Investment-Horizon Spillovers∗
Alex Chinco† and Mao Ye‡
February 16, 2017
Abstract
If short-run traders create risk for long-term investors, then the long-term returns of
stocks with more short-run trading activity should display a risk premium. We investigate whether this risk premium exists using a new statistical tool (the wavelet-variance
estimator) that measures horizon-specific trading activity. Stocks with more trading
activity at the 1-minute horizon have monthly abnormal returns of 0.49%. And, the
particular collection of stocks with the most short-run trading activity changes rapidly
from month to month, suggesting that the estimated risk premium isn’t due to some
persistent firm characteristic that’s both attracting short-run traders and increasing
long-term risk.
JEL Classification: C55, C58, G12, G14
Keywords: Investment Horizon, Trading Volume, Wavelet Variance
∗
We thank John Campbell, Bruce Carlin, Andrew Ellul, Fangjian Fu, Xavier Gabaix, Joel Hasbrouck,
Ohad Kadan, Andrew Karolyi, Robert Korajczyk, Pete Kyle, Lubos Pastor, Ronnie Sadka, Allan Timmermann, Jeff Wurgler, Amir Yaron, and Haoxiang Zhu as well as seminar participants at the University of
Illinois, the Northern Finance Association conference, the Financial Research Association conference, Washington University of St. Louis, and Michigan State for extremely helpful comments and suggestions. Tao
Feng, Rukai Lou, Xin Wang, Robbie Xu, Fan Yang, and Chao Zi provided excellent research assistance. This
research is supported by National Science Foundation grant 1352936 (joint with the Office of Financial Research at the U.S. Treasury Department). This work also uses the Extreme Science and Engineering Discovery
Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575. We
thank David O’Neal of the Pittsburgh Supercomputer Center for his assistance with supercomputing, which
was made possible through the XSEDE Extended Collaborative Support Service (ECSS) program.
Current Version: http://www.alexchinco.com/investment-horizon-spillovers.pdf
†
UIUC College of Business. www.alexchinco.com; [email protected]; (916) 709-9934.
‡
UIUC College of Business. www.yemaofin.com; [email protected]; (217) 244-0474.
1
1
Introduction
Long-term fundamentals don’t change fast enough to account for short-run fluctuations in
algorithmic traders’ demand. Long-term fundamentals get revealed in quarterly earnings
reports and annual financial statements, but algorithmic traders turn over “half their net
holdings in 137 seconds (Kirilenko et al., 2011).” And, these two timescales differ by 5
orders of magnitude:
60 seconds 60 minutes 24 hours 365 days
1 year
×
×
×
×
= 2 × 105 .
137 seconds
1 minute
1 hour
1 day
1 year
To put this number in perspective, note that there have only been around 2×105 generations
since the human/chimpanzee divergence (MacKay, 2003), so long-term fundamentals literally
evolve at the speed of speciation from an algorithmic trader’s point of view.
But, even if it can’t be explained by changes in the long-term “fundamentals of the
company that’s being traded,” 1 algorithmic traders’ demand can still affect long-term returns.
There are two opposing views on this idea. Some people are worried about that high-speed
traders are completely “rigging the market (Lewis, 2014)” and “destroying investor confidence
(Arnuk and Saluzzi, 2012).” If this is what’s going on, if algorithmic traders are making it
risky for other investors to trade on information about companies’ long-term fundamentals,
then less of this information is getting incorporated into prices, which is bad. But, it’s not
obvious that this is what’s going on. And, other people feel that the popular outcry against
high-speed traders is overblown, that “we shouldn’t get ourselves dragged into a hyped-up
war over a matter that doesn’t affect investors very much.” 2
If algorithmic traders make it risky for investors to trade on information about companies’
long-term fundamentals, then stocks with more trading activity at the 1-minute horizon
should command a risk premium in their monthly returns. In this paper, we empirically
investigate whether short-run trading activity affects investors operating at longer horizons
by measuring the monthly risk premium associated with trading activity at the 1-minute
horizon. In the past, measuring horizon-specific trading activity has meant finding data
on individual traders’ portfolio holdings, which is hard to come by. To get around this
obstacle, we introduce a new statistical tool (the wavelet-variance estimator) that recovers
horizon-specific trading activity from trading-volume data.
We present two main results. First, we find that the stocks with the most trading
activity at the 1-minute timescale earn a risk premium of 0.49% per month, suggesting that
trading activity at the 1-minute timescale creates some risk for long-term investors. And,
we document that the subset of stocks with the largest fraction of trading activity at the
1
2
SEC chairman Mary Schapiro at speaking engagement for the Christian Science Monitor. Feb 22, 2012.
Asness, C. and Mendelson, M. High-Frequency Hyperbole. Wall Street Journal. Apr 1, 2014.
2
Pr(1mint+1 = x|1mint = Hi)
Main Results
αHi-Lo , % per month
1.0
0.5
0.0
-0.5
-1.0
1min
0.4
0.3
0.2
0.1
0.0
Hi
1day
4
3
2
Lo
Figure 1: (Left) x-axis: timescale of trading activity. Black line: average difference between the monthly abnormal returns of the stocks with the largest fraction and the stocks with
the smallest fraction of their trading activity at each timescale. Grey band: 95% confidence
interval for this estimate at each timescale. Reads: “The quintile of stocks with the largest
fraction of trading activity at the 1-minute horizon in month t realize monthly abnormal
returns that are 0.49% higher than the quintile of stocks with the smallest fraction of trading
activity at the 1-minute horizon in month t.” (Right) Distribution of trading activity at the
1-minute timescale in month (t + 1) given that a stock belonged to the quintile of stocks with
the largest fraction of trading at the 1-minute timescale in month t. Reads: “If a firm is in
the quintile of stocks with the largest fraction of trading activity at the 1-minute horizon in
month t, then there is only a 37% chance that this firm is still in the quintile of stocks with
the largest fraction of trading activity at the 1-minute horizon in month (t + 1).”
1-minute horizon changes rapidly from month to month, suggesting that the risk premium
isn’t due to some persistent firm characteristic that’s both attracting algorithmic traders
and increasing long-term risk. In other words, our results strike a middle ground between
the two opposing viewpoints on the effects of algorithmic trading. There is a risk premium
associated with trading activity at the 1-minute horizon, but the size hardly suggests that
markets are rigged.
Wavelet-Variance Estimator. The wavelet-variance estimator, which was first introduced
to the statistics literature in Percival (1995), allows us to decompose changes in each stock’s
trading volume into horizon-specific components. Just like a pop song that sounds different
during the chorus and the verse (Bello et al., 2005), changes in trading volume are dominated
by different frequencies at different points in time. And, the wavelet-variance estimator
identifies which horizons dominate and when, it reveals what fraction of the changes in a
stock’s trading volume over the course of a month come from minute-to-minute fluctuations
rather than hour-to-hour or day-to-day fluctuations.
3
Volume is a natural place to infer traders’ investment horizons. Any time traders have
linear price impact—that is, convex transaction costs (Kyle, 1985; Huberman and Stanzl,
2004)—it will be optimal for them to space out their order flow as much as possible. Or,
in the words of Bertsimas and Lo (1998), “execution costs will be minimized at the point
where the marginal execution costs are equated across all periods. . . hence the trade sizes
should be set equal across all periods.” Thus, traders have information that will expire in
a day, then they will increase their demand throughout the day. If they have information
that will expire in a minute, then they will increase their demand for a minute. If they have
information that will expire in an hour, then they will increase their demand for an hour.
We use wavelets rather than traditional tools like Fourier analysis and autoregressions—
approaches that are, in fact, equivalent via the Wiener-Khinchin Theorem (Engelberg,
2007)—because these existing approaches impose periodic structure that we do not observe
in our data. In reality, trading volume is asynchronous. You might see a 1-hour burst of
trading activity starting at 9:37am, then another starting at 11:03am, and a third at 2:42pm;
but, you rarely see bursts of trading activity arriving and subsiding every hour like clockwork. Wavelet methods can handle these sorts of asynchronous shocks. Fourier analysis and
autoregressions cannot since they both require covariance stationarity.
Summary Statistics. We compute the fraction of trading activity at each timescale 1minute or longer using data on NYSE-listed stocks from the trade and quote (TAQ) database
aggregated at the 1-minute horizon from January 2002 to December 2010. We find substantial cross-sectional heterogeneity in horizon-specific trading activity. Among the quintile of
stocks with the largest fraction of short-run trading activity, 48.4% of the trading-volume
variance occurs at the 1-minute timescale; by constrast, among the quintile of stocks with
the smallest fraction of short-run trading activity, only 39.6% of the trading-volume variance
occurs at the 1-minute timescale. Thus, in our baseline specification, the fraction of tradingvolume variance at the 1-minute timescale typically varies by (48.4 − 39.6)/39.6 = 22.4%. This
cross-sectional dispersion persists even when we control for stock characteristics like volume,
size, and liquidity.
Main Results. This paper presents two main results. First, we show that the quintile of
stocks with the largest fraction of trading activity at the 1-minute timescale has abnormal
returns that are 0.49% per month higher than the quintile of stocks with the smallest fraction
at the 1-minute timescale. The pattern reverses itself at longer horizons. Abnormal returns
are monotonically decreasing in investment horizons as shown in the left panel of Figure
1. The quintile of stocks with the largest fraction of trading activity at the 1-day horizon
has abnormal returns that are 0.77% per month lower than the quintile of stocks with the
4
smallest fraction. And, the pattern is not explained by risk factors, liquidity measures, or
information risk. Thus, 2 stocks with the same fundamentals might realize different monthly
returns because 1 stock has more trading activity at the 1-minute horizon than the other.
Second, we document that the monthly risk premium associated with short-run trading
activity is unlikely to be explained by some persistent firm characteristic that’s both attracting algorithmic traders and also adding risk because the subset of stocks with the largest
fraction of trading activity at the 1-minute horizon changes rapidly from month to month. If
a firm is in the quintile of stocks with the largest fraction of trading activity at the 1-minute
horizon in month t, then there is only a 37% chance that this firm is still in the quintile of
stocks with the largest fraction of trading activity at the 1-minute horizon in month (t + 1)
as shown in the right panel of Figure 1. Put differently, algorithmic traders’ choice of which
stocks to trade can’t be explained by persistent firm characteristics or risk factors. We view
this rapid churn in where algorithmic traders choose to trade as a new stylized fact that’s
worthy of further investigation.
Measure Verification. Finally, we give supporting evidence that our measure of trading
activity at the 1-minute horizon is actually measuring short-run trading activity. To do this,
we examine a subset of 60 stocks where an exchange has collected microdata on individual
traders’ investment horizons from 2008 to 2009. We verify that the fraction of a stock’s
trading activity at the 1-minute horizon is highly correlated with this short-run trading
indicator. Thus, our wavelet-based measure is consistent with prior measures. But, we
emphasize that we can apply the wavelet-variance estimator to the full cross-section of NYSElisted stocks, not just this subset of 60 stocks.
1.1
Related Literature
This paper borrows from and brings together several different strands of literature.
Investment Horizon. Other papers have explored horizon-specific features of asset returns. For example, van Binsbergen et al. (2012) analyze the pricing of dividend strips,
Dew-Becker and Giglio (2016) use frequency analysis to decompose asset returns rather
than volume, and Giglio et al. (2014) look at the pricing of leaseholds that expire 100 years
or more into the future. Kamara et al. (2015) study the pricing of risk factors across investment horizons. Jagannathan and Wang (2007) find that the consumption-based capital
asset-pricing model (CCAPM) does a better job explaining the cross-section of asset returns
at the annual horizon than it does at the quarterly horizon. Cella et al. (2013) give evidence
that investors with short horizons amplify the effects of market-wide negative shocks by demanding liquidity at times when other potential buyers’ capital is scarce. Beber et al. (2012)
5
estimate asset-pricing models that allow for various investment horizons and find that the
cross-section of expected returns is better explained by exposures to risks that are measured
at long horizons.
We makes three contributions to the horizon literature. First, we propose a way to
measure the cross-section of traders’ investment horizons using only trading volume data.
For other asset-pricing applications of wavelets see Gençay et al. (2001), Hasbrouck (2015),
and Bandi et al. (2015). Second, we document substantial cross-sectional heterogeneity
in traders’ investment horizons, and demonstrate that variation in the amount of trading
activity at the 1-minute horizon is priced in monthly returns after controlling for existing
factors. Finally, the existing literature on investment horizons tends to consider relatively
long horizons, such as quarterly or yearly; however, our paper shows that trading at 1-minute
horizon affects monthly returns.
Additional Applications. The ability to measure traders’ investment horizons on a stockby-stock basis has a wide range of applications in finance. For instance, one of the key limits
on arbitrageurs in the limits-to-arbitrage literature is their investment horizon (De Long
et al., 1990). Moving from behavioral finance to corporate finance, there is a large literature
looking at the optimal responses of rational firm managers to biased shareholders (Baker and
Wurgler, 2013). This bias often times takes the form of short-term or myopic beliefs. More
generally, there is a large literature looking at feedback effects between financial markets and
the real economy (Chen et al., 2007). Differences in horizon might be an important driver for
these sorts of feedback effects. Our paper relates to the long-stranding literature examining
the impact of short-term market microstructure on long-term returns. We contribute to
this literature by showing that traders’ investment horizon directly matters after controlling
two existing channel: liquidity (Amihud and Mendelson, 1986; Pastor and Stambaugh, 2003;
Acharya and Pedersen, 2005) and information risk (Easley et al., 2002).
Trading Volume. Finally, this paper builds on a literature linking trading volume and
asset returns. For instance, Campbell et al. (1993), Lo and Wang (2000), and Lee and
Swaminathan (2000) look at the link between trading volume and return predictability. All
of this earlier work looks at a single horizon; whereas, in the current paper we study the
cross-horizon impacts of differences in trading activity on asset returns.
2
Wavelet-Variance Estimator
What fraction of the variation in a stock’s trading volume comes from changes in trading
activity from 1 minute to the next? What about from 1 hour to the next? Or, from 1
6
A Wavelet and Some Non-Wavelets
Haar Wavelet
Fixed Effect
Sine Wave
1
√
1/ 2
0
√
−1/ 2
−1
0
2
4
6
0
2
√
4
6
0
2
4
6
√
Figure 2: Haar Wavelet: variable that is 1/ 2 during the first period, −1/ 2 during the
second period, and 0 otherwise. Fixed Effect: variable that is 1 during the first period and 0
otherwise. It’s not a wavelet because it is not mean zero. Sine Wave: sin[τ/(2 · π)]. It’s not a
wavelet because it is not normalizable.
day to the next? We begin our analysis in this section by introducing the wavelet-variance
estimator to answer these questions. In this section we use τ = 1, 2, . . . , T to count periods.
2.1
What Is A Wavelet?
This subsection gives a brief introduction to wavelets. We refer interested readers to Mallat
(1999) or Percival and Walden (2000) for a more thorough discussion.
Regression Analysis. Using wavelets to analyze a stock’s trading volume means regressing
the demeaned trading-volume time series for each stock, vlm τ − hvlm τ i, on a collection of
right-hand-side variables, {w(h,`),τ }, which are indexed by both their horizon, h = 0, . . . , (H−
1), and their location on the timeline, ` = 0, . . . , (Lh − 1):
X
vlm τ − hvlm τ i =
θ̂(h,`) · w(h,`),τ + ετ .
(1)
h,`
We will return to this double-indexing scheme momentarily, but for now we want to emphasize a different point. Using wavelet methods does not require us to learn an entirely
new empirical technique; rather, it requires us to use a special collection of right-hand-side
variables in our regressions that satisfy the three key properties defined below.
Three Key Properties. Any right-hand-side variable with the following 3 properties is a
7
wavelet. First, a wavelet has to be orthogonal. That is,
Z ∞
w(h,`),τ · w(h0 ,`0 ),τ · dτ
0=
(2)
−∞
for any pair of wavelets indexed by (h, `) and (h0 , `0 ) with (h, `) 6= (h0 , `0 ). Orthogonality
means that each of the right-hand-side variables is capturing different information about
trading volume. For example, time fixed effects like the one illustrated in the middle panel
of Figure 2 satisfy this orthogonality condition.
Second, a wavelet has to be mean-zero:
Z ∞
w(h,`),τ · dτ.
(3)
0=
−∞
This is where the “wave” in “wavelet” comes from. Each right-hand-side variable has to have
a peak that reaches above zero and a trough that dips below zero, and it is this wave-like
property that allows wavelet coefficients to capture variation from 1 period to the next.
Note that time fixed effects do not satisfy this second property. They are not mean-zero
variables and can only reflect whether a particular time period has a higher-than-average
trading volume. It is possible to compare groups of time fixed-effect estimates after the
fact to see how much higher-than-average 1 time period is than another, but there is no
period-to-period comparison embedded in the coefficients themselves.
Third and finally, a wavelet has to be normalizable:
Z ∞
[w(h,`),τ ]2 · dτ.
(4)
1=
−∞
This is where the diminutive “let” in “wavelet” comes from. For example, the canonical sine
wave does not satisfy this condition because it is not square integrable. Wavelets are useful
tools precisely because they are not spread out across the entire timeline. To illustrate,
think about regressing trading volume on the sine wave from the right panel of Figure 2.
The estimated coefficient will reflect the average differences in a stock’s volume in the first
period and the second period, the third period and the fourth period, the fifth period and
the sixth period, and so on. . . But, what if there is only a positive event in the second period
and a negative event in the fifth period? A small number of non-localized waves cannot
capture these effects since they go in opposite directions. A small number of wavelets can.
Double Indexing. Wavelets have two indexes: h and `. Because they behave like waves,
we need to specify the frequency at which they oscillate with the index h = 0, . . . , (H − 1).
Because wavelets behave like time fixed effects, we also need to specify their location on the
timeline with the index ` = 0, . . . , (Lh − 1). For example, the wavelet in the left panel of
Figure 2 completes 1 cycle every 2 periods, so we need to specify its horizon. But, it only
ever completes 1 of these cycles: up in the first period and down in the second. Thus, in
8
addition to its horizon, we also need to specify a wavelet’s location on the timeline.
2.2
Haar Wavelets
This paper uses Haar wavelets because of their similarity to fixed effects and first differences.
Definition 1 (Haar Wavelet). A Haar wavelet is a piecewise-constant function of time,


√ 1

if
` · 2h+1 ≤ τ < (` · 2h+1 + 2h )

 2(h+1)
1
(5)
w(h,`),τ = − √ (h+1)
if
(` · 2h+1 + 2h ) ≤ τ < (` + 1) · 2h+1 ,
2




0
else
where h = 0, . . . , (H − 1) denotes the wavelet’s horizon and ` = 0, . . . , (Lh − 1) denotes the
wavelet’s location. The longest horizon, H, is the largest integer such that H < log2 (T ).
So, for example, if there are T = 8 observations in a time series, then the maximal horizon
would be H = 2. Similarly, the number of locations at each horizon is given by Lh = T /2h+1 .
For instance, at horizon h = 2 there is only 1 location, ` = 0, since L2 = 8/23 = 1; whereas,
at horizon h = 0 there are four different locations, ` = 0, 1, 2, 3, since L0 = 8/21 = 4.
8-Period Example. Examine the demeaned trading-volume time series with 8 observations
in Figure 3:
h
i>
vlm − hvlm τ i = 0 100 0 0 0 −100 0 0 .
(6)
The Haar wavelets involves running a regression with seven right-hand-side variables. The
first variable will be the Haar wavelet at horizon h = 2:
h
i
1
>
w(2,0)
=√ × 1
(7)
1
1
1 −1 −1 −1 −1 .
8
This variable captures how the first half of the time series differs from the second half. The
√
>
scaling factor of 1/ 8 ensures normality, 1 = w(2,0)
w(2,0) . The second and third variables will
be the wavelets at horizon h = 1:
"
#
>
w(1,0)
1
1 −1 −1
0
0
0
0
1
=√ ×
.
(8)
>
4
w(1,1)
0
0
0
0
1
1 −1 −1
These variables capture how the first quarter of the time series differs from the second quarter
and how the third quarter of the time series differs from the fourth quarter. Finally, the
9
fourth, fifth, sixth, and seventh variables will be the wavelets at horizon h = 0:


>
w(0,0)
1 −1
0
0
0
0
0
0


>
 0

w(0,1)
0
1
−1
0
0
0
0
1

.
√
=
×

>
2 
w(0,2)
0
0
0
0
1
−1
0
0


>
0
0
0
0
0
0
1 −1
w(0,3)
(9)
These variables capture how the first period differs from the second period, the third period
differs from the fourth period, and so on. . .
Now, let us estimate the coefficients by computing the solution to the equations below:


  √
√
√
1/ 4
1/ 2
1/ 8


0
0
0
0
0
θ̂(2,0)


  1√
√
√
1/ 4
 100  / 8
0
−1/ 2
0
0
0 
 θ̂(1,0) 

  √

√
√
 0   1/ 8 −1/ 4
1
0
0
/ 2
0
0 



 
θ̂(1,1) 
√

 0   1/√8 −1/√4
1


0
0
−/ 2
0
0 

 
(10)
 θ̂(0,0) 

= √
√
√
.
1/ 4
1/ 2
 0  −1/ 8
0
0
0
0 

 θ̂(0,1) 

 
√
√
−100 −1/√8
1/ 4
0
0
0
−1/ 2
0 



 
√
√  θ̂(0,2) 
 0  −1/√8
1/ 2
0
−1/ 4
0
0
0

 
0
√
−1/
8
0
√
−1/
4
0
0
0
√
−1/
2
θ̂(0,3)
The unique combination of coefficients that satisfies both vlm 1 − hvlm τ i = 100 and vlm 5 −
hvlm τ i = −100 while leaving the remaining volume levels at zero is
i>
h
√
1/2 −1/2 −1/√2
1/√2
.
(11)
θ̂ = 100 × 1/ 2
0
0
This solution has a natural interpretation. For example, θ̂(2,0) > 0 since the first half of the
time series is larger on average than the second half of the time series. Likewise, θ̂(0,0) < 0
since the trading volume in the second period is greater than the trading volume in the first.
Finally, θ̂(0,1) = 0 since there is no difference between the third and fourth periods. Thus, the
Haar-wavelet coefficients telescope down, capturing differences between successively smaller
and smaller chunks of the time series, adjusting the scale at each level so that the wavelets
remain normalized.
Waves vs. Wavelets. In addition to illustrating how wavelets work, this example also
makes for a nice case study showing why this paper uses wavelets rather than sine and
cosine waves to analyze each stock’s demeaned trading volume time series. Namely, wavelets
allow us to identify asynchronous horizon-specific shocks. To illustrate, imagine regressing
the trading-volume time series from Figure 3 on a sine wave:
h
i
s> =
(12)
1 −1
1 −1
1 −1
1 −1 .
In order to fit the positive shock in the second period, the estimated coefficient would have to
10
Demeaned Trading-Volume Time Series
vlm τ − hvlm τ i
100
50
0
-50
-100
Start
1
2
3
4
5
6
7
End
Figure 3: Simulated data for demeaned trading-volume time series with T = 8 observations
that has a +100 share shock in the second period and a −100 share shock in the sixth period.
be −100, which would imply that the prediction for period six would be +100 shares leading
to a prediction error of 200 shares. Conversely, in order to fit the negative shock in the
sixth period, the estimated coefficient would have to be +100, again leading to a 200-share
prediction error in period 1. Looked at in this light, wavelet analysis is a generalization
of Fourier analysis that allows the estimated coefficients to vary across different cycles at
the same horizon. Wavelets are much better tools than sine and cosine waves for capturing
these sorts of local asynchronous changes in a time series. We discuss our choice of wavelet
methods over Fourier analysis in more detail in Appendix A.
Irregular Length. Finally, in all the earlier examples, the length of the time series was
a power of 2; however, in real-world applications, this is not the case. This detail does not
affect the intuition behind using the wavelet-variance estimator to measure the amount of
trading activity at each horizon, but it is important to account for this complication in our
empirical analysis. So, we use a modified version of the discrete wavelet transform known as
the maximal-overlap discrete wavelet transform (MODWT). See Percival (1995).
2.3
Estimator Definition
We now define the wavelet-variance estimator, which decomposes each stock’s trading-volume
variance into horizon-specific components.
11
Derivation. Let us first rewrite the usual trading-volume variance using Haar wavelets:
Var[vlm τ ] = 1/T · (vlm)> (vlm)
> = 1/T · Wθ̂
Wθ̂
= 1/T · θ̂ > W> W θ̂.
(13a)
(13b)
(13c)
Since wavelets are orthogonal and normal, W> W = I. So, Equation (13c) simplifies to:
X
2
.
(14)
θ̂(h,`)
Var[vlm τ ] = 1/T · θ̂ > θ̂ = 1/T ·
h,`
Definition 2 (Wavelet-Variance Estimator). The wavelet-variance estimator at horizon h
is given by
Wvarh [vlm τ ] =
1/T
·
Lh
X
2
θ̂(h,`)
.
(15)
`=0
This estimator captures the amount of variation in a stock’s trading volume that can be
explained by comparing successive periods of length 2h . Adding up the wavelet variances at
P
each horizon gives back the total variance of the time series, Var[vlm τ ] = h Wvarh [vlm τ ].
8-Period Example, Continued. To make this definition concrete, we compute the wavelet
variance of the time series in Figure 3. First, the total variation of the time series is:
02 1002 02 02 02 (−100)2 02 02
+
+
+
+
+
+
+
= 2,500 sh2/hr.
(16)
Var[vlm τ ] =
8
8
8
8
8
8
8
8
A quarter of this variation comes from comparing the first half of the time series to the
√
2
second half since Wvar2 [vlm τ ] = 1/T · θ̂(2,0)
= 1/8 ·(100/ 2)2 = 625 sh2/hr. Another quarter of this
variation comes from comparing successive quarters of the time series since Wvar1 [vlm τ ] =
625 sh2/hr. Finally, the remaining half of the variation comes from comparing successive
periods since Wvar0 [vlm τ ] = 1,250 sh2/hr.
3
Summary Statistics
When we apply the wavelet-variance estimator to minute-by-minute data for NYSE-listed
stocks, we find substantial cross-sectional heterogeneity in trading-activity timescales. The
fraction of trading-volume variance at the 1-minute timescale typically varies by more than
20% across NYSE-listed stocks. In the analysis below we use t to count months and τ to
count minutes within each month. Because this distinction is important to our analysis,
we flag monthly variables in teletype font to help the reader keep track of the different
timescales. So, for example, vlm n,τ is stock n’s trading volume in minute τ and vlmn,t is
stock n’s trading volume in month t.
12
3.1
Data Description
Below we give a description of both where our data come from and also how we define our
key monthly variables of interest.
Data Sources. We study minute-by-minute trading-volume records from the Trade and
Quote (TAQ) database over the period from January 2002 to December 2010. Our sample
begins in 2002 because this is the first full year with decimal pricing and because on January
24th, 2002 the New York Stock Exchange launched OpenBook, which gave a real-time view
of the exchange’s limit-order book for all NYSE-listed securities. We restrict our sample to
observations of NYSE-listed stocks during regular trading hours, 9:30am to 4:00pm. So, for
each stock there are 60 min/hr × 6.5 hr/day = 390 minute-level trading-volume observations per
day. Appendix B gives a example of the output from the wavelet-variance estimator.
Data on stock returns and characteristics comes from the combined CRSP and Compustat
datasets available through WRDS. We exclude small and illiquid stocks from our sample by
removing stocks that have either a price less than $5 on the last day of the previous month
or a market capitalization in the bottom tercile of the market on the last day of the previous
month. Each month we only use stocks that have at least 17 days of data. This leaves us
with approximately 1,000 stocks in each month.
Variable Definitions. In the analysis below, we compute three variables that capture the
fraction of trading activity across a broad range of timescales.
Definition 3 (Trading Activity at the 1-Minute Timescale). Ratio of the wavelet-variance of
the minute-level trading-volume time series at the 1-minute timescale during a given month
to the total variance of the minute-level trading-volume time series during that month:
Wvar0 [vlm n,τ |τ ∈ t]
.
(17)
1minn,t =
Var[vlm n,τ |τ ∈ t]
Definition 4 (Trading Activity at 1-Hour Timescale). Ratio of the wavelet-variance of the
minute-level trading-volume time series at the 1-hour timescale during a given month to the
total variance of the minute-level trading-volume time series during that month:
Wvar6 [vlm n,τ |τ ∈ t]
.
(18)
1hourn,t =
Var[vlm n,τ |τ ∈ t]
Definition 5 (Trading Activity at 1-Day Timescale). Ratio of the wavelet-variance of the
minute-level trading-volume time series at the 1-day horizon during a given month to the
total variance of the minute-level trading-volume time series during that month:
Wvar9 [vlm n,τ |τ ∈ t]
1dayn,t =
.
(19)
Var[vlm n,τ |τ ∈ t]
13
The first variable, 1minn,t , reflects the fraction of a stock’s trading-volume variance in month
t that occurs at the 1-minute timescale. The subscript of h = 0 on the associated waveletvariance estimator indicates that the estimator is capturing differences in trading volume
from 20 = 1 minute to the next. Similarly, the variables 1hourn,t and 1dayn,t capture the
fraction of a stock’s trading-volume variance in a given month that occurs at roughly the
1-hour and 1-day horizons respectively.
3.2
Cross-Sectional Variation
Let’s now look at how the timescale of trading activity varies across stocks.
Portfolio Averages. Table 1 gives summary statistics for value-weighted portfolios sorted
by 1minn,t , 1hourn,t , and 1dayn,t . The rows labeled “Hi” are associated with the quintile
of stocks that has the largest fraction of trading-volume variance at a particular timescale.
Whereas, the rows labeled “Lo” are associated with the quintile of stocks that has the smallest
fraction of trading-volume variance at that timescale. Table 1 reveals substantial crosssectional heterogeneity in the fraction of trading activity at each timescale. For example,
Panel (A) indicates that the highest quintile has 8.87/39.57 = 22.42% more trading activity at
the 1-minute timescale than the lowest quintile.
The fraction of trading activity of at all horizons has to sum to unity for each stock in
each month. Stocks with the largest fraction of trading activity at the 1-minute timescale will
tend to have less trading activity at all longer timescales as a result. But, the relationship
between any pair of timescales is far from mechanical. For example, Table 2 shows that
stocks in the quintile with the largest fraction of trading activity at the 1-minute timescale
are in the quintile with the smallest fraction of trading activity at the 1-day timescale only
62% of the time. Therefore, a stock might have a large fraction of its trading activity at the
1-minute timescale, a moderate fraction of its trading activity at the 1-day timescale, and
very little activity at all timescales in between.
Economic Magnitude. How should we interpret a number like 48.43% in the first row
of Panel (A) in Table 1? Consider a short thought experiment. Suppose that, instead of
looking at real-world trading-volume data, we studied a simulated time series with eight
periods where trading value in each period was generated by a coin flip: +100 shares if
heads and 0 shares if tails. In this setting, we know that all of the variation in occurs at
the shortest timescale since trading volume in each period is independently drawn from an
identical distribution. There are no shocks that directly affect the first half of the time series.
But, notice that the probability that we draw the same number of heads in the first four
14
periods as in the second four periods is only
2 2 2 2 2 !
1
4
4
4
4
4
×
+
+
+
+
= 27.3%.
8
2
0
1
2
3
4
(20)
So, in spite of the fact that there is no shock directly affecting the first half of the time series,
72.7% of the time it is going to look like there is some variation at this timescale. Most of
the time this difference will only be 1 heads, but the difference will still exist.
Generalizing this example, we show in Appendix C that, if a stock’s trading-volume data
were generated by a white-noise process at the 1-minute horizon, then this stock would have
50% of its trading-volume variance at the 1-minute horizon, 25% of its trading-volume variance
at the 2-minute horizon, 12.5% of its trading-volume variance at the 4-minute horizon, and
so on. . . So, if we return to our original question, then we can now read 1minH,t = 48.43% as
telling us that the stocks with the largest fraction of their trading activity at the 1-minute
timescale have magnitudes that are quite close to those that would be generated by white
noise. These stocks are dominated by trading activity at the 1-minute timescale.
Full Spectrum. We primarily focus on how trading activity at the 1-minute timescale
bubbles up and affects realized returns at the 1-month timescale. But, we want to emphasize
that there are interesting features in the data at other horizons as well. To give one example,
notice that there is a peak in the fraction of trading activity at the 1-hour timescale during
the financial crisis in Panel (B) of Table 1. One possibility is that the peak comes from
stocks that had hour-long sell-offs. This nuance doesn’t play a role in our analysis, but it
does highlight the methodological contribution in this paper.
3.3
Related Variables
Other variables are related to trading-activity timescale, but these variables do explain the
variation in timescales across stocks.
Firm Characteristics. Table 3 shows that firms in the quintile of stocks with the highest
fraction of their trading activity at the 1-minute timescale are different than firms in the
quintile of stocks with the lowest fraction of trading activity at the 1-minute timescale.
Specifically, they tend to be larger, more frequently traded, and more liquid than those in
the quintile of stocks with the lowest fraction at the 1-minute horizon. Trading volume is
defined as the base-2 logarithm of the daily average trading volume for the stock in month t.
The illiquidity measure comes from Amihud (2002) and reflects the daily-average absolute
percent change in prices associated with each million dollars of traded volume in month t.
Sorting on Residuals. Nevertheless, we show in Table 4 that, even among stocks with the
15
same characteristics, there is still substantial cross-sectional heterogeneity in trading-activity
timescales. We do this by first running cross-sectional regressions that predict each stocks’
trading activity at the 1-minute timescale:
1minn,t = α̂t + β̂t · vlmn,t + evlm
n,t ,
1minn,t = α̂t + β̂t · mcapn,t + emcap
n,t ,
and 1minn,t = α̂t + β̂t · illiqn,t + eilliq
n,t .
(21a)
(21b)
(21c)
The regression residual evlm
n,t , for instance, then denotes the portion of a stock’s trading
activity at the 1-minute timescale in month t that is unexplained by its trading volume.
Table 4 reports the same summary statistics as in Table 1 except that the portfolios are
now sorted on these residuals. In the raw summary statistics in Table 1, we saw that the
fraction of trading at the 1-minute timescale was 8.87%-points higher for firms in the quintile
of stocks with the largest fraction of short run trading. Panel (A) of Table 4 shows that
controlling for total trading volume only drops this difference to 7.92%-points. Controlling for
each stock’s market capitalization and liquidity has an even smaller effect on this difference.
Table 5 also shows that we get similar results if we control for all three of these characteristics
simultaneously by using the residuals from the regression below:
1minn,t = α̂t + β̂tvlm · vlmn,t + β̂tmcap · mcapn,t + β̂tilliq · illiqn,t + emv
n,t .
(22)
This regression-based approach to controlling for stock characteristics is analogous to first
sorting stocks into quintiles based on a particular characteristic and then, within each bucket,
sorting stocks on the fraction of their trading-volume variance at each horizon. Table 6
verifies that the regression-based approach and double sorting both yield similar results.
The benefit of using the regression-based approach is that it makes multivariate controls,
like the ones in Table 5 possible.
4
Main Results
We now present our two main findings. First, we show that the quintile of stocks with the
largest fraction of trading activity at the 1-minute timescale has abnormal returns that are
0.49% per month higher than the quintile of stocks with the smallest fraction at the 1-minute
timescale. Second, we document that the monthly risk premium associated with short-run
trading activity is unlikely to be explained by some persistent firm characteristic that’s both
attracting algorithmic traders and also adding risk because the subset of stocks with the
largest fraction of trading activity at the 1-minute horizon changes rapidly from month to
month. Thus, 2 stocks with the same fundamentals can realize different monthly returns
16
because 1 stock has a larger fraction of its trading activity at the 1-minute horizon.
4.1
Risk Premium
First, we find that the quintile of stocks with the highest fraction of trading activity at the
1-minute horizon in month t has abnormal returns that are 0.49% per month higher than the
quintile of stocks with the lowest fraction at the 1-minute horizon in month t.
Abnormal Returns. In our main specification, we estimate abnormal returns by regressing the excess returns of value-weighted portfolios sorted by the fraction of trading-volume
variance at a given horizon in month t on the excess returns to the Fama and French (1993)
factors:
rxi,t = α̂i + β̂iMkt · rxMkt,t + β̂iSmB · rxSmB,t + β̂iHmL · rxHmL,t + ei,t .
(23)
If the spread in excess returns across the horizon-specific portfolios can be explained by
their differing exposures to the Fama and French (1993) risk factors, then the estimated α̂i ’s
should be zero. This is not what we find in Table 7. Panel (A) shows that the quintile
of stocks with the largest fraction of trading activity at the 1-minute horizon in month t
has abnormal returns that are 0.49% per month higher than the quintile of stocks with the
smallest fraction at the 1-minute horizon. Thus, differences in the fraction of trading activity
at the 1-minute horizon are priced in monthly returns.
This finding is inconsistent with the hypothesis that, because algorithmic traders that
rebalance their portfolios on a minute-to-minute basis usually close out their positions at
the end of each trading day, their demand doesn’t affect long-term investors much. Their
relative trading activity has a statistically measurable and economically meaningful effect
on monthly returns. Market-microstructure effects do not just average out at the monthly
horizon. Activity at each investment horizon does not exist in a bubble.
Portfolio Characteristics. We showed in Section 3 that the stocks with the largest fraction
of their trading-volume variance at the 1-minute horizon tend to be big, frequently traded,
liquid stocks. After viewing the results in Table 7, one might be concerned that these stock
characteristics are driving the abnormal returns. To address this concern, we again sort the
stocks into portfolios based on the residual fraction of trading-volume variance at the 1minute horizon as in Equations (21a), (21b), and (21c). As described before, this procedure
is tantamount to sorting first on 1 of these characteristics and then sorting within each
bucket on the fraction of trading-volume variance at the 1-minute horizon. We then re-run
the regression described in Equation (23) using these new characteristic-controlled portfolios.
17
Table 8 displays these results. We find that, after controlling for each stock’s characteristics, the abnormal returns earned by the quintile of stocks with the largest fraction
of trading-volume variance at the 1-minute horizon get larger, jumping to 0.92% per month
when conditioning on trading volume in the current month. After controlling for each stock’s
market capitalization, the difference jumps to 0.76% per month. When we control for each
stock’s Amihud (2002) liquidity in month t, the difference remains steady at 0.49% per month.
The observable characteristics of the stocks involved do not appear to explain the result.
1-Month vs. 1-Day Horizons. Panel (C) of Table 7 shows that the quintile of stocks
with the largest fraction of trading activity at the 1-day horizon in month t has abnormal
returns that are 0.77% per month lower than the quintile of stocks with the smallest fraction
at the 1-day horizon. Thus, the earlier result reverses itself when sorting by the fraction of
trading-volume variance at the 1-day horizon. What’s more, we already saw in the left panel
of Figure 1 that abnormal returns are monotonically decreasing in the timescale of trading
activity. The stocks with the largest fraction of trading activity at the 1-minute horizon
have higher abnormal returns than the stocks with the largest fraction of trading activity at
the 1-hour horizon, and both of these groups have higher abnormal returns than the stocks
with the largest fraction of trading activity at the 1-day horizon.
4.2
Algorithmic-Attention Puzzle
Second, we document that the monthly risk premium associated with short-run trading activity is unlikely to be explained by some persistent firm characteristic that’s both attracting
algorithmic traders and also adding risk because the subset of stocks with the largest fraction
of trading activity at the 1-minute horizon changes rapidly from month to month. We view
this rapid churn in where algorithmic traders choose to trade as a new stylized fact that’s
worthy of further investigation.
Transition Probabilities. We find that the fraction of trading at the 1-minute horizon
for a single stock varies considerably over time. Table 9 shows the transition probabilities
between portfolios sorted by the fraction of trading at the 1-minute horizon, 1-hour horizon,
and 1-day horizon. Panel (A) shows that, if a stock is among the quintile of stocks with
the largest fraction of trading activity at the 1-minute horizon in the current month, then
it only has a 37% chance of remaining among this group of stocks next month. This rapid
churn implies that factor exposures computed over several months are going to do a poor
job of explaining abnormal returns earned by portfolios sorted on horizon-specific trading.
And, this is exactly what we find when we try to explain the risk premium associated with
18
trading activity at the 1-minute horizon with monthly risk factors.
Liquidity Measures. Amihud (2002) defines liquidity as the absolute percent change in
prices associated with an additional $1 million in traded volume over the course of a day.
But, this is not the only way to define liquidity. We consider the liquidity factor proposed in
Pastor and Stambaugh (2003), which measures the average affect of an increase in demand
on day d on returns on day (d + 1) over the course of a month for each stock. The authors
compute a traded-factor, which is available through WRDS, by averaging these stock-specific
estimates each month. We include the Pastor and Stambaugh (2003) liquidity factor in our
regression from Equation (23),
rxi,t = α̂i + β̂iMkt · rxMkt,t + β̂iSmB · rxSmB,t + β̂iHmL · rxHmL,t + β̂iPS · rxPS,t + ei,t ,
(24)
and we find that exposure to this risk factor does not explain the difference in portfolios’
abnormal returns. These results are reported in Table 10.
This measure of liquidity is specifically focused on effects at the daily horizon, so there is
a good reason to believe, a priori, that exposure to this Pastor and Stambaugh (2003) traded
factor is only going to be loosely related to the abnormal returns earned by the portfolios
sorted on 1minn,t . After adding in the Pastor and Stambaugh (2003) liquidity factor, the
abnormal returns to a portfolio that is long the stocks with the most trading activity at the
1-minute horizon and short the stocks with the least trading activity at the 1-minute horizon
rises to 0.54% per month.
In contrast to Pastor and Stambaugh (2003), Acharya and Pedersen (2005) show that
the risk of becoming illiquid, not just illiquidity itself, is priced. To examine whether or not
illiquidity risk is driving our results, we first compute each stock’s exposure to the Amihud
(2002) liquidity factor over the previous two years using time-series regressions for each stock:
rxn,t = θ̂0,n + θ̂nMkt · rxMkt,t + θ̂nSmB · rxSmB,t + θ̂nHmL · rxHmL,t
+ θ̂nilliq · illiqn,t + en,t ,
(25)
Then, we sort stocks in month t based on their residual trading activity at the 1-minute
horizon after accounting for their exposure to liquidity risk, θ̂nilliq , as defined in the crosssectional regression below,
illiq
1minn,t = α̂t + β̂t · θ̂n,t
+ eAP
n,t .
(26)
We then re-estimate the regression from Equation (23) using these new portfolios as our test
assets. Panel (B) of Table 10 shows that illiquidity risk does not explain the difference in
abnormal returns between the quintile of stocks with the largest fraction of trading activity
at the 1-minute horizon and the quintile of stocks with the smallest fraction.
Finally, we explore a different measure of liquidity that decomposes price impact into
19
transient and permanent components found in Sadka (2006). Non-traded factors reflecting both the transient and permanent components of price impact are available on Ronnie
Sadka’s website.3 To examine whether or not the permanent and transient components of
price impact are driving our results, we first compute each stock’s exposure to these nontraded factors over the previous two years using time-series regressions for each stock, just
like before:
rxn,t = θ̂0,n + θ̂nMkt · rxMkt,t + θ̂nSmB · rxSmB,t + θ̂nHmL · rxHmL,t
+ θ̂nperm · permn,t + θ̂ntrans · transn,t + en,t ,
(27)
Then, we sort stocks in month t based on their residual trading activity at the 1-minute
horizon after accounting for the permanent and transient components of price impact, Sadka
n,t ,
as defined in the cross-sectional regression below,
perm
trans
1minn,t = α̂t + β̂tperm · θ̂n,t
+ β̂ttrans · θ̂n,t
+ eSadka
n,t .
(28)
If the abnormal returns earned by the quintile of stocks with the highest fraction of tradingvolume variance at the 1-minute horizon were due to differences in the transient component
of price impact, then creating portfolios using the residuals from these cross-sectional regressions should eliminate the abnormal returns. Panel (C) of Table 10 displays the results.
Accounting for the permanent and transient components of price impact does not explain
the difference between the abnormal returns in month t of the quintile of stocks with the
highest fraction of trading activity at the 1-minute horizon in month t and the quintile of
stocks with the least.
Information Risk. In addition to looking at measures of liquidity related to price impact,
we also investigate the role of informed trading. Specifically, we use the probability of
informed trading (PIN) measure used in Easley et al. (1996) and available on Jefferson
Duarte’s website.4 This measure is available at the stock-by-year level.
Like before, we sort stocks in month t based on their residual trading activity at the
1-minute horizon after accounting for their probability of informed trading in that month,
PIN
n,t , as defined in the cross-sectional regression below,
1minn,t = α̂t + β̂t · PINn,t + ePIN
n,t .
(29)
If the abnormal returns earned by the quintile of stocks with the highest fraction of tradingvolume variance at the 1-minute horizon were due to differences in the amount of informed
trading, then creating portfolios using the residuals from these PIN regressions should eliminate the abnormal returns. This is not what we find in Table 11. After conditioning on the
3
4
See http://www2.bc.edu/ronnie-sadka/Sadka-LIQ-factors-1983-2012-WRDS.xlsx.
See Duarte and Young (2009) and http://www.owlnet.rice.edu/~jd10/pins.zip.
20
amount of informed trading in each stock, the abnormal returns earned by the quintile of
stocks with the highest fraction of trading-volume variance at the 1-minute horizon become
even larger, jumping from 0.49% in Panel (A) of Table 7 to 2.12% in Table 11. Put another
way, the stocks with the highest monthly returns are the ones with the highest fraction of
their trading activity at the 1-minute horizon relative to what would be expected given their
informed trading.
Past Returns. Past performance can drive future rebalancing behavior. If a stock performs very well, then more traders will have to add or remove it from their portfolio. So, at
first, one might be concerned that trading-volume variance at the 1-minute horizon might
be driven by past performance and thereby proxying for momentum. We run two sets of
analyses to verify that our results are not being driven by momentum.
First, we re-run the analysis in Table 7 including a momentum factor (Carhart, 1997):
rxi,t = α̂i + β̂iMkt · rxMkt,t + β̂iSmB · rxSmB,t + β̂iHmL · rxHmL,t
+ β̂iMom · rxMom,t + ei,t .
(30)
If the momentum were driving the spread in abnormal returns across portfolios sorted by
the fraction of trading-volume variance at the 1-minute horizon, then we would expect the
estimated α̂i ’s to drop to zero. Panel (A) of Table 12 shows that this is not the case. Even
after adding in the momentum factor, the abnormal returns to a portfolio that is long the
stocks with the most trading activity at the 1-minute horizon and short the stocks with the
least trading activity at the 1-minute horizon stays at 0.49% per month. What’s more, the
loading on the momentum factor is extremely small.
In addition, we also sort stocks in month t based on their residual trading activity at the
1-minute horizon after accounting for their excess returns in the previous month, erxlag
n,t , as
defined in the cross-sectional regression below,
1minn,t = α̂t + β̂t · rxn,t−1 + erxlag
n,t .
(31)
If the abnormal returns earned by the quintile of stocks with the highest fraction of tradingvolume variance at the 1-minute horizon were due to changes in past returns, then creating
portfolios using the residuals from these cross-sectional regressions should eliminate the
abnormal returns. This is not what we find in Panel (B) of Table 12.
Trading-Strategy Returns. If the collection of stocks with lots of trading activity at the
1-minute horizon changes rapidly from month to month, then it should only be possible to
trade on this information over short horizons. This is exactly what we find in Table 13. We
study the abnormal returns in the first 1, 2, 3, and 4 weeks of month t to a trading strategy
is long the quintile of stocks with the largest fraction of trading activity at the 1-minute
21
horizon in month (t − 1) and short the quintile of stocks with the smallest fraction in month
(t − 1). When the holding period is only 1 week, the strategy generates an abnormal return
relative to the Fama and French (1993) 3-factor model of 1.01% per month. These abnormal
returns decay to 0 as the holding period increases out to a month.
5
Measure Verification
Finally, we give supporting evidence that our measure of trading activity at the 1-minute
horizon is actually measuring short-run trading activity.
Data Description. To do this, we use NASDAQ data on a stratified sample of 60 Russell
3000 stocks during 2008 and 2009 that are listed on the NYSE (Brogaard et al., 2014).
For these 60 NYSE-listed stocks, trading on the NASDAQ exchange accounts for a sizable
fraction of their total trading volume because the Unlisted Trading Privileges Act allows
stocks to be traded outside their listing venue (O’Hara and Ye, 2011). Thus, to the extent
that traders’ investment horizons are similar on both the NYSE and NASDAQ exchanges,
we can use data from the NASDAQ to characterize the distribution of investors’ horizons
for NYSE-listed stocks. For the 60 NYSE-listed stocks, NASDAQ categorizes 26 firms as
high-frequency traders (HFTs) based on their knowledge of customers and their analysis of
these firms’ trading activities. For example, NASDAQ tracks how often a firm’s net intra-day
position crosses zero, the length of a firm’s average order duration, and its order-to-trade
ratio (Brogaard et al., 2014). Note that the NASDAQ identifies HFTs that only engage in
proprietary trading. HFT desks in large institutions, such as Goldman Sachs and Morgan
Stanley, are excluded. Small HFTs that route their orders through these large institutions
are also ignored. So, the non-HFT category may contain HFTs.
Variable Construction. NASDAQ classifies each trade into 1 of 4 categories (HH, HN,
NH, and NN) based on how high-frequency traders (HFTs) were involved in the trade. HFTs
engage in one type of algorithmic trading (Jones, 2013; Hasbrouck and Saar, 2013), commonly
referred to “proprietary algorithmic trading.” The HH category denotes trades where one
HFT takes liquidity from another HFT. The HN category denotes trades where an HFT
takes liquidity from a non-HFT. The NH category denotes trades where a non-HFT takes
liquidity from an HFT. Finally, the NN category denotes traders where one non-HFT takes
liquidity from another non-HFT. Taking liquidity means using market orders and providing
liquidity means using limit orders.
We construct variables capturing the fraction of HFTs in each stock according to NASDAQ. In our main analysis, we consider the ratio of pure HFT trading volume to total
22
trading volume,
onlyHftn,t =
HHn,t
HHn,t + HNn,t + NHn,t + NNn,t
.
(32)
We also show that our results are quite similar if we examine the ratio of volume involving
an HFT in any capacity to total trading volume,
HHn,t + HNn,t + NHn,t
.
(33)
anyHftn,t =
HHn,t + HNn,t + NHn,t + NNn,t
Table 14 shows the summary statistics for each of these variables.
Regression Results. In our main specification—that is, the first and third columns in
Table 15—we regress the fraction of each stock’s trading-volume variance that comes at the
1-minute horizon on NASDAQ’s HFT fraction,
^ n,t + en,t
1]
minn,t = β̂ · onlyHft
(34a)
^ n,t + en,t ,
and 1]
minn,t = β̂ · anyHft
(34b)
after normalizing these variables to have zero mean and unit standard deviation, x̃ =
(x − µx )/σ . The first column of Table 15 reveals that a 1-standard-deviation increase in the
x
fraction of trading volume coming from HFTs according to NASDAQ is associated with a
0.41-standard-deviation increase in the fraction of trading-volume variance occurring at the
1-minute horizon. The third column of Table 15 shows that is point estimate is roughly the
same, 0.34, if we use the alternative definition of high-frequency trading from NASDAQ,
anyHft n,t . The second and fourth columns show that this cross-sectional relationship persists when we include lags of both variables. The fifth column represents an autoregression
of the normalized fraction of a stock’s trading-volume variance at the 1-minute horizon in
the current month on the normalized fraction of the stock’s trading-volume variance in the
last month at the 1-minute horizon.
6
Conclusion
There’s a big debate about how algorithmic traders’ trading activity affects long-term investors. This paper adds empirical content to this debate by notice that, if algorithmic
traders make it risky for investors to trade on information about companies’ long-term fundamentals, then stocks with more trading activity at the 1-minute horizon should command
a risk premium in their monthly returns. We find that the stocks with the most trading
activity at the 1-minute timescale earn a risk premium of 0.49% per month, suggesting that
trading activity at the 1-minute timescale creates some risk for long-term investors. And,
we document that the subset of stocks with the largest fraction of trading activity at the
23
1-minute horizon changes rapidly from month to month, suggesting that the risk premium
isn’t due to some persistent firm characteristic that’s both attracting algorithmic traders
and increasing long-term risk.
This paper also makes a more general methodological contribution. The ability to measure horizon-specific trading activity on a stock-by-stock basis has a wide range of applications in finance. For instance, one of the key limits on arbitrageurs in the limits-to-arbitrage
literature is their investment horizon (De Long et al., 1990). Moving from behavioral finance
to corporate finance, there is a large literature looking at the optimal responses of rational
firm managers to biased shareholders (Baker and Wurgler, 2013). This bias often times takes
the form of short-term or myopic beliefs. More generally, there is a large literature looking at feedback effects between financial markets and the real economy (Chen et al., 2007).
Differences in horizon might be an important driver for these sorts of feedback effects.
24
References
Acharya, V. and L. Pedersen (2005). Asset pricing with liquidity risk. Journal of Financial
Economics 77 (2), 375–410.
Amihud, Y. (2002). Illiquidity and stock returns: Cross-section and Time-series effects.
Journal of Financial Markets 5 (1), 31–56.
Amihud, Y. and H. Mendelson (1986). Asset pricing and the bid-ask spread. Journal of
Financial Economics 17 (2), 223–249.
Arnuk, S. and J. Saluzzi (2012). Broken Markets: How High Frequency Trading and Predatory
Practices on Wall Street are Destroying Investor Confidence and Your Portfolio (1 ed.).
FT Press.
Baker, M. and J. Wurgler (2013). Behavioral corporate finance: An updated survey. Handbook of the Economics of Finance 2, 357–424.
Bandi, F., B. Perron, A. Tamoni, and C. Tebaldi (2015). The scale of predictability. Working
Paper.
Beber, A., J. Driessen, and P. Tuijp (2012). Pricing liquidity risk with heterogeneous investment horizons. Working Paper.
Bello, J., L. Daudet, S. Abdallah, C. Duxbury, M. Davies, and M. Sandler (2005). A
tutorial on onset detection in music signals. IEEE Transactions on Speech and Audio
Processing 13 (5), 1035–1047.
Bertsimas, D. and A. Lo (1998). Optimal control of execution costs. Journal of Financial
Markets 1 (1), 1–50.
van Binsbergen, J., M. Brandt, and R. Koijen (2012). On the timing and pricing of dividends.
American Economic Review 102 (4), 1596–1618.
Brogaard, J., T. Hendershott, and R. Riordan (2014). High-frequency trading and price
discovery. Review of Financial Studies 27 (8), 2267–2306.
Campbell, J., S. Grossman, and J. Wang (1993). Trading volume and serial correlation in
stock returns. Quarterly Journal of Economics 108 (4), 905–939.
Carhart, M. (1997). On persistence in mutual fund performance. Journal of Finance 52 (1),
57–87.
Cella, C., A. Ellul, and M. Giannetti (2013). Investors’ horizons and the amplification of
market shocks. Review of Financial Studies 26 (7), 1607–1648.
Chen, Q., I. Goldstein, and W. Jiang (2007). Price informativeness and investment sensitivity
to stock price. Review of Financial Studies 20 (3), 619–650.
25
De Long, B., A. Shleifer, S. Lawrence, and R. Waldmann (1990). Noise trader risk in financial
markets. Journal of Political Economy 98 (4), 703–738.
Dew-Becker, I. and S. Giglio (2016). Asset pricing in the frequency domain: Theory and
empirics. Review of Financial Studies Forthcoming.
Duarte, J. and L. Young (2009). Why is pin priced? Journal of Financial Economics 91 (2),
119–138.
Easley, D., S. Hvidkjaer, and M. O’Hara (2002). Is information risk a determinant of asset
returns. Journal of Finance 57 (5), 2185–2221.
Easley, D., N. Kiefer, M. O’Hara, and J. Paperman (1996). Liquidity, information, and
infrequently traded stocks. Journal of Finance 51 (4), 1405–1436.
Engelberg, S. (2007). Random Signals and Noise: A Mathematical Introduction (1 ed.).
CRC Press.
Fama, E. and K. French (1993). Common risk factors in the returns on stocks and bonds.
Journal of Financial Economics 33 (1), 3–56.
Gençay, R., F. Selçuk, and B. Whitcher (2001). An Introduction to Wavelets and Other
Filtering Methods in Finance and Economics (1 ed.). Academic Press.
Giglio, S., M. Maggiori, and J. Stroebel (2014). Very long-run discount rates. Quarterly
Journal of Economics 130 (1), 1–53.
Hasbrouck, J. (2015). High-frequency quoting: Short-term volatility in bids and offers. SSRN
Working Paper.
Hasbrouck, J. and G. Saar (2013). Low-latency trading. Journal of Financial Markets 16 (4),
646–679.
Huberman, G. and W. Stanzl (2004). Price manipulation and quasi-arbitrage. Econometrica 72 (4), 1247–1275.
Jagannathan, R. and Y. Wang (2007). Lazy investors, discretionary consumption, and the
cross-section of stock returns. Journal of Finance 62 (4), 1623–1661.
Jones, C. (2013). What do we know about high-frequency trading? Working Paper.
Kamara, A., R. Korajczyk, X. Lou, and R. Sadka (2015). Horizon pricing. Journal of
Financial and Quantitative Analysis (Forthcoming).
Kirilenko, A., A. Kyle, M. Samadi, and T. Tuzun (2011). The flash crash: The impact of
high frequency trading on an electronic market. Working paper.
Kyle, A. (1985). Continuous auctions and insider trading. Econometrica 53 (6), 1315–1335.
Lee, C. and B. Swaminathan (2000). Price momentum and trading volume. Journal of
Finance 55 (5), 2017–2069.
26
Lewis, M. (2014). Flash Boys: A Wall Street Revolt (1 ed.). W. W. Norton & Company.
Lo, A. and J. Wang (2000). Trading volume: Definitions, data analysis, and implications of
portfolio theory. Review of Financial Studies 13 (2), 257–300.
MacKay, D. (2003). Information Theory, Inference, and Learning (1 ed.). Cambridge University Press.
Mallat, S. (1999). A Wavelet Tour of Signal Processing (1 ed.). Academic Press.
O’Hara, M. and M. Ye (2011). Is market fragmentation harming market quality? Journal
of Financial Economics 100 (3), 459–474.
Pastor, L. and R. Stambaugh (2003). Liquidity risk and expected stock returns. Journal of
Political Economy 111 (3), 642–685.
Percival, D. (1995). On estimation of the wavelet variance. Biometrika 82 (3), 619–631.
Percival, D. and A. Walden (2000). Wavelet Methods for Time Series Analysis (1 ed.).
Cambridge University Press.
Sadka, R. (2006). Momentum and post-earnings-announcement drift anomalies: The role of
liquidity risk. Journal of Financial Economics 80 (2), 309–349.
27
Sample Path
Power Spectrum
Wavelet Variance
-21
log2 (WVar )
-25
-22
log2 (S)
0.25
-30
0.00
-23
-35
-0.25
-24
-40
-0.50
Mon Tue Wed Thu Fri
60 min
vlm τ − hvlm τ i
0.50
-25
-6
-3
0
log2 (days)
3
-6
-3
0
log2 (days)
3
Figure 4: Simulated example illustrating benefits of wavelet methods over Fourier analysis.
(Left) One week of simulated trading-volume data with hour-long shocks. Data are generated
√
using Equation (37) with parameters λ = 0.05 × 104 sh/min and σ = 0.06 × 104 sh/ min.
(Middle) Power spectrum estimated using Equation (36) via the results from Equation
(35). (Right) Wavelet variance estimated using Equation (15). Reads: “There is no peak
at the 60-minute horizon in the power-spectrum plot showing that frequency analysis will
not pick-up asynchronous fluctuations; by contrast, there is a sharp peak at the 60-minute
horizon in the wavelet-variance plot.”
A
Why Not Fourier Analysis?
Fourier analysis might at first seem like a natural way of quantifying the amount of trading
activity at each horizon; however, it turns out to be rather unhelpful.
What Is Fourier Analysis? Suppose there is a minute-by-minute trading-volume time
series that realizes hour-long shocks. To recover the 1-hour horizon from this time series,
Fourier analysis tells us to estimate a collection of regressions at frequencies ranging from 1
cycle per month to 1 cycle per minute:
vlm τ − hvlm τ i = αf · sin(π · f · τ ) + βf · cos(π · f · τ ) + ετ ,
with f ∈ [1/8190, 1]
(35)
A frequency of f = 1/390 cycles per minute, for example, denotes the 1-day time horizon since
there are 6.5 hr/day × 60 min/hr = 390 minutes in a trading day. The amount of variation at a
particular frequency is then proportional to the power of that frequency, Sf , defined as:
αf2 + βf2
Sf =
(36)
2
If the time series realizes hour-long shocks, then we should find a peak in the power of the
series at the hourly horizon, f = 1/60, correct? This is what computing the power of a time
series at a particular frequency is designed to capture, right?
Asynchronous Shocks. Yes, but only if the shocks come at regular intervals. For instance,
if the first 60 minutes realized a positive shock, minutes 61 through 120 realized a negative
shock, minutes 121 through 180 realized a positive shock again, and so on. . . then Fourier
analysis would be the right approach. However, trading-volume shocks have irregular arrival
times and random signs. Fourier analysis cannot handle this sort of asynchronous structure.
Simulation-Based Example. Consider a short example to solidify this point. We sim28
ImmunoGen’s Trading-Volume Variance
log2 (StDev)
18
Wvarh /Var
1.00
1 day
2 hr
1 hr
30 min
15 min
8 min
4 min
2 min
1 min
0.75
0.50
16
0.25
14
0.00
2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
Figure 5: x-axis: times in months from January 2002 to December 2010. (Left) ImmunoGen’s average trading-volume volatility each month in units of shares per (day)1/2 on
a base-2 logarithmic scale. (Right) Fraction of Immunogen’s trading-volume variance each
month at horizons ranging from 1-minute to 1-day long. Reads: “ImmunoGen’s tradingvolume volatility varied by a factor of 219.4/213.6 ≈ 55 from January 2002 to December 2010,
and the fraction of these fluctuations occurring at the 1-minute horizon ranged from 31% to
58%.”
ulate a month-long time series of minute-by-minute trading-volume data with 60-minute
shocks by first randomly selecting J = 1000 minutes during which hour-long jumps begin,
iid
{ξ1 , ξ2 , · · · , ξJ }, and then adding white noise, ετ ∼ N(0, 1):
(
1000
X
+1 w/ prob. 50%
vlm τ − hvlm τ i = σ · ετ +
λ · 1{τ ∈[ξj , ξj +59]} ·
(37)
−1
w/
prob.
50
%
j=1
Each of the jumps has a magnitude of λ = 0.05 × 104 sh/min and is equally likely to be positive
√
or negative. The white noise has a standard deviation of σ = 0.06×104 sh/ min. The resulting
series is shown in the left-most panel of Figure 4.
By construction, this process only has white noise and 60-minute-long shocks. If Fourier
analysis were the correct tool for identifying horizon-specific trading-volume fluctuations,
then you would expect there to be a spike in the power of the time series at the 60-minute
horizon. But, what happens if we look for evidence of this 60-minute timescale by estimating
the power spectrum shown in the middle panel of Figure 4? Do we see any evidence of a
60-minute shock? No. There is nothing at the 60-minute horizon. Asynchronous shocks of
a fixed length do not show up in the Fourier power spectrum. They do, however, show up
in the Wavelet-variance plot as shown in the right-most panel of Figure 4 where there is a
clear spike at the 1-hour horizon.
B
ImmunoGen, Inc. Example
Figure 5 gives an example of the output from the wavelet-variance estimator for a single
NYSE-listed stock: ImmunoGen, Inc. The left panel shows the base-2 logarithm of ImmunoGen’s average trading-volume volatility each month from January 2002 to December 2010
in units of shares per (day)1/2 . The right panel then shows the fraction of ImmunoGen’s
29
Wavelet-Variance Spectrum of Simulated Data
White-Noise Process
Process with Daily Shocks
log2 (Wvar/Var)
50.00%
6.25%
0.78%
0.10%
1min
1hr
1day
1wk 1min
1hr
1day
1wk
Figure 6: Wavelet-variance spectrum on a log-log scale for one month (i.e., T = 8,190 minutes) of simulated data. (Left) Data generated by white-noise process described by Equation
iid
(38) where µ = 0, σ = 1, and τ ∼ N(0, 1). Reads: “50% of the variation in the process
occurs at the 1-minute horizon, 0.78% of the variation occurs at the 1-hour horizon, and
0.13% of the variation occurs at the 1-day horizon.” (Right) Data generated by process with
p
iid
daily shocks described by Equation (40) where µ = 0, σ = ς = 1/2, and τ , εd ∼ N(0, 1).
Reads: “Roughly 21% of the variation in the time series occurs at both the 1-minute and
1-day horizons.”
trading-volume variance at each horizon ranging from 1-minute to 1-day long.
To give a sense of scale, in January 2006 the company’s trading volume fluctuated by
around 216.4 ≈ 83k shares on a daily basis. This level has varied pretty wildly over our sample
period, peaking in June 2010 at around 219.4 ≈ 700k shares per (day)1/2 and bottoming out
during the financial crisis at around 213.6 ≈ 13k shares per (day)1/2 . The fraction of these
fluctuations occurring at the 1-minute horizon ranged from 31% in December 2010 to 58%
in May of the same year, but the pattern was quite different. For instance, the fraction of
trading-volume variance at the 1-minute horizon does not display a steady rise starting in
June 2008 like the total trading-volume volatility time series. This example shows that the
total amount of trading-volume volatility and the amount of trading-volume variance at each
horizon can move in different ways.
C
Wavelet-Variance Spectrum of White Noise
This section applies the wavelet-variance estimator to a pair of simulated data series. Our
goal is to further illustrate how the estimator works by decomposing the variance of processes
with known horizons.
White-Noise Process. We begin by analyzing a white-noise process where all of the variance occurs at the 1-minute horizon. Specifically, we look at a single month-long realization
of the minute-by-minute time series below,
vlm τ = µ + σ · τ ,
iid
(38)
where µ = 0, σ = 1, and τ ∼ N(0, 1). The time series contains T = 60min/hr × 6.5hr/day ×
21day/month = 8,190 minute-level observations corresponding to a single trading month. The
30
left panel of Figure 6 displays the resulting wavelet-variance spectrum for this process on a
log-log scale. For example, the plot says that 50% of the variation in the time series occurs
at the 1-minute horizon and 0.78% of the variation in the time series occurs at the 1-hour
horizon.
The key thing to notice in the left panel of Figure 6 is that the wavelet-variance estimator
only assigns 50% of the variance in the time series to the 1-minute horizon even though we
know that, by construction, all of the variation is occurring at this shortest horizon. At first,
it might seem like something is wrong. Should not the wavelet-variance estimator assign
all of the variance to the 1-minute horizon? It turns out that the answer is “No.” And, in
fact, this linearly decreasing pattern in the wavelet-variance spectrum on a log-log scale is a
characteristic property of white-noise processes.
The pattern occurs because, unlike in Subsection 2.3 above, we do not know the true
mean of the time series, and statistical fluctuations at the 1-minute horizon affect the sample
mean at longer horizons. For instance, by the law of total variance, we know that the sample
variance of the minute-by-minute trading-volume time series over the course of a day is the
sum of the true variance and the variance of the sample average,
P390
P390
P390
2
2
2
1
1
1
(vlm
−
hvlmi)
=
(vlm
−
µi)
+
(39)
·
·
·
τ
τ
τ
=1
τ
=1
τ =1 (µ − hvlmi) .
390
390
390
Since there are 390 minutes in a trading day, we should expect that the wavelet-variance
1
estimator finds 390
× 0.50 = 0.13% of the trading-volume variance at the daily horizon even
though there are no daily shocks. This is exactly what we find in the left panel of Figure
6. By affecting the sample mean, shocks at the 1-minute horizon can impact the sample
variance of the time series at longer horizons.
Process with Daily Shocks. Now that we have seen what the wavelet-variance spectrum
looks like for a time series that only has shocks at the 1-minute horizon, let us turn our
attention to a time series with shocks at multiple horizons. We now look at a single monthlong realization of the time series below that has shocks at both the 1-minute and 1-day
horizons,
vlm τ = µ + σ · τ + ς · εd ,
p
iid
(40)
where µ = 0, σ = ς = 1/2, and τ , εd ∼ N(0, 1). In other words, half of the variation in
the time series occurs at the 1-minute horizon and half occurs at the 1-month horizon. Once
again, the time series contains T = 8,190 minute-level observations corresponding to a single
trading month. Figure 6 displays the resulting wavelet-variance spectrum for this process
on a log-log scale. For example, the plot says that roughly 21% of the variation in the time
series occurs at both the 1-minute and 1-day horizons. Reassuringly, there are equal-size
peaks in the spectrum at the 1-minute and 1-day horizons corresponding to the 1-minute
and 1-day shocks built into Equation (40). But, just like in the white-noise simulation, all
of the spectral mass is not located at the 1-minute and 1-day horizons because fluctuations
at these horizons affect the sample averages at other horizons.
31
Summary Statistics for Portfolios Sorted by Horizon-Specific Trading
Sorted by 1minn,t
(A)
1mini,t
Hi
4
3
2
Lo
All
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µ
σ
48.43
0.56
46.35
0.83
Sorted by 1hourn,t
8.87
min
q25
q50
q75
max
47.03
48.01
48.50
48.91
49.38
44.43
45.80
46.48
46.94
48.16
42.49
44.24
44.82
45.41
47.07
44.82
0.95
43.13
1.05
40.39
42.37
43.07
43.78
45.77
39.57
1.27
35.84
38.71
39.51
40.39
43.36
45.90
0.93
44.04
45.16
45.91
46.61
47.71
min
q25
q50
q75
max
(B)
(0.12)
1houri,t
Hi
4
3
2
Lo
All
●
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diff
σ
2.92
0.30
2.29
2.69
2.94
3.09
4.32
2.22
0.22
1.61
2.07
2.26
2.37
2.83
1.86
0.19
1.38
1.73
1.86
1.99
2.40
1.54
0.15
1.19
1.45
1.54
1.65
1.91
1.14
0.09
0.96
1.08
1.12
1.22
1.35
1.79
0.28
1.30
1.56
1.78
2.01
2.44
min
q25
q50
q75
max
1.77
(0.03)
1dayi,t
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[%]
µ
(C)
Sorted by 1dayn,t
diff
[%]
diff
[%]
µ
σ
1.60
0.26
1.03
1.41
1.57
1.76
2.33
0.89
0.15
0.50
0.80
0.89
0.98
1.27
0.63
0.12
0.36
0.55
0.64
0.71
0.95
0.45
0.09
0.25
0.39
0.45
0.51
0.67
0.25
0.04
0.15
0.22
0.25
0.28
0.36
0.66
0.16
0.38
0.53
0.64
0.76
1.13
1.35
(0.03)
Table 1: Summary statistics for value-weighted portfolios sorted by horizon-specific trading
activity. Sample period: January 2002 to December 2010. µ and σ: mean and standard
deviation of the fraction of trading-volume variance at a particular horizon. diff: difference
between the means of the Hi and Lo portfolios with the standard error given in parentheses.
min, q25 , q50 , q75 , and max: minimum, maximum, median, and interquartile range of the
fraction of trading-volume variance at each horizon. Sparkline plots show portfolio-month
averages. Reads: “The quintile of stocks with the highest fraction of trading at the 1-minute
horizon has 8.87/39.57 = 22.5% more short-run trading than the quintile of stocks with the
lowest fraction of trading at the 1-minute horizon.”
32
Probability of Assignment to Portfolios Sorted By Trading Activity at
1-Day Horizon Given Assignment at 1-Minute Horizon
Sorted by 1minn,t
Sorted by 1dayn,t
Hi
4
3
2
Lo
Hi
0.01
0.02
0.10
0.25
0.62
4
0.04
0.15
0.28
0.34
0.20
3
0.12
0.28
0.29
0.22
0.09
2
0.29
0.33
0.22
0.12
0.05
Lo
0.55
0.22
0.12
0.07
0.05
Table 2: Probability that stock n is assigned to a particular quintile based on its trading
activity at the 1-day horizon in month t given its trading activity at the 1-minute horizon in
month t. Reads: “If a stock was among the 20% of stocks with the largest fraction of trading
activity at the 1-minute horizon in month t, then 62% of the time it is among the 20% of
stocks with the smallest fraction of trading activity at the 1-day horizon in month t.”
33
Characteristics of Portfolios Sorted by Horizon-Specific Trading
Sorted by 1minn,t
(A)
vlmi,t
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mcapi,t
[log2 (sh/day)]
µ
σ
22.34
1.36
21.71
0.46
21.18
20.55
19.68
0.52
0.68
1.31
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●
●
●
●●
●●
●
●
●●● ●
●
●
µ
σ
20.39
10.80
11.24
2.97
●
●
●●
●
●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
8.54
●●
●●
●
●●
●●
●
●
●●
●●●
●
●●●
●●
●
●
●
●●●
●●
●
●
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●
●
●
●
●
●
●
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●
●
●●
●●
●
●●● ●
●●●●●
●
●●
●●●
●
●●
●
●●●● ●
●●
●
●
●
6.44
●
●
●●
●● ●
●
● ●
●
●
● ●
●
● ●
●
●
●●●●
●
●
●●
●● ● ●● ●●● ●●
●●● ●
●●
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●
●
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●
●●●
●
●
● ●
●
●
●
●●
●
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●
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●
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● ●
●●
●●●
●
●
●
●
●
●
●
●●●●●
●●●●
●
●●
●●●
●●●● ● ●●●
● ●
●
●
●
5.05
●
●
●
●
● ●
●
● ●
●
● ● ●
● ●● ●
●
●● ●
●
●●
●
●
●●
●
●
● ● ● ●
●
●
●
●
●
●
●●
●
● ●●
●
●
●
●● ●
●
●● ●
●
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●
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● ●●
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●●●●●●● ●
●
●
●●●●
●
●●
●
●
● ●
●●
●
●
●
mcapi,t
µ
σ
20.29
1.40
20.93
0.85
21.19
0.55
21.59
●●● ●●●
●●●●●●●●
●●●
●●●
●
●
[log2 (sh/day)]
21.43
●
● ●
●
●●●
●●●
●
●
●
●
● ●●●
● ●●● ●
●
●
●
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●
●●
●
●
●●●● ●
●●
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●●●●
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●
1.03
2.20
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●
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● ●
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●
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●
●●
●●
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●
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●●●●● ●●●●
●
µ
σ
20.51
0.99
21.08
0.55
21.48
0.41
21.68
0.77
21.95
1.84
●
●
●
●
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●
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●
● ●●●●●●●
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●●
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●● ● ●● ●
●
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●
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●
●●● ●●●
●
●●● ●●●●● ●
●
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●
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8.77
4.07
●
●
●
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●
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● ●
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●
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●●●
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●●●
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●
2.95
3.97
11.28
●
●
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●
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●
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●
●
●●●
illiqi,t
[$1bil]
6.70
●
●
illiqi,t
4.66
16.67
●
●
●● ●
●
●
●
●●●●
●●●● ●●●●●●●
●
●
●
●●●●●●●
●●
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●●●
●●●
●●●●●●● ●●
●
●
1.83
7.77
9.89
●
●
●● ●●
●
●●
2.51
σ
8.56
●
●
●
●
[$1bil]
µ
●
●
●
2.86
µ
mcapi,t
[log2 (sh/day)]
illiqi,t
[$1bil]
σ
2.81
8.52
2.87
9.63
2.64
10.77
3.45
16.05
9.54
●
●
●
●
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●
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●
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●
●●●●
●●
●
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● ●
●
●
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●
●
●
µ
σ
0.06
0.06
0.05
0.05
0.05
0.04
0.08
0.12
0.14
0.20
% /$1mil]
[ day
0.24
●
●
●
% /$1mil]
[ day
0.16
●
●●
●●●
●
●●●●●
●●●●●
●
●
●
0.09
0.04
●
●
●
●
0.11
0.05
●●
●
●
0.06
●
●
●
● ●●●●● ●
●
● ●
●
●
●●●●●●●
●●●●●
●
●●
● ●●●●●●● ●●●●●●●●●●●●● ●
●●●●
●●●
●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●
●
●
0.06
0.03
●
●
●●
●
●
●● ●
0.04
0.04
●●●●●●
●
●
0.05
●
●
●●●
●●●●● ●
●●●●
● ●●
●●●
●
●●●●●
●
●
●
●
●
●
● ●●
●
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●
●
● ●
●●●●
●
● ●●●●●●●●
●
●●
●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●● ●●
●
0.04
0.03
●
● ●
●●●
●
●
●
0.04
0.04
●●
●
● ●
●
●
●
0.09
0.04
●
●
●
●
0.06
0.05
●
●
●
●
●
●
●
●
● ● ●●● ●
●●
●
●●●● ●●●●
● ●
●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●
●●●● ●●●●●●●●●● ●●
●
σ
σ
●
●
µ
µ
●
●
% /$1mil]
[ day
●
●
●
●
●
● ● ●
●
●
●
●
●
●
●●●● ●●
●● ●● ●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
Table 3: Summary statistics describing the characteristics of value-weighted portfolios
sorted on trading activity at the 1-minute horizon. Sample period: January 2002 to December 2010. µ and σ: mean and standard deviation. vlm i,t : daily-average trading volume
for the portfolio each month in units of shares per (day)1/2 on a base-2 logarithmic scale.
mcap i,t : market capitalization for the portfolio at the end of each month in units of billions
of dollars. illiq i,t : Amihud (2002)-illiquidity measure for the portfolio in units of % per
day per million dollars traded. Sparkline plots show portfolio-month averages. Reads: “The
quintile of stocks with the highest fraction of trading at the 1-minute horizon is almost twice
as liquid at the quintile of stocks with the lowest fraction of trading at the 1-minute horizon.”
34
Summary Statistics for Portfolios Sorted by Residual Trading Activity
at 1-Minute Horizon
evlm
n,t
(A)
Sorted by evlm
n,t
µ
Hi
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●
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●
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●
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●
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●
●●
●
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● ●
●
●
●●
●
●●●● ● ● ●
●● ●●
●
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●
●
2
●
●
●
●
●
●●
●
●
● ●
●
●●●
●
●
● ●
●●●
●
●
●
●
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●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
Lo
●
●
● ● ●
●●
●●●
●
●
●
●
●
●
●●
● ●●
●
●●
● ●
●
●
●
●
●
●●
●●
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●
●
●
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●
● ●
●●
●
●
●
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●
●●●
●
●● ● ● ●●●
●●
●
● ●● ●
● ●●
●
●
●
●
●●●
●●●●
● ● ●●
●
●
●●●
●● ●
●
●●
● ●●●
●●● ●
●
●
●●●
●
● ●●●●
● ● ●
●
●
●● ●
●●
● ●
●
●●
●● ● ●
●
●
● ●
●
●
●● ●
●
●
●
●
●
●
●
●
●
σ
diff
Hi
4
3
2
Lo
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●●
●
●
●
●● ●●
●
●
●
●
●
●
3.22
3.48
3.75
4.26
1.74
0.27
1.22
1.49
1.75
1.96
2.48
0.39
0.20
0.22
0.44
0.52
0.77
−1.13
0.17
−0.10
−4.43
7.92
(0.10)
−1.48 −1.24 −1.16 −1.03 −0.73
−8.66 −4.77 −4.41 −3.94 −3.23
0.78
●
●
●
●
●●
●
●● ●●
●
●●
●
0.20
1.37
1.69
1.81
1.94
2.51
●
●
0.42
0.17
0.32
0.42
0.52
0.94
−1.19
0.22
−0.06
●
●
●
●
●●●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
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●
●●
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●
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●
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●
●●
● ●
● ●
●
●●
●
●
●
●●
● ●
●
●
●
●●
●●
●
● ●
●
●
●
● ●
●
●●
● ●●
●
●
●
●
●
●●
●
●
●●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●●●
●● ●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●●●
●
●
● ●●
●
●
●
● ●
● ●
●
●
●
●
●●
●●
●
●
●
●
● ●● ●
●
●
●
●●
● ●●●● ●
●
●●
●●
●
●
● ●●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●●
●● ●
●●
●
●
●
●
●● ●
●●
●
●
●
●
●●
●
●
●
●●●
●
●
●●
●
●●
●●●●
●
●● ●
●●●
●
●●●
● ●●
●●
●
●
●
●
●●●
●
●●●●●● ●
●
● ● ●
●●●
●
●
● ●●
●●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●
●
●
−5.04
8.58
(0.14)
−1.77
−9.27
1.24
eilliq
i,t
(C)
µ
Hi
4
3
●
●
●
●
●●
●
●
●
●
●
●
●●
● ●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
● ●●●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●
●●
●●
●●●●●
●
●●
● ● ●
● ●
●
●
●
● ●●
●
●
●●
●
● ●
●
● ●●
●
● ●
●
●
●
●●
●
●
● ●●
● ●
●
●● ●●
●
●●●●
●
●
●
●
●
●
●
●
●
●
●●
●●●
●
● ●
● ●●
●
●
●
●
●
●●
●●
●
●
●
●
● ●●●
●
●
●●
●
●
●●
●
●
● ●
●
● ●
●●
●
●
●
●
●●
●
●●
●
●●●
● ●
●
●
●●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
● ●●
●
● ●● ●
●●
●
●●●
●
●● ●
●
●
●
●
●
●
●
●
2
●●●
Lo
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●●
●
●
●
●●
●
●●
●●●●
●●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
● ●
●
●●
●
● ●●
● ●
●
●
●
●
●
●●
●●
●●
●
● ●
●
●
●
●●
● ●
●●
●
●
●
● ●
● ●
●● ●●
●
●
●
● ●●●
●
●●●
●
● ●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●●
● ●
●
●●
●
●
●
●
●
●
● ●
●
●●●●
●
●
●
●
● ●● ●● ●
● ●
●●
●●
●●
●●
●
●●●
● ●●●●
●
●
●
●
●
●●●●
●●●
●
●●
●●● ● ●
●
●●●
● ● ●●
●
●●●●●
●
●
●
●
●
●
●
●
●●
●
●
●
max
1.83
●
●
q75
●
●
●
●
●
●
●
q50
4.33
●
●
●
●
● ● ● ● ● ●
●● ●
●●●● ●
● ●
●
● ●
●
●
●
● ●
●
●
●
●
●
●
●
● ●●
● ●
●●
●
●
●
●
●●●●
●
●
●●
●●●●●
q25
3.84
●
●
min
3.49
●
●
diff
3.34
●●
●
●
●
●●
σ
[%]
2.41
●
●
●
●
●
max
0.37
●
●●●●●● ●●●●
●●
●
q75
3.55
●
●
●
● ●
●
●
●
●●
●
●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●● ●
●●
●
●
●
●
●
●
●
●
●
●
●●●
●
●●
● ●
●
● ●
●
●
●
q50
2.62
emcap
i,t
●
q25
0.37
µ
Sorted by emcap
n,t
min
3.49
(B)
Sorted by eilliq
n,t
[%]
σ
diff
−1.34 −1.19 −1.07 −0.67
−5.21 −4.71 −4.28 −3.43
[%]
min
q25
q50
q75
max
4.04
0.60
2.76
3.66
4.03
4.46
5.46
1.96
0.24
1.47
1.79
1.97
2.12
2.57
0.43
0.15
0.12
0.35
0.43
0.53
0.77
−1.25
0.25
−4.82
8.86
(0.12)
−1.94 −1.41
−9.03 −5.11
0.78
−1.26 −1.10 −0.77
−4.74 −4.39 −3.18
Table 4: Summary statistics for value-weighted portfolios sorted by residual trading activity at the 1-minute horizon as defined in Equations (21a), (21b), and (21c). Sample period:
January 2002 to December 2010. µ and σ: mean and standard deviation of the residual fraction of trading-volume variance at a particular horizon. diff: difference between the means
of the Hi and Lo portfolios with the standard error given in parentheses. min, q25 , q50 ,
q75 , and max: minimum, maximum, median, and interquartile range of the residual fraction
of trading-volume variance at each horizon. Sparkline plots show portfolio-month averages.
Reads: “The fraction of trading at the 1-minute horizon is correlated with other stock characteristics like trading volume, size, and liquidity, but there is substantial heterogeneity in
traders’ horizons even for stocks with the same trading volume, size, and liquidity.”
35
Summary Statistics for Portfolios Sorted by Residual Trading Activity
at the 1-Minute Horizon Using a Multivariate Regression
emv
i,t
Sorted by emv
n,t
µ
Hi
4
3
2
Lo
●
●
●
●
●
●● ●
●
●●
●
● ●
●●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
● ●
●●
●●
●●
● ●
●
●
●
●
●
●
●●
●●
●
●●●
●
●
●
●
●
●
●
●
●
●●
●
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●
●
●
●
●●
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●
●
●
●●●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●●●●
●
●
●
●
●
●
●
●
●●●
●
●●●● ●
●●
●●
●●●
●●
●●●●●
● ●●
●
●
●
●
●
●
●
●
●●
●●
●●
●
●●
●●
●
●
●
●
● ●●
●● ●●●●
●
●
●●●●●●
●
●
●
●
●
●
● ●
● ●
●
● ●●
●●
●
●
●
●●
●●
●
●
●
●
●
●●
●
●
●●
●●
●●
●●
●●
●
●
●
●
●
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●
●
●
●●●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●●
●●
●●●●
●●●●●●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●●
●●●
●
●●
●
●
●
●
●
●
●
●●
●●
●
●
●
●●
●●
●
●
●
●
●
●
●
●●●●
●
●●
●
●
● ●●●
●
●
●
●
●●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
● ● ●
● ●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
● ●●●
●●
●●
● ●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
● ●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●● ●
●
●●●●●●●● ●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●
● ●
●
●●
●
●●
●●
●
●
●
●
●
●
●●
● ●
● ●
●●●
●●●
●
● ●●●
●
● ●●
●
●
● ●
●
● ●
●
●
●
●
●●
●
●
●●
●
●●
●
● ●●
●
●●
● ●
●
●●●
●
●
●
●●●●
●
σ
3.42
0.36
1.65
0.27
0.34
0.20
−1.13
0.19
−4.11
diff
[%]
min
7.52
(0.09)
q25
q50
q75
max
2.47
3.15
3.38
3.65
4.50
1.14
1.43
1.68
1.83
2.58
−0.12
0.22
0.35
0.47
0.80
−1.65 −1.25 −1.14 −1.03 −0.52
−8.85 −4.55 −3.98 −3.61 −2.85
0.77
Table 5: Summary statistics for value-weighted portfolios sorted by residual trading activity
at the 1-minute horizon from the multivariate regression defined in Equation (22). Sample
period: January 2002 to December 2010. µ and σ: mean and standard deviation of the
fraction of trading-volume variance at a particular horizon. diff: difference between the
means of the Hi and Lo portfolios with the standard error given in parentheses. min, q25 ,
q50 , q75 , and max: minimum, maximum, median, and interquartile range of the fraction
of trading-volume variance at each horizon. Sparkline plots show portfolio-month valueweighted averages. Reads: “The fraction of trading at the 1-minute horizon is correlated
with other stock characteristics like trading volume, size, and liquidity, but there is still
substantial heterogeneity in traders’ horizons even when simultaneously controlling for all
these variables.”
36
Summary Statistics for Portfolios Double Sorted by Trading Activity at
the 1-Minute Horizon Within Characteristic Quintiles
1mini,t
Hi
within vlmn,t
Sorted by 1minn,t
(A)
4
3
2
Lo
σ
min
q25
q50
q75
max
48.45
0.67
47.00
47.85
48.46
49.01
49.60
●●
46.72
0.79
45.29
46.07
46.66
47.45
48.23
●
45.38
0.88
43.82
44.71
45.31
46.13
47.22
●
43.87
0.87
42.25
43.19
43.74
44.64
46.13
40.61
1.04
36.71
30.01
40.58
41.27
43.33
min
q25
q50
q75
max
●
●
●
●
●● ●
●●
●●
●
●
●
● ● ●●
●●
●
●
●●
●
●
●
● ● ●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●● ●
●
●● ●
●
●
●
µ
●
●
●
●
●
●
●● ● ●
●
●
●
●●
●●
●
●
●
●
●
●●●
●
●
● ●
●
●● ●●
●
●●
●●
●●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●●●●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●●
●●
●
●
●
●
●
●●
●
●
● ● ●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
● ●●
● ●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●● ●●
●
●
●
●
●
●
●●●●
●
●
●
●
●
●
●●
●
●
●
●●
● ●●
●
●
●
●
●
●●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
● ●
●● ● ●●
●
●
●
●●
●
●
●
●●● ●
●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●● ●
●●
●
●
●
●
●
●
●
●
●●
● ●
●
●
●●
●
●
● ● ●
●
●●●
●
●●
●●
●
●
●
●
●
●●
●
● ●●●●●
●
●
●
●
●
●
●●
●
●●
●
●●●
●
●
●
●
●●
●
●
● ●●●
● ● ●
● ●
●●
●
●
●
●
●
●●●
●
●● ●
● ●
●
●
●●●●
●●
●
●
●
●
●●
●
●
●
●
●●
●● ●
●●
●
●
●●
●●
●
●
●●
●
●
●●
●
●●
●●
●
●
●●
●●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
diff
7.83
(0.09)
1mini,t
within mcapn,t
Sorted by 1minn,t
(B)
Hi
4
3
2
Lo
●●
● ● ●● ● ●●
●
●●
● ●
●
●
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diff
σ
48.51
0.71
46.91
47.90
48.53
49.08
49.80
46.85
0.86
45.36
46.10
46.77
47.59
48.52
45.61
0.95
43.95
44.85
45.51
46.39
47.58
44.21
1.01
42.26
43.45
44.10
45.07
46.40
41.03
1.09
36.91
40.46
40.92
41.66
43.81
min
q25
q50
q75
max
7.49
(0.08)
within illiqn,t
Sorted by 1minn,t
1mini,t
4
3
2
Lo
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[%]
µ
(C)
Hi
[%]
diff
[%]
µ
σ
48.47
0.73
46.98
47.86
48.50
49.07
49.77
46.78
0.89
45.20
46.04
46.68
47.55
48.53
45.51
0.95
43.70
44.78
45.39
46.25
47.48
44.05
0.99
41.99
43.39
43.90
44.90
46.37
40.78
1.11
36.94
40.13
40.73
41.53
43.30
7.69
(0.08)
Table 6: Summary statistics for value-weighted portfolios sorted trading activity at the 1minute horizon within each characteristic quintile. Sample period: January 2002 to December 2010. µ and σ: mean and standard deviation of the fraction of trading-volume variance
at a particular horizon. diff: difference between the means of the Hi and Lo portfolios with
the standard error given in parentheses. min, q25 , q50 , q75 , and max: minimum, maximum,
median, and interquartile range of the fraction of trading-volume variance at each horizon.
Sparkline plots show portfolio-month value-weighted averages. Reads: “The fraction of trading at the 1-minute horizon is correlated with other stock characteristics like trading volume,
size, and liquidity, but there is still substantial heterogeneity in traders’ horizons even for
stocks in the same trading volume, size, and liquidity quintiles. Moreover, the differences
between the averages of the Hi and Lo portfolios are quantitatively similar to the differences
in Table 4.”
37
Abnormal Returns of Portfolios Sorted by Horizon-Specific Trading
αi
Sorted by 1minn,t
(A)
Hi
4
Sorted by 1hourn,t
Sorted by 1dayn,t
0.24
1.04
(0.13)
0.12
0.11
(0.15)
(0.03)
(0.06)
(0.06)
−0.39
0.95
0.17
0.10
(0.19)
(0.04)
(0.08)
(0.07)
0.49
−0.11
−0.46
0.14
βiMkt
βiSmB
βiHmL
(0.24)
[%/month]
(0.05)
(0.10)
R2
91.7%
95.5%
91.1%
91.1%
89.9%
48.9%
0.94
0.12
0.07
(0.13)
(0.03)
(0.06)
(0.05)
0.12
1.01
0.08
(0.11)
(0.02)
−0.04
0.21
0.96
(0.12)
(0.03)
0.04
0.94
(0.13)
(0.03)
−0.20
0.88
(0.12)
(0.04)
(0.04)
−0.06
0.10
(0.05)
(0.05)
−0.08
0.11
92.1%
95.2%
93.1%
91.9%
(0.05)
0.26
(0.03)
−0.16
(0.05)
(0.05)
0.37
0.06
0.28
16.6%
(0.20)
(0.04)
(0.08)
−0.20
βiSmB
βiHmL
R2
βiMkt
●
●
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● ●
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●
1.01
0.16
0.14
(0.20)
(0.04)
(0.08)
(0.08)
0.23
1.02
0.00
0.04
(0.13)
(0.03)
(0.05)
(0.05)
0.19
0.99
0.14
(0.12)
(0.03)
−0.15
0.04
0.87
(0.12)
(0.03)
0.24
0.84
(0.13)
−0.77
(0.05)
(0.05)
−0.07
0.16
(0.05)
(0.05)
0.25
(0.03)
−0.29
(0.05)
(0.05)
0.18
0.45
(0.06)
(0.11)
−0.11
(0.10)
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92.5%
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86.6%
93.4%
93.5%
92.3%
90.3%
27.5%
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●
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●
●
(0.08)
−0.53
(0.26)
●
●
●
R2
(0.05)
[%/month]
●
●
●
●
(0.09)
0.17
αi
Hi − Lo
(0.02)
0.01
0.97
Hi − Lo
Lo
(0.09)
−0.18
0.20
2
2
1.00
(0.05)
(0.05)
3
3
0.07
(0.05)
(0.06)
4
4
(0.03)
0.24
(0.03)
Hi
Hi
(0.12)
−0.29
0.07
αi
(C)
0.84
βiHmL
−0.03
Hi − Lo
Lo
0.09
βiSmB
(0.04)
2
(B)
βiMkt
(0.04)
3
Lo
[%/month]
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Table 7: Abnormal returns to value-weighted portfolios sorted by horizon-specific trading
activity. Sample period: January 2002 to December 2010. Numbers in parentheses are
standard errors. Sparkline plots show portfolio excess returns. Reads: “The quintile of stocks
with the highest fraction of trading activity at the 1-minute horizon in month t has abnormal
returns that are 0.49% per month higher than the quintile of stocks with the lowest fraction
in month t.”
38
Abnormal Returns of Portfolios Sorted by Residual Trading Activity at
the 1-Minute Horizon
αi
(A)
Sorted by evlm
n,t
Hi
4
Sorted by emcap
n,t
Sorted by eilliq
n,t
Lo
Hi − Lo
(0.03)
0.09
0.91
(0.12)
0.16
0.96
0.00
0.30
(0.12)
(0.03)
(0.05)
(0.05)
−0.60
1.13
0.17
0.26
(0.18)
(0.04)
(0.08)
(0.07)
0.92
−0.22
−0.57
0.26
βiMkt
βiSmB
βiHmL
−0.18
0.22
(0.22)
[%/month]
(0.05)
0.24
0.97
(0.16)
(0.04)
0.12
1.02
(0.10)
(0.02)
0.22
0.93
(0.11)
(0.02)
−0.08
0.90
(0.12)
(0.03)
−0.53
0.82
(0.18)
(0.04)
0.76
0.15
(0.27)
(0.06)
αi
2
(0.13)
−0.19
0.08
Hi − Lo
3
0.92
(0.04)
(0.05)
3
4
0.06
(0.04)
(0.05)
4
Hi
(0.02)
0.00
(0.03)
Hi
(C)
(0.11)
−0.40
0.04
αi
Lo
0.91
βiHmL
−0.03
Hi − Lo
2
0.09
βiSmB
(0.05)
2
(B)
βiMkt
(0.05)
3
Lo
[%/month]
[%/month]
βiMkt
0.09
0.83
(0.20)
(0.04)
0.05
1.01
(0.13)
(0.03)
0.26
1.04
(0.12)
(0.09)
(0.07)
(0.06)
−0.22
0.10
(0.04)
(0.04)
−0.14
0.13
92.7%
94.2%
94.4%
87.0%
27.9%
89.6%
95.4%
94.4%
(0.04)
−0.08
0.23
(0.05)
(0.04)
−0.06
−0.04
82.2%
−0.12
0.26
11.9%
(0.07)
(0.12)
●●● ●
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●
−0.28
0.24
(0.08)
(0.08)
−0.18
0.03
0.07
(0.03)
−0.03
(0.05)
(0.05)
0.16
0.98
0.12
0.12
(0.12)
(0.03)
(0.05)
(0.05)
−0.40
0.94
0.17
0.09
(0.13)
(0.03)
(0.05)
(0.05)
0.49
−0.11
−0.45
0.15
(0.10)
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93.3%
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● ●
●
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●
(0.11)
βiHmL
●
● ●
●
●
●
●
●
●
●
●●
●
●
● ●
●
●
●
●
●
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●
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●
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● ●
●
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●
●
●
●
●
(0.07)
βiSmB
(0.11)
●
●
●
●
●●
R2
(0.04)
(0.05)
(0.06)
93.1%
(0.09)
(0.05)
(0.26)
R2
R2
92.9%
92.9%
94.5%
94.2%
86.7%
27.4%
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●●
●
Table 8: Abnormal returns to value-weighted portfolios sorted by residual trading activity
at the 1-minute horizon as defined in Equations (21a), (21b), and (21c). Sample period:
January 2002 to December 2010. Numbers in parentheses are standard errors. Sparkline
plots show portfolio excess returns. Reads: “Even among the stocks with the same trading
volume, size, and liquidity in month t, the quintile of stocks with the highest fraction of
trading activity at the 1-minute horizon in month t has higher abnormal returns.”
39
Transition Probabilities for Portfolios Sorted by Horizon-Specific
Trading
Sorted by 1minn,t
(A)
Sorted by 1minn,t+1
Hi
4
3
2
Lo
Hi
0.37
0.22
0.15
0.12
0.10
4
0.22
0.24
0.21
0.16
0.14
3
0.16
0.21
0.22
0.21
0.17
2
0.12
0.17
0.21
0.24
0.24
Lo
0.10
0.14
0.18
0.23
0.32
Sorted by 1hourn,t
(B)
Sorted by 1hourn,t+1
Hi
4
3
2
Lo
Hi
0.37
0.24
0.16
0.11
0.08
4
0.24
0.25
0.21
0.16
0.11
3
0.16
0.21
0.23
0.22
0.15
2
0.12
0.16
0.21
0.25
0.23
Lo
0.08
0.11
0.16
0.23
0.39
Sorted by 1dayn,t
(C)
Sorted by 1dayn,t+1
Hi
4
3
2
Lo
Hi
0.29
0.23
0.19
0.15
0.10
4
0.23
0.24
0.21
0.18
0.12
3
0.19
0.21
0.21
0.20
0.16
2
0.16
0.17
0.20
0.22
0.22
Lo
0.10
0.13
0.16
0.22
0.35
Table 9: Probability that a stock transitions between portfolios sorted by the fraction of
trading-volume variance at a particular horizon. Rows denote classification in the current
month. Columns denote classification in the subsequent month. Roughly 2% of stocks drop
from the sample due to market-capitalization or price restrictions, so the rows do not add up
to 1. Reads: “If a stock is among the quintile of stocks with the most trading at the 1-minute
horizon in the current month, then it has a 37% chance of remaining among the quintile of
stocks with the most trading activity at the 1-minute horizon next month.”
40
Abnormal Returns of Portfolios Sorted by Residual Trading Activity at
the 1-Minute Horizon After Controlling for Liquidity Measures
αi
Sorted by 1minn,t
(A)
Hi
4
3
2
Lo
Hi − Lo
Sorted by eAP
n,t
Hi
Sorted by eSadka
n,t
2
Lo
Hi − Lo
0.14
1.04
(0.12)
(0.03)
−0.03
93.2%
93.0%
(0.04)
−0.19
0.05
0.06
(0.05)
(0.05)
(0.03)
−0.05
0.13
0.09
(0.05)
(0.03)
(0.05)
0.23
(0.03)
0.10
0.97
0.10
0.17
0.09
(0.11)
(0.03)
(0.05)
(0.05)
(0.03)
−0.42
0.95
0.16
0.12
0.02
(0.19)
(0.04)
(0.08)
(0.08)
(0.04)
0.54
−0.11
−0.45
0.11
(0.09)
−0.05
[%/month]
(0.05)
βiMkt
(0.09)
−0.28
0.24
(0.05)
(0.05)
−0.14
−0.01
92.0%
0.07
93.1%
(0.13)
(0.03)
0.24
1.01
(0.13)
(0.03)
−0.06
(0.05)
(0.05)
0.15
0.97
0.08
0.12
(0.11)
(0.02)
(0.05)
(0.04)
−0.39
0.94
0.20
0.13
(0.18)
(0.04)
(0.08)
(0.07)
0.54
−0.07
−0.48
0.11
βiMkt
βiSmB
βiHmL
−0.28
0.19
(0.23)
[%/month]
(0.05)
0.08
0.86
(0.11)
(0.02)
0.07
1.03
(0.13)
(0.03)
0.37
1.09
(0.15)
(0.05)
(0.10)
(0.04)
(0.04)
−0.18
0.02
0.08
(0.03)
−0.09
(0.06)
(0.06)
0.31
1.04
0.07
0.13
(0.13)
(0.03)
(0.05)
(0.05)
−0.30
1.04
0.09
0.17
(0.18)
(0.04)
(0.08)
(0.07)
0.38
−0.18
−0.37
0.02
(0.09)
28.4%
91.6%
95.0%
87.3%
25.6%
(0.08)
●
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●● ● ●●
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●●●
●●●
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● ●●●●
● ●
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●●●
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●●
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●
(0.09)
(0.05)
(0.05)
87.0%
(0.05)
(0.06)
(0.20)
94.0%
R2
βiHmL
0.97
94.7%
(0.05)
βiSmB
0.03
αi
3
1.00
(0.03)
R2
(0.04)
(0.03)
Hi − Lo
4
0.00
(0.13)
βiPS
−0.28
(0.12)
2
Hi
0.84
(0.02)
βiHmL
0.87
3
(C)
0.12
βiSmB
0.15
4
Lo
βiMkt
(0.11)
(0.23)
αi
(B)
[%/month]
R2
93.4%
92.7%
92.1%
93.7%
89.2%
33.6%
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●
Table 10: Panel A: Abnormal returns to value-weighted portfolios sorted by trading activity
at the 1-minute horizon controlling for the Pastor and Stambaugh (2003) liquidity factor.
Panels B and C: Abnormal returns to value-weighted portfolios sorted by residual trading
activity at the 1-minute horizon as defined in Equations (26) and (28), which control for the
liquidity risk in Acharya and Pedersen (2005) and Sadka (2006) respectively. Sample period:
January 2002 to December 2010. Numbers in parentheses are standard errors. Sparkline
plots show portfolio excess returns. Reads: “The positive abnormal returns earned by stocks
with lots of trading activity at the 1-minute horizon are not explained by their exposure to
41
liquidity factors.”
Abnormal Returns of Portfolios Sorted by Residual Trading Activity at
the 1-Minute Horizon After Controlling Relative to PIN
αi
Sorted by ePIN
n,t
Hi
4
3
2
Lo
Hi − Lo
[%/month]
βiMkt
0.35
0.70
(0.25)
(0.06)
0.55
0.93
(0.27)
(0.06)
0.47
0.95
(0.17)
(0.04)
0.21
0.97
(0.19)
−1.77
βiSmB
βiHmL
−0.16
0.10
(0.09)
(0.11)
−0.33
0.03
(0.09)
(0.12)
−0.18
0.23
(0.06)
(0.08)
0.25
(0.04)
−0.04
(0.07)
(0.08)
1.09
0.07
0.27
(0.30)
(0.07)
(0.11)
(0.13)
2.12
−0.39
−0.23
−0.17
(0.34)
(0.08)
(0.12)
(0.15)
R2
83.4%
88.6%
94.8%
94.5%
89.5%
52.1%
●
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●
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●
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●
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●
●
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●
●
●
●
●
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●
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●
●
●
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●
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●
●
●
●
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●
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●
●
●
●
●
●
●
●
●
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●
●
●
●
●
●
●
●
●
●
●
●
●
●
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●
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●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
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●
●
●
●
●
●
●
●
Table 11: Abnormal returns to value-weighted portfolios sorted by residual trading activity
at the 1-minute horizon as defined in Equations (29), which controls for information risk
via the probability-of-informed-trading (PIN) measure in Duarte and Young (2009). Sample period: January 2002 to December 2004. Numbers in parentheses are standard errors.
Sparkline plots show portfolio excess returns. Reads: “The positive abnormal returns earned
by stocks with lots of trading activity at the 1-minute horizon persists after controlling for
the probability of informed trading.”
42
Abnormal Returns of Portfolios Sorted by Trading Activity at the
1-Minute Horizon Controlling for Momentum
αi
Sorted by 1minn,t
(A)
Hi
4
3
2
Lo
Hi − Lo
Sorted by erxlag
n,t
Hi
4
3
2
Lo
Hi − Lo
βiMkt
0.09
0.84
(0.11)
(0.03)
0.07
1.01
(0.13)
(0.03)
0.24
1.02
(0.12)
(0.03)
βiSmB
βiHmL
−0.29
0.24
0.02
(0.04)
(0.04)
(0.02)
−0.18
0.01
0.01
(0.05)
(0.05)
(0.03)
−0.02
0.07
−0.02
94.2%
(0.05)
94.4%
(0.05)
0.99
0.11
0.11
0.02
(0.03)
(0.05)
(0.05)
(0.02)
−0.40
0.96
0.16
0.10
0.02
(0.18)
(0.05)
(0.08)
(0.07)
(0.04)
0.49
−0.11
−0.46
0.14
0.00
(0.09)
(0.04)
[%/month]
βiMkt
0.08
0.83
(0.10)
(0.02)
0.08
0.99
(0.12)
(0.03)
0.23
1.06
(0.12)
(0.09)
βiSmB
βiHmL
−0.29
0.24
(0.04)
(0.04)
−0.18
0.01
(0.05)
(0.05)
0.09
(0.03)
−0.04
(0.05)
(0.05)
0.20
0.99
0.09
0.10
(0.12)
(0.03)
(0.05)
(0.05)
−0.40
0.92
0.19
0.14
(0.18)
(0.04)
(0.08)
(0.07)
0.47
−0.09
−0.47
0.10
(0.22)
(0.05)
(0.09)
(0.09)
93.1%
92.7%
(0.02)
0.20
(0.06)
R2
βiMom
(0.12)
(0.22)
αi
(B)
[%/month]
87.0%
27.9%
R2
93.3%
92.9%
94.1%
94.6%
86.2%
26.8%
●●● ●
●
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Table 12: Panel A: Abnormal returns to value-weighted portfolios sorted by trading activity at the 1-minute horizon controlling for the Carhart (1997) momentum factor. Panel
B: Abnormal returns to value-weighted portfolios sorted by residual trading activity at the
1-minute horizon as defined in Equation (31), which controls for each stock’s excess returns
in the previous month. Sample period: January 2002 to December 2010. Numbers in parentheses are standard errors. Sparkline plots show portfolio excess returns. Reads: “The high
monthly abnormal returns earned by stocks with lots of trading at the 1-minute horizon are
not associated with momentum factors.”
43
Abnormal Returns to Trading Strategy Based on Trading Activity at
the 1-Minute Horizon
Holding Period
1 Weeks
α
[%/month]
1.01
(0.53)
2 Weeks
0.48
(0.36)
3 Weeks
0.27
(0.27)
4 Weeks
0.06
(0.22)
β Mkt
β SmB
−0.08
(0.03)
−0.23
(0.07)
(0.07)
−0.11
(0.05)
−0.27
(0.09)
(0.08)
−0.11
(0.04)
−0.35
(0.09)
(0.07)
−0.12
−0.38
(0.08)
(0.05)
(0.08)
β HmL
0.20
0.13
0.05
0.05
R2
15.4%
17.4%
22.2%
28.6%
Table 13: Trading-strategy abnormal returns in the first 1, 2, 3, and 4 weeks of month t for
portfolios sorted by the fraction of trading activity at the 1-minute horizon in month (t − 1).
Trading strategy is long the quintile of stocks with the largest fraction of trading activity
at the 1-minute horizon in month (t − 1) and short the quintile of stocks with the smallest
fraction in month (t − 1). Regardless of the holding period, all abnormal returns are reported
in units of % per month. Sample period: February 2002 to December 2010. Numbers in
parentheses are standard errors. Reads: “Trading on the distribution of investment horizons
generated positive abnormal returns during the first week, but these returns decay to zero as
the holding period increases.”
44
Summary Statistics for NASDAQ’s Classification
µ
σ
min
q25
q50
q75
max
onlyHftn,t
[%]
8.36
7.51 0.00
2.39
5.26
13.35
38.82
anyHftn,t
[%]
50.88
19.62 0.22
34.89
50.61
68.08
87.77
Table 14: Summary statistics for NASDAQ’s trader classifications. onlyHftn,t : Ratio of
pure HFT volume to the total volume. anyHftn,t : Ratio of the amount of volume involving a
high-frequency trader in any capacity to the total volume. Sparkline plots show cross-sectional
distributions. Sample: monthly data on the 60 NYSE-listed stocks during 2008 and 2009.
Reads: “On average, for the 60 stocks where NASDAQ classifies trader horizons, only 8.36%
of the trades each month involve 2 high-frequency traders.”
45
Comparing 1minn,t to NASDAQ’s Classification
Dependent Variable: 1]
minn,t
^ n,t
onlyHft
0.41
0.21
(0.02)
(0.06)
0.12
^ n,(t−1)
onlyHft
(0.06)
^ n,t
anyHft
0.34
0.06
(0.02)
(0.06)
0.20
^ n,(t−1)
anyHft
(0.06)
1]
minn,(t−1)
Adj. R2
17.0%
0.23
0.28
0.37
(0.03)
(0.02)
(0.02)
19.1%
13.7%
22.2%
11.3%
Table 15: Results from regressions of the normalized fraction of trading-volume variance
at the 1-minute horizon on the NASDAQ’s normalized measure of short-run trading activity
] n,t : Normalized
at the stock-by-month level as described in Equations (34a) and (34b). 1min
^ n,t : Normalized ratio
fraction of trading-volume variance at the 1-minute horizon. onlyHft
^ n,t : Normalized ratio of the amount of
of pure HFT volume to the total volume. anyHft
volume involving a high-frequency trader in any capacity to the total volume. Numbers in
parentheses are standard errors. Sample: monthly data on the 60 stocks for which we have
data from NASDAQ during 2008 and 2009. Reads: “A 1-standard-deviation increase in
the fraction of trading volume coming from high-frequency traders according to NASDAQ is
associated with a 0.41-standard-deviation increase in the fraction of trading-volume variance
occurring at the 1-minute horizon.”
46