Price Clustering Discrepancies in Limit Order Flows

A B S T R A C T

This article provides evidence that price clustering can be explained, in part, by asymmetry in sell and buy limit order flows. Consistent with extant literature (e.g. Kraus and Stoll, 1972; Holthausen, Leftwich and Mayers, 1987 & 1990; and Gemmill, 1996), we report that buy orders are relatively more informative than sell orders. We report that, on average, limit sell orders cluster on round increments significantly more than that of limit buy orders. Empirical literature shows that the price impact of buy trades exceeds that of sell trades. We provide evidence that limit-buy orders are not always more informative than limit-sell orders, as measured by price clustering, and the informativeness is dependent upon the condition of the firm/market. Contrary to prior transaction level research (Harris, 1991), we report that limit order clustering does not significantly increase following extreme market downturns. Conversely, we find that limit order clustering either decreases, or remains unchanged, as prices decline.

1. Introduction

The purpose of this paper is to investigate how the frequency with which security prices fall on

round increments, or price clustering, responds to the broader market state. By using a comprehensive

intra-day limit order dataset that identifies liquidity supplier buy and sell orders, we are able to study

clustering frequencies of limit-buy and -sell orders separately. In aggregate, we find that executed limit-

sell orders cluster on five penny increments more frequently than that of limit-buy orders. If some

quantity of these limit orders are submitted on round increments in an effort to minimize cognitive

processing costs, the transactions occurring at these prices are less representative of the security’s

fundamental value. Thus, asymmetry in price clustering between executed limit-buy and –sell orders

will reduce the relative informativeness of sell orders.

Prior research shows that markets react differently to buy and sell orders and that the price impact

of purchases exceeds that of sales (Kraus and Stoll, 1972; Holthausen, Leftwich and Mayers, 1987 and

1990; and Gemmill, 1996). Saar (2001) provides a theoretical justification for this impact differential

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based on the way institutional portfolio managers optimally use information given a constrained set

of allowable strategies. The aggregate asymmetry in clustering frequency reported in this study

provides another plausible explanation for this price impact disparity.

Even though asymmetrical clustering frequency may be a mechanism through which the price

impact differential is generated, it is not obvious why investors would expend less cognitive resources

on limit-sell orders in aggregate. Closer examination reveals that the direction of the clustering

disparity is contingent on the ambient state of the market and the performance of the firm’s equity.

We document that limit-buy orders cluster significantly more during market and stock price declines,

while limit-sell orders cluster more during advances. Thus, buy trades are not universally more

informative than sell trades. In both univariate and multivariate analyses we find that limit-sell orders

cluster significantly more than limit-buy orders when either firm or market prices are increasing. When

prices are declining, however, we find that the converse is true.

In an environment where prices are steadily rising, information relevant to a particular security is

more likely to inspire optimism about future cash flows. Thus, the suppliers of liquidity, traders that

submit limit-sell orders in this scenario, are less likely to be motivated by information. Harris (1991)

contends that traders use discrete price sets to lower the cost of negotiating. Along this line of

reasoning, liquidity suppliers, without any superior information, might rationally submit orders

clustered on round increments to increase the probability that their orders execute. If the decision to

trade is not motivated by information, as in the case of portfolio rebalancing, expending additional

cognitive resources to derive a more precise assessment of a security’s value is suboptimal if doing so

would lower the potential of execution.

The foundational research on price clustering documents that security prices, across various

financial markets, tend to land on certain number sets more frequently than others (Wyckoff, 1963;

Osborne, 1962; Niederhoffer, 1965 and 1966; and Ball, Torous, and Tschoegl, 1985). Empirical

research prior to decimalization demonstrates that stock prices cluster most on whole integers,

followed by halves, then odd quarters, and lastly on odd eighths. Major U.S. stock exchanges reduced

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the minimum tick size to pennies in the early 2000s, and a variety of papers, including Ikenberry and

Weston (2008), provide significant evidence that the reduction in tick size from one sixteenth of a

dollar did not taper the clustering phenomenon. In the post-decimalization regime, stocks should

trade on five penny increments 20% of the time in the absence of price clustering. In agreement with

previous studies(e.g. Alexander and Peterson, 2007; Ikenberry and Weston, 2008), we also show that

transaction prices fall on nickels and dimes significantly more than the expected 20%.

Harris (1991) also provides evidence that the frequency of prices clustering significantly increases

following an extreme market decline. By analyzing the difference in transaction price clustering

between the week prior to the 1987 stock market crash and the week of the crash, he finds that

clustering increases across all price levels. The author notes that this finding is to some extent

troubling because a large body of theoretical and empirical work has documented a positive relation

between underlying security prices and the frequency with which those prices fall on discrete price

sets. According to this reasoning, a significant market decline should place downward pressure on

prices leading to a decline in overall clustering frequencies.

Contrary to Harris (1991), we find that limit-order clustering decreases in response to both firm

and market price declines. Asymmetry in price clustering among order flows expands with volatility.

