and timestamps were observable,
and (2) the measures that require either timestamps or trade direction, or both,
that I discussed previously. In other words, I want to examine the performance
of my estimators against those that require progressively more information. This
approach is similar to Goyenko, Holden, and Trzcinka (2009), who compare low-
frequency liquidity measures with their high-frequency counterparts for selected
U.S. stocks.
Now, ideally, I would like to employ data from an OTC market, since that is
where I expect my measures would naturally find applications, but to the best
of my knowledge, no time-stamped OTC trade and quote data are available that
would allow me to do this exercise. I thefore employ the widely-used TAQ data
for selected NYSE-listed stocks; the TAQ data are time stamped to the second
and contain information about m and q, so that the true effective spread can be
readily computed.
23
7.1 Data and descriptive statistics
The universe of NYSE-listed stocks is too broad to consider here in full. Because
my measures of effective spread would typically be applied to OTC-traded con-
tracts, which tend to be less liquid and trade less frequently than exchange-traded
instruments, I focus on stocks with small market capitalization. In particular, I
take all stocks in the TAQ database that satisfy two criteria. The first is that the
stock was included in the S&P Small-Cap 600 Index for the entire period between
January 2, 2005, and December 31, 2014. The second is that there are trade and
quote data available for this stock in the TAQ database for every trading day in
this period. These criteria select 147 stocks.
For each stock and day in my sample, I download from the Wharton Research
Data Services (WRDS) the WRDS-derived trades files (WCT data sets), which
contain trades matched with the prevailing National Best Bid and Offer quotes.
I then filter the data and retain only those trades with trade times between 9:35
a.m. and 4:00 p.m., positive transaction price, positive prevailing mid-quote, and
positive quoted spread. In addition, I drop all trades where the prevailing quoted
spread is greater than 50 times the median quoted spread for the same day, and
where the proportional effective spread is greater than 50 times the median propor-
tional effective spread for the same day; these rules are similar to those proposed
by Barndorff-Nielsen et al. (2008).
Table 2 reports some descriptive statistics for the data, separately for five two-
year periods that span my sample. The average daily number of trades varies
between 1,000 and 2,000 for a typical stock day. The average effective spread
varies between 12 and 16 basis points, while the average daily realized volatility
varies between 150 and 250 basis points. The average SNR, which I define here as
the ratio of effective spread and realized volatility (s/σ), fluctuated between 6 and
9 percent. Thus, despite being small cap, the typical stock in my sample traded
relatively frequently and with a fairly tight spread during my sample period. At
24
the same time, there are stocks and trading days with relatively little trading and
fairly large effective spreads as indicated by the 5th and 95th percentiles reported
in the table.
7.2 Results
My main empirical results are summarized in Panel A of Table 3. I run the effective
spread estimations separately for each stock month, stock quarter, stock half-year,
and stock year in the sample period and compare the estimates with the actual
effective spreads observed for a given stock in a given time period. Specifically, I
calculate the bias and RMSE associated with each estimator and the correlation
of the estimated spread with the actual effective spreads calculated from TAQ. I
consider my estimators ES(i)T , i = 1, 2, 3 and the Roll, RV-based, and regression-
based estimators for comparison. Recall that the latter three are infeasible in the
absence of timestamps.
I find that estimating the effective spread without timestamps is very chal-
lenging. The infeasible estimators are significantly less biased and an order of
magnitude more accurate in terms of RMSE than my estimators; they are also
much more closely correlated with the actual effective spreads. In line with theory,
the over-identified estimator ES(3)T is generally most accurate in terms of RMSE
out of my three estimators, although ES(2)T tends to be more closely correlated
with the actual spread. The results do not improve as the number of observations
used for estimation increases. As expected, all infeasible estimators deliver RMSE
that is an order of magnitude lower than that of my feasible estimators. The esti-
mator based on realized volatility performs remarkably well, exhibiting almost no
bias and having significantly lower RMSE than the Roll measure.
The relatively poor performance of the feasible estimators should not come as
a surprise: The average SNR for the 147 stocks in my sample is very small, and the
simulation results discussed previously clearly indicate that in such circumstances
25
all feasible estimators struggle. To shed more light on the relationship between
the SNR and RMSE, I perform the following experiment. Rather than using the
original transaction prices when computing effective spread estimates, I construct
a new set of transaction prices p where I set pi,t = mi,t + 10(pi,t−mi,t) for all i and
t—that is, I artificially inflate the actual effective spread by a factor of 10. This
procedure leaves the intraday volatility and time-series dynamics of the mid-quote
unchanged, but it increases the SNR tenfold. I then reproduce the results reported
in Panel A of Table 3 using the artificial transaction prices p in place of the actual
transaction prices p.
The results are reported in Panel B. I find that the relative performance of
my estimators improves significantly. Although they are still upward biased, their
RMSE is now much smaller relative to the actual spread. Also, while the two
estimators that utilize timestamps (Roll and RV all) still outperform my estimators,
the differences in terms of RMSE have become smaller. The regression-based
estimators that require trade direction perform better that either Roll or RV all.
Finally, the correlation between my estimates and the true spreads has increased
significantly.
In addition to the experiment with the SNR, I study how the intraday number
of transactions (n) affects performance. I do this by sampling sparsely from the
set of transaction prices, retaining only every 10th observation on a given day for
a given stock, and re-run all estimations on the sparsely sampled data. The results
are reported in Panels C and D of Table 3; the former reports results based on
the original data, while the latter shows results based on the artificial transaction
prices previously described (inflated true effective spreads).
Starting with Panel C, I find that the performance of my estimators is largely
unaffected by sparse sampling. This finding is in line with the theoretical result
that n has only a second-order effect on the RMSE of these estimators. In contrast,
the infeasible Roll and RV-based estimators exhibit a significant deterioration in
26
precision as n decreases. Notably, the RV-based estimator now exhibits a signifi-
cant upward bias and a substantial increase in RMSE. When I artificially inflate
the actual effective spreads by a factor of 10 (Panel D), the differences between
the RMSE of my estimators and the infeasible ones become even smaller.
In summary, the empirical results based on the 147 stock in my sample are
largely consistent with the theoretical and simulation results reported previously.
The key driver of the performance of my estimators is the SNR. To illustrate this
finding graphically, Figure 3 plots the RMSE expressed as a fraction of the true
spread separately for stock-month sorted into deciles by their SNR. The figure is
based on the same data as Panel D in Table 3. Clearly, as the SNR increases, the
performance improves, and gradually approaches the performance of the infeasible
estimators. This is very much in line with the behavior of the theoretical RMSE
shown in Figure 1 in Section 3.1.
7.3 Time-varying estimation
I now turn to the time-varying kernel estimation proposed in Section 6. I use
the Gaussian kernel and set the bandwidth parameter according to H =√T .
Figure 4 summarized the estimation results. Because it is impossible to show the
time-varying estimates for all 147 stocks here, I simply report the cross-sectional
averages over time. Panel A shows results based on the original data, while Panel
B reports results based on the (artificial) transaction prices obtained by inflating
the actual effective spreads by a factor of 10.
I find that my time-varying estimators perform well when the SNR is not very
low, which is hardly surprising given the findings of the previous subsection. When
the SNR is low, the estimators are significantly upward biased and do not capture
the dynamics of the actual effective spread accurately. In particular, ES(1)t,T drops
substantially during the financial turmoil of 2008 even though the actual effective
spread increased. While the range-based estimators do not suffer from this problem
27
and are generally more closely correlated with the true spreads, they tend to peak
a few months late. When I inflate the true effective spreads, the performance of all
estimators improves considerably, both in terms of bias and correlation with the
true spread.
8 Conclusion
In this paper, I have studied the problem of estimating transaction costs in the
absence of timestamps. Building on insights from the previous literature, I pro-
posed several measures of the effective spread and studied their sampling properties
within the simple framework of Roll (1984). I corroborated my theoretical findings
using a Monte Carlo simulation and assessed the performance of my estimators in
an empirical application to selected NYSE-listed small-cap stocks.
The theoretical, simulation-based, and empirical results show that the loss of
information due to missing timestamps is large. My estimators are suitable for
measuring transaction costs in illiquid OTC markets, where effective spreads tend
to be wide relative to the fundamental volatility, but not necessarily in highly liquid
exchange-based markets such as those for equities and futures contracts, where the
opposite is generally true. But in those cases, accurate transaction timestamps are
typically available, so my estimators would not be necessary.
Throughout the paper, I have worked in the widely used framework of Roll
(1984). Whereas the simplicity of this framework allows for straightforward analyt-
ical derivations, future work may consider more elaborate microstructure models,
such as those by Huang and Stoll (1997); Madhavan, Richardson, and Roomans
(1997); Bessembinder, Maxwell, and Venkataraman (2006); and Edwards, Harris,
and Piwowar (2007). These models allow for a richer relationship between order
flow and returns, and they relax some of the arguably restrictive assumptions of
the Roll (1984) model. It would be interesting to explore whether these models can
28
be reliably estimated when timestamps are missing. Last but not least, it would
be interesting to study whether the effective spread can be estimated consistently
in the presence of stochastic volatility of the efficient price.
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A Proofs
Proof of Proposition 1. Dropping the subscript t to simplify notation, we have
s2 − s2 = 2(3d− d)− s2, (36)
=3s2
2
[(1
n− 1
n∑i=1
(qi − q)2)− 1
]
+ 2s
[3
n− 1
n∑i=1
(mi − m)(qi − q)−1
n
n∑i=1
(mi −m0)qi
]
+ 2
[3
n− 1
n∑i=1
(mi − m)2 − 1
n
n∑i=1
(mi −m0)2
](37)
=: An +Bn + Cn. (38)
By construction, E(An) = E(Bn) = E(Cn) = 0, and it is easy to show that E(AnBn) =
E(AnCn) = E(BnCn) = 0 because E(miqj) = 0 for all i and j. Thus, Var(s2) =
E(A2n) + E(B2
n) + E(C2n). It is clear from the equation above that s2 does not depend on
m0, so we will set it to zero to simplify notation.
Starting with E(A2n), write
A2n =
9s4
4
1
(n− 1)2
n∑i=1
n∑j=1
(qi − q)2(qj − q)2 −1
n− 1
n∑i=1
(qi − q)2 + 1
. (39)
33
Because E(qiqj) = 0 if i 6= j and q2i ≡ 1, we have
E
n∑i=1
n∑j=1
(qi − q)2(qj − q)2 = n2 − 2E
n∑i=1
n∑j=1
qiqj
+1
n2E
n∑i=1
n∑j=1
n∑k=1
n∑l=1
qiqjqkql
,
(40)
= n2 − 2n+ 3− 2
n. (41)
This, together with E(∑n
i=1(qi − q)2)
= n− 1, gives after some algebra
E(A2n) =
9s4
2n(n− 1). (42)
Turning to E(B2n), write
B2n = 4s2
[9
(n− 1)2− 6
n(n− 1)+
1
n2
] n∑i=1
n∑j=1
mimjqiqj
+ 4s2[
6
n2(n− 1)− 18
n(n− 1)2
] n∑i=1
n∑j=1
n∑k=1
mimjqjqk
+36s2
n2(n− 1)2
n∑i=1
n∑j=1
n∑k=1
n∑l=1
mimjqkql. (43)
Because E(εiεj) = 0 if i 6= j,
E
n∑i=1
n∑j=1
mimjqiqj
=n∑i=1
n∑j=1
E(mimj)E(qiqj) =n∑i=1
E(m2i )q
2i =
n∑i=1
i∑p=1
E(ε2p) =σ2
2(n+ 1).
(44)
Similarly, E(∑
i
∑j
∑kmimjqjqk
)=∑
i
∑j E(mimj) and E
(∑i
∑j
∑k
∑lmimjqkql
)=
n∑
i
∑j E(mimj). Thus, it remains to derive
∑i
∑j E(mimj). The case i = j follows
from above, so we focus on the case when i 6= j:
n∑i=1
n∑j=1
i 6=j
E(mimj) = 2
n∑i=1
n∑j=i+1
E(mimj), (45)
34
= 2n∑i=1
n∑j=i+1
E
i∑p=1
(i∑
r=1
+
j∑r=i+1
)εpεr
, (46)
= 2
n∑i=1
n∑j=i+1
i∑p=1
E(ε2p), (47)
=2σ2
n
n∑i=1
n∑j=i+1
i, (48)
= σ2n(n+ 1)− 1
3σ2(n+ 1)(2n+ 1). (49)
Plugging (44) and (49) into the expectation of (43) and simplifying gives
E(B2n) =
2s2σ2(2n2 + 3n+ 1)
n2(n− 1). (50)
Finally, we derive E(C2n). Write
C2n = 4
[9
(n− 1)2− 6
n(n− 1)+
1
n2
] n∑i=1
n∑j=1
m2im
2j
+ 4
[6
n2(n− 1)− 18
n(n− 1)2
] n∑i=1
n∑j=1
n∑k=1
mimjm2k
+36
n2(n− 1)2
n∑i=1
n∑j=1
n∑k=1
n∑l=1
mimjmkml. (51)
We focus on the last term because the other two terms follow from the derivation of this
term. Observe that
n∑i=1
n∑j=1
n∑k=1
n∑l=1
mimjmkml =
n∑i=1
m2i + 3
n∑i=1
n∑j=1
i 6=j
m2im
2j + 4
n∑i=1
n∑j=1
i 6=j
m3imj
+ 12n∑i=1
n∑j=1
n∑k=1
i<j<k
m2imjmk + 12
n∑i=1
n∑j=1
n∑k=1
i<j<k
mim2jmk
+ 12
n∑i=1
n∑j=1
n∑k=1
i<j<k
mimjm2k + 24
n∑i=1
n∑j=1
n∑k=1
n∑l=1
i<j<k<l
mimjmkml.
(52)
35
To save space, we derive here only the expectation of the last term; the other terms
follow using the same approach:
E
n∑i=1
n∑j=1
n∑k=1
n∑l=1
i<j<k<l
mimjmkml
= E
n∑i=1
n∑j=i+1
n∑k=j+1
n∑l=k+1
i∑p=1
j∑r=1
k∑s=1
l∑t=1
εpεrεsεt
, (53)
= E
n∑i=1
n∑j=i+1
n∑k=j+1
n∑l=k+1
i∑p=1
(i∑
r=1
+
j∑r=i+1
) i∑s=1
+
j∑s=i+1
+
k∑s=j+1
×
i∑t=1
+
j∑t=i+1
+
k∑t=j+1
+
l∑t=k+1
εpεrεsεt
, (54)
=n∑i=1
n∑j=i+1
n∑k=j+1
n∑l=k+1
E
i∑p=1
i∑r=1
i∑s=1
i∑t=1
εpεrεsεt
+ 3E
i∑p=1
i∑r=1
j∑s=i+1
j∑t=i+1
εpεrεsεt
+E
i∑p=1
i∑r=1
k∑s=j+1
k∑t=j+1
εpεrεsεt
,(55)
=1
n2
n∑i=1
n∑j=i+1
n∑k=j+1
n∑l=k+1
3σ4i2 + κi+ 3σ4i(j − i) + σ4i(k − j), (56)
=σ4
3n4 +
(σ4 +
κ
5
)n3 +
(13σ4
12+κ
2
)n2 +
(σ4
2+κ
3
)n+
σ4
12− κ
30
1
n, (57)
where κ = E(ε41)− 3σ4. Above, we use the fact that
E
i∑p=1
i∑r=1
i∑s=1
i∑t=1
εpεrεsεt
=i∑
p=1
i∑r=1
i∑s=1
i∑t=1
E(εpεrεsεt), (58)
= 3
i∑p=1
i∑r=1
p 6=r
E(ε2pε2r) +
i∑p=1
E(e4p), (59)
=1
n2(3σ4i2 + κi), (60)
36
and
E
i∑p=1
i∑r=1
j∑s=i+1
j∑t=i+1
εpεrεsεt
= E
i∑p=1
i∑r=1
εpεr
E
(j∑
s=i+1
j∑t=i+1
εsεt
), (61)
=
i∑p=1
E(ε2p)
( j∑r=i+1
E(ε2r)
), (62)
=1
n2σ4i(j − i). (63)
The expectation of the other terms in C2n can be obtained analogously. We obtain
E(C2n) =
2(2σ4n+ σ4 + 2κ)(2n3 + 7n2 + 7n+ 2)
15n3(n− 1). (64)
The variance of s2 then follows after some algebra. �
Proof of Proposition 2. Write
T∑j=1
2wjt(3d2j − d2t ) =
T∑j=1
wjt
(1
2
[1
n− 1
n∑i=1
3(qi,j − qj)2 − 1
]s2j − s2t
)(65)
+ 2T∑j
wjtsj
[3
n− 1
n∑i=1
(mi,j − mj)(qi,j − qj)−1
n
n∑i=1
(mi,j −m0,j)qi,j
]
+ 2T∑j=1
wjt
[3
n− 1
n∑i=1
(mi,j − mj)2 − 1
n
n∑i=1
(mi,j −m0,j)2
], (66)
=: At,T +Bt,T + Ct,T . (67)
Define Qj,n = 12
[1
n−1∑n
i=1 3(qi,j − qj)2 − 1]
and write
At,T =T∑j=1
wjt(s2j − s2t ) +
T∑j=1
wjt(Qj,n − 1)s2j . (68)
Now, because Qj,n and sj are uncorrelated and E(Qj,n) = 1, the expectation of the
37
second term equals zero, and its variance is given by
E
T∑j=1
wjt(Qj,n − 1)s2j
2
= E
T∑j=1
w2jt(Qj,n − 1)2s4j
, (69)
≤ CE(Qj,n − 1)2T∑j=1
w2jt, (70)
= O((n2H)−1). (71)
Turning to the first term in (68), write
T∑j=1
wjt(s2j − s2t ) =
∑j:|j−t|<h
wjt(s2j − s2t ) +
∑j:|j−t|≥h
wjt(s2j − s2t ). (72)
Following Giraitis et al. (2016), we take h = bH log1/2H for some positive constant b.
Then the second term can be ignored, as it is of smaller order than the first term. Now
∣∣∣∣ ∑j:|j−t|<h
wjt(s2j − s2t )
∣∣∣∣ ≤ C supj:|j−t|<h
|sj − st|∑
j:|j−t|<h
wjt = Op
((H
T
)1/2), (73)
because |sj−st| ≤[supj:|j−t|<h(sj − sj)2
]1/2= Op((H/T )1/2) and
∑j:|j−t|<hwjt = O(1).
Thus, At,T = Op
((H/T
)1/2)+O
((n2H)−1/2
).
Turning to Bt,T , define Pj,n := 3n−1
∑ni=1(mi,j−mj)(qi,j−qj)− 1
n
∑ni=1(mi,j−m0,j)qi,j
and note that E(Pj,n) = 0 and E(Pj,nPk,n) = 0 unless j = k. Thus, E(Bt,T ) = 0. Its
variance satisfies
E(B2t,T ) = E(4
T∑j=1
w2jts
2jP
2j,n) ≤ CE(P 2
j,n)
T∑j=1
w2jt = CO
(n−1
)O(H−1
)= O
((nH)−1
),
(74)
as E(P 2j,n) = O(n−1). Thus, Bt,T = Op((nH)−1/2). Finally, by a similar argument, we
obtain Ct,T = Op(H−1/2
), which completes the proof. �
Proof of Proposition 3. For the sake of brevity, I present a proof for W = I.
In particular, I show that g′t,Tgt,T converges in probability, at appropriate rates and
38
uniformly in θ, to
[1
4(s2t − s2) +
1
2(σ2t − σ2)
]2+
[1
4(s2t − s2) +
1
6(σ2t − σ2)
]2+
[(4 log 2)(σ2t − σ2) + 2
√8
π(stσt − sσ) + (s2t − s2)
]2, (75)
which is clearly minimized at (st, σt). Starting with g(1)t,T , write
(g(1)t,T
)2=
T∑j=1
wjtd2j −
s2t4− σ2t
2
2
+ 2
T∑j=1
wjtd2j −
s2t4− σ2t
2
×(
1
4(s2t − s2) +
1
2(σ2t − σ2)
)+
(1
4(s2t − s2) +
1
2(σ2t − σ2)
)2
. (76)
Now
∣∣∣∣ T∑j=1
wjtd2j−
s2t4−σ
2t
2
∣∣∣∣ ≤ ∣∣∣∣ T∑j=1
wjt
(1
n
n∑i=1
(mi,j −m0,j)2 − σ2t
2
)∣∣∣∣+∣∣∣∣ T∑j=1
wjt
(1
n
n∑i=1
(mi,j −m0,j)qi,j
)∣∣∣∣.(77)
Define Mj,n = 1n
∑ni=1(mi,j −m0,j)
2 and write
T∑j=1
wjt(Mj,n −σ2t2
) =
T∑j=1
wjt
(Mj,n −
σ2j2
)+
1
2
T∑j=1
wjt(σ2j − σ2t
). (78)
By the same argument as in (72)–(73), the second term in (78) is Op((H/T )1/2). The
first term in (78) can be written as
T∑j=1
wjt
(Mj,n −
σ2j2
)=
T∑j=1
wjt
(Mj,n −
σ2j2
n+ 1
n
)+ op(1), (79)
where E(Mj,n −σ2j
2n+1n ) = 0 and Var(Mj,n −
σ2j
2n+1n ) = O(1), which can be shown
using the law of iterated expectations and the results in Section 4 of the Supplementary
Appendix. Thus, |∑T
j=1wjt(Mj,n − σ2j /2)| = Op(H−1/2). Turning to the second term
in (77), we apply similar arguments to arrive at |∑T
j=1wjt(1n
∑ni=1(mi,j −m0,j)qi,j
)| =
39
Op((nH)−1/2). Because st and σ2t are bounded, we have
(g(1)t,T
)2=
(1
4(s2t − s2) +
1
2(σ2t − σ2)
)2
+Op((H/T )1/2) +Op(H−1/2) (80)
uniformly in s and σ. (g(2)t,T )2 can be treated analogously. Turning to (g
(3)t,T )2, write
g(3)t,T =
T∑j=1
wjtr2j − k2σ2t − k1stσt − s2t
2
+ 2
T∑j=1
wjtr2j − k2σ2t − k1stσt − s2t
(k2(σ2t − σ2) + k1(stσt − sσ) + s2t − s2)
+(k2(σ
2t − σ) + k1(stσt − sσ) + s2t − s2
)2, (81)
where k1 = 2√
8/π and k2 = 4 log 2. Define Ij = {(i, k)|qi,j = 1, qk,j = −1]}, j = 1, ..., T ,
and observe that for every j, with probability 21−n,
max(i,k)∈Ij
(mi,j −mk,j) + s ≤ maxipi,j −min
ipi,j ≤ max
imi,j −min
imi,j + s. (82)
Thus, we can proceed by conditioning on the event {Ij 6= ∅ for all j}. Define zi,j =
mi,j/σj , j = 1, ..., T , and write
∣∣∣∣∣∣T∑j=1
wjt[σj(maxizi,j −min
izi,j) + sj ]
2 −T∑j=1
wjtr2j
∣∣∣∣∣∣≤
∣∣∣∣∣∣T∑j=1
wjt[σj(maxizi,j −min
izi,j) + sj ]
2 −T∑j=1
wjt[σj max(i,k)∈Ij
(zi,j − zk,j) + sj ]2
∣∣∣∣∣∣ (83)
≤ C
∣∣∣∣∣∣T∑j=1
wjt[(maxizi,j −min
izi,j)
2 − ( max(i,k)∈Ij
(zi,j − zk,j))2]
∣∣∣∣∣∣+ C
∣∣∣∣∣∣T∑j=1
wjt[(maxizi,j −min
izi,j)− max
(i,k)∈Ij(zi,j − zk,j)]
∣∣∣∣∣∣ . (84)
40
By the Markov inequality,
P
∣∣∣∣∣∣T∑j=1
wjt[(maxizi,j −min
izi,j)
2 − ( max(i,k)∈Ij
(zi,j − zk,j))2]
∣∣∣∣∣∣ > ε
≤ 1
ε(λ2,n − λ2,n), (85)
where λ2,n = E([maxi zi,j − mini zi,j ]2) and λ2,n = E([max(i,k)∈Ij (zi,j − zk,j)]
2). Be-
cause {zi,j} is a scaled symmetric Gaussian random walk, it follows that limn→∞ λ2,n =
limn→∞ λ2,n = k2 (Christensen, Podolskij, and Vetter, 2009). The second term in (84)
can be handled similarly. Thus, (83) is op(1) as n→∞, and it suffices to focus on
∣∣∣∣∣∣T∑j=1
wjt(σjRjn + sj)2 − k2σ2t − k1stσt − s2t
∣∣∣∣∣∣ (86)
≤
∣∣∣∣∣∣T∑j=1
wjt(σ2jR
2jn − k2σ2t )
∣∣∣∣∣∣+
∣∣∣∣∣∣T∑j=1
wjt(2sjσjRjn − k1stσt)
∣∣∣∣∣∣+
∣∣∣∣∣∣T∑j=1
wjt(s2j − s2t )
∣∣∣∣∣∣ (87)
=: Dt,T + Et,T + Ft,T , (88)
where Rjn = maxi zi,j −mini zi,j , j = 1, ..., T . Now Ft,T = Op((H/T )1/2) by (44)–(45),
and for Dt,T we have
Dt,T ≤ C
∣∣∣∣∣∣T∑j=1
wjt[R2jn − E(R2
jn)]
∣∣∣∣∣∣+ C∣∣E(R2
jn)− k2∣∣+ k2
∣∣∣∣∣∣T∑j=1
wjt(σ2j − σ2t )
∣∣∣∣∣∣ . (89)
By the central limit theorem, the first term on the right-hand side of (89) is Op(H−1/2),
the second term is o(1) as n → ∞, and the third term is Op((H/T )1/2) by the same
argument as in (72)–(73). Thus, Dt,T = Op(H−1/2)+Op((H/T )1/2). Following the same
steps, and noting that ||stσt − sjσj ||2 ≤ C||st − sj ||2 +C||σt − σj ||2 +C|st − sj ||σt − σj |
and hence sup|t−j|≤h ||stσt − sjσj ||2 = Op(h/t), we find that Et,T is of the same order as
Dt,T , which completes the proof. �
41
B Figures and Tables
1 2 3 4
-2
0
2
1 2 3 4
-2
0
2
ES1 Roll RVall RS1 RS2 RS3
1 2 3 4
-1
1
ES1 Roll RVall RS1 RS2 RS3
Figure 1: Simulated RMSE of alternative estimators of the effective spread as afunction of n on a log-log scale. The parameter values are σ = 35 bps, κ = 0,and s = 120 bps (left panel), 50 bps (middle panel), and 10 bps (right panel), andT = 1.
1.0 1.5 2.0 2.5 3.0 3.5 4.0
25
35
45
(a) σ= 35 bps, s = 10 bps
TrueSimulated approx. (28)Simulated approx. (27)Simulated approx. (29)
1.0 1.5 2.0 2.5 3.0 3.5 4.0
20
30
40
(b) σ= 35 bps, s = 5 bps
Figure 2: Expected squared range of p and its approximations as a functionof log10 n. The line labeled “True” shows the true expectation E[(maxj pj,t −minj pj,t)
2], “Simulated approx. (28)” shows the right-hand side of (28), “Simu-lated approx. (27)” shows the right-hand side of (27), and “Analytical approx.(29)” shows the right-hand side of (29). The various expectations are approxi-mated by simulation with 100,000 replications. The efficient price innovations arenormally distributed with a volatility of 35 bps.
42
SNR ES1 ES2 ES3 RS1 RS2 RS3 Roll RVall
1 2 3 4 5 6 7 8 9 10
0.2
0.4
0.6
0.8
1.0
1.2
1.4 SNR ES1 ES2 ES3 RS1 RS2 RS3 Roll RVall
Figure 3: Average RMSE expressed as a fraction of the true effective spread fordeciles based on the signal-to-noise (SNR) ratio. Every month, the stocks in thesample are sorted into deciles by their SNR. The RMSE for each decile is thencalculated by averaging across all stock-month observations in the decile.
True ES1 ES2 ES3
2006 2008 2010 2012 2014
50
100
150
True ES1 ES2 ES3
True ES1 ES2 ES3
2006 2008 2010 2012 2014
200
400 True ES1 ES2 ES3
Figure 4: Time-varying effective spread estimates averaged across the 147 stocksin the sample. The left panel shows results for the original data, and the rightpanel shows results based on transaction prices with artificially inflated effectivespreads. All time-varying estimations are performed using the Gaussian kernel,and the bandwidth is set equal to the square root of the sample size.
43
s=
50b
ps
s=
20
bp
ss
=10
bp
ss
=5
bp
s
nT
2550
100
250
25
50
100
250
25
50
100
250
25
50
100
250
10ES(1
)T
49.9
449
.90
49.9
650
.00
18.8
819.3
819.6
819.9
39.4
69.2
79.0
19.2
77.0
76.2
65.5
34.8
7[3
.59]
[2.5
7][1
.79]
[1.1
3][7
.38]
[5.1
8]
[3.4
0]
[2.0
3]
[7.8
8]
[6.7
1]
[5.7
2]
[4.2
8]
[7.6
3]
[6.4
0]
[5.4
2]
[4.4
1]
ES(2
)T
49.6
749
.89
49.9
549
.99
21.3
321.2
421.1
821.1
19.3
39.4
59.6
69.9
06.6
65.9
65.4
04.7
8[3
.22]
[2.1
3][1
.50]
[0.9
4][5
.69]
[4.7
9]
[3.9
5]
[3.0
4]
[7.0
2]
[6.3
5]
[5.5
1]
[3.9
8]
[6.7
5]
[5.9
4]
[5.1
0]
[3.8
8]
ES(3
)T
50.0
350
.04
50.0
250
.02
19.3
319.6
219.8
119.9
110.5
810.1
09.9
69.9
67.1
76.0
15.3
04.8
1[2
.92]
[2.0
0][1
.43]
[0.9
0][6
.18]
[4.4
4]
[3.1
2]
[1.9
8]
[6.7
0]
[5.3
7]
[4.0
5]
[2.4
7]
[6.7
8]
[5.3
1]
[4.2
0]
[3.1
9]
50ES(1
)T
49.9
349
.98
49.9
849
.99
19.2
119.6
019.7
919.9
39.2
39.0
79.1
29.4
46.5
15.9
05.2
24.6
4[2
.16]
[1.5
3][1
.07]
[0.6
8][5
.77]
[3.8
3]
[2.5
7]
[1.5
4]
[7.1
4]
[6.1
2]
[5.0
7]
[3.5
9]
[6.8
8]
[5.8
0]
[4.9
0]
[4.0
4]
ES(2
)T
48.7
949
.65
49.9
249
.98
19.6
720.0
219.7
019.4
910.4
310.1
110.4
610.4
35.5
04.6
44.8
45.0
0[7
.64]
[4.5
8][2
.11]
[0.4
4][3
.99]
[3.4
0]
[2.8
0]
[2.2
5]
[4.0
4]
[3.3
3]
[2.6
7]
[1.9
3]
[4.0
8]
[3.1
5]
[2.3
0]
[1.4
2]
ES(3
)T
49.9
550
.04
50.0
149
.99
19.7
619.7
119.7
419.8
511.5
510.7
110.5
710.2
86.3
45.1
15.0
55.0
8[1
.35]
[0.9
4][0
.67]
[0.4
2][4
.02]
[3.2
9]
[2.4
2]
[1.5
3]
[4.7
0]
[3.5
4]
[2.4
9]
[1.3
9]
[4.8
1]
[3.3
1]
[2.1
3]
[1.2
6]
250ES(1
)T
49.9
849
.96
49.9
849
.99
19.3
519.6
619.8
719.9
49.2
29.1
29.0
99.4
56.5
35.7
45.1
64.7
3[1
.86]
[1.3
2][0
.94]
[0.5
9][5
.33]
[3.6
2]
[2.3
5]
[1.4
7]
[7.0
4]
[5.9
8]
[4.9
5]
[3.4
0]
[6.7
1]
[5.6
8]
[4.8
3]
[3.9
8]
ES(2
)T
49.1
949
.73
49.9
950
.01
18.9
418.9
918.9
719.1
210.3
010.4
810.6
810.7
85.7
55.4
05.1
75.0
5[6
.36]
[3.7
7][1
.22]
[0.3
1][4
.58]
[3.9
8]
[3.5
4]
[3.0
0]
[3.6
7]
[3.2
5]
[2.9
7]
[2.7
0]
[3.8
2]
[2.9
3]
[1.9
8]
[1.2
0]
ES(3
)T
49.8
849
.92
49.9
449
.95
19.4
719.5
019.6
219.7
411.7
011.2
710.8
910.5
46.8
26.0
15.5
15.2
9[0
.98]
[0.6
9][0
.48]
[0.3
1][3
.94]
[3.0
9]
[2.2
4]
[1.3
8]
[4.5
4]
[3.5
3]
[2.6
2]
[1.6
6]
[4.7
6]
[3.2
4]
[1.8
8]
[0.9
8]
Tab
le1:
Sim
ula
tion
resu
lts
for
effec
tive
spre
ades
tim
ator
sunder
assu
mpti
ons
ofSec
tion
2.T
he
table
rep
orts
the
mea
neff
ecti
vesp
read
obta
ined
inth
esi
mula
tion
and
the
RM
SE
inbra
cket
s.T
he
effici
ent
pri
cein
nov
atio
ns
are
nor
mal
lydis
trib
ute
d,
and
the
dai
lyin
tegr
ated
vola
tility
ofth
eeffi
cien
tpri
ceis
set
to35
bps.
The
resu
lts
are
bas
edon
10,0
00M
onte
Car
lore
plica
tion
s.
44
2005–6 2007–8 2009–10 2011–12 2013–14
A. Number of transactionsMean 975 2098 1947 1697 1757Std. dev. 1023 2001 2635 2019 18365th percentile 174 409 283 228 26895th percentile 2740 5831 5867 5248 4874
B. Effective spread (bps)Mean 12.6 13.1 15.6 13.2 13.0Std. dev. 7.3 9.0 12.1 9.0 8.75th percentile 5.8 5.3 5.4 4.8 4.895th percentile 25.9 27.7 36.2 30.0 31.0
C. Realized volatility (bps)Mean 168.8 246.0 252.3 192.0 156.3Std. dev. 67.7 151.8 133.4 92.3 64.45th percentile 86.6 89.7 100.6 84.6 80.995th percentile 293.4 559.1 498.0 362.4 272.5
D. Signal-to-noise ratioMean 0.078 0.058 0.063 0.071 0.084Std. dev. 0.046 0.025 0.030 0.036 0.0475th percentile 0.039 0.030 0.031 0.033 0.04095th percentile 0.144 0.103 0.116 0.136 0.169
Table 2: The descriptive statistics are calculated over all stock days in a giventwo-year period. The effective spread and realized volatility were winsorized at the99.5% level, separately for each stock, before pooling and calculating the stock-daydescriptive statistics. The sample consists of 147 small-cap stocks over the periodfrom January 2005 to December 2014, spanning 2,517 business days.
45
Mon
thly
Qu
arte
rly
Sem
ian
nu
alA
nnual
Mea
nB
ias
RM
SE
Cor
rM
ean
Bia
sR
MS
EC
orr
Mea
nB
ias
RM
SE
Corr
Mea
nB
ias
RM
SE
Cor
r
A.
All
tran
sact
ion
s,ori
gin
alsp
read
sT
rue
13.5
2-
--
13.5
2-
--
13.5
2-
--
13.5
2-
--
ES(1)
T91.3
477
.82
95.0
10.
342
92.6
179
.09
92.3
90.
364
92.1
278
.60
90.8
70.
378
91.
51
77.
99
89.
29
0.410
ES(2)
T79.1
465
.62
85.5
90.
403
86.3
172
.79
97.0
80.
472
86.5
673
.04
95.8
80.
456
88.
57
75.
05
94.
96
0.434
ES(3)
T81.3
867
.86
86.6
10.
334
79.8
366
.30
85.6
10.
389
82.5
869
.06
85.6
90.
390
85.
44
71.
92
86.
09
0.395
Rol
l8.
731
-4.7
896.
007
0.95
58.
813
-4.7
105.
785
0.96
78.
874
-4.6
465.
599
0.9
69
8.9
88
-4.5
33
5.3
62
0.969
RVall
13.1
6-0
.357
2.3
45
0.9
6513
.25
-0.2
702.
154
0.96
913
.34
-0.1
812.
093
0.9
68
13.4
9-0
.027
2.0
46
0.967
RS(1)
T7.8
33
-5.6
87
10.7
30.4
557.
106
-6.4
1710
.07
0.50
16.
872
-6.6
489.
688
0.5
42
6.770
-6.7
529.1
65
0.621
RS(2)
T11.1
0-2
.426
6.4
44
0.7
6811
.00
-2.5
205.
317
0.84
110
.93
-2.5
914.
863
0.8
72
10.9
1-2
.617
4.5
07
0.893
RS(3)
T9.4
29
-4.0
92
5.0
95
0.9
749.
407
-4.1
165.
082
0.97
79.
352
-4.1
685.
156
0.9
75
9.318
-4.2
045.1
71
0.970
B.
All
tran
sact
ion
s,sp
read
sm
ult
ipli
edby
10T
rue
135.3
--
-13
5.3
--
-13
5.3
--
-13
5.3
--
-
ES(1)
T207.
472
.13
89.0
40.
937
209.
574
.24
87.3
50.
957
210.
875
.51
87.6
00.
964
212
.977.
63
89.
68
0.960
ES(2)
T210.
775
.44
101.2
0.73
221
3.2
77.8
710
0.2
0.75
521
4.9
79.6
310
0.6
0.763
218
.382.
99
102
.00.
769
ES(3)
T256.
412
1.1
136.6
0.77
524
3.6
108.
312
2.1
0.80
823
8.2
102.
911
4.5
0.837
232
.096.
69
106
.00.
866
Rol
l10
7.6
-27.
6539
.87
0.96
010
8.4
-26.
8837
.79
0.96
510
9.1
-26.
1536
.07
0.9
65
110
.4-2
4.88
33.
84
0.965
RVall
128.
9-6
.359
22.0
40.9
7112
9.8
-5.4
7020
.15
0.97
513
0.6
-4.6
1219
.63
0.9
74
132.
2-3
.065
19.
30
0.971
RS(1)
T128.
9-6
.375
15.0
70.9
8912
8.7
-6.6
2713
.41
0.99
412
8.0
-7.2
2015
.05
0.9
92
127.
5-7
.762
16.
45
0.988
RS(2)
T129.
5-5
.793
13.8
20.9
9312
9.3
-5.9
9513
.52
0.99
412
8.7
-6.5
7015
.25
0.9
92
128.
2-7
.026
16.
76
0.988
RS(3)
T119.
9-1
5.4
120.6
20.9
9511
9.5
-15.
7621
.13
0.99
511
8.8
-16.
4123
.14
0.9
92
118.
5-1
6.8
224.
48
0.986
Con
t.on
nex
tpa
ge.
46
Con
t.fr
om
pre
viou
spa
ge.
Mon
thly
Qu
arte
rly
Sem
ian
nu
alA
nnual
Mea
nB
ias
RM
SE
Cor
rM
ean
Bia
sR
MS
EC
orr
Mea
nB
ias
RM
SE
Corr
Mea
nB
ias
RM
SE
Cor
r
C.
Ever
y10
thtr
an
sact
ion
,ori
gin
al
spre
ads
Tru
e13
.52
--
-13
.52
--
-13
.52
--
-13
.52
--
-
ES(1)
T89.3
476
.44
93.5
00.
339
90.6
377
.72
90.9
50.
365
90.0
377
.13
89.3
90.
376
89.
46
76.
55
87.
85
0.406
ES(2)
T75.8
962
.98
81.9
60.
453
85.9
273
.01
97.0
50.
482
86.9
074
.00
97.0
60.
477
88.
96
76.
06
96.
90
0.446
ES(3)
T81.8
468
.94
85.5
90.
379
81.5
768
.66
85.2
10.
412
84.2
571
.35
85.4
60.
424
87.
23
74.
33
85.
91
0.444
Rol
l6.
901
-6.0
039.
607
0.62
16.
820
-6.0
879.
040
0.65
56.
835
-6.0
688.
702
0.6
75
6.9
11
-5.9
91
8.3
66
0.679
RVall
31.2
818
.37
22.5
90.
870
31.5
218
.61
22.6
30.
877
31.7
318
.82
22.7
20.
877
32.
08
19.
18
22.
87
0.874
RS(1)
T8.0
69
-5.2
95
11.1
80.4
227.
079
-6.2
8510
.27
0.45
06.
728
-6.6
339.
868
0.4
80
6.473
-6.8
929.4
77
0.539
RS(2)
T11.0
7-2
.292
7.1
87
0.7
1510
.93
-2.4
405.
619
0.80
710
.84
-2.5
184.
975
0.8
49
10.6
8-2
.681
4.6
72
0.873
RS(3)
T8.5
97
-4.7
67
6.0
67
0.9
258.
562
-4.8
025.
905
0.95
28.
505
-4.8
565.
922
0.9
60
8.470
-4.8
955.9
28
0.957
D.
Eve
ry10th
tran
sact
ion
,sp
read
sm
ult
ipli
edby
10T
rue
135.3
--
-13
5.3
--
-13
5.3
--
-13
5.3
--
-
ES(1)
T206.
372
.66
91.1
30.
933
208.
674
.88
88.9
70.
956
209.
876
.12
89.0
40.
964
211
.978.
21
90.
94
0.962
ES(2)
T183.
349
.64
76.6
90.
762
182.
949
.24
73.4
70.
782
183.
649
.97
71.8
70.
792
186
.753.
03
74.
19
0.777
ES(3)
T209.
776
.05
90.9
80.
855
202.
769
.01
82.6
60.
874
199.
766
.08
77.2
10.
899
197
.263.
53
71.
75
0.926
Rol
l16
1.5
27.8
441
.81
0.96
816
2.9
29.2
340
.13
0.97
816
4.0
30.3
341
.24
0.9
78
166
.032.
36
43.
53
0.975
RVall
170.
536
.81
48.7
50.
978
171.
838
.14
48.5
20.
984
172.
939
.24
50.0
00.
983
175
.041.
33
52.
45
0.979
RS(1)
T127.
7-6
.016
16.3
50.9
8512
7.5
-6.2
2513
.40
0.99
212
6.8
-6.8
1814
.73
0.9
91
124.
7-9
.049
18.
19
0.987
RS(2)
T128.
2-5
.454
13.5
40.9
9212
8.1
-5.5
9812
.59
0.99
412
7.5
-6.1
5514
.26
0.9
92
125.
5-8
.222
18.
21
0.986
RS(3)
T126.
4-7
.264
12.2
90.9
9512
6.2
-7.5
2311
.67
0.99
612
5.5
-8.1
7413
.96
0.9
94
125.
1-8
.558
16.
28
0.988
Tab
le3:
Em
pir
ical
resu
lts
for
the
vari
ous
effec
tive
spre
ades
tim
ator
sap
plied
to14
7sm
all-
cap
stock
sb
etw
een
Jan
uar
y20
05an
dD
ecem
ber
2014
.T
he
table
rep
orts
the
mea
neff
ecti
vesp
read
esti
mat
es(M
ean),
the
bia
s(B
ias)
wit
hre
spec
tto
the
actu
aleff
ecti
vesp
read
sob
serv
edin
the
TA
Qdat
a(T
rue)
,th
eR
MSE
,an
dth
eco
rrel
atio
nb
etw
een
esti
mat
edan
dac
tual
valu
esof
the
effec
tive
spre
ad(C
orr)
.T
he
resu
lts
are
obta
ined
by
pool
ing
acro
ssst
ock
s.
47