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Trading costs and priced illiquidity in highfrequency trading markets∗
YASHAR HEYDARI BARARDEHI†
Department of EconomicsUniversity of Illinois
214 David Kinley Hall1407 W. GregoryUrbana IL 61801
DAN BERNHARDT‡
Department of EconomicsUniversity of Illinois
214 David Kinley Hall1407 W. GregoryUrbana IL 61801
RYAN J. DAVIES§
Finance DivisionBabson College
220 Tomasso HallBabson Park MA 02457
September 14, 2014
∗©2013 Barardehi, Bernhardt, Davies. We are grateful for comments received from seminar participantsat the University of Illinois and the Boston Area Finance Symposium. Yashar Heydari Barardehi is grate-ful for support from Paul Boltz Fellowship. Ryan Davies acknowledges support from the Babson FacultyResearch Fund.†Tel: +1-217-819-6478. email: [email protected]‡Tel: +1-217-244-5708. email: [email protected]§Tel: +1-781-239-5345. email: [email protected]
Trading costs and priced illiquidity in high frequency trading markets
Abstract
Decimalization, regulatory changes, and technological advances have transformedstock markets, shrinking bid-ask spreads, driving down depth, and causing institutionalinvestors to employ order-splitting algorithms. We develop a stock-specific measure ofmarket activity for this environment by calculating the time it takes for sequencesof trades each worth a fixed percent of a firm’s market capitalization to execute—shorter durations indicate more active markets. As market activity rises so do tradesizes and signed order imbalances; but price impacts fall for small and mid-sized firms;and for large firms, price impacts exhibit a single-peaked relationship with marketactivity. Most variation in market activity reflects endogenous variation in liquidityprovision/consumption; and not information arrival. We instrument for this endogene-ity and uncover predictable, uniform trading costs/market activity relationships acrossstocks—with trading costs falling by 5–8 basis points as instrumented market activ-ity rises. Using these trading cost measures as illiquidity measures in an augmentedCAPM, we find that they outperform standard measures (bid-ask spreads, Amihud’smeasure), and that liquidity premia have not declined.
Equity markets have experienced revolutionary changes in the past two decades. Prior
to 1997, stock prices were quoted in eighths, and trading costs and liquidity provision were
driven by this large tick size. Today, the typical tick size is a penny, and limit order book
depth near the inside quotes has collapsed. Dedicated specialists no longer provide tradi-
tional market making services. Instead, those services are largely provided by “high frequency
traders” who use complex computer algorithms to submit thousands of small, fleeting orders
throughout the day. As a result, average trade sizes and inside bid-ask spreads are now tiny,
and traditional measures of trading costs and liquidity no longer describe the concerns of
institutional traders1 who seek to minimize trading costs by splitting their large orders into
many smaller orders that are dynamically submitted over time and across trading venues.
These changes lead us to ask: in today’s markets, what drives variations in market ac-
tivity? Is it costlier to establish or unwind positions when markets are active or inactive?
What market conditions predict when trading costs will be higher or lower? And, which
illiquidity measures for individual stocks explain their pricing?
A priori, the answers are unclear: More active markets may have more participants, but
markets may be more active precisely because some participants have acquired information;
markets may appear more active just because liquidity provision is unusually high, inducing
agents to consume it aggressively; and the nature of illiquidity risk about which institutional
traders care is unclear, as they may be able to time trades according to market conditions.
To answer questions such as these, one needs a measure of market activity that controls
for dynamic order splitting and division of orders against the book. Our new trade-based
measure of market activity groups consecutive trades together into a sequence of trades with
a cumulative dollar value of a fixed amount (e.g., 0.04% of a firm’s market capitalization).
Once a stock’s target dollar value is reached, we start over, grouping the next set of consec-
utive trades until their cumulative dollar value reaches its target value, continuing in this
way iteratively over a year. We measure market activity by the time it takes for a trade
1Blume and Keim (2012) report that the share of institutional ownership in U.S. equity markets was67% at the end of 2010.
1
sequence to reach its target dollar value—shorter durations indicate more active markets.
Controlling for the dollar value of a sequence lets us isolate the effects of market activity
from that of trading volume. That is, while volume and price volatility are highly positively
correlated (as Andersen and Bondarenko (2014) emphasize), this relation does not inform
an institutional trader about how the costs of unwinding a given position vary with market
activity. By choosing target dollar values that are neither too large nor too small, we address
concerns both about aggregating over multiple activity levels due to excessively large targets,
and mis-attributing variation in market activity due to excessively small targets.
With our stock-specific market activity measure in hand, we establish how fundamentals
vary with market activity. We first show that both trade size and signed trade imbalance
rise with market activity. In particular, on average, as the duration of a trade sequence for
a stock falls, so does the number of trades comprising the sequence—and given the same
dollar-volume, fewer trades reflect larger trades. So, too, using the Lee and Ready (1991)
algorithm to identify the share of value-weighted trades in a trade sequence that is buyer- or
seller-initiated (whichever is highest), we find that as market activity rises, trade becomes
less balanced. These changes are large, especially in highly active markets. For example,
going from least to most active markets, signed trade imbalances rise by roughly 20%, and
for smaller stocks, mean trade sizes double.
It is not surprising that more active markets correspond with more aggressive trading.
The classical models of speculation in financial markets (Kyle (1985), Glosten (1993)) pre-
dict precisely this: speculators with substantial private information will establish larger
positions, submitting larger, liquidity consuming, orders that cause market activity, signed
trade imbalance, and trade sizes all to rise.
But, a very different explanation for these phenomena exists. Liquidity provision may
be time-varying and certain times may feature more depth available near inside quotes than
others. When the market is unusually liquid on one side, institutional traders respond ag-
gressively, submitting larger market orders that are filled at these “good” prices. As a result,
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the corresponding trade sequences have shorter durations, with fewer trades and less bal-
anced signed trade. In contrast, with little depth, institutional traders consume less liquidity
via market orders, and supply more via limit orders in order to reduce price impacts.2 The
result is longer durations, more balanced signed trade (reflecting a more even mix between
demanding and supplying liquidity), and more trades comprising a trade sequence.
To distinguish between these very different explanations, we look at price impacts. Clas-
sical models of speculation predict that more aggressive trading should result in larger price
impacts; but, if endogenous consumption of unusually high depth at “good” prices is the
primary driver, then price impacts should fall as market activity rises. We measure price
impact by both the average absolute value of the return over a trade sequence at a given
market activity level, and the average annual standard deviation of those returns.
For small and mid-sized stocks, trade sizes and percent trade imbalances rise with mar-
ket activity, but both measures of price impacts fall sharply—declines on the order of 40%.
These relationships are especially steep at very high market activity levels. Thus, even though
volatility is positively associated with dollar volume and market activity, the opposite rela-
tionship holds once one controls for the level of volume. In contrast, for larger stocks, price
impacts peak at intermediate market activity levels, and, indeed, the standard deviation of
returns is smallest when markets are least active.
To investigate these price impact patterns more deeply, we calculate percentile statistics
of returns by activity level and then compute return percentile differences, rπ− r1−π, e.g., the
75th-25th percentile difference. For small and mid-sized stocks these differences fall uniformly
as market activity rises. In contrast, for larger stocks, extreme return percentile differences
(e.g., r.95−r.05) widen as market activity rises, but differences of less extreme percentiles (e.g.,
r.75−r.25) either shrink with market activity or are single-peaked functions of market activity.
The uniformly negative relationship between price impacts and market activity for small
2See, for example, Bloomfield et al. (2005), Goettler, Parlour and Rajan (2005), Hollifield, Miller andSandas (2004), and Hollifield et al. (2006) and Easley et al. (2012).
3
and mid-sized stocks indicates that time-varying endogenous consumption and provision of
liquidity is the primary driver of variation in market activity, not information. In contrast,
the fat tails of the return distribution in active markets of large stocks suggests that, in
unusual times, significant information arrival drives some higher market activity, but, in
normal times, market activity mostly reflects variation in liquidity. Cumulatively, these
findings indicate that the classic models of informed trade no longer describe outcomes in
today’s stock markets. Rather, high frequency variation in market fundamentals for a given
stock are driven primarily by endogenous responses of traders to the time-varying levels of
liquidity extant.
These important findings still leave one seeking more. An institutional investor seeking to
minimize the costs of establishing or unwinding a large position cannot exploit information
arrival to which it is not privy, and may have difficulty forecasting when liquidity provision
will be “unusually high”. One wants to control for endogeneity and information arrival to
make causal inferences about how market activity affects trading costs. That is, one wants to
isolate primitive market activity/trading cost relationships that are stable and predictable.
To do this, we use the number of trades filling the previous trade sequence to instrument
for the duration of the current trade sequence (current market activity). The observation
underlying our choice of instrument is that liquidity persists over time—present and past
states of the order book are positively correlated. The endogenous composition of current
orders, however, hinges only on the contemporaneous state of the order book, which strongly
covaries with current activity. Effects of past order book states on current price impacts
only transmit via the current state. Using instrumented market activity also filters out the
variation associated with information arrival which is, presumably, unpredictable.
For each stock, we first regress log duration on a polynomial of lagged trade frequency,
annually. Over 95% of these regressions have significant F statistics, with large positive
coefficients, indicating that the fitted values of these regressions can serve as “instrumented”
levels of activity. In contrast, the analogous regressions using lagged trade frequency to pre-
4
dict signed trade imbalance yield far fewer significant F statistics; and signed trade imbalance
is far less correlated with instrumented market activity than with raw market activity.
Instrumented market activity yields very stable and uniform trading costs/market ac-
tivity relationships across all firm sizes. Instrumenting flattens the price impact/activity
relationship for small stocks by roughly 75%; but for larger stocks, price impact/activity re-
lationship becomes monotone and, indeed, it becomes steeper. The result is that, regardless
of firm size, average price impacts fall uniformly by five to eight basis points as predicted
market conditions rise from least active to most active levels. The costs of trading smaller
stocks are more sensitive to predicted market activity levels, but the relative (percentage)
variations in trading costs are qualitatively identical for all stock sizes. Our regression results
indicate that institutional traders can successfully time when they establish or unwind posi-
tions to reduce trading costs by meaningful amounts. We also find that while bid-ask spreads
have fallen uniformly in more recent years, the predictable components of trading costs have
not: the costs of unwinding positions in large market capitalization stocks have fallen, but
those for small and medium sized stocks have risen, especially in very active markets.
Our ability to relate trading costs to market activity and the cross-sectional attributes of
firms leads us to investigate whether we can shed light on how illiquidity is priced. We ask:
which illiquidity measure best captures the concerns of traders? and, how have the return
premia demanded by traders for taking positions in less liquid stocks evolved?
We start by creating a new measure of individual stock liquidity, BBD, by scaling our
price impact measure by the dollar amount traded. The measure is based on a rolling average
of the per-dollar price impact of trading a fixed proportion of a stock’s market capitalization
at a given level of market activity to account for the varying nature of trading costs. In
essence, BBD is a trade-time analogue of Amihud’s (2002) temporal measure of liquidity
that isolates different market conditions.
A priori, it is not clear how institutions will price trading cost risks in equity markets.
For very illiquid securities such as corporate debt, concerns that markets might dry up likely
5
dictate liquidity premia. After all, it was such dry ups that led to the failures of Long-Term
Capital Management and Lehman Brothers. Equity markets, however, are so liquid that
the liquidity considerations of institutional traders may reflect expected price impacts when
unwinding positions in “standard times”, where they can time trades in accordance with
predicted market activity.
Regardless of which raw or instrumented market activity levels we use to calculate BBD,
it is priced in an augmented capital asset pricing model. Furthermore, these liquidity mea-
sures are robust: similar liquidity premia obtain when we use different market activity levels
to constructBBD. We compareBBD with standard liquidity measures—average percentage
bid-ask spreads and Amihud’s per dollar price impact. We also consider a high frequency
version of Amihud’s measure constructed at hourly rather than daily frequencies. In the
2001–2012 post-decimalization era, each of these measures is priced on its own. To compare
the measures, we employ the residuals from the regression of one measure on a second in the
pricing regressions. The BBD residual is always priced, whereas the other residuals are not.
Splitting the sample over time reveals that the superior performance of BBD reflects
more recent years following the implementation of RegNMS and the resulting increase in
high frequency trading, dynamic order splitting and trading volumes. Even though bid-ask
spreads have fallen in recent years, implied liquidity premia have increased, regardless of the
liquidity measure used. The cumulative evidence indicates that our trade-time based notion
of liquidity better captures liquidity in today’s markets.
Our paper is organized as follows. We next discuss related research. Section I develops our
market activity measure. Section II establishes how different fundamentals vary with market
activity. Section III evaluates different illiquidity measures in an asset pricing model. Sec-
tion IV concludes. An appendix explores calendar time relationships between volatility and
market activity, and shows our results are robust to different error structure specifications.
6
A Related Literature
Trade time and market dynamics. Mapping asset market dynamics onto trade-time
space is not a new idea. Dufour and Engle (2000) examine the duration between successive
trades and show that price adjustments are faster when trading intensity is high (i.e., du-
rations are short). They argue that periods of high trade intensity indicate the presence of
informed trading and conclude that these periods correspond to periods of market illiquidity.
Engle and Russell (1998) and Engle (2000) develop related models of autoregressive condi-
tional duration (ACD), which provide semi-parametric estimates of trade intensities based
on trade arrival rates. ACD models are designed primarily to estimate the expected cost
and time to execute a single order, submitted within a very short time frame, and not the
cost of a series of orders submitted over longer periods of time.