We document a significant difference in clustering between buy and sell limit orders during volatility

shifts, with limit-sell orders clustering more during price advances and limit-buy orders clustering more

during price declines. These findings are both economically and statistically significant.

This paper contributes to two important areas of the finance literature: investor behavior and

market efficiency. We report that investors submitting buy- and sell-limit orders have conflicting

preferences for discrete price sets, which is particularly true during market/firm price movements.

Liquidity suppliers initiating buy-limit orders become increasingly interested in finding equilibrium

values during price inclines, while those submitting sell-limit orders are more willing to deal on discrete

price sets. With the roles reversing during firm and/or market price declines. We also contribute to

prior literature by providing a meaningful explanation as to why price clustering decreases during

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market/firm declines and increases during advances. We find that limit-sell orders play a significant

role in driving clustering frequency down during declining markets, and up during advancing markets.

The rest of the paper is organized as follows. Section 2 discusses the data used throughout the

analysis. Section 3 reports the results from our empirical examinations along with motivational

literature. Section 4 provides a summary and conclusions.

2. Data description

Throughout the trading day, the NASDAQ TotalView-ITCH data feed provides a real-time view

of the order book for all NASDAQ market participants. A tick-by-tick record of the order and trade

transaction history appearing on the NASDAQ TotalView-ITCH dataset, subsequently referred to as

(ITCH), is made available by NASDAQ to researchers and analysts. From January 3, 2012 to

December 31, 2012, we obtain from ITCH, intra-day trading volume, order prices (buy and sell

stamped), and executed shares. The ITCH dataset consists only of orders and trades occurring on the

NASDAQ exchange, however, the sample firms can be listed on any exchange. Order messages are

attached to each order on the book. We extract all executed orders by isolating message types 'E', 'C',

and 'P'. The message 'E' is sent on an order that is executed in whole or in part at the initial display

price. The message 'C' is sent on an order that is executed in whole or in part at a price different from

the initial display price. Lastly, message type 'P' is sent on an order that matches between non-

displayable order types, i.e. "hidden liquidity". From the Center for Research in Security Prices (CRSP)

we gather daily prices, volume, shares outstanding, and closing bid and ask prices. Only common

stocks (i.e. share codes 10 & 11) whose share prices are above $5 and less than $500, and exhibit a

minimum of two trades per day are retained in the sample. In addition, we require our sample firms

to trade in at least 80% of the total number of trading days within our one year sample period.1 We

apply these restrictions on the data to avoid infrequently traded firms and noise introduced by the

bid/ask bounce. Our final sample consists of 1,243 NYSE/AMEX listed stocks and 1,383 NASDAQ

listed stocks.

1 Ikenberry and Weston (2008) make similar data restrictions when examining price clustering in U.S. stock prices

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Cluster% is the variable of interest throughout the empirical analysis. We follow previous studies

that examine limit-order price clustering (e.g. Niederhoffer, 1965; Ahn, Cai, and Cheung, 2005; Chiao

and Wang, 2009, etc.). We measure the weekly frequency with which buy (sell) limit-order prices

execute on $0.05 increments2. For each firm, we summate the number of times orders fall on an

increment of five cents, i.e. $.00, $.05, $.10, $.15, by week. We then divide that total by the number

of limit orders executed within that same week time period. We denote this ratio as Cluster%, which

captures the frequency with which limit orders fall on a discrete price set.

3. Empirical results

The summary statistics for our sample of firms are reported in Table 1. As expected, the

NYSE/AMEX listed stocks have a mean market value, $9.05 billion, which is much larger than the

$3.24 billion average size for NASDAQ firms. The NYSE/AMEX listed stocks also account for more

daily trading volume on the NASDAQ exchange than that of the NASDAQ-listed firms. However,

the average NYSE/AMEX firm in our sample executes 277,680 shares per day, or approximately 138

shares per trade, whereas the average NASDAQ listed stock executes 197,105 shares per day, or

approximately 140 shares per trade. The NASDAQ-listed firms appear to be slightly more volatile,

experience higher percentage bid-ask spreads, and have a much lower average share price. The average

number of limit-sell orders executed daily does not seem to be significantly different from that of the

number of limit-buy orders, for all firms in the sample.

3.1. Univariate results

We begin by partitioning the intra-day ITCH dataset into executed limit-buy and limit-sell orders,

with the intent of investigating whether the order prices differ significantly in the frequency with which

they fall on increments of $0.05. Chao and Wang (2009) make the argument that limit-order data

reflects investors’ intentions more realistically than trade and quote data. We therefore focus on limit

2 Alexander and Peterson (2007) examine price clustering in the post-decimalization period (post 2001) and find that

prices fall on nickels and dimes more frequently than any other price increment.

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orders because it provides a unique setting in which examining investor transaction behavior can be

more directly observed.

