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`Short Selling and the Price Discovery Process
`
`Ekkehart Boehmer
`EDHEC Business School
`
`Juan (Julie) Wu
`University of Georgia
`
`We show that stock prices are more accurate when short sellers are more active. First, in
`a large panel of NYSE-listed stocks, intraday informational efficiency of prices improves
`with greater shorting flow. Second, at monthly and annual horizons, more shorting flow
`accelerates the incorporation of public information into prices. Third, greater shorting
`flow reduces post-earnings-announcement drift for negative earnings surprises. Fourth,
`short sellers change their trading around extreme return events in a way that aids price
`discovery and reduces divergence from fundamental values. These results are robust
`to various econometric specifications, and their magnitude is economically meaningful.
`(JEL G14)
`
`The consequences of short selling for share prices, market quality, and
`information flow are still fervently debated by academics, securities market
`regulators, and politicians. The informational efficiency of prices, a public
`good, is a key attribute of capital markets that can have significant implications
`for the real economy.1 Short sellers account for more than 20% of trading
`volume and are generally regarded as traders with access to value-relevant
`information (Boehmer, Jones, and Zhang 2008). This suggests that they play
`an important role in the price discovery process. However, being informed
`does not necessarily imply that their trading instantaneously impounds this
`
`We thank Kerry Back, David Bessler, George Jiang, Sorin Sorescu, Matt Spiegel (the editor), Heather
`Tookes, an anonymous referee, and seminar participants at the All-Georgia Finance Conference, EDHEC, First
`Annual Academic Forum for Securities Lending Research, HEC Lausanne, HEC Paris, Indiana University,
`Rice University, Texas A&M University, University of Georgia, University of Houston, University of Oregon,
`University of South Carolina, and the 2007 FMA Doctoral Consortium for helpful comments. Juan (Julie)
`is grateful for financial support through the Mays Business School Postdoctoral Fellowship program. This
`article was previously circulated under the title “Short Selling and the Informational Efficiency of Prices.” Send
`correspondence to Ekkehart Boehmer, EDHEC Business School, 393 Promenade des Anglais, 06202 Nice,
`France; telephone: + 33 644 27 88 71. E-mail: ekkehart.boehmer@edhec.edu. Juan (Julie) Wu, Department
`of Finance, Terry College of Business, University of Georgia, Athens, GA 30602; telephone: (706) 542-0934.
`E-mail: juliewu@terry.uga.edu.
`1 More efficient stock prices more accurately reflect a firm’s fundamentals and can guide firms in making
`better-informed investment and financing decisions. Related theoretical work focusing on the link between
`the informativeness of market prices and corporate decisions includes, among others, Tobin (1969), Dow and
`Gorton (1997), Subrahmanyam and Titman (2001), and Goldstein and Guembel (2008). Also related are recent
`empirical studies on seasoned equity offerings (Giammarino et al. 2004), mergers and acquisitions (Luo 2005),
`and investments in general (Chen, Goldstein, and Jiang 2007).
`
`© The Author 2012. Published by Oxford University Press on behalf of The Society for Financial Studies.
`All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.
`doi:10.1093/rfs/hhs097
`Advance Access publication September 3, 2012
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`information into prices—in fact, informed traders often have incentives to
`trade in a way that minimizes information leakage. In this article, we use daily
`data on short-selling flow and various dimensions of informational efficiency
`to systematically quantify the effect of daily short-selling flow on the price
`discovery process.
`Financial theory takes different views on short sellers and the consequences
`of their trading decisions on price discovery and, more generally, on market
`quality. In some models, short sellers are rational informed traders who
`promote efficiency by moving mispriced securities closer to their fundamentals
`(see, e.g., Diamond and Verrecchia 1987). In other models, short sellers
`follow manipulative and predatory trading strategies that result
`in less
`informative prices (Goldstein and Guembel 2008) or cause overshooting of
`prices (Brunnermeier and Pedersen 2005). Most empirical studies suggest that
`short sellers are informed traders. Using either monthly short interest data (see,
`e.g., Asquith and Meulbroek 1995; Dechow et al. 2001; Desai et al. 2002;
`Asquith, Pathak, and Ritter 2005) or shorting flow data (see, e.g., Christophe,
`Ferri, and Angel 2004; Boehmer, Jones, and Zhang 2008; Diether, Lee, and
`Werner 2008), these authors document that short sellers have value-relevant
`information and suggest that their trading helps correct overvaluation.
`Our article connects to this point. In line with previous work, we agree that
`short sellers’ information will eventually be incorporated into prices; going
`beyond previous work, we use higher-frequency daily data on short-selling
`flow to characterize more precisely how and when short sellers impact price
`discovery. Most prior work uses monthly short interest reports to examine
`whether short sellers anticipate future returns or changes in firm fundamentals
`(see, e.g., Dechow et al. 2001; Karpoff and Lou 2010; Henry, Kisgen, and
`Wu 2011). We take a different approach by directly focusing on short sellers’
`daily trading activity and its impact on price discovery at different horizons.
`This allows us to systematically examine whether short sellers’ information is
`incorporated into prices and how quickly this takes place. Our daily flow data
`are more appropriate for this analysis than are monthly snapshots of short
`interest data when short sellers adopt short-term trading strategies.Indeed,
`recent empirical evidence suggests that many short sellers are active short-
`term traders. Between November 1998 and October 1999, Reed (2007) finds
`that the median duration of a position in the equity lending market is three
`days, and the mode is only one day. Diether, Lee, and Werner (2008) estimate
`an average days-to-cover ratio of four to five days for a shorted stock in 2005.
`These findings indicate that a large portion of recent short-selling activity is
`short-term and indeed often limited to intraday horizons. Daily shorting flow
`data allow us to capture the effect of these shorting activities on prices and
`facilitate a more detailed analysis than monthly short interest data.
`We use four distinct approaches when analyzing the effect of shorting on
`informational efficiency. First, following Boehmer and Kelley (2009), we
`construct transaction-based high-frequency measures of efficiency. Second,
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`Short Selling and the Price Discovery Process
`
`we adopt Hou and Moskowitz’s (2005) lower-frequency price-delay measure,
`an estimate of how quickly prices incorporate public information. Third,
`we use the well-established post-earnings-announcement drift anomaly (see
`Ball and Brown 1968) as a measure of inefficiency and test whether short
`sellers influence its magnitude. Fourth, we examine short selling around large
`price movements and price reversals. By design, these four approaches are
`complementary in their assumptions and allow us to examine the effects of
`short selling on efficiency from different perspectives. Together, analyzing the
`influence of short selling along those four distinct dimensions of informational
`efficiency provides a detailed and integrated view on the role short sellers play
`in equity markets.
`Each of the four approaches suggests that short sellers improve the
`informational efficiency of prices. First, more shorting flow reduces the
`deviation of intraday transaction prices from a random walk, so more
`shorting makes prices more efficient. Second, more shorting flow is associated
`with shorter Hou-Moskowitz price delays, suggesting that prices incorporate
`public information faster when short sellers are more active. Third, for the
`most negative quartile of earnings surprises, an above-median increase in
`shorting immediately after the earnings announcement eliminates the drift.
`Fourth, we find no evidence that short sellers exacerbate large negative price
`shocks. Conversely, their trading patterns seem to facilitate more accurate
`pricing even on extreme return days. All these results are robust to different
`econometric methods and specifications, and as we discuss in Section 7,
`they are difficult to explain by reverse causality. Further analysis reveals that
`the efficiency-enhancing effect of short selling is economically meaningful.
`Overall, these findings suggest that short sellers play a critical role in facilitating
`rational price discovery, a major function of capital markets, along several
`dimensions.
`Our article is related to several earlier studies on short selling. We
`complement earlier work by Dechow et al. (2001), Desai et al. (2002),
`Hirshleifer, Teoh, and Yu (2011), and others that examines the relation between
`monthly short interest and variables related to firm fundamentals. These studies
`typically benefit from long monthly time series. Our shorting flow data cover
`only three years, but we can zoom in on the daily horizon and directly evaluate
`the impact that shorting has on prices. As Richardson (2003) points out, higher
`data frequency is important in identifying the impact of short selling on firm
`fundamentals. We also complement earlier work by Boehmer, Jones, and Zhang
`(2008) and Diether, Lee, and Werner (2008), who find that daily shorting
`flows predict returns over horizons up to several months. Both of these sets
`of studies suggest that the presence of short sellers is linked to some correction
`of overvaluation. However, neither focuses on the questions of when exactly
`short selling affects prices or how quickly their information is incorporated
`into prices. Our first contribution is to complement these studies by focusing
`directly on the link between daily shorting flows and four different measures
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`of informational efficiency. We show that shorting flow indeed makes prices
`more efficient and that this process begins at intraday horizons.
`Our study is also related to prior work on short-selling constraints. Proxies
`for shorting constraints include indicators for the practice and prohibition
`of short selling across equity markets (Bris, Goetzmann, and Zhu 2007),
`addition/removal of short-sale restrictions in certain stocks in Hong Kong
`(Chang, Cheng, and Yu 2007), loan rates from a large U.S. security lender in the
`late 1990s (Reed 2007), and data on share lending supply and borrowing fees
`from U.S. and other equity markets (Saffi and Sigurdsson 2011). Focusing on
`variation in shorting constraints and directly linking short sellers’actual trading
`decisions to price efficiency are complementary approaches, and the latter
`allows us to examine the consequences of short sellers’ decisions more directly.
`Our finding that
`short
`sellers enhance efficiency around earnings
`announcements informs the growing body of literature on post-earnings-
`announcement drift (initially documented in Ball and Brown 1968). Although
`there is mounting evidence that post-earnings-announcement drift (PEAD) is
`one of the more persistent anomalies in financial markets, empirical work on
`shorting behavior in this context is quite limited. Using monthly short interest
`data, Cao et al. (2007) find relatively weak evidence that short sellers reduce
`drift, but Lasser, Wang, and Zhang (2010) argue that short interest is not
`related to PEAD in the expected manner. Even with intraday shorting flows,
`Zheng (2009) finds no evidence that short sellers affect PEAD. Berkman and
`McKenzie (2012) find that short selling (proxied by loaned shares in the equity
`lending market) increases after negative earnings shocks but conclude that it
`does not remove long-term PEAD measured over the quarter following the
`earnings announcement. We contribute detailed daily evidence to this debate.
`Consistent with Berkman and McKenzie (2012), we find that shorting increases
`after negative earnings surprises. Boehmer, Jones, and Zhang (2008) show that
`the ability of daily short selling to predict future returns dissipates roughly one
`month after portfolio formation. Thus, short sellers’ ability to exploit PEAD
`should be strongest during the month after the earnings announcement. We find
`that short selling eliminates PEAD over this horizon in the stocks with negative
`surprises where short sellers are most active. Overall, these tests also benefit
`from the daily nature of our data on shorting flows, which allows us to create
`more powerful tests than would be possible based on monthly short interest
`reports. Different from the above studies, our new result in this respect is that
`the activity of short sellers eliminates PEAD at least in some stocks, and this
`happens fairly quickly, further supporting the positive role of short sellers in
`promoting efficient pricing.
`Finally, our analysis provides important guidance for current worldwide
`debates regarding the optimal regulation of short selling.2 Our article
`
`2 Anecdotal evidence indeed goes both ways. Jim Chanos, president of Kynikos Associates (the largest fund
`specializing in short selling), is best known for being one of the first to spot problems with Enron. However,
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`Short Selling and the Price Discovery Process
`
`contributes to these debates by systematically highlighting how short sellers
`help increase market quality and illustrating specific ways in which this
`occurs.
`The remainder of the article is organized as follows. Section 1 describes
`the data and our sample. Section 2 introduces our measures of relative
`informational efficiency. Section 3 analyzes the relation between short selling
`and high-frequency measures of efficiency, whereas Section 4 looks at
`the relation between shorting and low-frequency measures of efficiency. In
`Section 5, we describe our event-based analysis that relates post-earnings-
`announcements drift to shorting activity, and in Section 6, we examine short
`selling around extreme return events. In Section 7, we describe several
`robustness tests and provide some evidence on causality. Section 8 concludes
`the article.
`
`1. Data and Sample
`
`The shorting flow data used in this article are published by the NYSE under the
`Regulation SHO pilot program and are available from January 2005 through
`June 2007.3 We augment the shorting data by identical, proprietary data
`obtained from the NYSE that cover the remaining six months of 2007. For
`each trade, our data include the size of the portion transacted by short sellers,
`if any. We aggregate the intraday shorting flow that is executed during normal
`trading hours into daily observations. This sample is limited to a three-year
`period, but the daily frequency of these flow data allows for more powerful and
`more accurate tests than those constructed from the monthly short interest data
`that are used in many of the earlier studies.
`We match the daily shorting flow data with the Center for Research in
`Security Prices (CRSP) database to obtain daily returns, consolidated trading
`volume, closing prices, and shares outstanding. We include only domestic
`common stocks (share codes 10 and 11) in the analysis but exclude Berkshire
`Hathaway Class A and B shares, which are priced around $3,000 and near
`$100,000, respectively, during the sample period. We compute daily liquidity
`and price efficiency measures from the NYSE’s Trades and Quotes (TAQ) data.
`On an average day, our final sample covers 1,361 stocks.
`
`2. Measuring Price Discovery
`
`We employ four different approaches to measure how efficiently prices
`incorporate information. First, our most powerful tests focus on high-frequency
`
`some high-profile lawsuits, including Biovail, a Canadian pharmaceutical company suing hedge fund SAC, and
`Overstock.com, suing Rocker Partners, accuse short sellers of manipulating their stock prices.
`3 Regulation SHO initiated by the SEC aims to “study the effects of relatively unrestricted short selling on
`market volatility, price efficiency, and liquidity” (see Regulation SHO-Pilot Program, April 19, 2005, at
`www.sec.gov/spotlight/shopilot.htm).
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`measures of the relative informational efficiency of prices. We measure how
`closely transaction prices move relative to a random walk and conduct tests
`at the daily frequency to relate these measures to short-selling flow. Second,
`we use a longer-horizon measure based on daily and weekly returns. These
`tests consider the speed with which public information is incorporated into
`prices over horizons ranging from one month to one year. Third, we exploit
`the well-documented post-earnings-announcement drift to study the effect
`of short selling in an event-based context. Fourth, we identify unusually
`large price changes that are later reversed, and we look at short selling
`around these changes. Extreme price movements are useful in evaluating
`the motivation for short selling, because they shed light on whether short
`sellers trade with the intention to exacerbate or reduce and reverse large price
`declines.
`
`2.1 High-frequency informational efficiency
`We use two different measures to capture the relative efficiency of transaction
`prices—the pricing error as suggested in Hasbrouck (1993) and the absolute
`value of intraday return autocorrelations. Both measures are computed from
`intraday transactions or quote data, and both capture temporary deviations from
`a random walk (see Boehmer and Kelley 2009). Recent empirical evidence in
`Chordia, Roll, and Subrahmanyam (2005) supports this short-term view. Their
`analysis suggests that “astute traders” monitor the market intently and most
`information is incorporated into prices within 30 minutes through their trading
`activities. As a result, transaction-based efficiency measures capture temporary
`deviations from fundamental values well.
`We follow Hasbrouck (1993) and Boehmer and Kelley (2009) in computing
`pricing errors (see the Appendix for details). We decompose the observed (log)
`transaction price, pt , into an efficient price (random walk) component, mt , and
`a stationary component, the pricing error st . The efficient price is assumed to
`be nonstationary and is defined as a security’s expected value conditional on all
`available information, including public information and the portion of private
`information that can be inferred from order flow. The pricing error, which
`measures the temporary deviation between the actual transaction price and the
`efficient price, reflects information-unrelated frictions in the market (such as
`price discreteness, inventory control effects, and other transient components
`of trade execution costs). To compute the pricing error, we use all trades and
`execution prices of a stock. We estimate a vector autoregression (VAR) model to
`separate changes in the efficient price from transient price changes. Because the
`pricing error is assumed to follow a zero-mean covariance-stationary process,
`its dispersion, σ (s), is a measure of its magnitude. In our empirical analysis,
`we standardize σ (s) by the dispersion of intraday transaction prices, σ (p),
`to control for cross-sectional differences in price volatility. Henceforth, this
`ratio σ (s)/σ (p) is referred to as the “pricing error” for brevity. To reduce the
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`influence of outliers, the dispersion of the pricing error is required to be less
`than dispersion of intraday transaction prices.4
`Our second short-term measure of relative price efficiency is the absolute
`value of quote midpoint return autocorrelations. The intuition is that if the quote
`midpoint is the market’s best estimate of the equilibrium value of the stock at
`any point in time, an efficient price process implies that quote midpoints follow
`a random walk. Therefore, quote midpoints should exhibit less autocorrelation
`in either direction and a smaller absolute value of autocorrelation indicates
`greater price efficiency. To estimate quote midpoint return autocorrelations,
`we choose a thirty-minute interval (results are qualitatively identical for five-
`and ten-minute return intervals) based on the results from Chordia, Roll, and
`Subrahmanyam (2005). We use |AR30| to denote the absolute value of this
`autocorrelation.
`In the context of price discovery, pricing errors are easier to interpret than
`autocorrelations, because only pricing errors differentiate between information-
`related and information-unrelated price changes. By construction, pricing errors
`only attribute information-unrelated price changes to deviations from a random
`walk, whereas autocorrelations incorporate all price changes. For example,
`splitting a large order by an informed trader would produce zero pricing error
`because prices change to reflect information from the informed order flow,
`but it would generate a positive autocorrelation. As price adjustments due to
`new information are not reflections of inefficiencies, pricing errors are a more
`sensible measure of the relative informational efficiency of prices.
`
`2.2 Low-frequency informational efficiency
`Hou and Moskowitz (2005) introduce price delays—a low-frequency measure
`of relative efficiency that relies on the speed of adjustment to market-wide
`information.5 We replicate their annual delay measure and, additionally, create
`an analogous monthly measure. For the annual measure, we follow their
`approach and compute weekly Wednesday-to-Wednesday returns for each
`stock. We regress these returns on contemporaneous and four weeks of lagged
`market returns over one calendar year. Specifically, we run the following
`regression:
`
`4(cid:2)
`
`rj,t = αj + βj Rm,t +
`
`δj nRm,t,n + εj,t ,
`
`(1)
`
`n=1
`
`4 Boehmer, Saar, and Yu (2005) apply Hasbrouck’s (1993) method to study the effect of the increased pretrade
`transparency associated with the introduction of OpenBook on the NYSE on stock price efficiency. Boehmer and
`Kelley (2009) find that institutions contribute to price efficiency using similar approaches. Hotchkiss and Ronen
`(2002) examine the informational efficiency of corporate bond prices using a simplified procedure suggested by
`Hasbrouck (1993).
`5 See, for example, Griffin, Kelly, and Nardari (2010) and Saffi and Sigurdsson (2011) for applications in an
`international context.
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`where rj,t is the return on stock j and Rm,t is the value-weighted market return
`in week t. Then, we estimate a second regression that restricts the coefficients
`on lagged market returns to zero. The delay measure is calculated as 1−[(R2
`(restricted model) /R2 (unrestricted model)].6 Similar to an F-test, this measure
`captures the portion of individual stock return variation that is explained by
`lagged market returns. A larger delay means a less efficient stock price, in the
`sense that it takes longer for the stock to incorporate market-wide information.
`Relative to the high-frequency efficiency measures, a stock’s price delay
`describes the price discovery process over a much longer horizon. Instead of
`transaction-to-transaction return dynamics, the delay measure assesses week-
`to-week return patterns. Yet, the (untabulated) correlation between annual price
`delays and the annual averages of daily efficiency measures ranges from 0.2
`to 0.3, suggesting that these measures mostly capture different aspects of
`efficiency but also have a common component.7
`Our analysis covers three years of data, so using an annual variable limits the
`precision with which we can estimate relations between short selling and price
`delays. To construct a more powerful test, we modify Hou and Moskowitz’s
`approach and compute monthly price delays using daily, rather than weekly,
`observations and five days of lagged market returns in regression (1). We require
`a minimum of fifteen observations per firm per month to compute a monthly
`price delay. We obtain qualitatively and statistically similar results using annual
`and monthly delays and report only the latter because of our short sample period.
`Finally, we exploit the potential asymmetry in price adjustment speed.
`Because short sellers primarily benefit from price declines, we expect that
`information gets incorporated faster with more shorting when market-wide
`information is negative. We modify the above unrestricted models to isolate
`negative market returns:
`
`5(cid:2)
`
`(2)
`
`,t−n + εj,t ,
`
`δ−n
`j R
`
`−m
`
`,t +
`
`−m
`
`rj,t = αj + βj R
`
`n=1
`,t equals the daily market return when it is negative. We then use
`where R
`the R2 from the modified unrestricted model in the denominator to calculate a
`modified delay measure that captures price adjustment to negative information.
`
`−m
`
`2.3 Post-earnings-announcement drift
`Post-earnings-announcement drift is a well-established financial phenomenon
`that indicates some degree of informational inefficiency in the capital markets.
`
`6 To reduce noise in this measure, we require a stock to have at least twenty weekly returns during a calendar year.
`7 Another low-frequency relative efficiency measure is the R2 from a market model regression as suggested in
`Morck, Yeung, and Yu (2000). They argue that lower R2 indicates more firm-specific information and can
`thus be used as a measure of information efficiency of stock prices. However, recent work casts doubt on this
`interpretation and suggests that R2 does not capture information well (Griffin, Kelly, and Nardari 2010; Saffi
`and Sigurdsson 2011).
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`Ball and Brown (1968) first document that abnormal returns of stocks with
`positive earnings surprises tend to remain positive for several weeks following
`the earnings announcement and remain negative for stocks with negative
`surprises. This return pattern generates an arbitrage opportunity for savvy
`traders. If short sellers are sophisticated traders who attempt to exploit this
`opportunity, we expect shorting to increase immediately following negative
`earnings surprises. If short sellers make prices more informationally efficient,
`the increased shorting activity following negative surprises should attenuate the
`post-earnings-announcement drift. We use this event-based test to supplement
`our previous two measures of informational efficiency.
`Battalio and Mendenhall (2005) and Livnat and Mendenhall (2006) show that
`earnings surprise measures based on analyst forecasts are easier to interpret than
`those obtained from a time-series model of (Compustat) earnings, because the
`former are not subject to issues, such as earnings restatement and special items.
`We compute earnings surprises as the difference between actual earnings and
`the most recent monthly I/B/E/S consensus forecasts, scaled by the stock price
`two days before the announcement date. We construct abnormal returns as a
`stock’s raw returns net of value-weighted market returns and measure the drift
`as the cumulative abnormal return following each earnings surprise.
`
`2.4 Return reversals at the daily frequency
`Opponents of unrestricted short selling often allege that short selling puts
`excess downward pressure on prices.8 As a result, these opponents claim,
`prices are too low relative to fundamental values when short sellers are active.
`A related allegation is that short sellers can manipulate prices by shorting
`intensely, thereby driving prices down below their efficient values. Once these
`stocks are undervalued, the short sellers could then cover their positions as the
`true valuations are slowly revealed and prices reverse toward their efficient
`values. Both of these scenarios imply that short sellers are more active on
`days when prices decline and especially so when these declines are not related
`to fundamental information. We provide evidence on this issue by selecting
`large price moves and looking at short sellers’ behavior around these extreme
`return days.
`
`3. Shorting Flow and the Short-horizon Efficiency of Transaction Prices
`
`Relative short-horizon efficiency describes how closely transaction prices
`follow a random walk, and we estimate how short-selling flow affects the
`degree of short-term efficiency. We regress daily measures of efficiency on
`lagged shorting and control variables. As the relevant measures of efficiency
`and shorting are available at the daily frequency, these tests are quite powerful.
`
`8 For public concerns or issuers’ comments, see SEC Release No. 34-58592 or the NYSE survey on short selling
`(“Short selling study: The views of corporate issuers,” October 17, 2008).
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` at University of Georgia Libraries on August 7, 2015
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`The Review of Financial Studies / v 26 n 2 2013
`
`We use the following basic model to test hypotheses about the effect of short
`selling on efficiency:
`Efficiencyi,t = αt + βt Shortingi,t−1 + γt Controlsi,t−1 + εi,t .
`(3)
`The dependent variable is either the pricing error, σ (s)/σ (p), or the absolute
`value of midquote return autocorrelation, |AR30|. Following Boehmer, Jones,
`and Zhang (2008), we standardize daily shorting flow by the stock’s daily
`share trading volume. This standardization makes shorting activity comparable
`across stocks with different trading volumes. If more shorting systematically
`contributes to greater price efficiency, stock prices should deviate less from
`a random walk, implying a negative β. We lag all explanatory variables by
`one period to mitigate possible effects of changes in price efficiency on these
`contemporaneous explanatory variables.9
`Extant research suggests several control variables that are potentially
`associated with price efficiency. We include measures of execution costs, order
`imbalances, share price, market capitalization, and trading volume as controls
`in our base regressions. To measure execution costs, we use relative effective
`spreads (measured as twice the distance between the execution price and the
`prevailing quote midpoint scaled by the prevailing quote midpoint).10 Higher
`execution costs make arbitrage less profitable and therefore deter the entrance
`of sophisticated traders, whose trading helps keep prices in line with their
`fundamentals. This reasoning suggests that stocks with higher trading costs
`tend to deviate more from their fundamental values and thus are less efficiently
`priced. Another variable that may be closely related to price (in)efficiency is
`one-sided trading pressure. If excess demand is not immediately absorbed by
`liquidity providers on the other side of the imbalance, less efficient prices
`will result, at least temporarily. If short selling is related to the degree of
`one-sided trading pressure in either direction, our results may not reflect the
`effect of shorting but rather the liquidity needs of other traders. We control
`for this possibility by including the absolute value of order imbalances in the
`regressions. We use Lee and Ready’s (1991) algorithm to classify trades into
`buy-signed trades, where buyers are more aggressive than sellers, and sell-
`signed trades, where sel