In modern markets, the information content of one trade is likely small; and consecu-
tive trades may represent a single trading decision, where a larger order has been crossed
against multiple smaller orders in the book. To account for these features, Gourieroux et
al. (1999) measure market activity by the time it takes the market to execute an exogenous
level of volume or dollar volume. They focus on the evolution of market activity over the
trading day on Paris Bourse for a single stock over 18 trading days, exploring the average
time duration to unwind a fixed position at different times. They find activity peaks at
times corresponding to the opening of the London Stock Exchange and the NYSE. Easley et
al. (2012) are similarly motivated to group consecutive trades across time according to the
time required to trade an exogenous level of order flow. Within each group, they use the
sign of price changes over one-minute periods to infer buyer-initiated versus seller-initiated
trades, creating a measure of volume “toxicity”, VPIN, the high frequency analogue to their
probability of informed trading (PIN) measure.
Our findings highlight a tension between concerns about using and consuming liquidity,
and identification of the probabilities of informed trade off of a competitive (zero-profit) limit
order pricing construction. If liquidity provision is competitive, then traders might as well
7
always consume liquidity—it is because markets are sometimes more illiquid than others that
traders choose sometimes to consume and sometimes to supply liquidity (see also Andersen
and Bondarenko (2014)).
Feldhutter (2012) develops the notion of an imputed roundtrip trade, based on the em-
pirical feature that trades in corporate bonds often occur infrequently, but when they do
trade, there are often several trades in a short period of time. He argues that these trades
are likely linked and should be considered together when measuring trading costs.
Relation between market activity and liquidity. There has been a long debate about
the relation between market activity and liquidity. Demsetz (1968) finds that more active
stocks tended to be more liquid, and argues that more frequent trading is associated with
low dealer costs. Lippman and McCall (1986) argue that active markets have a lower op-
portunity cost of searching and are thus more liquid. Jones (2002) and Fujimoto (2004) find
no relation between liquidity measures and changes, or shocks, in turnover. Johnson (2008)
develops a model in which volume is positively related to liquidity risk, rather than liquidity,
and provides supporting evidence from U.S. government bond markets.
Liquidity Measures. Amihud and Mendelson (1986) show that when the standard tick
size was one-eighth, bid-ask spreads were priced and were a reasonable measure of liquidity.
Other quote-based liquidity measures include effective bid-ask spreads, the Roll (1984) mea-
sure, and Hasbrouck’s Gibbs estimate.3 Lesmond et al. (1999) argue that periods of zero
returns may represent an absence of informed trading, and as such, these periods tend to
correspond with a high cost of informed trading.
Variants of Kyle’s λ (Kyle (1985)) have also been used as a liquidity measure (see, e.g.,
Glosten and Harris (1988), Brennan and Subrahmanyam (1996), or Pastor and Stambaugh
(2003)). Bernhardt and Hughson (2002) estimate price impacts using a structural model
whose equilibrium, unlike Kyle’s, explicitly incorporates orders that are not pooled.
3Effective bid-ask spreads measure the spread between the trade price and quote midpoint. Roll (1984)focuses on how the tick size induces negative correlation in prices. Hasbrouck (2009) proposes a Bayesianplatform to generate a Gibbs estimate of Roll’s effective trading cost indicator.
8
Chordia et al. (2010), Angel et al. (2011), Kim and Murphy (2013), and others have ques-
tioned the accuracy and feasibility of many existing liquidity measures in a high frequency
trading world with tiny spreads and little depth at the inside quote. Holden and Jacobsen
(2014) document realized dollar spreads and share depth as low as 0.88¢ and 564 shares,
respectively, for a sample of randomly selected stocks in 2008. Today, liquidity concerns do
not revolve around avoiding minimum price ticks, so spreads do not capture liquidity from
that perspective. Order-splitting makes consecutive trades dependent (Hasbrouck (2009),
Easley et al. (2012)) and interpretations of estimates of λ heroic. Moreover, the high vol-
ume results in few instances of zero returns (Mazza (2013)). Further, the increased use of
hidden orders makes it difficult to accurately measure depth—a key parameter for many
order-based liquidity measures. Deuskar and Johnson (2012) propose a measure of liquidity
that uses information from the cumulative depth in the limit order book, rather than just
the inside quote. Unfortunately, using limit order book information is not feasible for most
applications. Moreover, this measure imperfectly captures dynamic order splitting.
In light of these issues, a new generation of liquidity measures has been developed based
on volume, rather than orders. These measures are less affected by dynamic order-splitting.
The most widely used is Amihud’s (2002) measure, which is based on the ratio of absolute
daily returns divided by daily dollar volume. This measure has been truly useful at explain-
ing the cross-section of returns, and indeed in time series (Amihud (2002) and Jones (2002)).
Goyenko et al. (2009) show that Amihud’s measure remains useful in the post-decimalization
era. With increased high frequency trading, however, such low frequency measures may lose
valuable information by aggregating across heterogeneous market activity conditions.
Acharya and Pedersen (2005) incorporate a portfolio-level aggregate of the Amihud liq-
uidity measure in a market model and argue that premia can be explained by covariations of
stock liquidity with market-wide return and this liquidity proxy. Akbas et al. (2011) extend
Acharya and Pederson (2005) by adding a measure of idiosyncratic liquidity volatility. They
show it is priced in the presence of systematic liquidity risk. These papers are related to
9
research on the commonality of market-wide liquidity (e.g., Chordia et al. (2000) or Sadka
(2006)). In contrast, our focus is developing a new, robust measure of firm-specific liquidity.
Finally, we note that while conventional liquidity measures such as bid-ask spreads have
likely become flawed, they may still be sufficiently correlated with “true” liquidity that they
can serve as useful empirical proxies. As such, a contribution of our paper is to develop a
liquidity measure that relates liquidity to market activity, and to explore the extent to which
conventional measures can proxy for market-activity based liquidity measures.
I Measuring Market Activity
To start, we define a trade sequence and explain how we use the duration of a trade
sequence to measure market activity. Each year, we number trades in stock j sequentially,
using index nj. For trade nj, we use τj(nj), Qj(nj) and Pj(nj) to denote respectively, (i) its
time measured in seconds from the beginning of the year, (ii) its size (in shares), and (iii) its
price. We construct our market activity measure by calculating the time it takes to execute
a sequence of consecutive trades that have an aggregate value of at least Vj,t for stock j in
month t. Thus, a shorter duration indicates a more active market. For the reasons detailed
below, we set Vj,t to be proportional to stock j’s market capitalization at the end of the
previous month, Mj,t−1. The first trade sequence begins with the first trade of the year, and
each subsequent trade sequence begins with the first trade following the previous sequence.
Figure 1 illustrates a typical pattern.
Formally, we iteratively solve for the last trade of the kth trade sequence, k = {1, 2, 3, . . .}, as:
nkj = argminn∗
n∗∑
n=nk−1j +1
PCj (n)×Qj(n)
∣∣∣∣∣∣∣n∗∑
n=nk−1j +1
PCj (n)×Qj(n) ≥ Vj,t
, (1)
where n0j = 0 and the value of aggregate trades is measured using the previous day’s closing
10
A: Dollar volume
0
Dollar
volu
me
Time
B: Cumulative dollar volume
dur(k−1) dur(k) dur(k+1)
V
0
Cum
ultiv
e d
olla
r volu
me
Time
Figure 1: An illustration of how we measure the durations of trade sequenceswith an aggregate value of at least Vj,t.
price, PCj (nj).
4 Then we obtain the duration of the kth trade sequence,
durj(k) = τj(nkj )− τj(nk−1j + 1), (2)
the corresponding return,
rj(k) =Pj(n
kj )
Pj(nk−1j + 1)
− 1, (3)
and the dollar value of the sequence,
DV OLj(k) =∑nk
j
n=nk−1j +1
PCj (n)×Qj(n). (4)
4The last quoted bid-ask midpoint is used when the closing price is not available.
11
The per-dollar price impact for stock j of the kth trade sequence is
DIMPj(k) =|rj(k)|
DV OLj(k). (5)
Notice that we adjust returns by DV OLj(k), rather than Vj,t, since the size of the last trade,
nkj , typically exceeds the quantity required to deliver a cumulative value of Vj,t.
We construct trade sequences that span multiple trading days. However, since overnight
price adjustments could bias our empirical results, we exclude any sequence that spans mul-
tiple trading days (our estimates are robust to relaxing this exclusion). Constructing, but
excluding, overnight trade sequences ensures a random starting (ending) point for the first
(last) trade sequence of a given day, precluding any possible systematic bias associated with
always beginning (ending) the first (last) trade sequence at the open (close).
We sort the remaining trade sequences for stock j in month t into ten equally-sized
groups i according to their duration, where i = 1 has the shortest durations (highest market
activity) and i = 10 has the longest durations (lowest market activity). Each year, we also
sort firms into market capitalization deciles.
We can now compare the market impact of unwinding a given position at different market
activity levels. We employ two market impact measures:
• Price impact: Each year, for each stock, we calculate the mean of the absolute return
of each trade sequence in a given duration group. We then average these annual mean
absolute returns across years for all stocks in a given size decile.5
• Return volatility: Each year, for each stock, we calculate the standard deviation of the
return of trade sequences in each duration group. We then average these annual return
standard deviations across years for stocks in a given size decile. Computing annual
standard deviations before averaging minimizes temporal aggregation concerns.
Later, we use a scaled 3-month rolling average of the price impact measure, DIMPj(k), for
5Averaging stock-specific moments makes the average independent of number of sequences per stock.
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different subsets of trade sequences, as a priced liquidity measure, BBDj,t.
Choice of Vj,t. We focus on the market impact and the time to execute a sequence of
orders that for a firm j have an aggregate value Vj,t equal to θ% of its market capitalization
at the end of the previous month, Mj,t−1. Our base formulation using θ = 0.04 generates
a median duration (across stocks and years) of roughly 20 minutes. A robustness analysis
verifies that our qualitative findings are not driven by the choice of θ: qualitatively similar
findings obtain using θ = 0.03 or θ = 0.05.
Several considerations enter our choice of specification: (i) the positions are large enough
to be relevant for institutional traders, and to control for dynamic order splitting and divi-
sion of orders against the book; (ii) the positions are not so large that a single trade sequence
spans different market activity levels; (iii) the proportional specification captures the feature
that institutional traders tend to establish larger positions in larger firms, and it facilitates
cross-stock comparisons; (iv) the proportional specification flattens out the distribution of
durations across market capitalizations (see Figure 2). As a result, durations of small market
capitalization stocks are not too much longer, delivering enough observations for our empir-
ical analysis: we drop stock-year observations for stocks without enough trade sequences in
the year to avoid small sample issues, and setting θ = 0.04 ensures that we do not lose too
many stock-year observations, even in early years. In contrast, a fixed dollar-amount chosen
based on a medium-sized stock’s market capitalization would deliver too few observations
for small-cap firms, and far too many for large-cap firms.6
Comparisons with Gourieroux et al. (1999) and Easley et al. (2012). As in these
papers, we group individual trades together to deal with division of orders against the book,
dynamic order submission strategies, and so on. There are, however, important distinctions
between these approaches and ours. Easley et al. (2012) divide daily volume for a stock into
6Easley, et al. (2012) bundle trades into volume groups with size equal to 1/50 of annual daily tradevolume. This approach comes close to equalizing the number of bundles across stocks within and acrossyears, but bundle sizes expand sharply over time due to the explosion in trading volume. In contrast, ourapproach roughly preserves bundle size across years.
13
50 roughly equally-sized bundles—meaning that they have different volume cutoffs on differ-
ent trading days. Importantly, with the explosion of trading volumes over the past decade,
the dollar values of these bundles have also exploded. This creates problems if one wants to
assess how trading costs have evolved over time, or to aggregate over time. Similar issues
arise with comparisons of stocks with different levels of market capitalization, or, of stocks
with similar market capitalizations, but different levels of market activity. In contrast, we
set our target dollar-volume to be a proportion of the firm’s market-capitalization. Thus, the
time durations of our trade sequences control for variations in clock-time speed; we find much
shorter time durations of trade sequences in more recent years with temporally comparable
dollar values. This feature facilitates temporal comparisons and temporal aggregation across
roughly homogeneous entities; and by controlling for market capitalization, one can, for ex-
ample, convert our characterizations into analyses of the costs of trading fixed dollar amounts.
Gourieroux et al. (1999) compute the time durations of fixed dollar positions. One con-
cern is that they use contemporaneous price to measure individual trade values, so their
notion of trade duration of fixed dollar positions is affected by intra-day price movements.
Ceteris paribus, this means that, on average, durations are shorter when prices move up,
and longer when they move down. To deal with this, we use the previous day’s closing price
to calculate trade values, so that durations of trade sequences are independent of contem-
poraneous price movements, allowing clean comparisons of the relationship between market
activity and trading costs.
A Data
Our sample period runs from January 1, 2001 to December 31, 2012. Each year, we con-
struct a sample of U.S.-based NYSE-listed ordinary common shares (CRSP share codes 10
and 11). We drop a stock-year observation if it has less than 800 trade sequences in the year,
or if the stock had a closing price7 below $1.00 on any trading day in the year. Monthly
7The average of the closing bid and ask prices is used if a last trade price is not available.
14
market capitalizations of firms come from the CRSP monthly stock file. We sort stocks into
size-based deciles using their market capitalizations on the first trading day of a year. We
use the previous day’s closing price from the CRSP daily stock file to compute dollar volumes
so that duration measurements are not biased by any associated price impact.8
We calculate CAPM beta, βj,t, for stock j in month t by regressing excess weekly stock
returns (versus the weekly secondary market 6-month Treasury bill rate) against the excess
weekly market return, over the previous two-year period (t−24 to t−1). The market return,
rmktt is based on an equally-weighted portfolio of the stocks in our sample in that month. If a
stock is not listed over the previous 24 months, we employ the longest time period available,
requiring at least 26 weekly observations, where we drop the initial observation to avoid a
listing effect.9 Firm j’s book-to-market ratio, BMj,t−1, is given by its book value per share,
obtained from Compustat, divided by its share price, at the end of month t− 1.
We obtain trade prices, quantities and time stamps from the consolidated trade history in
the NYSE TAQ database. We consider all stock trades on U.S.-based trading venues during
regular market hours, i.e., between 9:30AM and 4:00PM (EST), together with trades corre-
sponding to delayed reporting of market-on-close orders that are executed after 4:00PM.10
We exclude trade sequences with absolute returns that exceed 10%. We calculate percentage
bid-ask spreads using the TAQ National Best Bid and Offer (NBBO) series on WRDS.
Table I presents annual summary sample statistics. Median stock prices largely mimic
overall market performance. The median annual number of trade sequences increases three-
fold over the sample period, reaching a peak of 6,305 in 2009 due to the lower market
capitalizations after the 2008 financial crisis. Notice that median market capitalizations are
almost equal in 2001 and 2009—the values of Vj,t are similar in those years. Despite this, the
median number of trade sequences is much higher in 2009, highlighting the large increase in
8Since we exclude trade sequences spanning more than one trading day, we do not need to adjust rj(k)for stock splits or dividend distributions.
9We discard the few observations where the estimated CAPM beta exceeds 10 or is less than −3.10These trades are flagged“MOC” in TAQ data.
15
Table I: Year-by-year summary statistics of the final sample. The closing shareprice, book-to-market ratio and log of market capitalization are reported as the mediansof their monthly values. $-spreads are the median time-weighted dollar bid-ask spread atthe NBBO. The last column reports the median number of trade sequences (each with anaggregate value of at least Vj,t) in a year across stocks.
Year # of Stocks Share Price Book-to-Market ln(Market Cap) $-spreads # of Sequences2001 1,091 26.50 0.47 21.14 0.072 2,2882002 1,181 24.78 0.49 21.11 0.052 2,5632003 1,170 23.62 0.53 20.98 0.035 2,7522004 1,277 28.86 0.45 21.28 0.031 2,9682005 1,268 30.95 0.43 21.41 0.030 3,3172006 1,277 32.75 0.41 21.53 0.030 3,8552007 1,255 33.70 0.41 21.63 0.029 4,9072008 1,277 24.28 0.53 21.31 0.030 6,2832009 1,198 19.62 0.67 21.11 0.023 6,3052010 1,192 24.95 0.55 21.37 0.020 5,2002011 1,183 28.66 0.50 21.54 0.022 5,0502012 1,169 29.09 0.52 21.52 0.024 4,296
trade volumes. Median dollar-spreads fall from 7.2¢ in 2001 to 2.4¢ in 2012.
B Overview of market activity measure
We first overview basic features of our market activity measure. Figure 2 illustrates how du-
rations of trade sequences vary over time and firm size. In a given year, duration and firm size
have a U-shaped relationship: while larger stocks are more actively traded, Vj,t is proportional
to firm size, which flattens the duration/firm size relationship, especially in recent years. The
very large positions for which we calculate durations for the largest stocks drive their longer
durations; and the modest trading activity for small stocks drives their longer durations.
Overall, the number of trade sequences does not vary wildly with market capitalization.
Durations tend to fall over time, consistent with the rise in market activity. The average
median duration for the second size decile of stocks falls from 33 minutes in 2001 to 26
minutes in 2012; while that for the largest stocks goes from 42 to 26 minutes. Durations are
shortest in 2009 due to the depressed market valuations in that year.
16
5
15
25
35
45
Size decile
Av
era
ge
me
dia
n d
ura
tio
n (
min
ute
s)
Year
Figure 2: Duration of trade sequences by year and by firm size. The figurereports the average median duration (in minutes) of a sequence of trades with a cumulativeaggregate value of at least 0.04% of a firm’s market capitalization. The median duration isfirst calculated on a stock-by-stock basis, and then the average is taken across stocks withina size decile in a year. Size decile 1 contains the smallest firms; decile 10 contains the largest.
We next investigate how the distribution of durations is related to firm size. Figure 3 plots
the cumulative distribution functions (CDFs) of durations over the entire sample period for
representative small, medium and large market capitalization stocks: Hanger, Inc. (HGR),
Archer Daniels Midland Co. (ADM), and International Business Machines Corp. (IBM). All
three stocks have durations that are as short as a few seconds and as long as one hundred
minutes. The large firm (IBM) has far more evenly-distributed durations (its PDF has thin-
ner tails). In contrast, ADM and HGR have many more short durations (about 80% are less
than 40 minutes). HGR has fatter probability density function tails than ADM, indicating
greater contributions from very short and very long durations.
Because a single stock may be unrepresentative, we next compute CDFs of durations at
the stock size decile level, averaging across stocks in the decile. Figure 4 presents the esti-
mated CDFs for the three smallest and three largest size-deciles. Consistent with Figure 3,
Figure 4 shows the distributions for large stocks are far less skewed than those for small
stocks. To address the possibility that the distribution of the full sample could mix the tem-
17
0
.2
.4
.6
.8
1
Cum
ulat
ive
Pro
babi
lity
0 20 40 60 80 100Duration (minutes)
ADM HGR IBM
Figure 3: Distribution of durations of trade sequences for three representativestocks. The figure plots the cumulative distribution functions of durations (in minutes)for International Business Machines Corp. (IBM), Archer Daniels Midland Co. (ADM),and Hanger, Inc. (HGR) from January 1, 2001 to December 31, 2012. Durations are basedon the elapsed time for a sequence of trades with a cumulative aggregate value of at least0.04% of the firm’s market capitalization.
poral effects of increasing market activity with the cross-sectional properties, we decompose
the sample into early and late years. The distributions of durations are more skewed in early
years. More generally, the qualitative pattern that CDFs of durations are more skewed for
smaller firms holds on a year-by-year basis, indicating greater variation in their activity levels.
II Analysis
Market activity, signed trade imbalances and trade size. We begin our substantial
analysis by documenting a well-known, but poorly understood, relationships between mar-
ket activity, signed trade imbalances and trade sizes. We use the Lee and Ready (1991)
algorithm to identify buyer- and seller-initiated trades.11 We define signed trade imbalance
11A trade is marked buyer-initiated (seller-initiated) if the execution price is above (below) the mid-pointquoted price. If the execution and mid-quote prices are the same, the trade is marked buyer-initiated(seller-initiated) if the execution price increased (decreased) with respect to the most recent trade.
18
A-1
:S
mal
lfi
rms,
2001
-201
2A
-2:
Sm
all
firm
s,20
01–2
006
A-3
:S
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2007
–201
2
020406080100
Percentile
05
01
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15
0T
rad
e d
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min
ute
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e d
ecile
1S
ize
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ize
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3
020406080100
Percentile
05
01
00
15
0T
rad
e d
ura
tio
n (
min
ute
s)
Siz
e d
ecile
1S
ize
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020406080100
Percentile
05
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020406080100
Percentile
02
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Tra
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ratio
n (
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ute
s)
Siz
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ize
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cile
10
Fig
ure
4:E
mp
iric
al
cum
ula
tive
densi
tyfu
nct
ions
of
dura
tion
by
size
deci
le.
Dura
tion
sar
eco
mpute
dbas
edon
θ=
0.04
and
thei
rC
DF
sar
epre
sente
dfo
rtw
osu
bse
tsof
larg
ean
dsm
all
stock
s,ov
erth
een
tire
per
iod
(200
1–20
12),
and
the
earl
y(2
001-
2006
)an
dla
ter
(200
7–20
12)
sub-p
erio
ds.
To
calc
ula
teth
eC
DF
ofdura
tion
sfo
rst
ockj
inye
ary,
ever
ym
onth
,w
efirs
tso
rtdura
tion
sin
to20
buck
ets,
each
conta
inin
g5%
ofob
serv
atio
ns.
Then
we
pool
buck
ets
acro
ssdiff
eren
tm
onth
s.T
he
med
ian
ofth
ese
dura
tion
sco
rres
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ds
toth
equan
tile
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isti
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llin
gat
the
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nt
ofth
atdura
tion
buck
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For
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ple
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edia
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the
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edbuck
etof
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ates
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tile
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nex
tbuck
etes
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ates
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tile
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isti
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For
each
dura
tion
buck
et,
we
aver
age
the
stock
-sp
ecifi
cquan
tile
stat
isti
ces
tim
ates
over
year
san
dst
ock
sin
size
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ilex
.P
lott
ing
thes
eav
erag
esag
ainst
thei
rre
leva
nt
per
centi
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oints
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rea
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ortf
olio
,yie
lds
the
empir
ical
CD
Fof
dura
tion
sco
ndit
ional
onfirm
size
.
19
as the share of value-weighted trades out of the total dollar volume in a trade sequence that
is buyer- or seller-initiated (whichever is highest). We also look at how trade size relates to
market activity. To do this, we examine the relation between the number of trades that fill
a trade sequence and market activity—given the same dollar-volume comprising durations
at different levels of market activity, fewer trades reflect larger trades.
Figure 5 shows that (i) average signed trade imbalances rise with market activity—more
active markets feature more aggressive trading on one side of the market; (ii) average signed
trade imbalances rise especially sharply in highly active (short duration) markets; (iii) trade
sizes sharply rise with market activity; (iv) average signed trade imbalances decline with firm
size; and (v) changes in these measures become especially sharp in highly active markets.
The changes in these measures are large: going from least to most active markets, signed
trade imbalances rise by roughly 20%, and for smaller stocks, mean trade sizes double.
By themselves, these facts seem innocuous. Classical models of speculation (e.g., Kyle
(1985), Glosten (1993)) would suggest that signed trade imbalances and trade sizes should
rise with market activity: speculators with substantial private fundamental information will
establish substantial positions based on that information, demanding liquidity by submitting
larger market orders, which will cause market activity, signed trade imbalance, and trade
sizes all to rise.
But, those fundamentals can covary for very different reasons. If liquidity provision is
time varying—if markets are more liquid (“deeper” near good inside quotes) at some times
than others—then traders will respond endogenously to the extant liquidity in their order
composition. If liquidity is unusually high on one side, large traders submit larger market
orders that are filled at these prices, resulting in (i) higher market activity, (ii) fewer trades
comprising a trade sequence, and (iii) higher signed trade imbalances. In contrast, if depth
is modest, an institutional trader seeking to establish or unwind a large position must weigh
how to consume the limited liquidity available via market orders, and how much liquidity to
provide via limit orders in order to reduce price impacts, and the extent to which to delay
20
Small firms Large firms
.55
.6.6
5.7
.75
.8
Ave
rag
e s
ign
ed
tra
de
im
ba
lan
ce
Low HighActivity level
Size decile 1 Size decile 2 Size decile 3
.55
.6.6
5.7
.75
.8
Ave
rag
e s
ign
ed
tra
de
im
ba
lan
ce
Low HighActivity level
Size decile 8 Size decile 9 Size decile 10
02
55
07
51
00
Ave
rag
e n
um
be
r o
f tr
ad
es
Low HighActivity level
Size decile 1 Size decile 2 Size decile 3
05
00
10
00
15
00
Ave
rag
e n
um
be
r o
f tr
ad
es
Low HighActivity level
Size decile 8 Size decile 9 Size decile 10
Figure 5: Average signed dollar-volume and average number of trades by durationgroup. Average signed trade imbalance and average number of trades for trade sequenceswith cumulative value of 0.04% of market capitalization in 2001–2012 versus market activitylevel for the three deciles of smallest and largest stocks. Reported average signed tradeimbalance and average number of trades are, respectively, the average of firm-specificannual mean estimated signed trade imbalance and number of trades (realized over tradesequences), computed per duration group. The average of means is taken across all firms ina given size decile, over the entire sample period by activity level.
submitting orders until liquidity improves. As a result: (i) durations are longer, (ii) more
trades comprise a trade sequence, and (iii) signed trade is more balanced.
Market activity and price impacts. Thus, both information arrival and endogenous liq-
uidity provision choices can lead to higher signed trade imbalances and larger trades in more
active markets. The question becomes: which is it? Reflection suggests how to get at an-
swers: if higher signed trade imbalances reflect information arrival, then price impacts should
rise with market activity; but if high signed trade imbalances reflect endogenous consumption
of unusually high depth at good prices, then price impacts should fall with market activity.
21
Small firms Large firms
.3.4
.5.6
.7.8
Ave
rag
e a
bso
lute
re
turn
(%
)
Low HighActivity level
Size decile 1 Size decile 2 Size decile 3
.24
.26
.28
.3.3
2.3
4
Ave
rag
e a
bso
lute
re
turn
(%
)
Low HighActivity level
Size decile 8 Size decile 9 Size decile 10
.6.8
11
.2
Ave
rag
e r
etu
rn s
tan
da
rd d
evia
tio
n (
%)
Low HighActivity level
Size decile 1 Size decile 2 Size decile 3
.4.4
2.4
4.4
6.4
8
Ave
rag
e r
etu
rn s
tan
da
rd d
evia
tio
n (
%)
Low HighActivity level
Size decile 8 Size decile 9 Size decile 10
Figure 6: Average absolute return and return volatility by duration group.Market impacts of θ = 0.04% of market capitalization in 2001–2012 versus market activitylevel for the three deciles of smallest and largest stocks. The reported average price impactis the average of firm-specific monthly means of absolute returns (realized over tradesequences), computed per duration group. Reported average return volatility is the averageof firm-specific monthly standard deviations of returns (realized over trade sequences),computed per duration group. The averages of means and standard deviations are takenover all firms in a given size decile, over the entire sample period by activity level.
Figure 6 shows how our two market impact measures vary with market activity. For small
and mid-sized stocks, both mean absolute returns and the standard deviation of returns12 fall
sharply as market activity rises.13 This activity-impact relationship is quite pronounced for
smaller firms. For the decile of smallest firms, the price impact of the same dollar volume rises
by almost 50% as markets go from highest to lowest activity levels. This finding indicates
12In the Appendix, we calculate the implied calendar time volatility were the market activity to stayconstant over a fixed time interval (e.g., a trading day). This approach is a better way to measure activityover time because it does not mix different levels of activity that might arise over a non-trivial interval oftime. We show that the implied calendar time volatility rises with market activity.
13The unreported patterns for mid-sized stocks (4 ≤ x ≤ 7) are similar to those of smaller stocks.
22
that for small and mid-sized firms, time varying liquidity provision and consumption drive the
market activity/trade size/signed trade imbalance relationships, and not information arrival.
In contrast, for larger stocks, market impacts do not fall monotonically as market activity
rises. Instead, they peak at relatively high market activity levels. For the decile of largest
firms, market impacts are smallest when markets are least active. Volatility falls off sharply
in less active markets, dropping over 20% from its peak. The patterns for larger stocks reflect
that the tails of their return distributions are quite fat in active markets—suggesting that
for large stocks, information arrival sometimes boosts market activity.14
To affirm these conclusions, we order observed returns for a stock within each duration
group over a year. We then find annual percentile statistics for the return within each dura-
tion (activity) group, on a stock-by-stock basis. Next, we average these percentile statistics
across all stock-years by duration group to obtain the average return, rπ for a percentile π.
Finally, we calculate inter-percentile ranges (e.g., r.99−r.01). Figure 7 depicts the relationship
between inter-percentile ranges and market activity for the smallest and largest firms.
For larger stocks, (i) the interpercentile range r.99−r.01 is highest when markets are most
active—extreme returns are most likely in active markets, and least likely in inactive mar-
kets, reflecting that substantial information arrival results in more active markets; but (ii)
the interpercentile ranges of intermediate percentile statistics (e.g., r.75 − r.25) rise slightly
as activity rises from low to intermediate activity levels, before declining non-trivially as
activity rises at high activity levels, indicating that endogenous liquidity consumption also
drives these price impacts. This combination underlies the single-peak patterns of average
mean returns and return volatility in market activity for large stocks. In contrast, for small
firms, all inter-percentile ranges narrow uniformly as market activity rises.
Thus, endogenous liquidity provision and consumption seems to drive market activity
14In unreported results, we find that trade sequences with longer durations are more likely to start or endnear midday (between 11:45AM and 2:00PM), whereas those with shorter durations are more likely near thebeginning and end of trading days. To ensure that observed patterns are not a time-of-day phenomenon, werepeat the analysis conditioning on the start and end times of trade sequences, for three equally-long windowsover a trading day corresponding to early-, mid-, and late-day. We find similar patterns in each window.
23
Smallest size decile (x = 1) Largest size decile (x = 10)
02
46
Ave
rage
ret
urn
(%)
Low Medium HIghActivity level
r.99−r.01 r.95−r.05 r.75−r.25
.51
1.5
22.
53
Ave
rage
ret
urn
(%)
Low Medium HighActivity level
r.99−r.01 r.95−r.05 r.75−r.25
Figure 7: Estimated interpercentile ranges for realized returns by duration group.Plot of the interpercentile ranges of the distribution of realized returns for different marketactivity levels. The percentile statistics of returns (realized over trade sequences) are firstcomputed stock-by-stock, annually, for each duration group, and then averaged acrossstocks/years. Finally, the differences between selected average percentile statistics (r.99−r.01,r.95− r.05, and r.75− r.25) are computed by activity group. Durations are based on θ = 0.04.
patterns for small and mid-sized firms. For these stocks, price impacts are uniformly smaller
in active markets. This is inconsistent with information arrival driving market impacts. In-
stead, this evidence together with the evidence that the signed trade imbalances also rise
with market activity indicates that active markets are driven by traders choosing to quickly
consume unusually high depth near good inner quotes. In contrast, for large stocks, time-
varying liquidity provision seems to drive the market impact-market activity relationship in
“normal” times (captured by less extreme percentile differences); while in “unusual” times,
high information arrival results in active markets and high price impacts.
To affirm these conclusions, Figure 8 presents the relationship between signed trade im-
balances and (i) mean absolute returns, (ii) return volatility, and (iii) mean number of trades
in a trade sequence. Save for the largest stocks, price impacts and mean number of trades
fall as signed trade imbalance rises.15
In sum, for small and mid-sized stocks, more aggressive trading on one side of the market
is associated with higher market activity, but smaller price impacts. These patterns are
15In unreported results, we verify that these results hold within each level of market activity.
24
Small firms Large firms
.3.4
.5.6
.7
Ave
rag
e a
bso
lute
re
turn
(%
)
.24
.26
.28
.3.3
2.3
4
Ave
rag
e a
bso
lute
re
turn
(%
)
.6.7
.8.9
11
.1
Ave
rag
e r
etu
rn s
tan
da
rd d
evia
tio
n (
%)
.4.4
2.4
4.4
6
Ave
rag
e r
etu
rn s
tan
da
rd d
evia
tio
n (
%)
025
50
75
100
Avera
ge n
um
ber
of tr
ades
Low HighSigned trade imbalance
Size decile 1 Size decile 2 Size decile 3
0500
1000
1500
Avera
ge n
um
ber
of tr
ades
Low HighSigned trade imbalance
Size decile 8 Size decile 9 Size decile 10
Figure 8: Average absolute return, return volatility, and average number oftrades by signed trade imbalance group. Average absolute return, average returnvolatility, and mean number of trades for a trade sequence with cumulative value equalto 0.04% of market capitalization in 2001–2012 versus signed trade imbalance for decilesof smaller and larger stocks. We first sort trade sequences into ten equally-sized groupsof percent signed trades, every month. The reported averages are the sample averages offirm-specific annually-computed mean absolute return, return standard deviation, and meanrealized depth (all assessed over sequences), computed per signed trade imbalance group.The averages of means are taken across all firms in a given size decile, over the entire sampleperiod by signed trade imbalance level.
inconsistent with the predictions of classical models of speculation—Kyle-style competitive
dealership markets and Glosten-style competitive limit order markets. In the former, in-
formed traders optimally choose market order sizes, and dealers set an asset’s price equal to
25
its expected value given the information in net order flow. In the latter, each informed trader
chooses the size of his limit order when competitive liquidity providers price limit orders to
break even conditional on the information contained in the fact that their limit orders were
filled. In both of these settings, larger orders or greater net order flows, i.e., greater signed
trade imbalance, have greater price impacts—the opposite of what we document.
A key premise of these models is that market liquidity is invariant: traders do not have to
choose between consuming and supplying liquidity when establishing or unwinding positions.
In contrast, our findings highlight the impact of time-varying, imperfectly competitive liq-
uidity provision, and the consequences for traders’ choices of how much liquidity to consume
and provide (see, for example, Goettler, Parlour and Rajan (2005), Hollifield, Miller and
Sandas (2004), and Hollifield et al. (2006)). For most stocks, the impacts of the endogenous
choice of order type in response to extant liquidity dominate the predicted relationships
between signed flow, market activity and price impact of standard models of speculation.
Controlling for information arrival and endogenous liquidity consumption. An in-
stitutional investor seeking to establish or unwind a large position would like to time trading
according to market activity in order to minimize trading costs. Such an investor, however,
presumably cannot exploit information arrival to which it is not privy, nor predict when
liquidity provision on the other side of the market will be “unusually high”. This leads us
to try to separate the impacts of information arrival and the endogenous consumption and
provision of liquidity, in order to isolate more primitive market activity/trading cost rela-
tionships that are more stable and predictable. We seek to identify predictable variation in
“real” market activity that may reflect, for example, variation in general investor interest in
trading a stock, and see how trading costs vary with predicted market activity conditions.
To control for endogeneity and information arrival, we use the number of trades filling
the previous trade sequence to instrument for the duration of current trade sequence (cur-
rent market activity). The observation underlying our choice of instrument is that liquidity
persists over time—present and past states of the order book are positively correlated. The
26
endogenous composition of orders, however, hinges only on the contemporaneous state of the
order book, which strongly covaries with current activity as a result. Any effects of past states
of the order book on the current price impact only transmit via the current state of the book.
Using instrumented market activity also filters out the variation associated with information
arrival which is, presumably, unpredictable.16 We next present evidence that cumulatively
indicates that the number of trades in the previous sequence is a valid instrument.
For each stock, we first regress the natural logarithm of duration on a quadratic polyno-
mial of the number of trades in the previous trade sequence, annually (fitting these regres-
sions annually accounts for temporal changes in order sizes).17 The fitted values of these
regressions serve as instrumented levels of activity. We then sort the monthly samples by
instrumented activity and investigate relationships just as we did with raw durations.
To document the instrument’s merits, we start by estimating the linear correlation be-
tween current duration and the number of trades filling the previous trade sequence for each
stock-year. The empirical distribution of this statistic reveals the significance of the correla-
tion parameter for a typical stock. The top row of Figure 9 shows this empirical distribution
for different-sized firms. The correlation coefficient is positive and large for a typical stock in
each size category. Its average rises from 0.2 for small stocks to 0.25 for large stocks. Thus,
past liquidity predicts current activity, especially for larger stocks.
The bottom row of Figure 9 presents the empirical distributions of the F -statistics for the
significance of the first stage log-polynomial regressions at the stock-year level. Over 95%
of the F-statistics exceed 10, indicating that the past number of trades is a valid statistical
instrument for market activity (Staiger and Stock (1997) suggest that F = 10 is a reasonable
cutoff when there is one instrument).
One might be concerned that this persistence could be mirrored by equal persistence
16Recall that we provide evidence that information arrival is more pronounced for larger firms in moreactive market conditions.
17Linear or higher order polynomial specifications lead to similar results; but the quadratic specificationhas the strongest first stage fit.
27
Sm
all
stock
s(D
ecil
es1
to3)
Mid
-siz
edst
ock
s(D
ecil
es4
to7)
Lar
gest
ock
s(D
ecil
es8
to10
)01234
Density
−.5
0.5
1C
orr
ela
tio
n b
etw
ee
n d
ura
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n a
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la
gg
ed
eff
ective
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01234Density
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.4.6
.8C
orr
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n b
etw
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n d
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01234Density
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.4.6
.8C
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to3)
Mid
-siz
edst
ock
s(D
ecil
es4
to7)
Lar
gest
ock
s(D
ecil
es8
to10
)
0.001.002.003.004.005Density
01
00
20
03
00
40
05
00
F−
sta
tistic
0.001.002.003.004.005Density
01
00
20
03
00
40
05
00
F−
sta
tistic
0.001.002.003.004.005Density
01
00
20
03
00
40
05
00
F−
sta
tistic
Fig
ure
9:E
mp
iric
al
dis
trib
uti
on
of
linear
corr
ela
tions
and
log-p
oly
nom
ialF
-sta
tist
ics
by
size
deci
le.
Inth
eto
pgr
aphs,
for
each
stock
-yea
rob
serv
atio
n,
we
find
the
the
corr
elat
ion
bet
wee
ncu
rren
tdura
tion
and
lagg
ednum
ber
oftr
ades
.T
he
kern
elden
sity
ofth
ese
corr
elat
ion
coeffi
cien
tsis
esti
mat
edby
size
grou
ps.
Inth
eb
otto
mgr
aphs,
for
each
stock
-yea
rob
serv
atio
n,
the
curr
ent
log
dura
tion
isre
gres
sed
ona
quad
rati
cp
olynom
ial
ofth
ela
gged
effec
tive
dep
than
dth
eco
rre-
spon
din
gF
-sta
tist
icis
obta
ined
.W
eth
enes
tim
ate,
by
size
grou
ps,
the
kern
elden
sity
for
the
empir
ical
dis
trib
uti
ons
ofth
eF
-sta
tist
ics.
The
vert
ical
das
hed
lines
indic
ateF
-sta
tist
ics
equal
to10
.V
alues
grea
ter
than
250
are
omit
ted
for
vis
ual
clar
ity.
28
in the endogenous provision and consumption of liquidity, which would render futile our
efforts to distinguish between them. To establish that this is not so, we first calculate the
analogous first stage regressions with signed trade imbalance, rather than market activity,
as the endogenous variable. In these unreported results, only about 75% of the associated
F -statistics exceed 10, indicating a weaker relationship between past number of trades and
signed trade imbalance than that with activity.
Our next validation step explores how signed trade imbalances vary with the levels of
activity and instrumented activity. If the endogenous consumption and provision of liquidity
can be predicted in the same way as market activity then these two patterns should be sim-
ilar. To see whether this is so, we first sort the monthly samples of each stock into deciles of
instrumented activity. In each decile, we find the annual mean signed trade imbalance, on
a stock-by-stock basis. Lastly, we calculate the averages of those means across stocks and
years by instrumented activity level. We do the same exercise sorting initially on activity,
in order to compare the patterns of interest. To ensure this comparison is statistically valid,
we restrict the analysis to stocks whose first stage regression F -statistics exceed 10.
Figure 10 shows that signed trade imbalances still co-vary with instrumented market ac-
tivity. However, this association is much weaker than that with the level of market activity
itself. Concretely, the relative change in average signed trade imbalance as market activity
rises from its lowest level to its highest is at least twice what it is as instrumented market
activity goes from its lowest level to its highest. In sharp contrast, we now show that instru-
mentation has dramatic effects on trading cost/market (instrumented) activity relationships.
Figure 11 presents the relationship between mean price impacts and instrumented market
activity—detailing how trading costs vary with predictable components of market activity.
Instrumentation delivers consistent patterns between trading costs and predicted activity for
all stock size groups. For smaller stocks, the relationship between trading costs and predicted
market activity becomes far flatter than the relationship between price impact and actual
market activity—the difference in price impacts between inactive and active markets falls
29
Siz
ed
ecil
e1
Siz
ed
ecil
e2
Siz
ed
ecil
e3
.55.6.65.7.75.8
Average signed trade imbalance
Low
High
Activity/I
nstr
um
en
ted
activity
No
in
str
um
en
atio
nW
ith
in
str
um
en
tatio
n
.55.6.65.7.75.8
Average signed trade imbalance
Low
High
Activity/I
nstr
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
.55.6.65.7.75.8
Average signed trade imbalance
Low
High
Activity/I
nstr
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
Siz
ed
ecil
e8
Siz
ed
ecil
e9
Siz
ed
ecil
e10
.55.6.65.7.75.8
Average signed trade imbalance
Low
High
Activity/I
nstr
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
.55.6.65.7.75.8
Average signed trade imbalance
Low
High
Activity/I
nstr
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
.55.6.65.7.75.8
Average signed trade imbalance
Low
High
Activity/I
nstr
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
Fig
ure
10:
Avera
ge
signed
trade
by
inst
rum
ente
dact
ivit
y(d
ura
tion)
gro
up.
Ave
rage
sign
edtr
ade
imbal
ance
of0.
04%
ofm
arke
tca
pit
aliz
atio
nin
2001
–201
2ve
rsus
inst
rum
ente
dac
tivit
yle
vel
for
the
dec
iles
ofsm
alle
ran
dla
rger
stock
s.T
he
rep
orte
dav
erag
esi
gned
-dol
lar
volu
me
isth
eav
erag
eof
firm
-sp
ecifi
can
nual
mea
ndol
lar-
wei
ghte
dpro
por
tion
ofbuy-
(sel
l-)
orie
nte
dtr
ades
(rea
lize
dov
ertr
ade
sequen
ces)
,co
mpute
dp
erin
stru
men
ted
acti
vit
ygr
oup.
The
aver
age
ofm
eans
ista
ken
acro
ssal
lfirm
sin
agi
ven
size
dec
ile,
over
the
enti
resa
mple
per
iod
by
inst
rum
ente
dac
tivit
ygr
oup.
Ave
rage
sign
edtr
ade
atea
chgr
oup
ofac
tivit
yis
dis
pla
yed
tohig
hligh
tdiff
eren
ces.
Sto
ck-y
ear
obse
rvat
ions
wit
hfirs
tst
age
regr
essi
onF
-sta
tist
ics
less
than
10ar
edro
pp
ed.
30
Siz
ed
ecil
e1
Siz
ed
ecil
e2
Siz
ed
ecil
e3
.3.4.5.6.7.8Average absolute return (%)
Low
High
Activity/I
nstr
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
.3.4.5.6.7.8Average absolute return (%)
Low
High
Activity /
In
str
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
.3.4.5.6.7.8Average absolute return (%)
Low
High
Activity /
In
str
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
Siz
ed
ecil
e8
Siz
ed
ecil
e9
Siz
ed
ecil
e10
.25.27.29.31.33.35Average absolute return (%)
Low
High
Activity /
In
str
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
.25.27.29.31.33.35Average absolute return (%)
Low
High
Activity /
In
str
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
.25.27.29.31.33.35Average absolute return (%)
Low
High
Activity /
In
str
um
en
ted
activity
No
in
str
um
en
tatio
nW
ith
in
str
um
en
tatio
n
Fig
ure
11:
Avera
ge
ab
solu
tere
turn
by
inst
rum
ente
dact
ivit
ygro
up.
Ave
rage
abso
lute
retu
rnof
0.04
%of
mar
ket-
cap
in20
01–2
012
vers
us
inst
rum
ente
dac
tivit
yle
vel
for
the
dec
iles
ofsm
alle
ran
dla
rger
stock
s.T
he
rep
orte
dav
erag
eab
solu
tere
turn
isth
eav
erag
eof
firm
-sp
ecifi
can
nual
mea
nab
solu
tere
turn
(rea
lize
dov
ertr
ade
sequen
ces)
,co
mpute
dp
erin
stru
men
ted
acti
vit
ygr
oup.
The
aver
age
ofm
eans
ista
ken
acro
ssal
lfirm
sin
agi
ven
size
dec
ile,
over
the
enti
resa
mple
per
iod
by
inst
rum
ente
dac
tivit
yle
vel.
Ave
rage
abso
lute
retu
rnve
rsus
acti
vit
yis
also
dis
pla
yed
tohig
hligh
tdiff
eren
ces.
Sto
ck-y
ear
obse
rvat
ions
wit
hfirs
tst
age
regr
essi
onF
-sta
tist
ics
bel
ow10
are
dro
pp
ed.
31
from about 30 basis points to about 7. This finding, in part, highlights the apparently more
pronounced market-chasing of liquidity providing orders in less active markets of smaller
stocks that lead to large cumulative price impacts. For larger stocks, instrumentation has
dramatic effects: the non-monotone relationship between activity and price impact vanishes.
Instead, the average price impact falls as instrumented market activity rises, just like with
smaller stocks. Moreover, the difference between the peak and trough of the trading cost ac-
tivity relationship rises. Although trading costs are higher for smaller stocks, the percentage
decline in trading costs as we go from less active to more active predicted market conditions
is qualitatively the same for all stock size groups.
This uniform relationship indicates that the instrumentation removes the effects of infor-
mation arrival and endogenous liquidity consumption, which varies sharply with stock size.
Indeed, by contrasting the activity/price impact relationships with and without instrumenta-
tion, one can identify the key roles that the contemporaneous choice of order type and infor-
mation arrival play to shape the relationships between market impacts and trading activity.
Temporal changes. Figure 12 plots the ratio of trading costs during the later (2007–2012)
and earlier (2001–2006) time periods at different levels of instrumented market activity. This
time decomposition corresponds to the massive structural changes in U.S. equity markets
which occurred after the implementation of RegNMS in 2006. While bid-ask spreads are
smaller in later years, the evolution of the costs of trading large positions of relevance to in-
stitutional traders is more nuanced. First, while the costs of trading larger stocks have fallen
in more recent years, at each predicted market activity level, the costs of trading small stocks
have risen (the ratio of costs uniformly exceeds one for the decile of smallest stocks, and it ex-
ceeds one in active markets for small and mid-sized firms). Second, the sensitivity of trading
costs to predicted market activity has fallen for small stocks, but risen for very large stocks.
Regression Analysis. We now augment this non-parametric analysis with a formal regres-
sion analysis, estimating the relationship between trading costs and instrumented market ac-
tivity. To standardize measurements across stocks and simplify interpretations, each month
32
Avera
ge
pri
ce
imp
acts
A-1
:S
mall
stock
sA
-2:
Mid
-siz
edst
ock
sA
-3:
Lar
gest
ock
s.75.85.951.051.15
Ratio of trading costs
Low
High
Instr
um
en
ted
activity le
ve
l
Siz
e d
ecile
1S
ize
de
cile
2S
ize
de
cile
3
.75.85.951.051.15
Ratio of trading costs
Low
High
Instr
um
en
ted
activity le
ve
l
Siz
e d
ecile
4S
ize
de
cile
5S
ize
de
cile
6S
ize
de
cile
7
.75.85.951.051.15
Ratio of trading costs
Low
High
Instr
um
en
ted
activity le
ve
l
Siz
e d
ecile
8S
ize
de
cile
9S
ize
de
cile
10
Avera
ge
retu
rnvola
tility
B-1
:S
mall
stock
sB
-2:
Mid
-siz
edst
ock
sB
-3:
Lar
gest
ock
s
.75.85.951.051.15
Ratio of trading costs
Low
High
Instr
um
en
ted
activity le
ve
l
Siz
e d
ecile
1S
ize
de
cile
2S
ize
de
cile
3
.75.85.951.051.15
Ratio of trading costs
Low
High
Instr
um
en
ted
activity le
ve
l
Siz
e d
ecile
4S
ize
de
cile
5S
ize
de
cile
6S
ize
de
cile
7
.75.85.951.051.15
Ratio of trading costs
Low
High
Instr
um
en
ted
activity le
ve
l
Siz
e d
ecile
8S
ize
de
cile
9S
ize
de
cile
10
Fig
ure
12:
Tra
din
gco
sts
in2007–2012
rela
tive
totr
adin
gco
sts
in2001–2006.
The
plo
tssh
owth
era
tio
ofav
erag
etr
adin
gco
sts
in(2
007–
2012
)to
aver
age
trad
ing
cost
sin
(200
1–20
06),
atdiff
eren
tle
vels
ofin
stru
men
ted
mar
ket
acti
vit
yfo
rdec
iles
ofsm
alle
ran
dla
rger
stock
s.T
he
aver
age
abso
lute
retu
rnis
the
aver
age
offirm
-sp
ecifi
can
nual
mea
nab
solu
tere
turn
(rea
lize
dov
ertr
ade
sequen
ces)
,co
mpute
dp
erin
stru
men
ted
acti
vit
ygr
oup.
The
aver
age
ofm
eans
ista
ken
acro
ssal
lfirm
sin
agi
ven
size
dec
ile,
wit
hin
each
sub-p
erio
dby
inst
rum
ente
dac
tivit
ygr
oup.
Tra
de
sequen
ces
are
calc
ula
ted
bas
edon
0.04
%of
mar
ket
capit
aliz
atio
n.
33
we sort a stock’s trade sequences from longest to shortest durations to generate percentiles
of instrumented activity on a monthly basis.18 We employ two types of regression analyses:
1. We regress the absolute return (in basis points) realized over a trade sequence on the
corresponding instrumented activity percentile, per stock-year. To make inferences,
we use the empirical distributions of the slope coefficients from these regressions. We
treat these point estimates as a random sample. Relying on the central limit theorem,
we use the sample mean and standard deviation of slope point estimates to construct
confidence intervals for the average slope for a given size decile.
2. We pool observations across stocks in a size decile over time, inversely weighting ob-
servations for a stock by the number of trade sequences it has in the year (to account
for heterogeneity across stocks in the number of trade sequences). For each decile, we
then regress absolute returns (in basis points) on instrumented activity percentile.
The mean stock-specific slope in the first approach and the point estimate of the slope in
the second, measure by how much (in basis points) trading costs change as instrumented
activity rises from the 1st percentile to the 99th.
Table II shows that both estimation methods deliver qualitatively identical sensitivities
of trading costs to market activity at a given size decile. The estimates indicate that trading
costs fall as instrumented activity rises—the trading costs of establishing or unwinding a
position equal to 0.04% of a stock’s market capitalization fall by 4–8 basis points going from
a market with the least predicted activity to one with the greatest predicted activity. In it
relative terms, price impacts rise by about 12% as one goes from markets with the greatest
predicted activity to the those with the least. The estimates also reveal that the predicted
cost saving from trading in active markets is greater for smaller stocks—the sensitivity of
trading costs to activity roughly doubles as one goes from the largest decile of stocks to the
18Thus, if a stock has 200 trade sequences in a month, the longest duration becomes percentile 0.5, andthe shortest duration becomes percentile 100.
34
Table II: Sensitivity of trading costs to the variation in instrumented activity. For
each stock-year observation, the absolute return realized over a trade sequence is regressed on
the corresponding activity percentile. The empirical distributions of these point estimates within
each size decile are employed to draw statistical inference. The sampling distribution is assumed
to follow a student t distribution (panel A). Observations are pooled across stocks within a size
decile. Absolute return is regressed on the corresponding activity percentile; partial regression
outputs are reported (panel B). The point estimates reflect the amount by which trading costs
change as instrumented market activity rises from lowest to highest percentile.
Panel A: Stock level estimates Panel B: Size group level estimatesSize decile Mean slope Std. Dev. 95% CI Estimate Std. Err. 95% CI
1 −7.91 13.90 (−8.64,−7.18) −7.60 0.14 (−7.86,−7.33)2 −8.23 11.27 (−8.82−7.65) −7.92 0.09 (−8.10,−7.74)3 −7.29 8.43 (−7.73,−6.86) −7.13 0.07 (−7.26,−7.00)4 −6.99 7.25 (−7.37,−6.62) −6.93 0.06 (−7.05,−6.81)5 −6.29 6.40 (−6.62,−5.95) −6.23 0.05 (−6.33,−6.13)6 −5.60 5.75 (−5.90,−5.30) −5.58 0.05 (−5.67,−5.48)7 −5.37 5.50 (−5.65,−5.09) −5.41 0.04 (−5.49,−5.32)8 −5.19 5.45 (−5.47,−4.91) −5.20 0.04 (−5.28,−5.12)9 −4.95 5.25 (−5.22,−4.68) −4.97 0.04 (−5.05,−4.88)10 −3.98 5.79 (−4.28,−3.68) −4.15 0.05 (−4.25,−4.04)
smallest. These findings are consistent with our non-parametric findings illustrated in Fig-
ure 11. Moreover, we predict market activity using a relatively simple prediction procedure.
Thus, these trading cost savings presumably constitute a lower bound on potential savings,
as investors may employ more sophisticated methods to predict market activity than our
simple procedure.
III Priced liquidity
The revolutionary changes in the design of equity markets and the way institutions opti-
mally trade large positions may mean that traditional liquidity measures no longer capture
the impact of individual stock liquidity on asset prices. We now investigate whether our
trade time liquidity measures better measure the impact of liquidity on asset prices, and, if
so, which one is best. A priori, it is not obvious which market activity levels matter “most”
35
to traders who care about price impacts of unwinding positions—price impacts are higher
in less active markets, but institutional traders may be able to time trades in accordance
with market activity. We now show that our trade time liquidity measures better measure
the impact of liquidity on asset prices than traditional measures, and that estimates do not
qualitatively hinge on which of our trade time liquidity measures is used.
We consider nine variants of our liquidity measure, based on different sorts of the trade
sequences on a monthly, stock-by-stock basis. The first sort focuses on only the 40% of trade
sequences with medium levels of signed trade imbalance, which crudely controls for the level
of signed trade imbalance. Within these trade sequences, we sort by activity level (duration).
The other two sorts are conducted over all trade sequences: we sort observations into low,
medium, and high levels of instrumented market activity or raw market activity.
Sort Category Sort Criteria for Trade Sequences
ζ = 1 Lowest 30% of activity levels, medium signed trade imbalanceζ = 2 Middle 40% of activity levels, medium signed trade imbalanceζ = 3 Highest 30% of activity levels, medium signed trade imbalance
ζ = 4 Lowest 30% of instrumented activity levelζ = 5 Middle 40% of instrumented activity levelζ = 6 Highest 30% of instrumented activity level
ζ = 7 Lowest 30% of activity levelζ = 8 Middle 40% of activity levelζ = 9 Highest 30% of activity level
Our trade-time liquidity measure using sort category ζ for stock j in month t, BBDζj,t, is
the average per-dollar absolute returns (scaled by 106) of stock j’s trade sequences in that
category, during the 3-month period ending the last day of month t.
We estimate an augmented CAPM that includes book-to-market, size, and the BBD
liquidity measure as pricing factors:
rj,t − rft = γ0 + γ1βj,t(rmktt − rft ) + γ2BMj,t−1 + γ3Ln(Mj,t−1) + γ4BBD
ζj,t−1 (6)
+120∑m=4
γm5 Dumm + γ6∆(Y IR)j,t + εj,t,
where ∆(Y IR)j,t denotes the change (from month t − 1) in the difference between the in-
36
dustry and market return rmktt , where stock j is excluded when computing the return of the
industry to which stock j belongs. Dumm = 1 if j is observed in month m and is 0 other-
wise.19 These monthly dummies capture any common variation in excess returns resulting,
for example, from changes in the trading environment.
We use GLS estimation with stock fixed effects in the error term. This strategy helps
identify excess return differences caused by unobserved temporally-constant firm attributes.
We allow for firm clusters and estimate robust standard errors to avoid downward biases in
standard error estimates that might result from firm-specific serial correlations in the error
term.20 We control for time-specific error term autocorrelations using ∆(Y IR): ∆(Y IR)
captures common industry shocks that could lead to contemporaneously correlated errors
for stocks in the same industry. In the appendix, we establish robustness of our findings to
alternative ways of dealing with the error term structure. As in Amihud (2002), we drop the
top 1% of observations according to the liquidity factor(s) each month.21 We only drop the
highest values since the measure is positive by construction and has a positively skewed dis-
tribution. Trimming the upper tail helps avoid spurious results driven by the high variability
of extreme observations.22
Table III presents estimates of (6) using each of our nine sort criteria (ζ). All coefficients
have the expected signs and are statistically significant. Regardless of which sort criteria is
used, the coefficient of BBDt−1 is positive and significant, and the overall explanatory power
of each regression is similar. BBD reflects the price impact per million dollars. Substituting
in the median level of BBD yields the monthly return premium that investors demand on
average for facing the cost of possibly unwinding a million dollar position of a typical stock
19The sample period begins in April 2001, due to our three-month rolling window construction.20Petersen (2009) discusses the benefits of this estimation strategy.21Amihud (2002) also trims the bottom 1%; this has no qualitative effects on estimates. We also obtain
similar estimates with alternative trimming thresholds (e.g., top 2% or top 5%).22We also perform these estimates within subsamples of size and time. Each year, we decompose the
samples into three size categories according to their beginning-of-the-year market capitalizations. Then weestimate (6) for each size category in sub-periods 2001–2006 and 2007–2012. The results are qualitativelysimilar to those reported here, with the caveat that the coefficient of illiquidity for the subsample of largestocks becomes insignificant (likely due to the small sample size).
37
Tab
leII
I:A
ugm
ente
dC
AP
Min
cludin
gtr
ade-t
ime
liquid
ity
measu
re.
The
table
rep
orts
esti
mat
ion
resu
lts
for
(6)
usi
ng
the
trim
med
sam
ple
,w
her
eth
eer
ror
term
allo
ws
for
stock
fixed
effec
ts.
Sta
ndar
der
rors
are
clust
ered
atth
est
ock
leve
l.M
onth
dum
mie
sca
ptu
rean
yco
mm
onva
riat
ion
inex
cess
retu
rns.
β(rmkt
t−rf t
)is
the
pro
duct
ofth
efirm
’sm
arke
tb
eta
and
exce
ssm
onth
lyre
turn
bas
edon
mon
thly
hol
din
gp
erio
dre
turn
sfr
omC
RSP
and
1-m
onth
trea
sury
bills
.BM
t−1
isth
epre
vio
us
per
iod’s
Book
-to-
Mar
ket
rati
o.Ln
(Mt−
1)
isth
enat
ura
llo
gari
thm
ofth
efirm
’sm
arke
tca
pit
aliz
atio
nat
the
end
ofpre
vio
us
mon
th.BBDt−
1is
the
pre
vio
us
mon
th’s
trad
e-ti
me
liquid
ity
mea
sure
com
pute
dac
cord
ing
todiff
eren
tζ’s
bas
edon
θ=
0.04
.Sym
bol
s∗ ,∗∗
,an
d∗∗∗
den
ote
sign
ifica
nce
at10
%,
5%,
and
1%le
vels
,re
spec
tive
ly.
Sta
ndar
der
rors
are
rep
orte
din
par
enth
eses
.P
rem
ium
isth
epro
duct
ofth
eBBD
coeffi
cien
tan
dth
eav
erag
eof
the
corr
esp
ondin
gliquid
ity
mea
sure
.
Act
ivit
yat
med
ium
ord
erfl
owIn
stru
men
ted
acti
vit
yA
ctiv
ity
Low
Med
ium
Hig
hL
owM
ediu
mH
igh
Low
Med
ium
Hig
hC
o-va
riat
eζ
=1
ζ=
2ζ
=3
ζ=
4ζ
=5
ζ=
6ζ
=7
ζ=
8ζ
=9
β(rmkt
t−rf t
)0.
704∗∗∗
0.70
2∗∗∗
0.70
4∗∗∗
0.70
9∗∗∗
0.70
5∗∗∗
0.70
3∗∗∗
0.70
7∗∗∗
0.70
2∗∗∗
0.70
8∗∗∗
(0.0
20)
(0.0
20)
(0.0
20)
(0.0
20)
(0.0
20)
(0.0
20)
(0.0
20)
(0.0
20)
(0.0
20)
BM
t−1
0.00
5∗∗∗
0.00
5∗∗∗
0.00
5∗∗∗
0.00
5∗∗∗
0.00
5∗∗∗
0.00
5∗∗∗
0.00
5∗∗∗
0.00
5∗∗∗
0.00
4∗∗∗
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
Ln
(Mt−
1)
−0.
021∗∗∗−
0.02
0∗∗∗−
0.02
1∗∗∗−
0.02
1∗∗∗−
0.02
0∗∗∗−
0.02
1∗∗∗−
0.02
1∗∗∗−
0.02
0∗∗∗−
0.02
1∗∗∗
(0.0
01)
(0.0
01)
(0.0
01)
(0.0
01)
(0.0
01)
(0.0
01)
(0.0
01)
(0.0
01)
(0.0
01)
BBDζ t−
10.
144∗∗∗
0.18
8∗∗∗
0.18
8∗∗∗
0.20
9∗∗∗
0.22
6∗∗∗
0.22
4∗∗∗
0.16
4∗∗∗
0.21
3∗∗∗
0.21
4∗∗∗
(0.0
26)
(0.0
29)
(0.0
35)
(0.0
36)
(0.0
36)
(0.0
37)
(0.0
28)
(0.0
32)
(0.0
38)
Log
-lik
elih
ood
1377
7913
7899
1381
6713
7574
1376
3213
7673
1375
1913
7769
1385
42R
20.
320.
320.
320.
320.
320.
320.
320.
320.
32O
bse
rvat
ion
s16
1187
1611
8016
1172
1600
5116
0047
1600
4616
0560
1611
8416
1189
Pre
miu
m(b
ps)
6.4
7.7
6.7
8.1
8.1
7.4
7.1
8.4
6.8
38
relative to a hypothetical “perfectly liquid” stock. The average monthly premium across
different BBD versions is 7.4bp, and the premium is similar for each BBD measure.23
Comparisons with other liquidity measures. We next compare the improvements in
asset pricing models that obtain from including our liquidity measure with the improvements
obtained from including commonly-used liquidity measures—Amihud (2002), dollar spreads,
and percentage spreads. The standard Amihud illiquidity measure aggregates information
over a trading day and hence may not capture the valuable intraday information in our mea-
sure. This property leads us to consider a high-frequency version of Amihud based on hourly
observations of price changes and dollar volume. Using this alternative measure ensures
that the superior performance of BBD that we document is not due to excessive temporal
aggregation in the daily Amihud measure.
To calculate the standard (low-frequency) Amihud measure (AML), we first compute the
daily per-dollar price impact for each stock j on each day q, DIMPj(q) =|rj(q)|
DV OLj(q), where
rj(q) is the return on day q, and DV OLj(q) is the daily volume. The low-frequency Amihud
measure for stock j in month t is given by the average of DIMPj(q) over trading days q in
the 3-month period ending on the last day of month t, scaled up by 106.
The high-frequency analogue of the Amihud measure, AMHj,t, is calculated as follows.
First, we calculate the hourly per-dollar price impact for a stock as the absolute hourly re-
alized return divided by the corresponding executed hourly dollar-volume.Daily averages of
these hourly price impacts produce daily per-dollar price impacts. The high-frequency Ami-
hud measure for stock j in month t equals the average of the daily per-dollar price impacts
over trading days q in the 3-month period ending on the last day of month t, scaled by 106.
Average dollar bid-ask spreads (DSP ) and percentage bid-ask spreads (PSP ) are calcu-
lated using a time-weighted average of spreads based on the NBBO at a one-second frequency,
23In unreported results, we find that the log-likelihood is maximized by liquidity measures that placemore weight on active markets. This finding suggests that traders may have the ability to time some oftheir trading to occur in more active markets. The log-likelihood, however, is quite flat, indicating that themeasures contain roughly the same amount of information.
39
0.1
.2.3
.4S
pre
ads a
nd A
mih
ud (
low
fre
q)
.002
.004
.006
.008
.01
.012
BB
D a
nd A
mih
ud (
hig
h fre
q)
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
BBD(ζ = 3)
BBD(ζ = 2)
BBD(ζ = 1) Amihud (high frequency)
Amihud (low frequency)
Percentage bid−ask spreads
Figure 13: Monthly median values of alternative liquidity measures over time.The BBD measure (based on θ = 0.04) is calculated over long (ζ = 1), medium (ζ = 2)and short (ζ = 3) durations, when signed order flow is in the middle 40% of a month’sobservations. Also shown is the high frequency Amihud measure (AMH), the low frequencyAmihud measure (AML) and percentage bid-ask spreads (PSP ).
on a daily basis. We calculate a rolling 3-month average of spreads to be consistent with
the other measures. Hence, for stock j, the spread used in month t is the simple average of
DSP or PSP observed in months t− 1, t− 2, and t− 3, denoted DSPj,t−1 and PSPj,t−1.
Figure 13 plots the monthly median value of each liquidity measure. The measures are
highly correlated and display similar temporal variation. The simple average sample corre-
lations between BBD and PSP , AML, and AMH are 0.77, 0.80, and 0.66, respectively.24
Notably, the BBD measures vary more than the other measures.
Table IV summarizes estimates of (6) using these alternative liquidity measures in place
24Average correlations are computed across different versions of BBD.
40
of BBD (using the same error term structure and trimming criteria). Dollar bid-ask spreads
is the only alternative liquidity measure that is not significant (at α = 0.1), and thus our
subsequent analysis compares BBD with the priced measures: AML, AMH, and PSP .
Table IV: Augmented CAPM using standard liquidity measures. The table reportspanel estimates of (6) using AML, AMH, PSP , and DSP . Month dummies capturecommon variation caused by market condition changes, i.e., by common variation in excessreturns over time. Standard errors are clustered at the stock level. Stock fixed effectscapture any fixed heteroskedasticity. Observations in the top 1% tail of the liquiditymeasure are excluded if levels of independent variables are used. Symbols ∗, ∗∗, and ∗∗∗
denote significance at 10%, 5%, and 1% levels, respectively.
Co-variate Estimates
β(rmktt − rft ) 0.708∗∗∗ 0.710∗∗∗ 0.701∗∗∗ 0.709∗∗∗
(0.020) (0.021) (0.020) (0.021) )BMt−1 0.006∗∗∗ 0.005∗∗∗ 0.005∗∗∗ 0.006∗∗∗
(0.002) (0.002) (0.002) (0.002)Ln(Mt−1) −0.024∗∗∗ −0.025∗∗∗ −0.020∗∗∗ −0.026∗∗∗
(0.001) (0.001) (0.001) (0.001)AMLt−1 0.155∗∗∗
(0.041)AMHt−1 0.011∗∗∗
(0.003)PSPt−1 0.049∗∗∗
(0.005)DSPt−1 0.008
(0.027)
Log-likelihood 136897 136499 137913 134589Within–R2 0.31 0.31 0.32 0.31
Observations 161264 160998 161054 160992
The liquidity measures are highly correlated. A pricing regression that includes BBD
and another liquidity measure will have multicollinearity issues.25 To address this, we de-
compose orthogonally each alternative measure, F ∈ {AML,AMH,PSP}, with respect to
25In unreported results, we find that if both BBD and another liquidity risk measure are included inthe same regression, the coefficient on BBD is positive and statistically significant, while that on the otherliquidity measure is negative and insignificant. However, the high correlation in the measures means thatone should interpret such results with caution.
41
BBD, by obtaining the residuals zF
tj and zF
tj from the regressions:
BBDj,t = α1 + α2Fj,t + zF
tj, (7)
Fj,t = α′1 + α′2BBDj,t + zF
tj. (8)
We then estimate (6) with zF
tj or zF
tj as an explanatory variable in place of the liquidity factor.
Table V: Augmented CAPM with orthogonal components of our trade-time liquiditymeasures and low-frequency Amihud measure. The table reports panel estimation resultsfor (6) using the full sample. Standard errors are clustered at the stock level. Stock fixed effectscapture any fixed heteroskedasticity. Month dummies capture any common variation caused bymarket condition changes. We separately present results based on BBD measures correspondingto high, medium and low market activity levels, when order flow is medium (ζ = 1, 2, 3). Durationsare generated based on θ = 0.04. Observations in the upper 1% tail of BBD or AML in amonth are excluded. Stocks with less than 800 trade sequences in a year are excluded from thatyear’s data. Robust standard errors are reported in parentheses. Symbols ∗, ∗∗, and ∗∗∗ denotesignificance at the 10%, 5%, and 1% level, respectively.
Activity at medium order flowCo-variate Low (ζ = 1) Medium (ζ = 2) High (ζ = 3)
β(rmt − rft ) 0.706∗∗∗ 0.708∗∗∗ 0.705∗∗∗ 0.708∗∗∗ 0.706∗∗∗ 0.709∗∗∗
(0.020) (0.021) (0.021) (0.021) (0.020) (0.021)
BMt−1 0.006∗∗∗ 0.006∗∗∗ 0.006∗∗∗ 0.006∗∗∗ 0.005∗∗∗ 0.006∗∗∗
(0.002) (0.002) (0.002) (0.002) (0.002) (0.002)
Ln(Mt−1) −0.023∗∗∗ −0.025∗∗∗ −0.023∗∗∗ −0.025∗∗∗ −0.023∗∗∗ −0.025∗∗∗
(0.001) (0.001) (0.001) (0.001) (0.001) (0.001)
zAML
t−1 0.187∗∗ 0.230∗∗∗ 0.221∗∗∗
(0.046) (0.051) (0.049)
zAML
t−1 −0.183∗∗ −0.199∗∗ −0.158∗∗
(0.079) (0.080) (0.068)
Within-R2 0.32 0.32 0.32 0.32 0.32 0.32Observations 160640 160640 160668 160668 160585 160585
Tables V–VII indicate that BBD is more informative about stock liquidity than either the
Amihud measure or percentage spreads. By construction, Cov(zF
jt, Fjt) = Cov(zF
jt, BBDjt) =
0. Thus, the positive and significant estimates of the coefficients on zF
t−1, given different lev-
els of activity, imply that the illiquidity components that are not measured by F , but are
42
Table VI: Augmented CAPM with orthogonal components of our trade-time liquiditymeasures and high-frequency Amihud measure. The table reports panel estimation resultsfor (6) using the full sample. Standard errors are clustered at the stock level. Stock fixed effectscapture any fixed heteroskedasticity. Month dummies capture any common variation caused bymarket condition changes. We separately present results based on BBD measures correspondingto high, medium and low market activity levels when order flow is medium (ζ = 1, 2, 3). Durationsare generated given on θ = 0.04. Observations in the upper 1% tail of BBD or AMH in amonth are excluded. Stocks with less than 800 trade sequences in a year are excluded from thatyear’s data. Robust standard errors are reported in parentheses. Symbols ∗, ∗∗, and ∗∗∗ denotesignificance at the 10%, 5%, and 1% level, respectively.
Activity at medium order flowCo-variate Low (ζ = 1) Medium (ζ = 2) High (ζ = 3)
β(rmt − rft ) 0.708∗∗∗ 0.711∗∗∗ 0.706∗∗∗ 0.711∗∗∗ 0.707∗∗∗ 0.712∗∗∗
(0.021) (0.021) (0.021) (0.021) (0.021) (0.021)
BMt−1 0.005∗∗∗ 0.006∗∗∗ 0.005∗∗∗ 0.006∗∗∗ 0.005∗∗∗ 0.006∗∗∗
(0.002) (0.002) (0.002) (0.002) (0.002) (0.002)
Ln(Mt−1) −0.023∗∗∗ −0.025∗∗∗ −0.023∗∗∗ −0.025∗∗∗ −0.022∗∗∗ −0.025∗∗∗
(0.001) (0.001) (0.001) (0.001) (0.001) (0.001)
zAMH
t−1 0.111∗∗ 0.141∗∗∗ 0.150∗∗∗
(0.025) (0.029) (0.032)
zAMH
t−1 −0.002 −0.003 −0.003(0.003) (0.003) (0.003)
Within-R2 0.32 0.32 0.32 0.32 0.32 0.32Observations 160283 160283 160281 160281 160229 160229
captured by BBD, reflect valuable information for investors. In contrast, the negative or in-
significant coefficients for zF
t−1 in all pricing regressions suggest that AML, AMH, and PSP
contain little information about the illiquidity component that is not already captured by
BBD. The differences between the coefficients of zF
t−1 and zF
t−1 indicate that BBD is corre-
lated with the unobserved liquidity factor more strongly than the other measures are. These
results reveal the value-added of our trade-time liquidity measure vis a vis these widely-used
standard liquidity measures. The robustness of γ1, γ2, and γ3 when we substitute zF
t−1 for
zF
t−1, implies that BBD is nearly orthogonal to the other asset pricing factors in the model.26
26In unreported results, we verify that the residual analysis yields qualitatively similar findings for theother six versions of BBD, i.e., those corresponding to ζ ∈ {4, . . . , 9}.
43
Table VII: Augmented CAPM with orthogonal components of our trade-time liquiditymeasures and percentage bid-ask spreads. The table reports panel estimation results for(6) using the full sample. Standard errors are clustered at the stock level. Stock fixed effects areintroduced to capture any fixed heteroskedasticity. Month dummies are included to capture anycommon variation caused by market condition changes. We separately present results based onBBD measures corresponding to high, medium and low market activity levels when order flow ismedium (ζ = 1, 2, 3). Durations are generated given on θ = 0.04. Observations in the upper 1%tail of BBD or PSP in a month are excluded. Stocks with less than 800 trade sequences in a yearare excluded from that year’s data. Robust standard errors are reported in parentheses. Symbols∗, ∗∗, and ∗∗∗ denote significance at the 10%, 5%, and 1% level, respectively.
Activity at medium order flowCo-variate Low (ζ = 1) Medium (ζ = 2) High (ζ = 3)
β(rmt − rft ) 0.702∗∗∗ 0.704∗∗∗ 0.702∗∗∗ 0.704∗∗∗ 0.701∗∗∗ 0.704∗∗∗
(0.020) (0.020) (0.020) (0.020) (0.020) (0.020)
BMt−1 0.006∗∗∗ 0.006∗∗∗ 0.006∗∗∗ 0.006∗∗∗ 0.006∗∗∗ 0.006∗∗∗
(0.002) (0.002) (0.002) (0.002) (0.002) (0.002)
Ln(Mt−1) −0.024∗∗∗ −0.025∗∗∗ −0.024∗∗∗ −0.025∗∗∗ −0.024∗∗∗ −0.025∗∗∗
(0.001) (0.001) (0.001) (0.001) (0.001) (0.001)
zPSP
t−1 0.069∗∗ 0.106∗∗∗ 0.125∗∗∗
(0.035) (0.039) (0.046)
zPSP
t−1 0.010 0.004 0.002(0.008) (0.008) (0.009)
Within-R2 0.32 0.32 0.32 0.32 0.32 0.32Observations 160407 160407 160398 160398 160345 160345
We posited at the outset that in recent years, issues with standard liquidity measures have
grown, making it more important to employ alternative liquidity measures such as our trade-
time liquidity measures. To investigate this, we first explore the changes in priced illiquidity
by estimating (6) across two sub-samples: (i) from April 2001 to December 2006, and (ii)
from January 2007 to December 2012. We consider three versions of the BBD measure, com-
puted based on ζ = 1, 2, 3, and three standard liquidity measures, AML, AMH, and PSP .
Table VIII shows that the BBD coefficients are statistically significant in both sub-
periods, and even more strongly in the most recent sub-period. In contrast, each Amihud
measure ceases to be significant in one of the sub-periods. As mentioned earlier, each of these
44
Tab
leV
III:
Su
bsa
mp
leest
imati
on
of
au
gm
ente
dC
AP
Mw
ith
diff
ere
nt
liqu
idit
ym
easu
res.
Th
eta
ble
rep
orts
pan
eles
tim
atio
nre
sult
sof
(6)
for
two
sub
sam
ple
s:A
pri
l20
01–D
ec20
06an
dJan
2007
–Dec
2012
.S
tan
dar
der
rors
are
clu
ster
edat
the
stock
leve
l.S
tock
fixed
effec
tsar
ein
trod
uce
dto
cap
ture
any
fixed
het
eros
ked
asti
city
.M
onth
du
mm
ies
are
incl
ud
edto
cap
ture
any
com
mon
vari
atio
nca
use
dby
mark
etco
nd
itio
nch
ange
s.T
he
firs
tth
ree
colu
mn
sof
agi
ven
sub-s
amp
lep
rese
nt
resu
lts
for
ourBBD
mea
sure
sb
ase
don
hig
h,
med
ium
,an
dlo
wle
vel
sof
mar
ket
acti
vit
yw
hen
ord
erfl
owis
med
ium
(ζ=
1,2,3
)an
dar
eb
ased
onθ
=0.
04.
Th
efi
nal
thre
eco
lum
ns
ofa
pan
elp
rese
nt
esti
mat
ion
resu
lts
for
the
stan
dar
dli
qu
idit
ym
easu
res,AML
,AMH
,an
dPSP
.O
bse
rvat
ion
sin
the
upp
er1%
tail
ofth
ere
leva
nt
liqu
idit
ym
easu
rear
eex
clu
ded
.S
tock
sw
ith
less
than
800
trad
ese
qu
ence
sin
aye
arar
eex
clu
ded
from
that
year
’sd
ata
.R
ob
ust
stan
dard
erro
rsar
ere
por
ted
inp
aren
thes
es.
Th
esy
mb
ols∗ ,∗∗
,an
d∗∗∗
den
ote
sign
ifica
nce
at10
%,
5%
,an
d1%
leve
ls,
resp
ecti
vel
y.
Co-
vari
ate
Apri
l20
01–D
ecem
ber
2006
Jan
uar
y20
07–D
ecem
ber
2012
β(rm t−rf t
)0.
685∗∗∗
0.68
1∗∗∗
0.68
6∗∗∗
0.68
7∗∗∗
0.68
7∗∗∗
0.68
4∗∗∗
0.70
2∗∗∗
0.70
1∗∗∗
0.70
0∗∗∗
0.70
5∗∗∗
0.70
8∗∗∗
0.69
6∗∗∗
(0.0
26)
(0.0
26)
(0.0
27)
(0.0
26)
(0.0
27)
(0.0
26)
(0.0
27)
(0.0
27)
(0.0
26)
(0.0
27)
(0.0
27)
(0.0
26)
BM
t−1
0.00
8∗∗∗
0.00
8∗∗∗
0.00
9∗∗∗
0.00
9∗∗∗
0.00
9∗∗∗
0.00
8∗∗∗
0.00
4∗∗
0.00
4∗∗
0.00
4∗∗
0.00
5∗∗
0.00
4∗∗
0.00
4∗∗
(0.0
04)
(0.0
04)
(0.0
04)
0.00
4)(0
.004
)(0
.004
)(0
.002
)(0
.002
)(0
.002
)(0
.002
)(0
.002
)(0
.002
)
Ln
(Mt−
1)
−0.
042∗∗∗−
0.04
2∗∗∗−
0.04
3∗∗∗−
0.04
3∗∗∗−
0.04
4∗∗∗−
0.03
8∗∗∗−
0.04
8∗∗∗−
0.04
7∗∗∗−
0.04
7∗∗∗−
0.05
2∗∗∗−
0.05
3∗∗∗−
0.04
7∗∗∗
(0.0
03)
(0.0
03)
(0.0
03)
(0.0
03)
(0.0
03)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
03)
BBD
(ζ=
1)t−
10.
083∗∗
0.18
2∗∗∗
(0.0
34)
(0.0
45)
BBD
(ζ=
2)t−
10.
118∗∗∗
0.21
0∗∗∗
(0.0
41)
(0.0
47)
BBD
(ζ=
3)t−
10.
066∗
0.25
0∗∗∗
(0.0
39)
(0.0
58)
AMLt−
10.
049
0.22
3∗∗
(0.0
41)
(0.1
03)
AMHt−
10.
005∗
0.03
7(0
.003
)(0
.025
)
PSPt−
10.
040∗∗∗
0.07
2∗∗∗
(0.0
07)
(0.0
16)
Wit
hin
-R2
0.26
0.26
0.26
0.26
0.26
0.26
0.36
0.37
0.37
0.36
0.36
0.36
Obse
rvat
ions
7918
579
183
7917
479
208
7919
679
082
8200
281
997
8199
882
056
8180
281
972
Pre
miu
m(b
ps)
4.2
5.5
2.7
0.7
0.2
55.0
6.3
7.3
7.5
1.9
0.9
74.1
45
Tab
leIX
:Il
liqu
idit
ycoeffi
cie
nts
inC
AP
Mest
imate
s,com
pare
din
the
two
sub
-peri
od
s,aft
er
ort
hogon
al
decom
posi
-ti
on
.T
he
tab
lere
por
tsp
an
eles
tim
atio
np
arti
alre
sult
sof
(6)
for
two
sub
sam
ple
s,A
pri
l20
01–D
ec20
06an
dJan
2007
–Dec
2012
.S
tan
-d
ard
erro
rsar
ecl
ust
ered
atth
est
ock
level
.S
tock
fixed
effec
tsar
ein
trod
uce
dto
cap
ture
any
fixed
het
eros
ked
asti
city
.M
onth
du
mm
ies
are
incl
ud
edto
cap
ture
any
com
mon
vari
atio
nca
use
dby
mar
ket
con
dit
ion
chan
ges.
Th
eco
effici
ents
ofzF t−
1an
dzF t−
1ar
ere
por
ted
,w
her
eth
ree
vers
ion
sofBBD
(ζ=
1,2,3
)ar
eco
ntr
aste
dag
ain
stth
eth
ree
stan
dar
dli
qu
idit
ym
easu
res.
Ob
serv
atio
ns
inth
eu
pp
er1%
tail
ofth
ere
leva
nt
liqu
idit
ym
easu
res
are
excl
ud
ed.
Sto
cks
wit
hle
ssth
an80
0tr
ade
sequ
ence
sin
aye
arar
eex
clu
ded
from
that
year
’sd
ata.
Ro-
bu
stst
and
ard
erro
rsar
ere
por
ted
inp
aren
thes
es.
Th
esy
mb
ols∗ ,∗∗
,an
d∗∗∗
den
ote
sign
ifica
nce
at10
%,5%
,an
d1%
leve
ls,re
spec
tive
ly.
BBD
vsAML
Co-
vari
ate
Apri
l20
01–D
ecem
ber
2006
Jan
uar
y20
07–D
ecem
ber
2012
ζ=
1ζ
=2
ζ=
3ζ
=1
ζ=
2ζ
=3
zAM
L
t−1
0.18
4∗∗∗
0.19
9∗∗∗
0.08
9∗0.
144
0.23
1∗∗
0.33
2∗∗∗
(0.0
58)
(0.0
68)
(0.0
47)
(0.0
94)
(0.0
97)
(0.1
00)
zAM
L
t−1
−0.
243∗∗∗
−0.
210∗∗
−0.
059
−0.
100
−0.
219
−0.
382∗
(0.0
85)
(0.0
90)
(0.0
60)
(0.2
27)
(0.2
21)
(0.2
03)
BBD
vsAMH
Co-
vari
ate
Apri
l20
01–D
ecem
ber
2006
Jan
uar
y20
07–D
ecem
ber
2012
ζ=
1ζ
=2
ζ=
3ζ
=1
ζ=
2ζ
=3
zAM
H
t−1
0.05
5∗∗
0.06
9∗∗
0.02
70.
169∗∗
0.23
9∗∗∗
0.30
1∗∗∗
(0.0
26)
(0.0
36)
(0.0
29)
(0.0
66)
(0.0
65)
(0.0
72)
zAM
H
t−1
0.00
10.
001
0.00
3−
0.04
5−
0.06
7∗−
0.07
9∗∗
(0.0
02)
(0.0
02)
(0.0
02)
(0.0
41)
(0.0
37)
(0.0
37)
BBD
vsPSP
Co-
vari
ate
Apri
l20
01–D
ecem
ber
2006
Jan
uar
y20
07–D
ecem
ber
2012
ζ=
1ζ
=2
ζ=
3ζ
=1
ζ=
2ζ
=3
zPSP
t−1
−0.
092∗∗
−0.
093∗∗
−0.
149∗∗∗
0.17
9∗∗∗
0.25
3∗∗∗
0.33
6∗∗∗
(0.0
45)
(0.0
47)
(0.0
46)
(0.0
60)
(0.0
63)
(0.0
77)
zPSP
t−1
0.03
9∗∗∗
0.03
7∗∗∗
0.04
2∗∗∗
−0.
045∗∗
−0.
062∗∗∗
−0.
83∗∗∗
(0.0
09)
(0.0
09)
(0.0
07)
(0.0
22)
(0.0
21)
(0.0
24)
46
measures is, to some extent, correlated with the true idiosyncratic stock illiquidity, and as
such, with one another. Accordingly, we estimate (6) using the residuals zF
t−1 and zF
t−1 from
the orthogonal decompositions in the two sub-periods of early and recent years. We consider
three versions of BBD, based on ζ = 1, 2, 3, and three standard liquidity measures, AML,
AMH, and PSP . Since the coefficients of the other pricing variables remain stable across
the sub-periods, Table IX reports only the coefficients on the illiquidity variables.
After filtering BBD’s covariation with commonly-used liquidity measures, BBD con-
tains significant incremental information about liquidity, particularly in more recent years.
In the period 2007-2012, the BBD residuals are priced, while the residuals from the standard
illiquidity measures (both Amihud measures, and percentage bid-ask spreads) are not. In
the earlier period, the only residual liquidity measure priced is percentage bid-ask spreads
(PSP ). These findings indicate that the relative qualities of standard liquidity measures
have deteriorated over time as trading volume has exploded, bid-ask spreads have fallen
and trading strategies have taken on dynamic dimensions. The residual analysis provides
convincing evidence that our BBD trade-time liquidity measures best capture liquidity risk
in today’s trading environments. We also find that despite huge increases in trading volume,
liquidity premia have not fallen post RegNMS.
IV Conclusion
In today’s high frequency trading world, bid-ask spreads are tiny and depth modest. Tra-
ditional measures of trading costs and liquidity no longer describe concerns of institutional
traders who have responded to the massive changes in market structure by dividing orders
into many smaller orders that are dynamically submitted over time and across trading venues.
Our first contribution is to develop a measure of market activity that accounts for dy-
namic order splitting and division of orders against the book. We first characterize the
relative activity of a market for a stock by calculating the duration to unwind a position
worth a fixed share of the market capitalization for that stock—a more active market is one
47
where durations are shorter.
We establish that, consistent with classic models of speculation, as market activity rises,
so do both mean trade size and signed order imbalance—market activity is positively as-
sociated with more aggressive trading. But, the endogenous choices by traders of whether
to consume or provide liquidity may also account for such patterns—traders respond to
more depth near good inside quotes by submitting larger market orders. To distinguish
between these competing explanations we characterize how price impacts vary with mar-
ket conditions. We find that for small and mid-sized firms, price impacts fall with market
activity—indicating that endogenous order composition drives their market activity pat-
terns. In contrast, for larger firms, price impacts first rise and then fall with market activity.
Further investigation reveals that for large stocks, information arrival drives activity and
market impacts at “unusual” times, but endogenous order composition is the primary force
at other times. This evidence suggests that classical models of speculation no longer describe
trading in today’s high frequency financial markets.
We then isolate market activity-trading cost relationships that are stable and predictable.
To do this, we instrument for the duration of the current trade sequence (current market
activity) using the number of trades filling the previous trade sequence. We establish the
validity of this instrument—it controls for both information arrival and the endogenous com-
position of orders. We find that, regardless of firm size, trading costs uniformly fall as instru-
mented market activity rises. The decline in trading costs of 5 to 8 basis points as trading
activity goes from its lowest predicted level to its highest is meaningful, and can be exploited
by institutional traders. The percentage savings on trading costs from trading in more active
markets are comparable for different firm sizes, with higher absolute gains for smaller stocks.
We then use per-dollar versions of this price impact at different activity levels as market
time measures of liquidity (BBD), and explore whether it captures the liquidity concerns of
institutional traders. We estimate an augmented CAPM asset pricing model that includes
book-to-market, size, and our trade activity-based liquidity measures as additional pricing
48
factors. These BBD liquidity measures are consistently priced, and deliver qualitatively
similar liquidity premia. Further, BBD better captures priced liquidity concerns than clas-
sical measures (low- and high-frequency versions of Amihud’s measure or bid-ask spread),
especially in recent years where trading volumes have exploded. This finding highlights the
importance of using measures that capture market conditions, and the success of our measure
at doing so.
Despite increases in trading volumes driven by algorithmic and high frequency trading,
liquidity premiums have increased in recent years. It is possible that some market partici-
pants perceive higher liquidity risk due to the fleeting nature of automated market making,
with little human intervention and largely inaccessible quotes that last for less than a second.
Understanding liquidity provision and risk in this context is especially relevant to the ongoing
debate about the role of high frequency trading and recent calls to limit its role in the market.
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52
VI Appendix
Calendar time relationships. A positive relationship between volatility and market ac-
tivity exists in calendar time. To illustrate, we estimate the return volatility that would
result if a given level of market activity was maintained for an entire trading day. We first
estimate implied return volatilities for a given level of market (instrumented) activity.27 For
a given year y and stock j, we calculate the return variance and average duration (in sec-
onds) of trade sequences belonging to (instrumented) activity level i. Using these values,
and assuming independent price movements across different trade sequences, we estimate the
corresponding calendar day return standard deviation for a given market activity level as:
SDi (y, j) =
√(Return variance of trade sequences)× (number of seconds in trading day)
Average duration in seconds of trade sequences.
Size decile estimates are obtained by averaging SDi (y, j) across all stocks in size decile x.
Figure 14 shows that estimated daily return volatility rises sharply at high levels of mar-
ket activity.28 Daily return volatility, however, only rises with instrumented activity at about
20% of the rate with raw market activity. This finding suggests that the strong relation be-
tween market activity and daily volatility largely reflects the endogenous nature of market
activity. We also see that average daily return volatility varies with firm size—even using
instrumented activity, average daily return volatility is over twice as high for the smallest
stocks as for the largest.
Robustness to Error Structure. We impose a robust standard error structure with firm
level clusters. Our approach follows Petersen (2009) and accounts for potential cross-firm
and cross-time correlations that may be present in the panel data error terms. Failing to
account for these correlations would lead to underestimated standard errors. To account for
common fixed firm and time effects, we estimate a fixed-effect model using generalized least
27To ensure consistency, we only consider stocks whose first stage regressions when instrumenting haveF -statistics that exceed 10.
28An analogous exercise for yearly subsamples reveals similar patterns. Daily return volatility is generallyhigher in 2001 and 2009.
53
B: Small firms A: Large firms
05
10
15
Ave
rag
e d
aily
re
turn
vo
latilit
y (
%)
Low HighActivity level
Size decile 1 Size decile 2 Size decile 3
12
34
5
Ave
rag
e d
aily
re
turn
vo
latilit
y (
%)
Low HighActivity level
Size decile 8 Size decile 9 Size decile 10
2.5
33
.54
4.5
Ave
rag
e d
aily
re
turn
vo
latilit
y (
%)
Low HighInstrumented activity level
Size decile 1 Size decile 2 Size decile 3
1.6
1.8
22
.22
.4
Ave
rag
e d
aily
re
turn
vo
latilit
y (
%)
Low HighInstrumented activity level
Size decile 8 Size decile 9 Size decile 10
Figure 14: Average daily return volatility by market activity group. Averagereturn volatility of 0.04% of market capitalization in 2001–2012 versus market activitylevel for the three deciles of smallest and largest stocks. The reported average daily returnvolatility is the average of firm-specific annually computed daily standard deviations ofreturns SDi (y, j), computed per duration group. The average of daily standard deviations istaken across all firms in a given size decile, over the entire sample period, by activity level.
squares (GLS) with month dummies.
We also consider two other potential error term correlations: non-fixed time and non-fixed
firm effects. Non-fixed firm effects are those causing temporal autocorrelations in a given
firm’s error term. Such effects can arise when a shock to stock j persists (e.g., the effect of a
technology shock on stock’s j’s performance, which decays over time). Non-fixed time effects
make error terms of different firms related in a given period of time where the correlation
differs for different pairs of firms (see Petersen (2009) or Cameron et al. (2008)). These
effects can arise when industry-specific shocks affect most firms in an industry similarly,
54
leaving firms in other industries unaffected.
Fama and MacBeth (1973) adjust for non-fixed time effects by running cross-sectional
regressions period by period. Point estimates from different cross-sections give the empirical
distributions of parameters of interest. Further statistical inference using the Fama-MacBeth
method is based on the averages and standard deviations of empirical distributions, and
hence requires temporal independence of cross-sectional point estimates to obtain unbiased
estimates. Alas, extensive time dependence in our sample precludes using this approach.29
Some estimation strategies call for multi-dimensional clustering when a researcher sus-
pects errors to correlate in multiple dimensions (Cameron, et al. (2008)). To the best of our
knowledge, however, techniques that allow for multi-dimensional and non-nested clustered
error terms have been developed only for ordinary least squares (OLS) estimation. OLS
estimation using panel data is equivalent to pooling the information content of different pe-
riods about a given individual. Such aggregation is inappropriate here. Instead, we employ
the fixed-effects GLS approach as our main estimation strategy. In unreported results, we
estimate model (6) using OLS while clustering by firm, time, and both. Standard errors of
estimates given two dimensional clustering are only modestly larger (less than 40% higher)
than when we account for firm clusters only.
It is inappropriate to try to control for non-fixed time effects by clustering by time when
we use fixed-effects GLS, since the unbalanced nature of the panel makes the clusters non-
nested. Most significantly, the set of stocks in a given market decile varies non-trivially
over time. With non-nested clusters it is not clear which periods’ cross-stock error term
autocorrelations may cause upward biases in estimated standard errors. These concerns lead
us to introduce industry-specific shocks to control for non-fixed time effects associated with
shocks that affect firms within an industry similarly.
Using four-digit GIC industry codes, we identify about 65 industries per month. For each
29Petersen (2009) highlights downward biases in Fama-MacBeth estimated standard errors in the presenceof firm effects.
55
firm j, we compute the associated average industry returns on a monthly basis excluding its
own return. To de-trend, we find the excess industry return against the equally weighted
monthly average return. The first difference of the monthly industry excess return is added
to model (6), as a proxy of non-fixed time effects. Adding this variable to the model de-
creases the coefficient of the CAPM component by about 10%, but does not qualitatively
alter other parameter estimates or standard errors.30
An alternative approach to dealing with non-fixed time effects is to model the serial
correlation of error terms parametrically. Accordingly, we consider error terms that follow
an AR(1) process, εj,t = ρεj,t−1 + νj,t−1, with νit ∼ iidN(0,Σ2ν). We estimate model (6)
with this additional error structure using a fixed-effects GLS strategy. As when we cluster
at the firm level, the autoregressive specification of errors does not affect point estimates.
Furthermore, regardless of the liquidity measure employed, the associated standard errors
are slightly smaller than those reported in Table III.
Overall, these findings support our modeling approach of allowing for firm clustering,
along with stock fixed effects and time dummies. This approach mitigates the downward
biases in estimated standard errors. One need not cluster in the time dimension since the
CAPM term and time dummies capture most commonalities in the time variation of stock
excess returns.
30Nichols and Schaffer (2007) note that with differently-sized clusters, inference could be an issue. Whilehave differently-sized clusters, we have so many firm level clusters that asymptotic properties should hold.
56