3.1.1. Asymmetry in price clustering between limit order flows

Table 2 provides univariate evidence that mean clustering percentages between limit-buy and

limit-sell orders significantly differ across several specified price levels. We find that limit-sell orders

cluster consistently more than that of limit-buy orders. These results are convincingly significant and

robust to exchange listings. In aggregate, limit-sell orders falls on round numbers 0.87 percentage

points more frequently than that of limit-buy orders. These results are not only statistically significant,

but also economically meaningful. If prices are assumed to be uniformly distributed, the expected

clustering percentage on nickels and dimes in the post decimalization regime is equal to 20%;

therefore, clustering in excess of 20% is considered unexpected. The average firm in the sample

displays a weekly abnormal clustering percentage of 5.26%. This implies that the 0.87 percentage

point difference in clustering between limit-buy and -sell orders is no trivial amount.

In order to make more valid inferences on the difference in clustering percentage between limit-

buy and limit-sell orders, we partition the data into price levels. We find that price clustering exhibits

a u-shaped relation with underlying prices. Prices tend to cluster substantially more among the most

highly priced securities, over $200. Many researchers argue that discrete price sets are used to trade

more highly priced assets. Empirical research showing a positive relation between clustering

percentage and nominal share prices is extensive (e.g. Niederhoffer, 1965; Ikenberry and Weston,

2008; Alexander and Peterson, 2007). The purchase of a home is a commonly used example in which

a highly priced commodity is generally transacted on discrete sets, such as $1,000s or $5,000s. The

average unexpected clustering percentage among the most highly priced firms is 13.16%, compared

to that of the lowest priced firms, 6.52%. This pattern is well documented in prior literature (e.g.

Harris 1991); therefore, the more interesting part is what happens in-between the extreme price levels.

Clustering percentage decreases as prices rise between $20 and $200. This decrease is more difficult

to explain as theory implies that clustering is increasing with the underlying security price.

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Chan and Lakonishok (1993) make the argument that there are several possible explanations as

to why one might dispose of stock, but the choice to buy a particular stock is likely to carry positive

firm-specific news. There are several reasons for why an investor may seek to sell stock, so the

cognitive choice to set an order on a desired round number may not have a tremendous impact on

fundamental values. However, the decision to submit a limit-buy order on a round increment might

have a larger rippling effect on price resolution. If buy trades indeed result in greater price impact

(Saar, 2001), we would expect to witness similar findings within the clustering of limit order prices.

That is, limit-buy orders should transact on finer prices that are more indicative of the stock's intrinsic

value. Therefore, investors submitting limit-buy orders may be more concerned with transacting on

finer prices and less concerned with limiting negotiation costs (Harris, 1991). The results in Table 2

confirm this idea that the decision to submit a limit order on a round increment is contingent upon

whether the limit order is sell- or buy-initiated.

Ball, Torous, and Tschoegl (1985) present the price resolution hypothesis and propose that price

clustering is a result of investor uncertainty about the underlying value of a security. Coupling the price

resolution hypothesis with the theoretical arguments of Saar (2001), in which buy trades are shown to be

more informative than sell trades, we document that less informative limit-sell orders are more apt to

be placed on round numbers, and therefore increasing the percentage of price clustering. Conversely

we find limit-buy orders to be more informative as they cluster significantly less, resulting in a more

accurate reflection of fundamental prices. In general, we contend that investors submitting limit-sell

orders are more uncertain about fundamental values and thus more likely to place orders on round

increments.

Observable in Table 2, we pose that the decrease in clustering is a result of differing limit-order

flows within the specified price levels. We find a monotonic decrease in the difference in abnormal

clustering percentage between limit-buy and -sell orders as the nominal security price is increasing.

The difference between limit-buy and -sell orders is 1.05% for orders submitted at $10 or less, and

insignificantly different from zero for orders placed at a price above $200. In unreported results, the

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amount of limit-order volume in the quintile below $10 far exceeds that of the volume exhibited for

orders in the quintile above $200.

We continue our examination of buy- and sell-limit order differences by separating the data by

exchange listing following empirical research showing that prices cluster more among NASDAQ-

listed firms (Christie and Schultz, 1994; Christie, Harris, and Schultz, 1994; Bessembinder, 1999). The

results in panels B and C display clustering percentages for stocks segmented by exchange listing. We

group NYSE and AMEX firms together and compare those with stocks listed on the NASDAQ

exchange. We find strong univariate evidence that NASDAQ-listed stock order prices cluster more

frequently than NYSE/AMEX stock prices. In particular, the average firm listed on the NASDAQ

exhibits a mean clustering percentage of 26.73%, compared to 23.70% for NYSE/AMEX stocks.

In addition, the difference between limit-buy and -sell order clustering is more pronounce among

NASDAQ-listed firms. For the NASDAQ sample, limit-sell orders cluster on $0.05 increments 1.23%

more frequently than limit-buy orders, on average. For the NYSE/AMEX sample, limit-sell orders

only cluster 0.48% more often than limit-buy orders. The observed differences between limit-buy and

-sell orders across both exchanges are statistically significant at the 1% level. Again, we interpret these

results as evidence that limit-buy orders are relatively more informative than limit-sell orders, which

is particularly true among NASDAQ-listed stocks. NASDAQ stocks are generally smaller and more

difficult to value, which may explain why the observed difference in clustering frequencies between

limit-buy and -sell orders are more pronounce.

3.1.2. Price clustering in altering price conditions

Our next set of tests examine clustering differences in limit-order flows during periods of high

price volatility. Harris (1991) shows evidence that transaction level price clustering significantly

increases following the stock market crash of 1987. However, bearish markets lead to downward price

pressure which, as discussed previously, should reduce overall price clustering. We examine whether

price clustering exhibits a significant increase during periods of extreme price volatility. According to

our theoretical understanding of price clustering, we would expect to see a decrease in clustering as

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prices are falling, and conversely an increase in clustering as prices are rising. As a measure of

completeness, we categorize advancing and declining periods using three separate methods.

The first approach is to examine the differences between extreme market advances and declines.

We define a market as declining (advancing) when the NASDAQ index experiences a sizeable decrease

(increase). Again, we are concerned with the NASDAQ exchange only because the ITCH database

consists of firms that trade solely on the NASDAQ. Within our sample time frame, we identify

periods when the NASDAQ dropped by 11.93% (April 2, 2012-June 1,2012) and rose by 17.78%

(January 3, 2012-April 1, 2012). Secondly, we measure declining versus advancing markets as weekly

fluctuations in average returns on the S&P 500. Therefore we denote a declining period as a week in

which the average return on the S&P 500 is negative, and an advancing week when the average return

is positive. Our final method for approximating price volatility is measured on the firm level. We

define a declining period as a week in which a stock experiences negative average raw returns, and an

advancing period in which a stock exhibits positive average raw returns. Together, we can disentangle

what level of price volatility is driving clustering fluctuations, if any.

Panel A of Table 3 presents the univariate results for price clustering around extreme market

movements. We find that the average limit-order clusters significantly more, 0.49 percentage points ,

during the extreme advancing period, relative to the extreme declining period. This difference is

significant at the 1% level. We document that this finding does not seem to be isolated to stocks listed

on a particular exchange. The difference in price clustering between limit orders is not significant

when the measure for market movements is the average weekly return on the S&P 500. Panel C

reports results similar to that of Panel A, in that limit-order price clustering is higher during firm-level

advancing periods, but only marginally. These results provide evidence that when price clustering

behavior differs among market conditions, it is in the direction theory predicts (Ball, Torous, and

Tschoegl; 1985). Contrary to the findings of Harris (1991), this study provides evidence that limit-

order clustering is significantly greater during advancing periods as opposed to declining periods. As

security prices increase, i.e. advancing period, limit-order clustering also increases. The converse is

also true with regards to declining periods.

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Baker and Wurgler (2007) and Delong et al. (1990) argue that investors are subject to general

belief about future cash flows and risks not justifiable by available information. Therefore, investors

can drive prices to unjustifiable levels by their inability to determine the informativeness of a limit

order. We contend that price clustering distracts market participants from finding “true” equilibrium

values. Thus during price advances and declines, limit orders submitted on discrete sets continue to

drive prices away from fundamental values. Limit orders may also act as price barriers, for example,

hindering additional upward movement. (Ohta, 2006; Chiao and Wang, 2009).

3.1.3. Asymmetry in price clustering between limit order flows within altering price

conditions

Investors setting limit orders on discrete price sets well outside the bid-ask spread, may be hopeful

that when the market turns in their desired direction, their submitted orders will have the greatest

probability of being executed. Cooney, Van Ness and Van Ness (2003) show that execution rates of

even-priced limit orders exceed those of odd-priced limit orders. The authors make the argument that

investors setting limit orders may prefer to use round prices to increase the probability of their order

being executed. Chiao and Wang (2009) provide evidence that limit orders submitted by individual

investors tend to cluster on integer and even prices more often than those of other investor classes.

Therefore, individual investors may be arbitrarily setting limit orders on round increments around the

best bid and best offer in hopes of capitalizing on sudden price changes.

3.1.4. Limit-sell orders

Similar to Tables 1 and 2, we are interested in the discrepancies in price clustering effects among

limit-order flows. We therefore separate the data into limit-buy versus limit-sell orders. The results

in Table 3 suggest that limit-sell orders cluster on round increments significantly more during

advancing markets than that of limit-buy orders. During an extreme market advance, limit-sell orders

cluster 26.07% of the total weekly limit-sell orders executed, compared to 25.02% for limit-buy orders.

This difference is significant at the 1% level3. We find similar patterns when observing the clustering

3 We conduct a simple t-test comparing means between limit-buy and limit-sell orders. We find that the difference

1.05% is statistically significant at the 1% level.

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differences under the alternative definitions of price advances. Limit-sell orders cluster 1.37% more

frequently than buy orders during weeks in which the return on the S&P 500 is positive4. In addition,

limit-sell orders cluster 1.38% more often than that of limit-buy orders during weeks in which

individual firm returns are positive5. We conclude, from a univariate standpoint, that limit-sell orders

cluster considerably more during advancing periods than limit-buy orders.

We also report that the clustering percentage of limit-sell orders experiences a significant decrease

when moving from an advancing period to a declining period. In Panel A we find that limit-sell orders

cluster 1.26% more frequently during extreme market advances relative to extreme market declines.

In Panel B we report that the clustering percentage for limit-sell orders during weeks in which the

S&P 500 raw returns are positive is 0.65% greater than the clustering frequency when the S&P 500

raw returns are negative. Similarly, Panel C reports that the difference between sell-limit order

clustering frequencies between firm-level price inclines and declines is 0.70%. We evaluate the

difference in means using sample t-tests and find that each of the reported differences are significant

at the 1% level. As before, we partition the data by exchange listings and report qualitatively similar

results. Therefore, limit-sell orders appear to cluster substantially more during advancing periods

relative to declining periods.

We pose that limit-sell orders that are sitting on the limit-order book above the best ask will begin

to transact as prices rise. Investors setting limit-sell orders on round prices well above the asking price,

are causing limit-order clustering frequencies to increase as prices advance. Therefore price clustering

during market increases may be explained, in part, by liquidity suppliers’ sell limit orders. In addition,

if there is a plethora of closely priced sell limit orders waiting above the ask price, advances will be

decelerated when moving through order barriers (Ohta, 2006; Chiao and Wang, 2009). Therefore,

investors’ tendencies to set orders on round increments may make it increasingly more difficult to

evaluate fundamentals, in an already uncertain environment. Harris (1991) expounds upon the price

resolution hypothesis and states that price clustering depends on how well known is the intrinsic value of

4 A simple t-test shows that the difference in means is significant at the 1% level. 5As done previously, we conduct a t-test showing that this difference is significant at the 1% level.

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the security. If the value is not well know, then the frequency with which prices cluster will increase.

Thus limit-order clustering can prolong market uncertainty about fundamental values.

Limit-sell orders that are set above the asking price transact more frequently when prices are on

an upswing. When the market is bullish and prices are on the rise, investors may view it as an optimal

time to sell due to theuncertainty about future prices. One feasible explanation can be attributed to

loss aversion. This well documented economic decision theory assumes that losses and disadvantages

have a greater impact on preferences than gains and advantages (Tversky and Kahneman, 1991).

Therefore during advancing periods, investors submitting limit-sell orders may be more concerned

with the possibility of the market turning and losing profits than the possibility of prices continuing

in their upward fashion. Consistent with Cooney, Van Ness, and Van Ness (2003) investors may be

submitting limit-sell orders on round numbers more frequently during price advances in hopes of

increasing the probability of their orders being executed. We therefore contend that limit-sell orders

set above the asking price, and on a discrete price set, will execute at an increasing rate.

3.1.5. Limit-buy orders

In Table 3 we find that the clustering percentage differences between limit-buy orders and limit-

sell orders within each of the defined declining periods are insignificantly different. In other words,

limit-buy orders do not appear to cluster more frequently when prices are declining relative to limit-

sell orders. However, limit-buy orders do appear to cluster more frequently during declining periods

compared to advancing periods. These differences are statistically significant at the 1% level and

robust to each of the defined periods. Empirical research shows that the price impact of buy trades

exceeds that of sell trades (e.g. Kraus and Stoll, 1972; Holthausen, Leftwich and Mayers, 1987 & 1990;

Gemmill, 1996, etc.). We contribute to this strand of research by showing that the informativeness

of the limit-buy order is dependent upon the condition of the market and stock. An increase in price

clustering is argued to be a result of price uncertainty (Alexander and Peterson, 2007). Therefore the

increase in limit-buy order clustering during declining periods is evidence that buy orders may be less

informative when prices are decreasing.

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We propose that limit-buy orders that are sitting on the book well below the best bid will execute

only when prices begin to decline. Liquidity suppliers setting limit-buy orders well below the bid price

have the intention of buying stock when they believe prices are at their lowest. Investors may be

arbitrarily setting limit orders on discrete sets under the bid price as a mechanism to take advantage

of temporarily undervalued stocks. An investor looking to buy stock, whether correctly or incorrectly,

expects that the price of the security will increase following the purchase.

During a market downturn, investors are aware that any particular stock price will be deflated,

but whether that be temporary or permanent is unknown. Therefore, an investor looking to buy a

security may be less concerned about finding the equilibrium price because he/she assumes that the

stock is already underpriced. This could cause prices to deviate even further from fundamental

values. Inaccurate information may be imbedded into stock prices, from a swarm of limit-buy orders

simply being executed because they were set on discrete price sets before the market began to decline.

Within a univariate setting, we report that the difference in price clustering among order flows is

primarily driven by the rise in stock prices. Although limit-buy orders seem to demonstrategreater

price clustering frequency during declining periods, the magnitude is fairly trivial in most cases.

3.2. Multivariate Analysis

We recognize the need to control for variables that might influence limit-order clustering when

attempting to determine whether differences exist among order flows and whether those differences

are affected by market conditions. We begin by estimating the following equation using weekly pooled

cross-sectional limit order data.

𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡 = 𝛼𝑖 + 𝛿𝑡 + 𝛽1𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽2𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 + 𝛽31/√𝑁𝑇𝑖,𝑡

+ 𝛽4𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡+𝛽5𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽6𝑆𝑒𝑙𝑙𝑖,𝑡 + 𝜀𝑖,𝑡 1

The dependent variable, Cluster%, is the frequency with which limit-orders fall on increments of

$0.05. We follow several papers that examine price clustering in a multivariate framework and include

the following independent variableswhich are mainly motivated by the price resolution hypothesis (e.g.

Harris, 1991; Ikenberry and Weston, 2008; Alexander and Peterson, 2007). Size, is the market

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capitalization or closing price times number of shares outstanding. Size should be negatively related

to price clustering because more information is available for large firms (more analyst coverage) and

large firms are better diversified. Price is the daily closing price obtained from CRSP. Price is included

because, as reported in Table 2, clustering is increasing with stock price. 1/√𝑁𝑇is the inverse of the

square root of the total number of transactions for each firm by week. We expect clustering frequency

to decrease as transaction activity increases. Therefore we should find a positive relation between our

inverse transaction measure and clustering frequency.

Two other measures used as explanatory variables are the bid-ask spread and trade size. Spread%

is calculated as the difference in closing bid-ask prices divided by the midpoint6. In accordance with

the price resolution hypothesis, as the spread increases (widens) price uncertainty also increases. We

therefore expect to see a positive relation between the bid-ask spread and limit-order clustering.

TradeSize is the average trade size, and is constructed by dividing the number of executed shares by

the total number of transactions. Trade size should be positively related to clustering frequency

because large size trades are generally more informed.Finally, an indicator variable that captures

whether a limit order is buy or sell is also included as a regressor. We include both firm fixed effects

and time fixed effects in each model specification.

Column 1 of Table 4 reports the results from estimating equation (1). We find that limit-order

clustering is increasing in price, trade size, and spread and decreasing with transaction frequency.

These results are consistent with previous findings (e.g. Ikenberry and Weston, 2007) and support the

price resolution hypothesis. We therefore focus our efforts on the independent variable, Sell, which displays

a significant positive relation with the dependent variable, clustering percentage. Therefore limit-order

clustering is significantly affected by the type of order flow, whether it be buy or sell, holding all other

factors constant. This supports our univariate findings that limit-sell orders cluster more frequently

than that of limit-buy orders. Because the indicator variable Sell takes on the value of 1 for a limit-sell

order and 0 for a limit-buy order. This suggests, holding all else constant, a one unit increase in limit-

6 This simple measure of spread has been shown to be a reliable approximation for microstructure spreads (Chung

and Zhang, 2013; Roll and Subrahmanyam, 2010). We replicate the analysis using Corwin and Schultz (2012) simple spread approximation and find qualitatively similar results.

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buy orders leads to a 0.87 percentage point decrease in price clustering. To the extent that price

clustering is negatively related to price resolution (Ball, Torous, and Tschoegl, 1985), this coincides

with the theory that buy orders carry more information about fundamental values.

Our next set of tests examine the affects of price volatility on limit-order clustering. We expand

equation (1) to incorporate such effects. Therefore we estimate the following equation:

𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡 = 𝛼𝑖 + 𝛿𝑡 + 𝛽1𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽2𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 + 𝛽31/√𝑁𝑇𝑖,𝑡

+ 𝛽4𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡+𝛽5𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽6𝑆𝑒𝑙𝑙𝑖,𝑡 + 𝛽7𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡

+ 𝛽8𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 + 𝛽9𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 + 𝛽10𝑆𝑒𝑙𝑙 × 𝐷𝑒𝑐𝑖,𝑗,𝑡 + 𝜀𝑖,𝑗,𝑡

2

We construct indicator variables which take on the value of 1 for each of the three-defined

declining periods. FirmDecline captures firm-level price movements. MarketDecline measures average

weekly market movements (estimating using S&P 500 weekly returns). ExtremeDecline indicates

extreme unprecedented market movements discussed in detail previously. We also include interaction

terms, Sell*Declineij, which captures the effects of each specified market condition and sell orders on

clustering frequency. Within each regression we include both firm- and time- fixed effects.

The results in Columns 2, 3, and 4 report evidence that the frequency of limit-order clustering

decreases in declining markets. This is perhaps surprising given the evidence provided by Harris

(1991), in which trade price clustering increases sharply following a significant price decline. We do

not observe a significant increase in limit-order clustering following a severe economic downturn. To

the contrary, we find that limit-order clustering is significantly decreasing during both firm-declines

and severe market declines. The coefficients on FirmDecline and ExtremeDecline are statistically

significant, and negative; however the economic effect may be trivial. Our results therefore

consistently indicate that clustering is either negatively affected by price declines or unaffected. We

fail to find any evidence of clustering increases following price declines when examining limit orders.

Most theoretical and empirical models of price clustering however do predict a positive relation

between security prices and the frequency with which they cluster (Niederhoffer and Osborne, 1966;

Ball, Torous, and Tschoegl, 1985; Harris, 1991; Alexander and Peterson, 2007).

16

The results in Table 4, columns 5, 6, and 7, also provide evidence that there exists a significant

difference in limit-order clustering among order flows, which is amplified during price movements.

We find that the interaction terms between limit-sell orders and the dummy variables for price declines

are significantly negative. The results are qualitatively similar regardless of the measure of

declining/advancing periods. Not only are the coefficients statistically significant but they are also

economically meaningful. We interpret these results as evidence that a significant difference exists

between price clustering among limit-order flows, which in turn, appears highly correlated to market

and firm conditions. Limit-sell orders cluster substantially more during advancing markets, while

limit-buy order cluster on round increments more during declining markets, whether it be evaluated

at the firm-level or market level These results suggest that limit-sell orders are less informative than

limit-buy orders during price advances, as clustering frequency is increasing Therefore, limit-buy

orders appear more indicative of fundamental values during price upswings. The converse is also true

during price declines, in that limit-buy orders cluster more frequently than limit-sell orders and can

therefore be considered less informative about fundamental values.

We run post-hoc tests on the findings reported in Table 5 by partitioning the data into limit-buy

and -sell orders. The fact that the interaction terms between sell orders and declining periods are

significant, implies that further investigation is needed. We therefore estimate equation (2) using

separate samples for limit-buy orders and limit-sell orders. Again, we control for both firm- and time-

fixed effects.

The first three columns examine the effects of pricing conditions on limit-sell order clustering.

We find that sell limit orders cluster significantly less during declining periods and, inversely, more

during advancing periods. Column 1 shows that clustering among sell orders is decreasing as firm-

level prices are increasing. Similarly, Columns 2 and 3 report that limit-sell orders cluster substantially

less as market prices are on the rise.

The final three columns estimate equation (2) using the sample of limit-buy orders. We report

that the effects of price movements on limit-buy order clustering is an almost mirror image of that of

17

sell-order clustering. We find that limit-buyorders cluster significantly more when prices are declining.

This suggests that limit-buy orders may not always be more informative than limit-sell orders, when

taking into account price conditions. To the extent that price clustering is inversely related to price

discovery, limit-buy orders appear to become less informative as prices are decreasing, both aggregate

market prices and firm-level prices. To reiterate, these results hint that limit-buy orders may not always

be more informative than limit-sell orders. This interpretation of order flow needs to be done with

caution, as significant differences exist among altering price conditions.

As a measure of robustness, we partition the data by exchange listing (i.e. NYSE/AMEX and

NASDAQ) and re-evaluate the main results of this paper. Prior literature documents a significant

difference in price clustering between firms listed on the NASDAQ versus those listed on the

NYSE/AMEX (Christie and Schultz, 1994; Christie, Harris, and Schultz, 1994; Bessembinder, 1999;

and others). We find that limit-buy orders and limit-sell orders exhibit similar clustering behavior

across exchanges. The frequency of limit order clustering increases during declining markets and falls

during advancing markets. As before, limit-sell orders cluster more often than limit-buy orders,

however that relationship is conditioned upon the state of the market.

4. Concluding remarks

Extant empirical research shows that markets react differently to buy and sell orders and that the

price impact of purchases exceeds that of sales (e.g. Kraus and Stoll, 1972; Holthausen, Leftwich and

Mayers, 1987 & 1990; and Gemmill, 1996) We seek to provide evidence that limit order flows can

explain a significant portion of the observed price clustering in equity markets. If some quantity of

these limit orders are submitted on round increments in an effort to minimize cognitive processing

costs, the transactions occurring at these prices are less representative of the security’s fundamental

value. Thus, asymmetry in price clustering between executed limit-buy and –sell orders will reduce

the relative informativeness of sell orders.

Consistent with our expectations, we find that there exists a significant difference in the frequency

with which buy and sell limit orders cluster on increments of $0.05. From our multivariate analysis,

18

we find that buy/sell limit orders can explain a significant portion of the observed price clustering

within our sample, after controlling for factors that have been hypothesized to explain the variability

in clustering frequency. We report that limit-sell orders cluster significantly more, on average, than

limit-buy orders. However, this result is conditional upon the state of the market and the firm. Prior

literature has found that price clustering is negatively related to price discovery.

To the extent that clustering is negatively related to price resolution, we interpret the initial results

as limit-buy orders being more informative than that of limit-sell ordersbecause of the observed lower

clustering frequency. However, we find that limit-buy orders do not always appear more informative

than limit-sell orders. For example, as firm-level prices, and/or market prices, are falling, limit-buy

orders cluster significantly more than that of limit-sell orders. The reverse is also true during price

advances, with limit-sell orders clustering more.

We provide several explanations as to why limit-sell orders cluster more during bullish

market/firm price conditions and limit-buy orders cluster more during bearish market/firm price

conditions. For instance, we contend that limit-buy (sell) orders sitting on the limit order book well

below (above) the best bid (ask) will exhibit an increase in price clustering as prices decline (advance).

These limit-orders may have been set on round numbers strategically to increase the probability of

execution (Cooney, Van Ness and Van Ness, 2003) or less strategically to reduce cognitive processing

costs (e.g. Wyckoff, 1963; Niederhoffer 1965, 1966). Whether it be the former or the latter, we witness

significant differences in price clustering for limit-order flows during market/firm price shifts

(advances and declines).

We contribute to previous literature by showing that limit-order clustering is clearly dependent

upon the type of order that is being submitted, whether it be a buy or sell. If increased levels of

clustering are associated with lower price resolution, we provide evidence that limit-buy orders are not

always more informative than limit-sell orders, and the informativeness can be explained in part by

the overall state of the market. In other words, the frequency with which limit-buy and -sell orders

cluster is conditional upon the state of the market and firm. Limit-buy orders cluster relatively more

19

during price declines and limit-sell order cluster relatively more during price advances. In addition,

we do not find a significant increase in limit-order clustering following extreme market declines. In

fact, we witness a slight decrease in the frequency with which limit order prices fall on round

increments. .

5. References

Ahn, H. J., Cai, J., & Cheung, Y. L. (2005). Price clustering on the limit-order book: Evidence from

the Stock Exchange of Hong Kong. Journal of financial Markets, 8(4), 421-451.

Alexander, G.J. and M.A. Peterson, 2007, An Analysis of Trade-Size Clustering and its Relation to

Stealth Trading. Journal of Financial Economics, 84, 435-471.

Baker, M., & Wurgler, J. (2007). Investor sentiment in the stock market. Journal of

EconomicPerspectives, 21(2): 129-151.

Ball, C.A., W.N. Torous, and A.E. Tschoegl, 1985, The Degree of Price Resolution: The Case of the

Gold Market. Journal of Futures Markets, 5, 29-43.

Bessembinder H., 1999, Issues in assessing trade execution costs. Journal of Financial Markets, 6,233-

257.

Chan, L. K., & Lakonishok, J. (1993). Institutional trades and intraday stock price behavior.Journal

ofFinancial Economics, 33(2), 173-199.

Chiao, C., & Wang, Z. M. (2009). Price Clustering: Evidence Using Comprehensive Limit‐

OrderData. Financial Review, 44(1), 1-29.

Christie W.G. and P.H. Schultz, 1994, Why do NASDAQ Market Makers Avoid Odd-Eight Quotes?.

The Journal of Finance, 49, 1813-1840.

Christie, W.G., J.H., Harris, and P.H. Schultz, 1994, Why Did NASDAQ Market Makers

StopAvoiding Odd-Eighth Quotes?. The Journal of Finance, 49, 1841-1860.

20

Cooney Jr, J. W., Van Ness, B., & Van Ness, R. (2003). Do investors prefer even-eighth

prices?Evidence from NYSE limit orders. Journal of banking & finance, 27(4), 719-748.

Delong, J. B., Shleifer, A., Summers, L, and Waldmann, R. (1990). Noise Trader Risk in

FinancialMarkets." Journal of Political Economy, 98(4): 703-738.

Gemmill, G. (1996). Transparency and liquidity: A study of block trades on the London

StockExchange under different publication rules. The Journal of Finance, 51(5), 1765-1790.

Harris, L., 1991, Stock Price Clustering and Discreteness. Review of Financial Studies, 4, 389-415.

Harris, L., 1994, Minimum price variations, discrete bid-ask spreads, and quotation sizes. Reviewof

Financial Studies, 7, 149-178.

Holthausen, R. W., Leftwich, R. W., & Mayers, D. (1987). The effect of large block transactions on

security prices: A cross-sectional analysis. The Journal of Finance, 19, 237-267.

Holthausen, R. W., Leftwich, R. W., & Mayers, D. (1990). Large-block transactions, the speedof

response, and temporary and permanent stock-price effects. Journal of Financial

Economics, 26(1),71-95.

Ikenberry, D., and J.P. Weston, 2008, Clustering in US Stock Prices after Decimalization.European

Financial Management, 14, 30-54.

Kraus, A., and Stoll, H. R. (1972). Price impacts of block trading on the New York StockExchange. The

Journal of Finance, 27, 569-588.

Niederhoffer, V., 1965, Clustering of Stock Prices, Operations Research, 13, 258-265.

Niederhoffer, V., 1966, A New Look at Clustering of Stock Prices. Journal of Business, 39, 309-313.

Niederhoffer, V., and M.F.M. Osborne, 1966, Market Making and Reveral on the Stock

Exchange.Journal of the American Statistical Association, 61, 897-916.

Ohta, W. (2006). An analysis of intraday patterns in price clustering on the Tokyo

StockExchange. Journal of Banking & