allen carrion university of memphis madhuparna kolay
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Bulk Volume Trade Classification and Informed Trading∗
Allen Carrion
University of Memphis
Madhuparna Kolay
University of Portland
This Draft: January 2020
Abstract
We document that the existing evidence that bulk volume trade classification (BVC) can
measure informed trading arises largely due to misspecified tests. In particular, simulations show
that these tests detect spurious relationships in data containing only uninformed liquidity trades.
We also assess the performance of BVC order imbalances in the NASDAQ HFT dataset,
showing that BVC order imbalances underperform conventional order imbalance measures that
are based on aggressor flags in detecting informed trading. When we isolate the component of
order flow that BVC designates as passive informed trading, we find that this component of
order flow fails to predict future returns with the correct sign. On balance, our evidence supports
the use of conventional order imbalance measures to identify informed trading.
∗ We thank NASDAQ and Frank Hatheway for supplying the HFT dataset.
1
1. Introduction
A standard approach to identifying informed trading in microstructure data is to use order
imbalances (or unexpected order imbalances) as a proxy. Order imbalances measure the
difference between aggressive buying volume and aggressive selling volume in an interval, and
sometimes normalize this value by total volume traded.1 This has been a long-standing practice
and has been validated empirically, but this approach has a well-known shortcoming. Order
imbalances measure the amount of aggressive trading in an interval, and their use as a proxy for
informed trading rests on the assumption that, on balance, informed traders tend to demand
liquidity. However, as pointed out by Harris (1998), Kaniel and Liu (2006), Baruch, Panayides,
and Venkataraman (2017), and many others, there are conditions that can motivate informed
traders to trade passively. This has been confirmed empirically (Kaniel and Liu (2006), others).
The validity of the use of order flow imbalances as a proxy for informed trading rests on the
assumption that, while some informed traders my trade passively, they generally demand more
liquidity than they supply. O’Hara (2015) and Easley, Lopez de Prado, and O’Hara (2016) (ELO
(2016) hereafter) suggest that this assumption has weakened in modern markets, as informed
traders presumably have increased their use of algorithms that trade passively. ELO (2016) argue
that, as a result of increases in passive trading by informed traders, “the notion of the active side
of the trade signaling underlying information is undermined.”
Regardless of whether aggressive trading remains correlated with informed trading in
modern markets, more accurate methods of identifying informed trading that outperform order
imbalances by detecting both aggressive and passive informed trading would be desirable. ELO
1 Some studies calculate imbalances using number of trades or dollar volumes instead of share volumes.
Examples include Chordia, Roll and Subrahmanyam (2002), Chordia and Subrahmanyam (2004), and Kim and Stoll
(2014).
2
(2016) introduce the bulk volume classification technique (or BVC), which potentially addresses
this issue. As describe in ELO (2016), BVC “aggregates trades over short time or volume
intervals and then uses a standardized price change between the beginning and end of the interval
to approximate the percentage of buy and sell order flow.” The authors suggest that this
technique identifies imbalances from informed traders whether they trade aggressively or
passively. They provide the following interpretation of their main empirical result: “What
matters for our purposes is that order imbalance created from bulk volume works, in the sense
that it is positively related to the high-low spread [signifying a response to informed trading], but
that order imbalance created from the tick rule (or even derived directly from the aggressor flag)
does not work.”
If BVC indeed has these properties, this technique offers an invaluable tool for
researchers. However, the empirical evidence supporting these claims is limited. ELO (2016)
provide strong arguments motivating BVC, but their empirical validation of BVC is limited to a
single test asset and single BVC parameterization. We are only aware of two other studies that
conduct similar analyses (Panayides, Shofi, and Smith (2019) and Chakrabarty, Pascual, and
Shkilko (2015)), and they primarily rely on the methodology employed in ELO (2016) and a
similar test specification that was proposed in an earlier working paper version of ELO (2016)
and later abandoned.2,3 Both studies qualitatively reproduce the main ELO (2016) result in
samples of equity data using many stocks, but Chakrabarty, Pascual, and Shkilko (2015) express
2 The working paper version of ELO (2016) used a test with a high-low spread (HL) as a liquidity proxy.
The final version of ELO (2016) states that the HL spread is contaminated by “fundamental variance” and replaces it
with the Corwin-Shultz in this test. The final version of their test is discussed in detail in Section 3. 3 Panayides, Shofi, and Smith (2019) also conduct an alternate test that finds BVC predict returns around
corporate events. Andersen and Bondarenko (2014a, 2014b, and 2015) also test or discuss BVC. However, they
focus on its accuracy in identifying the aggressor side of trades or its use in VPIN calculations.
3
qualifications.4 Chakrabarty, Pascual, and Shkilko (2015) also use an older version of BVC
based on the ELO working paper, which is similar but not identical.
In this paper, we further investigate the ability of BVC to detect informed trading. First,
we revisit the initial empirical evidence provided by ELO (2016). We first discuss some
conceptual concerns with their tests. We then repeat their tests in simulated data with no
informed trading, and find that their tests are severely misspecified. The ELO (2016) regressions
never fail to reject the null hypothesis of no informed trading at the 1% level in the simulated
data. It appears that these tests are measuring something other than the relationship between
BVC and informed trading. While we believe that the original evidence in support of BVC’s
ability to detect informed trading is unreliable, this does not rule out the possibility that BVC
may still have the properties claimed by ELO (2016). Therefore, we propose two alternative tests
and conduct them on a sample of equity trading data. Both of these tests are motivated by the
idea that informed trading should positively predict future returns, and are related to tests
previously conducted in the literature (cites). In the first set of tests, we compete order
imbalances constructed with BVC with conventional order imbalances using the known trade
signs (true order imbalances). In multiple specifications with several return horizons and
parameterizations of BVC, true order imbalances outperform BVC order imbalances in every
case, and the coefficients on BVC order imbalances are often insignificant or predict future
returns with the wrong sign. In a second set of tests, we derive a measure from BVC and the true
order imbalance that should measure passive informed trading if BVC performs as claimed, and
test the ability of this measure to predict future returns. This measure fails to do so in all
4 Chakrabarty, Pascual, and Shkilko (2015) note that the relationship between BVC and the HL spread may
be driven by correlations with volatility. They also replace HL with returns and alternate liquidity proxies and find
mixed results; BVC is generally found to be positively related to returns and contemporaneous liquidity proxies but
does not uniformly outperform conventional order imbalance measures.
4
specifications examined, and often predicts returns with the wrong sign. We repeat these tests
using order imbalances constructed using trades signed with the Lee and Ready (1991) method
instead of true trade signs, and find similar results (available next version).
Our results suggest that researchers should be wary of employing BVC based on
expectations of its superior ability to measure informed trading. We find no evidence that BVC
outperforms conventional order imbalances constructed from true trade signs or Lee and Ready
(1991) trade signs in this regard; it actually underperforms significantly in our tests.
The rest of this paper is organized as follows. Section 2 describes and discusses the BVC
algorithm. Section 3 reviews the existing evidence used to support the claim that BVC identifies
informed trading. Section 4 presents results from alternate tests of the ability of BVC to identify
informed trading. Section 5 concludes.
2. The Bulk Volume Classification Algorithm
The bulk volume classification (or BVC) method does not classify individual trades as do
other conventional trade classification algorithms such as the Tick rule or the Lee and Ready
method. Instead, BVC aggregates trades into bars, by either volume or time.5 ELO (2012) and
ELO (2016) argue for the importance of aggregation by volume in modern markets, so we focus
on volume bars in this paper. The CDF of the price change between the close of a bar and the
close of the prior is then used to calculate the percentage of buys versus sells per bar. Since
trade-by-trade classification is replaced by a bar-by-bar probability of buys versus sells, the
aggregation may lead to greater efficiency with respect to data usage.
Specifically, the buyer-initiated volume is calculated as follows (Eq. (1), ELO (2016)):
5 Trade bars have also been used in Chakrabarty, Pascual and Shkilko (2015) and a working paper version
of ELO (2016).
5
���� = �� × � �∆��∆ , ��� (1)
where ���� is the estimated buyer initiated volume during bar �, �� is the aggregated volume, ∆��is
the price change between consecutive bars, σ∆p is the standard deviation of price changes, and t
is the cumulative distribution function of student’s t-distribution, with �� degrees of freedom.
The buyer initiated volume is then converted into an order imbalance with the following formula
(adapted from Eq. (5), ELO (2016)):
���_��� = 2 ������ − 1 (2)
The usefulness of BVC can be viewed from two perspectives. First, BVC is a potential
competitor to conventional trade signing algorithms. In this context, its performance can be
measured by its accuracy in reproducing the true trade signs (or “aggressor flags”) in the
underlying data. The empirical evidence on BVC’s accuracy as a trade signing algorithm is
mixed. In various tests in the literature, it does not generally outperform other popular trade
signing algorithms, but there is some evidence that BVC may have accuracy advantages over the
competing algorithms in certain settings, and it does correlate with true trade signs.6 However,
BVC has other advantages such as computational efficiency and minimal data requirements that
may make it attractive for specific applications. Second, and more importantly for this study,
BVC may outperform conventional order imbalances in measuring the information in order flow
regardless of its performance from a trade signing perspective. Stated alternately, BVC may
disagree with the true trade signs but still more accurately reflect the information in order flow
than the true trade signs. Conventional order imbalances are a proxy for informed trading under
the assumption that informed trading tends to be aggressive, while it is possible that BVC
6 See Panayides, Shofi, and Smith (2019) and Chakrabarty, Pascual, and Shkilko (2015), ELO (2016), and
Andersen and Bondarenko (2015).
6
actually measures informed trading whether it is aggressive or passive. We discuss the relevant
evidence for this potential property of BVC in Section 3 below.
BVC may be particularly advantageous in the context of markets where high frequency
trading is common. ELO (2016) assert that high speeds of trading, increased fragmentation, and
higher rates of order submissions and cancellations have changed the mechanics of trading in
ways that reduce the classification accuracy of conventional trade signing algorithms.7 ELO
(2016) also suggest that in the modern market environment the aggressor side of a trade is now
less likely to correlate with the informed side of the trade.
One potential drawback of BVC is that its output is not entirely deterministic. The
grouping of trades into bars can differ if the starting point changes. For example, consider two
researchers working on partially overlapping trade samples with different starting dates. If they
chose to fill bars continuously over the full sample (as in ELO (2016)), they would potentially
arrive at different bars for the overlapping trades. The situation is similar if two researchers use
the same sample but on chooses to fill bars continuously while the other restarts fresh with an
empty bar periodically (possibly every stock-month as in Panayides, Shofi, and Smith (2019) or
every trading day). Volatility estimation can also cause uncertainty in the output. σ∆p in Eq. (1) is
estimated in-sample using the full sample period in ELO (2016). Therefore, the volatility and
BVC classifications for a specific group of trades could differ depending on the starting or
ending dates of the samples they are included in. Further, if the researcher is concerned with
time-varying volatility, it would clearly be reasonable to re-estimate volatility over shorter
intervals. This would also potentially alter the output. Another related issue is that there is little
7 There is disagreement in the literature regarding this assertion. Chakrabarty, Pascual, and Shkilko (2015)
and Carrion and Kolay (2019) both report evidence that conventional trade signing algorithms perform well in data
from fast and fragmented markets, and Panayides, Shofi, and Smith (2019) report mixed evidence.
7
guidance in the literature on how to select the bar size for BVC. These problems could be
mitigated if researchers coordinated on a set of implementation details, but this has not happened
as of this writing. With the exception of some sensitivity analysis around bar size, these effects
have not been quantified in the literature so it is not clear if they are significant in practice.8
3. Existing Evidence on BVC and Informed Trading
First, we describe and discuss the evidence presented in ELO (2016). Next, we report the
results of a simulation exercise where we assess the statistical properties of the test specification
ELO (2016) by repeating their test in simulated data with no informed trading.
3.1 The ELO (2016) Test
The main evidence provided by ELO (2016) in support of the ability of BVC to identify
informed trading is provided by a regression designed to associate time variation in liquidity with
order flow imbalance measures.9 This is based on the idea that market makers will withdraw
liquidity when informed traders are active. The regression specification is:
��_� !"# = % + '(|���_���#| + '*|���_+��,#| + '-��_� !"#.( (3)
where CS_SPRD is the Corwin and Schultz (2012) high-low spread (Corwin-Schultz spread
hereafter), OIB_BVC is the order flow imbalance constructed using BVC, OIB_TICK is the
order flow imbalance constructed using the tick rule, and t indexes volume bars. The lagged
value of CS_SPRD is included as a control for autocorrelation, and t-statistics are Newey-West
adjusted.
8 See Chakrabarty, Pascual, and Shkilko (2015), Masot, Nawn, and Pascual (2018), and Panayides, Shofi,
and Smith (2019). 9 ELO (2016) also present related evidence graphically in their Figure 5, but without an accompanying
statistical test.
8
The results of this test are presented in Table 6 of ELO (2016). Three versions of the
regression model are estimated: one with OIB_BVC, one with OIB_TICK and a third with both.
All three versions include the lagged CS_SPRD. The coefficient on |OIB_BVC| is positive and
significant and the coefficient on |OIB_TICK| is negative and significant. These results are
consistent across models. ELO (2016) interpret these results as evidence that BVC works to
measure informed trading, and that the tick rule order imbalances identify aggressive traders that
tend to be uninformed. We discuss several observations and concerns with this test below.
First, we note that the main conclusion is only supported by the result of a single time
series regression, using a single test asset and a single BVC parameterization. The rationale for
this choice is not explained in the paper, and a separate analysis in ELO (2016) uses three test
assets and ten BVC parameterizations for each. Regardless of the rationale, it seems difficult to
draw strong conclusions from these results without evidence of robustness.
Second, this test relies on reductions in liquidity provision by market makers to detect
informed trading. In contrast to other tests in the literature that use permanent price impacts,
future returns, or other price-discovery based measures, this test relies on a near-immediate
market maker response to detect informed trading. As Kacperczyk and Pagnotta (2016) state,
“most theory-motivated information measures, such as the bid–ask spread and the price impact
of trades (Glosten and Milgrom, 1985; Kyle, 1985), rely on the notion that the presence of
informed traders is common knowledge to other market participants.” This is essentially a joint
hypothesis problem, where the test relies not only on the presence of informed trading but also
on the nature of the response by market makers. This immediate response is inconsistent with
some theories where market makers learn over a series of trades or informed traders attempt to
9
hide their informed status.10 If market makers respond slowly to informed imbalances, or
withdraw liquidity while patient liquidity demanders continue to post quotes at the old prices,
these tests will misclassify the imbalances as uninformed. We do not suggest that this issue
invalidates this test, as it is based on a theoretically sound response to informed trading. It seems
reasonable, however, to take the nature of this test into account when determining how much
weight to place on these results and as motivation to seek confirmation from other tests.
Third, it is not clear that the Corwin-Schultz spread is well-suited to this test. This test is
designed to associate time variation in liquidity provision with the time variation in informed
trading captured by the explanatory variables. The Corwin-Schultz spread relies only on trade
data, and ELO (2016) motivate their choice of this liquidity measure with concerns about the
reliability of quote data from fast markets. The Corwin-Schultz spread as originally introduced,
however, estimates an average spread over a series of observations and treats time variation
between adjacent observations as noise to be averaged out rather than as meaningful time
variation. In their monthly analysis, Corwin and Schultz (2012) discard all stock-months where
they are not able to average over at least 12 trading days. Corwin and Schultz (2012) also find
time variation in individual observations when using simulated data with a constant bid-ask
spread, including negative spread observations.11 Similarly, in our analysis below we find
significant time variation in the Corwin-Schultz spread, including negative observations, in
simulated data with a constant trading cost function.12 ELO (2016) use the individual spread
observations directly without smoothing or aggregation.
10 For example, in Kyle (1985) the informed trader trades slowly to blend in with liquidity traders, and its
private information is only revealed gradually in prices. 11 Corwin and Shultz (2012) investigate alternate adjustments for negative spreads and find that setting
negative spreads to 0 before averaging results in the best performance. 12 Not tabulated, available upon request.
10
Fourth, the BVC order imbalance is only competed against the tick rule order imbalance
and not the true order imbalance. One of the most important hypothesized properties of BVC is
its potential advantage over order imbalances constructed from true trade signs in identifying
informed trading.
Fifth, as others have pointed out in similar tests, there is a possibility that there is a
mechanical bias in regressions of this form.13 The Corwin-Schultz spread, used as the dependent
variable, is essentially a non-linear transformation of the high-low range in a bar. The BVC order
imbalance is essentially a non-linear transformation of the return in the bar, and the regressions
use the absolute value of this variable as an explanatory variable. The high-low range in a bar is
clearly mechanically related to the absolute value of the return in the bar, regardless of the
presence of informed trading.14 Therefore, one might expect regressions of the high-low range on
the contemporaneous absolute value of the return to be misspecified. It is not clear if the
transformations of the high-low range into the Corwin-Shultz spread and the absolute value of
the return into BVC break this mechanical relationship. While many of the other concerns we
raise in this section are qualitative and somewhat subjective, this issue is testable. We investigate
the properties of this regression specification in the next section.
3.2 Simulations
In this section, we conduct a simulation exercise to investigate the properties of the ELO
(2016) regression tests. We simply reproduce these regressions repeatedly in simulated data with
no informed trading. If these tests are well specified, the coefficients of interest will be zero on
13 See Chakrabarty, Pascual, and Shkilko (2015) and Andersen and Bondarenko (2015). 14 We are not aware of an analytical proof of this relationship, but it is easily demonstrated with
simulations.
11
average and their test statistics will reject the null hypothesis at an appropriate rate for the
specified significance level.
ELO (2016) only conduct their tests on BVC order imbalances constructed with 10,000
contract volume bars. However, they advance no arguments that this bar size is optimal and
provide no cautions that their results should be sensitive to bar size. They use bar sizes from
1,000 – 25,000 contracts in other sections of their paper. Therefore, we reproduce tests on order
imbalances constructed with 1,000, 10,000, and 25,000 contract bars. In each test, we follow
ELO (2016) in using the same bar size to calculate the Corwin-Shultz spread as that used to
calculate the BVC order imbalances.
To generate data with no informed trading, we use a version of the Glosten and Harris
(1988) model with the adverse selection component of the spread set to zero. Our
implementation of the model can be summarized as follows:
/# = /#.( +0# (4)
0#~2(0, 56) (5)
# = /# + 8#(9: + 9((�# − 1)) (6)
Where mt is the pre-trade midpoint (or, equivalently in this setting, the fair value of the security)
for trade t, Pt is the trade price for trade t, Qt is a trade sign indicator variable for trade t taking
the value of 1 for a buy trade and -1 for a sell trade, Vt is the unsigned volume of trade t, c0 and
c1 are trading cost parameters, and εt is the change in pre-trade midpoint due to public
information. We subtract one contract from Vt so c0 will match the half bid-ask spread exactly
for a one contract trade. Note that we simulate the model in event time rather than calendar time.
In this data generated from this model, trades have no permanent price impact at all.
12
For our main simulations, we generate signed trade volumes from a normal distribution,
round to the nearest contract, and use these signed volumes to calculate Qt and Vt. We also adjust
small trade sizes up to one contract before rounding to eliminate zero contract trades. This
procedure generates less skewness in trade size than that reported by ELO (2016), which we
believe is conservative in that it will likely lead to data with better-behaved statistical properties.
The price dynamics in these simulations contain no features resembling informed trading
and can be described simply as follows. Future midpoint price changes are uncorrelated with
trades. Trade prices only deviate from the pre-trade midpoints by temporary price impacts which
disappear by the next trade. These temporary price impacts include half the bid-ask spread plus a
linear function of the volume traded, which can be viewed as the effect of larger trades “walking
the book.” One could employ richer models that simulate other microstructure effects such as
autocorrelated order flow induced by order-splitting, inventory effects, time-varying volatility, or
imperfect resiliency where temporary price impacts decay slowly instead of disappearing by the
next trade. A well-specified test for informed trading should be robust to these effects, which
could generate patterns that share some features of informed trading and can be present in a
market with no informed trading. However, given the results for the simple simulation tests we
report below, we consider richer simulations unnecessary in this case.
For each bar size, we simulate 1,000 trials. For each trial we generate a dataset of
128,579,415 trades, matching the ELO (2016) ES sample. We choose our other simulation
parameters to approximately match the summary statistics reported by ELO (2016) for their
sample and other known characteristics of the ES market. Table 1 reports the simulation
parameters. Figure 1 shows the price path for a randomly selected trial. Next, in each trial we
form volume bars and calculate the BVC order imbalances and Corwin-Schultz spreads as
13
described above. In place of tick rule order imbalances, we calculate true order imbalances for
each bar using our simulation-generated trade signs. The true order imbalances are defined as:
���_+!;<# = �=>_�?@=ABC–EB@@_�?@=ABC�?@=ABC (7)
where BUY_VOLUMEt and SELL_VOLUMEt are the share volumes designated as buys and
sells in bar t and VOLUMEt is the total share volume in bar t.
Table 2 reports selected summary statistics from the simulated trades. In general, these
are similar to the ELO (2016) sample in the main dimensions. Of particular interest, the mean
Corwin-Schultz spread for the 10,000 contract bar size simulations is 22 bps, compared to 23 bps
in ELO (2016). The one notable difference is that the trade size distribution is less skewed than
the ELO (2016) sample as mentioned above; while the means are similar, the median in our
sample is 4 contracts compared to 1 contract in the ELO (2016) sample. While we attempt to
simulate data similar to that used in ELO (2016), we emphasize that this is for comparability
only; the validity of their regression model is not purported to be contingent on specific features
of the test asset or market, so we should not require a close match to their data in order to
investigate the regression specification properties of interest.
In each trial, we run variations of the following regression:
��_� !"# = % + '(|���_���#| + '*|���_+!;<#| + '-��_� !"#.( (8)
where all variables are as defined above. This regression is similar to Eq. (2), which corresponds
to Eq. (6) in ELO (2016). Aside from the inclusion of an intercept, our only deviation from Eq.
(2) is the replacement of the tick rule order imbalance with the true order imbalance. This choice
is motivated by the discussion in Section 3.1 above. We estimate three versions of the model that
vary only in the included imbalance variables: one version with only BVC order imbalance, one
14
with only the true order imbalance, and another with both. All versions include the lagged
Corwin-Shultz spread. We calculate Newey-West t-statistics.
Table 3 reports the average coefficients from Eq. (7) across all 1,000 trials for each bar
size/regression model combination, and t-statistics that test whether the average coefficient is
different from 0. Panel A reports results for 1,000 contract bars. Models 1 and 3 show that the
coefficient on |���_���#| is significantly positive, and Models 2 and 3 show that the average
coefficient on |���_+!;<#|is significantly negative. These results are qualitatively consistent
with the results reported in ELO (2016) for the 10,000 contract bar size, although the estimated
coefficients are smaller in magnitude. Panel B reports results for 10,000 contract bars. Models 1
and 3 show that the coefficient on |���_���#| is significantly negative, and Models 2 and 3
show that the average coefficient on |���_+!;<#|is insignificant. Panel C reports results for
25,000 contract bars. The results are similar to those for the 10,000 contract bars. Again Models
1 and 3 show that the coefficient on |���_���#| is significantly negative, and Models 2 and 3
show that the average coefficient on |���_+!;<#|is insignificant.
For all three parameterizations of BVC, these tests indicate a statistically significant
relationship between BVC and informed trading, despite being run on data with no informed
trading. Further, the relationship is not stable across BVC parameterizations. Taken literally,
when we divide the data into 1,000 contract bars, the tests indicate that BVC measures the
informed side of the order flow. However, for 10,000 and 25,000 contract bars, the tests indicate
that BVC measures the uninformed side. While these are clearly incorrect interpretations from
spurious results, it is noteworthy that the conclusions a researcher employing this test might form
are so sensitive to the BVC parameterization choices.
15
Table 4 provides an alternate perspective on these regressions. While Table 3 considers
results aggregated across all trials, Table 4 considers the distribution of Newey-West t-statistics
on the individual trials and reports associated rejection rates for the null hypotheses of no
relationship between the imbalance variables and informed trading. The data presented in Table
4 allows us to evaluate the size of the tests; given the lack of informed trading in the data, well-
specified tests should reject the null at approximately the specified significance level. Panels A
and B report rejection rates for hypothesis tests on the coefficient on |���_���#|. Panel A uses
tests from the regression model that omits |���_+!;<#| (corresponding to Model 1 in Table 3),
and Panel B uses tests from the regression model that controls for |���_+!;<#| (corresponding
to Model 3 in Table 3). Mean t-statistics are similar for both specifications and rejection rates are
identical. We use a 1% significance threshold.
The results in Table 4 Panels A and B are extreme. The two-sided test of the hypothesis
that the coefficient on |���_���#| is equal to zero is rejected for 100% of trials for all
parameterizations, with or without the |���_+!;<#| control. Every one of the 1,000 trials
incorrectly rejects the null in every specification tested. Similarly, the one-sided tests all reject
with at 100% or 0% rates. Recall that we are using a 1% significance level, so a well-specified
test would reject in approximately 1% of trials.15 In summary, in both models and for all three
bar sizes, we observe a Type 1 error in every single trial for the two-sided test and for one of the
one-sided tests. The asymmetry in the one-sided tests points to a bias, but the direction of the
bias changes with the parameterization of BVC and matches the direction suggested by the
aggregated results in Table 3.
15 Brown and Warner (1980) point out that, when the null is true and the test statistic is well-specified, the
rejection rate will not be exactly equal to the significance level; the rejection rate is a random variable with a
Bernoulli distribution. In our setting, this corresponds to a 95% confidence interval for rejection rates of (0.47% and
1.53%).
16
Table 4 Panels C and D report rejection rates for hypothesis tests on the coefficient on
|���_+!;<#|. Panel C uses tests from the regression model that omits |���_���#| (corresponding to Model 2 in Table 3), and Panel D uses tests from the regression model that
controls for |���_���#| (corresponding to Model 3 in Table 3). For 1,000 contract bars, the two-
sided tests reject 100% of the time in both models. Again, the one-sided tests reject the at 100%
or 0% rates in the directions suggested by the results in Table 3. For 10,000 and 25,000
contracts, however, the rejection rates in both models are much closer to the specified
significance level of 1%. For 10,000 contract bars, the rejection rates are all greater than 1% but
none are significantly different from 1% at the 5% level (i.e. all fall within the 95% confidence
interval around 1% given in Footnote 11). For 25,000 contract bars, the rejection rates are all
lower than 1%, but only significantly lower in one case (the 1-sided test of the null hypothesis of
|���_+!;<#| >= 0 without controlling for |���_���#|). The result that the rejection rates for
|���_+!;<#| are much more reasonable than those for |���_���#| for all but the small bar size
suggests that concerns of a mechanical relationship between |���_���#|and the Corwin-Shultz
spread have merit. However, the extreme misspecification observed for the 1,000 contract bars
indicates that this explanation is incomplete.
Note that for the 10,000 contract bar size the coefficient on |���_���#| is uniformly
significantly negative, while ELO (2016) uses this bar size and reports a significantly positive
coefficient. One might wonder whether this means the test is biased against the ELO (2016)
result and strengthens their conclusions. We do not believe that this is the correct interpretation.
First, given that the bias in this test is not stable across parameterizations, it is not likely to be
stable across characteristics of the data. As mentioned above, our simulated data omits many
features that may be present in real data such as order splitting, inventory effects, and imperfect
17
resiliency. Therefore, it would be an aggressive interpretation to claim that this negative result
holds uniformly in the absence of informed trading when bar sizes of 10,000 contracts or shares
are used to estimate BVC in actual market data. Second, it is important that the coefficients are
unbiased in this type of regression. In this specification, a negative coefficient has the specific
economic interpretation that BVC is measuring the trading of the uninformed counterparties of
informed order flow. It is therefore misleading to observe these negative coefficients when there
is no informed trading. Third, and perhaps most importantly, our results are so extreme that it is
difficult to rationalize using regressions of this form to measure the relationship between order
imbalances and informed trading. In our view, 100% rejection rates in these simulations indicate
a completely spurious regression rather than one containing a statistical issue that can be
overcome with an adjustment for a predictable bias or through inflation of the standard errors.
A reasonable question is whether the temporary price impacts in our simulations drive
our results by mimicking some dimension of informed trading. To investigate this possibility we
modify the simulations to omit the temporary price impacts and repeat the experiment.
Mechanically speaking, we set the c0 and c1 trading cost parameters in Eq. (6) to 0 and leave the
rest of the procedure unchanged. The midpoint prices then follow a random walk and all trades
take place at the midpoint, regardless of trade direction and size. The results are presented in
Tables A1 and A2 of Appendix A. Table A1 shows that the mean coefficients on |���_���#| are lower in magnitude than those in Table 3, but are all significantly negative. For the 1,000
contract bars this coefficient switches from the positive value shown in Table 3, and the signs
match for the larger bar sizes. The mean coefficients on |���_+!;<#| are close to zero and are
only significant for 1,000 contract bars. Table A2 reports extreme rejection rates for all
hypothesis tests involving |���_���#|. The lowest two-sided rejection rate is 38.7% for 25,000
18
contract bars. These rejection rates are generally lower than 100% rates observed in the
simulations that included temporary price impacts, but still indicate extreme misspecification.
Similarly, the one-sided tests either grossly over-reject or never reject. None of the rejection rates
for |���_+!;<#|are significantly different from 1%. This analysis shows that, while temporary
price impacts exacerbate the misspecification in the ELO (2016) test, the test is still grossly
misspecified when prices follow a random walk and are completely unrelated to trades.
Overall, we conclude that evidence in ELO (2016) in support of a relationship between
BVC and informed trading is unreliable. Our simulations show that the tests employed grossly
over-reject the null in the hypothesis test of central importance and are severely biased. Further,
the bias is unstable across BVC parameterizations and is not likely to be corrected with a simple
statistical adjustment.
4. Alternate Tests of the Relationship between BVC and Informed Trading
The evidence presented in Section 3 calls into question the previous evidence in the
literature supporting the relationship between BVC and informed trading. However, even if the
tests used in prior studies were flawed, BVC may still be a useful technique to measure informed
trading. In this section, we propose and implement what we believe to be improved tests to
assess this issue. We use the NASDAQ HFT dataset to conduct our tests, which we describe
below.
4.1 Motivation and Regression Specifications
We believe that the problems identified above with the tests conducted in ELO (2016)
primarily stem from timing issues. There is potentially a mechanical relationship between the
contemporaneous Corwin-Shultz spread and BVC order imbalance that mimics the hypothesized
19
informed trading effects. There are also theoretical arguments that this type of specification
could either miss information effects that occur with a delay if market makers recognize
informed order flow gradually or could confuse temporary price impacts with information
effects.
These issues can be avoided by employing tests that exploit the simple property that
informed trading will be in the direction of future returns. Informed buying (selling) will tend to
be followed by positive (negative) returns. We use tests that regress returns in intervals of
various lengths on signed order imbalances variables from the prior volume bar. We use
midpoint returns to mitigate the effects of bid-ask bounce. We interpret positive coefficients on
order imbalance variables as evidence that those variables are positively correlated with
informed trading. Related specifications have been used often in prior literature, see for example
Chordia and Subrahmanyam (2004), Kaniel and Liu (2006), and Kim and Stoll (2014).
We note a limitation to our tests. Information could be incorporated into prices too
rapidly after informed trades to be captured with this design. If prices do adjust almost
immediately after informed trades, then the information effects will be largely realized before the
end of the bar in which the trades occur. Other studies using similar designs address this issue by
combining contemporaneous and future price changes or returns into a single dependent variable
(Huang and Stoll (1996), Kaniel and Liu (2006), and others). Given that BVC uses
contemporaneous returns in its calculation, it is not obvious how a valid test can be designed that
include contemporaneous returns in the dependent variable. However, BVC cannot be calculated
until the end of a bar, so its relationship to information revealed after the bar is complete (which
our tests are designed to capture) is probably of greater interest. Regardless, we believe that
positive results in these tests could potentially provide reliable evidence of a relationship
20
between BVC and informed trading in the literature, while negative results would not rule out a
relationship between BVC and very short-lived information in our data.
We design our tests to address two questions. First, do BVC order imbalances improve on
the ability of true order imbalances to measure informed trading? And second, do BVC order
imbalances capture passive informed trading?
In our first set of tests, we simply compete the ability of BVC order imbalances against
true order imbalances to predict returns in post-bar intervals ranging from 15 seconds to 5
minutes.
!<+# = % + '*���_���#.( + '(���_+!;<#.( (9)
where RETt is the midpoint return computed in the interval from the first midpoint after the end
of the bar in which the order imbalance is measured until the first midpoint after the specified
time has elapsed, and other variables are as previously defined.
For our second set of tests, we construct a variable that is designed to isolate the
component of order flow attributable to passive informed trading if the claims ELO (2016) are
correct. We describe the construction of this variable as follows.
Trades in an interval (whether a volume, trade, or time bar) can be classified into the
categories VBB, VBS, VBN, VSS, VSB, and VSN, where V indicates the proportion of the interval’s
volume traded in that category, the first subscript denotes whether the aggressive side of the
trade bought or sold, and the second subscript denotes whether the more informed side of the
trade bought or sold. The second subscript is set to N when two uninformed traders trade with
each other. For example, VBS is the volume in the interval where aggressive uninformed buyers
trade with passive informed sellers.
21
A conventional order imbalance measure, using either true trade signs or a trade
classification algorithm, measures the buyer-initiated volume less the seller-initiated volume in a
block of trades. For a given block of trades, this measure can be represented as:
���_+!;< = (��� + ��E + ��G) − (�EE + �E� + �EG) (10)
Note that this measure does not require the researcher to identify the informed side of the trade,
only the aggressive side. We present this formula assuming true trade classifications are
observable, as when “aggressor flags” are provided. If trades are classified with error, this
relationship can be re-written with estimated variables and classification errors.
While OIB_TRUE mechanically measures the aggressive trading imbalance, it is often
used as a proxy for the informed trading imbalance. This is based on the assumption that
informed traders tend to trade more aggressively than uninformed traders. This assumption can
be stated as:
��� > �E� (11)
�EE > ��E (12)
BVC is used to construct a similar order imbalance variable. ELO (2016) argue that order
imbalances calculated using BVC capture information, and that when these order imbalances
differ from conventional order imbalances, the differences are due to information rather than
error. Therefore, if these claims are correct, we can represent a BVC order imbalance as:
�����I = (��� + �E�) − (�EE + ��E) (13)
Note that this representation ignores classification errors; we do not mean to imply that ELO
(2016) claim that this classification is perfect.
22
Using this representation, we can see that the difference between a BVC order imbalance
and a conventional order imbalance should contain information. Taking this difference and
simplifying, we find:
�����I − ���JK=B= L(��� + �E�) − (�EE + ��E)M − L(��� + ��E + ��G) − (�EE + �E� + �EG)M
= 2(�E� −��E) + (�EG − ��G) (14)
For brevity we refer to this difference variable as OIB_DIFF hereafter. The first term is twice the
imbalance between informed traders buying passively and those selling passively. The second
term is an imbalance between uninformed traders selling aggressively and those buying
aggressively. Therefore, the first term should positively predict future returns while the second
should have no effect, so the total difference term should positively predict future returns.16 With
perfect BVC classification, all passive informed trading in the bar should be isolated in this
difference variable and no aggressive informed trading should be included. If BVC classification
is noisy but on balance adequate, the sign of this variable should still tend to indicate the
direction of passive informed trading. This motivates the following regression specification for
our second set of tests:
!<+# = % + '(���_"�NN#.( (15)
For both tests, we estimate the regressions one stock at a time, and average the
coefficients from these time series regressions across stocks in a reverse Fama and MacBeth
(1973) procedure. For each coefficient, we calculate the t-statistic on the cross-sectional mean
and also calculate Newey-West t-statistics for each stock.
16 For now we are ignoring the reversals of temporary price impacts for simplicity. If material, this effect
should bias the test in favor of associating OIB_DIFF with informed trading.
23
4.2 Data
We conduct our tests in the NASDAQ HFT dataset, which contains trade and quote data
for a sample of 120 stocks over a subset of dates in 2008-2010. This dataset is also used in
Brogaard (2012), Carrion (2013), and Brogaard, Hendershott, and Riordan (2014), O’Hara, Yao,
and Ye (2014), Carrion and Kolay (2019), and others. The 120 stocks in this sample were chosen
with a stratified random sampling approach along the dimensions of market capitalization and
listing venue.17 Market capitalization is evenly split between small, medium and large firms.
Listing venues are split equally between NASDAQ and NYSE. We utilize two subsets of data
from this dataset: trade reports and the NASDAQ Inside Quotes (BBO). The trade sample covers
each trading date during 2008 and 2009 and the week of Feb 22 – 26, 2010. For each sample
stock and date, each trade executed on the NASDAQ exchange is shown, excluding trades done
in the opening, closing, and intraday crosses. Trades include a millisecond timestamp and are
signed to indicate whether they were initiated by a buyer or seller. The trade signs are based on
records of fee and rebate payments used by the exchange. The NASDAQ BBO data is available
only for the following subset of dates: 1) the first full week of the first month of each quarter
during 2008 and 2009; 2) the crisis week of Sept 15 – 19, 2008; and 3) the week of Feb 22 – 26,
2010. We apply a filter to the sample to remove trades and quotes occurring before 9:30 am and
after 4:00 pm.
There are several benefits of using this dataset for our analysis. First, it contains true
trade signs, which we use to create true order imbalances. Second, NASDAQ notes that trade and
quote sequencing is of high quality in this dataset, which is particularly relevant for our analysis.
17 Sample was chosen by Ryan Riordan and Terrence Hendershott. See Brogaard (2013) and Carrion (2013)
for further details.
24
Third, this data is drawn from a market characterized by pervasive high-frequency trader (HFT)
participation and short durations between trades and quotes, which is the type of environment
where ELO (2016) argue that BVC should have advantages over conventional techniques.
Carrion and Kolay (2019) study the same data and report that HFTs participate in 75.16% of the
sample trades, the median time elapsed between quotes is 0.024 seconds, and the median time
elapsed between trades is 0.001 seconds.
This dataset is described in more detail in Carrion (2013), Brogaard, Hendershott, and
Riordan (2014), and Carrion and Kolay (2019).
4.3 BVC Implementation Details
Implementing BVC requires the researcher to make a number of decisions. The most
obvious are the basic parameterization – the bar size and volatility estimation. It is also necessary
to decide how to handle the overnight trading period and trades that overflow bars. There is little
relevant guidance or precedent in the literature. ELO (2016) note that implementation in equity
markets involves addressing new issues that they did not face in their futures data. Chakrabarty,
Pascual and Shkilko (2015) and Panayides, Shohfi, and Smith (2019) are the only other studies
we are aware of that have implemented BVC in equity data, and we follow their procedures
where appropriate.
We design bar sizes to be roughly equivalent to the bar sizes use by ELO (2016). In their
ES sample they use a range of bar sizes from 1,000 contracts to 25,000 contracts, and focus on
10,000 contract bars for their main test. The number of contracts per bar itself has very little
economic meaning across instruments. Therefore, instead of directly using these numbers as
share volume per bar, we scale bar sizes for each stock that approximately match the bars per
25
hour pace of the three bar sizes in ELO (2016). Considering the trading hours and average daily
volume, the ELO (2016) 1,000, 25,000, and 25,000 contract bar sizes approximately correspond
to 100, 10, and 4 bars per hour on average.18 After adjusting for 6.5 hour equity trading day,
these sizes correspond to averages of 650, 65, and 26 bars per day. For each stock, we convert
these to “bars per day” rules which are then applied to find fixed numbers of shares per bar every
month using the average in-sample daily volume for each stock-month. For brevity we refer to
these sizes as small, medium, and large bars hereafter.
We drop stock-months for a specific bar size where this approach would give a bar size
of less than 500 shares. Very small bar sizes do not result in well-formed bars. With bar sizes
around a typical trade size, BVC would not result in much aggregation. With bar sizes smaller
than the typical trade size, BVC would actually result in disaggregation where most trades are
split over multiple bars, which is clearly contrary to the spirit of BVC. This filtering leaves us
with a sample of 56 usable stocks for small bars, 112 stocks for medium bars, and 117 stocks for
large bars.
It is also necessary to estimate volatility for each stock to compute the CDF in Eq (1).
ELO (2016) use the in-sample standard deviation of bar-to-bar price changes over the whole
sample. In our data we have a longer sample period and each stock’s volatility may vary
significantly, so we use in-sample standard deviations of bar-to-bar price changes within each
stock-month.
18 The ES market is open an average of 23.55 hours per day and the ELO (2016) one-year sample contained
128,579,415 trades with an average size of 4.5 contracts.
26
To handle the overnight period, we discard the partially filled bar at the end of each day
and start a fresh bar with the first trade of the next day. We also use the close-open within-bar
price change for the first bar of the day rather than using the closing bar from the previous day.
Chakrabarty, Pascual and Shkilko (2015) and Panayides, Shohfi, and Smith (2019) note a
problem with large trades that span multiple bars. There may be a valid price change to use in Eq
(1) for the first bar, but for subsequent bars the open and the close are the same price. With no
price change, BVC evenly splits the volume in these bars between buys and sells. We apply a
correction for this issue proposed by Chakrabarty, Pascual, and Shkilko (2015). When a bar is
filled solely by volume from a single trade, we assign the same buy/sell split calculated for the
last bar which contained volume from multiple trades. This assumes that the first bar in each
cluster of bars with volume from a single trade generally contains volume from previous trades
and the beginning of the large trade. At the end of the cluster, we treat the first bar with volume
from multiple trades normally. This issue is likely to be more severe when using small bar sizes,
and partially motivates our filtering criteria of a required minimum bar size of 500 shares
discussed above.
We only use the subperiod that includes NASDAQ BBO data for our main tests, but we
use the full sample period to construct BVC order imbalances. This is important because the
BVC estimation benefits from using the longer period of contiguous data to compute the
standard deviations of bar-to-bar price changes.
4.4 Results
Our first set of tests use the regression model in Eq. (9) to compare the predictive power
of OIB_BVC and OIB_TRUE for future returns. The results are reported in Table 5. Panel A
27
reports results for the small bar size, which corresponds to an average of 650 bars per day. The
regressions only use 56 of the stocks, because reasonably well-formed BVC bars could not be
calculated for the remaining stocks at this small of a bar size.19 Using returns measured at all
three post-bar horizons (15 seconds, 1 minute, and 5 minutes), the coefficients on both order
imbalance variables are positive, indicating that these variables positively predict future returns
and suggesting that they are both correlated with information in the order flow. The coefficient
on OIB_TRUE is larger than that on OIB_BVC in all three models. Based on the t-statistic that
tests whether the mean coefficient for all stocks is equal to 0, OIB_TRUE is significant at all
three horizons, while OIB_BVC is significant at 15 seconds and marginally significant at 1
minute. Inspecting the distributions of t-statistics from the 56 individual regressions reveals a
similar pattern. The coefficients on OIB_TRUE have higher mean t-statistics, are more often
individually significantly positive, and less often significantly negative than the coefficients on
OIB_BVC. However, the coefficients on OIB_BVC are more often significantly positive than
negative. The predictive power of both variables weakens as the return horizon lengthens, but
predictive power of OIB_TRUE weakens more slowly and retains statistical significance at 5
minutes. Overall, for the small bar size, OIB_TRUE outperforms OIB_BVC in predicting future
returns, but the coefficients on OIB_BVC have the correct sign and are statistically significant at
shorter horizons.
Table 5 Panels B and C report results for medium and large bars. After filtering, 112
(117) stocks are usable with medium (large) bars. For all three return horizons and both bar sizes,
the coefficients on OIB_TRUE are positive and significant based on t-tests of the mean. The
distributions of the individual t-statistics show some weakening of significance with large bars
19 The relevant filtering criteria is described in Section 4.3.
28
and longer return horizons, however. OIB_BVC has no significant marginal predictive power for
future returns at these bar sizes. The mean coefficients are negative in four of the six
specifications and are never significantly different from zero. The distribution of individual t-
statistics tells a similar story; the means are always below 0.25 and often negative, and the
frequencies of significantly positive individual coefficients are much lower than observed with
small bars and sometimes lower than the number of significantly negative coefficients in the
same test.
Our second set of tests assesses the predictive power of OIB_DIFF for future returns. If
BVC successfully measures passive informed trading, this variable should isolate this component
of the order flow and positively predict returns. The results are reported in Table 6. Panel A
reports results for small bars, Panel B reports results for medium bars, and Panel C reports results
for large bars. The results are consistent across all bar sizes and return horizons. The mean
coefficients on OIB_DIFF are uniformly negative and statistically significant. Turning to the
distributions of t-statistics from individual regressions, the means are uniformly negative and less
than -2.0 for all small bar regressions and the medium bar regression with the 15 second horizon.
Few coefficients are positive and significant (ranging from 0/56 with small bars and 15 second
and 1 minute horizons to 4/117 with large bars and 5 minute horizons), while many are negative
and significant (ranging from 16/117 with large bars and 5 minute horizons to 55/56 with small
bars and 15 second horizons). In short, we see no evidence that OIB_DIFF can positively predict
returns, which is inconsistent with the hypothesis that OIB_BVC captures passive informed
trading.
How should we interpret the often-strong negative relationship between OIB_DIFF and
future returns? Mechanically, order flow only enters into the OIB_DIFF variable when
29
OIB_BVC differs in its buy-sell classification from OIB_TRUE. Therefore, regardless of
whether the difference is related to information or not, it takes on positive values for aggressive
selling and negative values for aggressive buying on the disagreed upon order flow. If neither
OIB_BVC nor OIB_TRUE capture informed trading and the disagreements between them are
random, then we should expect OIB_DIFF to have no predictive power for future returns.
However, if BVC does not capture passive informed trading and the disagreements are random,
but aggressive trading tends to be more informed than passive trading, then we should expect
OIB_DIFF to have negative predictive power for future returns. Therefore, what we observe is
consistent with OIB_TRUE capturing informed trading and the disagreed upon order flow
consisting of more random misclassification than passive informed trading.
Considering the hypothesized properties of BVC, we consider OIB_TRUE the most
interesting benchmark of OIB_BVC’s ability to identify informed trading. However, researchers
often use data that do not contain true trade signs and must rely on trade signing algorithms such
as the Lee and Ready method to estimate order imbalances. Therefore, in a robustness test we
repeat the analysis presented in Table 5 using order imbalances constructed from Lee and Ready
trade signs (OIB_LR) in place of OIB_TRUE.20 We describe the procedure in Appendix B and
report the results in Appendix Table B1. The results are very similar to those in Table 5. The
coefficients on OIB_LR are positive in every specification and generally significant, while the
coefficients on OIB_BVC have mixed signs and are only positive and significant for small bars
with a 15 second return horizon.
20 We do not repeat the OIB_DIFF tests from Table 6 using Lee and Ready order imbalances. These tests
would lose their interpretation as tests of BVC’s ability to measure passive informed trading if OIB_DIFF was
constructed using trades signed with error instead of true trade signs. This can be illustrated by modifying Eq. (14)
to incorporate trade signing errors.
30
Overall, we find no evidence that OIB_BVC outperforms OIB_TRUE or OIB_LR in
measuring informed trading and no evidence that OIB_BVC captures passive informed trading.
From our first set of tests, OIB_BVC does have some positive predictive power for future returns
when controlling for OIB_TRUE for small bar sizes and short horizons, but even in these cases it
underperforms OIB_TRUE and cannot be calculated for many stocks if one requires reasonably
well-formed bars.
5. Conclusions
Our analysis calls into question the use of BVC to measure informed trading. We identify
several issues with the evidence used to support this relationship in prior literature. Most
importantly, we conduct a simulation exercise that reproduces the main test used in the literature
in data with no informed trading, and find that the results are similar to those previously thought
to show a relationship between BVC and informed trading. Our simulation exercise shows that
these tests are severely misspecified, are sensitive the BVC parameterization, and are consistent
with a mechanical relationship between BVC and the liquidity measure used as a dependent
variable. We conduct independent tests of this relationship in the NASDAQ HFT dataset using a
research design that we believe avoids these problems. We find that conventional order
imbalances constructed from both true trade signs and Lee and Ready (1991) trade signs
uniformly outperform BVC in predicting future returns across multiple combinations of BVC
parameterizations and return measurement horizons. Additional tests uniformly fail to show that
the disagreements between BVC and true trade signs are driven by informed passive trading as
suggested by ELO (2016).
31
We note that BVC has many degrees of freedom, with little guidance from the literature
on how to best parameterize it. BVC is also arguably non-deterministic; even with a fixed
parameterization the same group of trades could be classified differently depending on the length
of the sample they are included in. These issues make it difficult to make conclusive statements
about the properties of BVC. It is possible that BVC can be improved by some implementation
strategy that we are not aware of.
We focus on the relationship between BVC and informed trading only, and our results
should not be interpreted to imply that BVC does not have other useful applications. ELO (2016)
and other studies have established that BVC can be used as a trade signing algorithm, and
nothing in this paper cautions against this application when the goal is unrelated to the
information content of the trade. Prior research has shown that BVC does not generally
outperform other algorithms with regard to trade signing accuracy, but it has computational
advantages, does not require quote data, and may be more robust to data sequencing errors than
other methods.
32
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34
Table 1
Summary of Simulation Parameters. Signed trade volumes are generated from a normal distribution and are used
to calculate the unsigned trade volume and the trade sign indicator as per Eqs. 4, 5 and 6 in Section 3. Trades are
aggregated into volume bars of 1,000, 10,000, or 25,000 contracts. The number of trades per trial is matched to the
number of trades in the ELO (2016) E-mini S&P futures sample while other simulation parameters are chosen to
approximately match either the ELO (2016) E-mini S&P 500 futures sample characteristics or the E-mini S&P
futures' market characteristics. The number of trades per trial is matched to that in ELO's sample while the starting
E-mini S&P 500 futures midpoint is the settlement price of E-mini S&P futures on the day prior to the start of ELO's
sample period. The midpoint volatility corresponds to the closing value of VIX on the day prior to the start of ELO's
sample period (18.26) scaled to a per-trade value. The signed volume volatility is calibrated to match ELO's mean
trade size after adjustments. The Glosten & Harris (GH) trading costs parameter c0 is half of E-mini S&P futures
tick size while c1 is calibrated to match the mean Corwin-Shultz spread in ELO's sample with 10,000 contract bar
size. The bar volatility for each of the three bars is the midpoint volatility scaled to bar size.
Simulation Parameter Value
Number of trials for each bar size
1,000
Number of trades/trial 128,579,415
Starting E-mini S&P 500 futures midpoint 1,222
Midpoint volatility (per trade) 0.02
Signed volume volatility (contracts/trade) 5.6
Glosten & Harris trading cost parameter c0 0.125
Glosten & Harris trading cost parameter c2 0.09
Bar volatility, 1k bar size 2.98
Bar volatility, 10k bar size 9.43
Bar volatility, 25k bar size 14.91
35
Table 2
Simulation summary statistics. Mean order imbalances and Corwin-Schultz spread calculated from volume bars in
each trial. The calculation of BVC order imbalances and Corwin-Schultz spreads is as described in Section 4.3. As
in ELO, the Corwin-Schultz spreads are reported as a proportion of the price of a contract and order imbalances are
the absolute value of the estimated order imbalance as a fraction of the total volume in a bar.
Statistics Value
Trade Size, mean
4.53
Trade Size, median 4
Bar size = 1,000 contracts
Mean number of bars 582,896.7
Mean Corwin-Schultz spread 0.0019
Mean |OIB_BVC| 0.111
Mean |OIB_TRUE| 0.066
Bar size = 10,000 contracts
Mean number of bars 58,289.2
Mean Corwin-Schultz spread 0.0022
Mean |OIB_BVC| 0.050
Mean |OIB_TRUE| 0.021
Bar size = 25,000 contracts
Mean number of bars 23,315.4
Mean Corwin-Schultz spread 0.0023
Mean |OIB_BVC| 0.041
Mean |OIB_TRUE| 0.013
36
Table 3
Regressions of Corwin-Schultz spreads on order imbalances in simulated trade data. This table reports the
results of variations of the following regression:
��_� !"# = % + '(|���_���#| + '*|���_+!;<#| + '-��_� !"#.(
in simulated trade data. The simulation parameters are given in Table 1 and summary statistics of the simulated data
are presented in Table 2. CS_SPRD is the Corwin-Schultz spread, OIB_BVC is the BVC order imbalance, and
OIB_TRUE is the order imbalance constructed from true trade signs. The regression is estimated separately for each
of 1,000 draws of the data. Reported coefficients are means over all trials, and t-statistics that test the null that the
mean coefficient is equal to 0.
Variable Model
Panel A: Bar size = 1,000 contracts 1 2 3
Intercept 0.0015 0.0015 0.0015
(287.79) (289.73) (288.37)
|OIB_BVC| 0.0001 0.0001
(253.42) (253.40)
|OIB_TRUE| -0.0001 -0.0001
(-198.08) (-197.72)
CS_SPRDt-1 0.1774 0.1775 0.1774
(70.37) (70.41) (70.38)
Panel B: Bar size = 10,000 contracts 1 2 3
Intercept 0.0022 0.0021 0.0022
(329.14) (329.78) (329.00)
|OIB_BVC| -0.0018 -0.0018
(252.1) (252.24)
|OIB_TRUE| -0.00001 -0.00001
(-1.13) (-1.27)
CS_SPRDt-1 0.0532 0.0532 0.0532
(43.46) (43.43) (43.46)
Panel C: Bar size = 25,000 contracts 1 2 3
Intercept 0.0024 0.0023 0.0024
(312.61) (314.68) (312.76)
|OIB_BVC| -0.0044 -0.0044
(-246.93) (-246.91)
|OIB_TRUE| 0.00002 0.00002
(0.92) (1.06)
CS_SPRDt-1 0.0319 0.0319 0.0319
(41.11) (41.21) (41.11)
37
Table 4
Rejection rates for significance tests of order imbalances. This table reports the proportion of times the null
hypotheses for the coefficients on the order imbalances in the regression in Table 3 are rejected. For each of the
three bar sizes, 1000 regressions are estimated using simulated data. The simulation parameters are given in Table 1
and summary statistics of the simulated data are presented in Table 2. |OIB_BVC| is the absolute value of the
calculated contemporaneous BVC order imbalance and |OIB_TRUE| is the absolute value of the true order
imbalance from the simulations. The mean t-statistics reported are the cross-sectional averages of the 1,000 t-
statistics obtained for each bar size.
Panel A: Without controlling for |OIB_TRUE|
Rejection rates for significance tests of |OIB_BVC|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts 19.16 100.0% 0.0% 100.0%
10,000 contracts -17.37 100.0% 100.0% 0.0%
25,000 contracts -16.14 100.0% 100.0% 0.0%
Panel B: Controlling for |OIB_TRUE|
Rejection rates for significance tests of |OIB_BVC|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts 19.15 100.0% 0.0% 100.0%
10,000 contracts -17.37 100.0% 100.0% 0.0%
25,000 contracts -16.14 100.0% 100.0% 0.0%
Panel C: Without controlling for |OIB_BVC|
Rejection rates for significance tests of |OIB_TRUE|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts -8.36 100.0% 100.0% 0.0%
10,000 contracts -0.03 1.3% 1.2% 1.2%
25,000 contracts 0.03 0.6% 0.4% 0.8%
Panel D: Controlling for |OIB_BVC|
Rejection rates for significance tests of |OIB_TRUE|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts -8.33 100.0% 100.0% 0.0%
10,000 contracts -0.04 1.2% 1.2% 1.3%
25,000 contracts 0.03 0.7% 0.6% 0.8%
38
Table 5
Regressions of returns on order imbalances in equity trading data from the NASDAQ HFT dataset. This table
reports the results of the following regression:
!<+# = % + '*���_���#.( + '(���_+!;<#.(
for a sample of stocks selected for NASDAQ by Terrence Hendershott and Ryan Riordan. The full sample consists
of 61,271,087 trades for 120 stocks over the time periods January 2008 – December 2009 and February 22, 2010 –
February 26, 2010. !<+# is the midpoint-to-midpoint return from the end of the volume bar to a post-bar midpoint
after the return horizon has elapsed. ���JK=B,#.(is the lagged true order imbalance determined by rebate payments
from the data. ���_���#.( is the lagged order imbalance estimated using the BVC methodology as described in
Section 4.3. Panel A reports the result from 650 average bars per day, Panel B from 65 average bars per day and
Panel C 26 average bars per day. The number of stocks used drops below 120 in each panel since stocks which
cannot be distributed into well-formed bars get dropped from the sample. The coefficients and t-stats (raw and
Newey West corrected) reported are the averages across all stocks used in the regression. All coefficients are
multiplied by 1,000. Num pos sig (Num neg sig) represent the number of stocks for which the coefficient is positive
(negative) and significant.
Panel A: Average Bars per Day=650
N= 56 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0171 0.94 0.09 5 6
OIB_BVC 0.1104 2.71 2.59 34 3
OIB_TRUE 0.2518 6.62 9.90 56 0
1 minute Intercept 0.0406 1.21 0.27 4 1
OIB_BVC 0.1682 1.82 2.48 35 2
OIB_TRUE 0.2654 4.11 5.14 47 0
5 minutes Intercept 0.0498 1.1 0.07 3 2
OIB_BVC 0.0492 0.46 0.69 13 4
OIB_TRUE 0.2367 3.17 2.09 28 1
Panel B: Average Bars per Day=65
N= 112 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0234 0.99 -0.12 5 0
OIB_BVC -0.0374 -0.58 0.22 8 4
OIB_TRUE 0.3261 6.81 2.78 68 0
1 minute Intercept 0.0301 1.42 0.16 8 4
OIB_BVC -0.0564 -0.54 0.12 5 3
OIB_TRUE 0.4564 4.97 2.01 55 1
5 minutes Intercept 0.1113 1.93 0.17 7 0
OIB_BVC -0.0698 -0.54 -0.13 8 11
OIB_TRUE 0.4554 4.41 1.14 34 2
39
Panel C: Average Bars per Day=26
N= 117 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0219 1.12 0.12 4 8
OIB_BVC 0.0479 0.62 0.20 4 2
OIB_TRUE 0.3300 4.76 1.46 36 0
1 minute Intercept 0.0127 0.62 -0.09 2 3
OIB_BVC -0.0736 -0.65 0.02 4 4
OIB_TRUE 0.4628 3.76 1.09 30 0
5 minutes Intercept 0.0759 1.54 0.04 2 4
OIB_BVC 0.1427 0.99 0.18 7 2
OIB_TRUE 0.4315 2.76 0.67 18 3
40
Table 6
Regressions of returns on BVC – True order imbalance disagreement variable in equity trading data from
the NASDAQ HFT dataset. This table reports the results of the following regression:
!<+# = % + '(���_"�NN#.(
for a sample of stocks selected for NASDAQ by Terrence Hendershott and Ryan Riordan. The full sample consists
of 61,271,087 trades for 120 stocks over the time periods January 2008 – December 2009 and February 22, 2010 –
February 26, 2010. !<+# is the midpoint-to-midpoint return from the end of the volume bar to a post-bar midpoint
after � seconds have elapsed. ���_"�NN#.(is the lagged difference between the order imbalance estimated using the
BVC methodology and the true order imbalance, as described in Section 4.3. Panel A reports the result from 650
average bars per day, Panel B from 65 average bars per day and Panel C 26 average bars per day. The number of
stocks used drops below 120 in each panel since stocks which cannot be distributed into well-formed bars get
dropped from the sample. The coefficients and t-stats (raw and Newey West corrected) reported are the averages
across all stocks used in the regression. All coefficients are multiplied by 1,000. Num pos sig (Num neg sig)
represent the number of stocks for which the coefficient is positive (negative) and significant.
Panel A: Average Bars per Day=650
N= 56 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0170 0.94 0.07 5 6
OIB_DIFF -0.2653 -6.9 -10.64 0 55
1 minute Intercept 0.0405 1.2 0.26 4 1
OIB_DIFF -0.2821 -4.41 -5.65 0 46
5 minutes Intercept 0.0497 1.09 0.06 3 2
OIB_DIFF -0.2494 -3.37 -2.29 1 29
Panel B: Average Bars per Day=65
N= 112 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0228 0.95 -0.14 0 5
OIB_DIFF -0.2989 -6.16 -2.43 0 61
1 minute Intercept 0.0293 1.38 0.15 6 4
OIB_DIFF -0.4360 -4.72 -1.87 2 52
5 minutes Intercept 0.1111 1.91 0.16 5 0
OIB_DIFF -0.4718 -4.63 -1.22 2 37
41
Panel C: Average Bars per Day=26
N= 117 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0204 1.06 -0.13 2 8
OIB_DIFF -0.2671 -4.03 -1.10 1 31
1 minute Intercept 0.0117 0.57 -0.10 2 3
OIB_DIFF -0.4220 -3.51 -0.95 1 27
5 minutes Intercept 0.0747 1.52 0.03 2 4
OIB_DIFF -0.3903 -2.62 -0.60 4 16
43
Appendix A: Simulations without Temporary Price Impacts
In this section we repeat the analysis reported in Table 3 and Table 4 in simulated data
with no temporary price impacts. We generate data with no informed trading and no temporary
price impacts using following model:
# = /# = /#.( +0# (A1)
0#~2(0, 56) (A2)
Where mt is the pre-trade midpoint (or, equivalently in this setting, the fair value of the security)
for trade t and Pt is the trade price for trade t. Note that we simulate the model in event time
rather than calendar time. In this data generated from this model, trades have no temporary or
permanent price impact. This is equivalent to the modified Glosten and Harris (1998) model
described in Section 3.2. with the trading cost parameters c0 and c1 set to 0. The rest of the
procedure is identical to that described in Section 3.2, and we continue to use the relevant
parameters given in Table 1.
44
Table A1
Regressions of Corwin-Schultz spreads on order imbalances in simulated trade data with no temporary price
impacts. This table reports the results of variations of the following regression:
��_� !"# = % + '(|���_���#| + '*|���_+!;<#| + '-��_� !"#.(
in simulated trade data. The simulation parameters are given in Table 1, with the exception of the trading cost
parameters c0 and c1 which are set to 0. CS_SPRD is the Corwin-Schultz spread, OIB_BVC is the BVC order
imbalance, and OIB_TRUE is the order imbalance constructed from true trade signs. The regression is estimated
separately for each of 1,000 draws of the data. Reported coefficients are means over all trials, and t-statistics that test
the null that the mean coefficient is equal to 0.
Variable Model
Panel A: Bar size = 1,000 contracts 1 2 3
Intercept 0.0001 0.0001 0.0001
(286.83) (286.91) (286.68)
|OIB_BVC| -0.0001 -0.0001
(-207.35) (-207.35)
|OIB_TRUE| -0.0000 -0.0000
(-4.47) (-4.73)
CS_SPRDt-1 0.0251 0.0251 0.0251
(158.3) (158.3) (158.3)
Panel B: Bar size = 10,000 contracts 1 2 3
Intercept 0.0003 0.0003 0.0003
(286.46) (284.96) (285.57)
|OIB_BVC| -0.0002 -0.0002
(-104.27) (-104.26)
|OIB_TRUE| 0.0000 0.0000
(0.2) (0.17)
CS_SPRDt-1 0.0174 0.0174 0.0174
(83.35) (83.36) (83.37)
Panel C: Bar size = 25,000 contracts 1 2 3
Intercept 0.0005 0.0005 0.0005
(285.28) (285.17) (284.14)
|OIB_BVC| -0.0004 -0.0004
(-70.24) (-70.22)
|OIB_TRUE| 0.0000 0.0000
(0.04) (0.05)
CS_SPRDt-1 0.0163 0.0163 0.0163
(58.6) (58.66) (58.61)
45
Table A2
Rejection rates for significance tests of order imbalances in simulated trade data with no temporary price
impacts. This table reports the proportion of times the null hypotheses for the coefficients on the order imbalances
in the regression in Table A1 are rejected. For each of the three bar sizes, 1000 regressions are estimated using
simulated data. The simulation parameters are given in Table 1, with the exception of the trading cost parameters c0
and c1, which are set to 0. |OIB_BVC| is the absolute value of the calculated contemporaneous BVC order imbalance
and |OIB_TRUE| is the absolute value of the true order imbalance from the simulations. The mean t-stats reported
are the cross-sectional averages of the 1,000 t-stats obtained for each bar size.
Panel A: Without controlling for |OIB_TRUE|
Rejection rates for significance tests of |OIB_BVC|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts -10.04 100.0% 100.0% 0.0%
10,000 contracts -3.61 83.3% 89.9% 0.0%
25,000 contracts -2.29 38.7% 47.9% 0.0%
Panel B: Controlling for |OIB_TRUE|
Rejection rates for significance tests of |OIB_BVC|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts -10.04 100.0% 100.0% 0.0%
10,000 contracts -3.61 83.2% 89.9% 0.0%
25,000 contracts -2.29 38.7% 48.1% 0.0%
Panel C: Without controlling for |OIB_BVC|
Rejection rates for significance tests of |OIB_TRUE|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts -0.14 1.2% 1.3% 1.1%
10,000 contracts 0.01 1.1% 1.2% 0.9%
25,000 contracts 0.00 1.1% 1.0% 0.7%
Panel D: Controlling for |OIB_BVC|
Rejection rates for significance tests of |OIB_TRUE|,
significance level = 1%
Bar size Mean t
two-sided
|OIB_BVC| = 0
one-sided
|OIB_BVC| >= 0
one-sided
|OIB_BVC| <= 0
1,000 contracts -0.15 1.2% 1.3% 1.1%
10,000 contracts 0.01 1.1% 1.2% 0.8%
25,000 contracts 0.00 1.1% 0.9% 0.7%
46
Appendix B: Regressions of Returns on BVC and Lee and Ready Order Imbalances
In this section we repeat the analysis reported in Table 5 using order imbalances
constructed from trades signed with the Lee and Ready method instead of true trade signs. We
first sign trades using the Lee and Ready method. The Lee and Ready method applies the quote
rule where possible and the tick rule otherwise. In our implementation of the quote rule, we
match trades with the most recent pre-trade quote midpoint that is not in the same millisecond as
the trade. Trades above the midpoint are classified as buys, and trades below the midpoint are
classified as sells. The tick rule is used when trade occurs at the midpoint of the matched quotes,
or in a small number of cases where no pre-trade quote is available. The tick rule compares the
current trade price with the preceding trade price and classifies the trade as a buy if the current
price is higher or a sell if the current price is lower. If the preceding trade price is identical to the
current price, the tick rule looks back to the most recent different price for a benchmark.
Next, we aggregate the signed trades in each bar into order imbalances as follows. The
true order imbalances are defined as:
���_O!# = �=>_�?@=ABC–EB@@_�?@=ABC�?@=ABC (B1)
where BUY_VOLUMEt and SELL_VOLUMEt are the share volumes designated as buys and
sells by the Lee and Ready method in bar t and VOLUMEt is the total share volume in bar t. The
bars are volume bars formed as described in Section 4.3.
For our regressions, we modify Eq. (9) as follows:
!<+# = % + '*���_���#.( + '(���_O!#.( (B2)
47
where OIB_LR is the Lee and Ready order imbalance defined in Eq. (B1) and other variables are
as previously defined. We estimate this regression one stock at a time and report the results in
Table B1 below.
48
Table B1
Regressions of returns on order imbalances in equity trading data from the NASDAQ HFT dataset. This table
reports the results of the following regression:
!<+# = % + '*���_���#.( + '(���_LR#.(
for a sample of stocks selected for NASDAQ by Terrence Hendershott and Ryan Riordan. The full sample consists
of 61,271,087 trades for 120 stocks over the time periods January 2008 – December 2009 and February 22, 2010 –
February 26, 2010. !<+# is the midpoint-to-midpoint return from the end of the volume bar to a post-bar midpoint
after the return horizon has elapsed. ���_O!#.(is the lagged order imbalance estimated using a modified Lee and
Ready (1991) algorithm that matches trades with the last quote in the prior millisecond. ���_���#.( is the lagged
order imbalance estimated using the BVC methodology as described in Section 4.3. Panel A reports the result from
650 average bars per day, Panel B from 65 average bars per day and Panel C 26 average bars per day. The number
of stocks used drops below 120 in each panel since stocks which cannot be distributed into well-formed bars get
dropped from the sample. The coefficients and t-stats (raw and Newey West corrected) reported are the averages
across all stocks used in the regression. All coefficients are multiplied by 1,000. Num pos sig (Num neg sig)
represent the number of stocks for which the coefficient is positive (negative) and significant.
Panel A: Average Bars per Day=650
N= 56 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0164 0.93 0.08 4 6
OIB_BVC 0.0903 2.15 2.29 30 3
OIB_LR 0.2626 6.76 10.54 55 0
1 minute Intercept 0.0399 1.21 0.27 4 1
OIB_BVC 0.1294 1.36 2.17 28 2
OIB_LR 0.2901 4.37 5.80 48 0
5 minutes Intercept 0.0491 1.10 0.07 4 2
OIB_BVC -0.0075 -0.07 0.43 11 3
OIB_LR 0.2751 3.53 2.60 30 0
Panel B: Average Bars per Day=65
N= 112 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0246 1.02 -0.11 2 6
OIB_BVC -0.0606 -0.84 0.15 5 4
OIB_LR 0.3638 7.19 3.00 71 0
1 minute Intercept 0.0269 1.21 0.17 6 4
OIB_BVC -0.0527 -0.52 0.04 5 4
OIB_LR 0.4699 5.25 2.29 62 0
5 minutes Intercept 0.1128 1.93 0.17 6 0
OIB_BVC -0.1422 -1.02 -0.21 4 13
OIB_LR 0.5599 5.19 1.46 43 0
49
Panel C: Average Bars per Day=26
N= 117 stocks
Variable Coefficient
t-statistic
Num
pos sig
Num
neg sig Return Horizon t(mean) mean(t )
15 seconds Intercept 0.0228 1.16 -0.10 4 8
OIB_BVC 0.0156 0.22 0.14 2 2
OIB_LR 0.3781 5.45 1.66 49 0
1 minute Intercept 0.0125 0.62 -0.07 2 3
OIB_BVC -0.0892 -0.71 -0.03 4 4
OIB_LR 0.5041 3.94 1.30 34 0
5 minutes Intercept 0.0788 1.57 0.05 3 4
OIB_BVC 0.0959 0.35 0.13 6 3
OIB_LR 0.0505 3.51 0.87 24 4
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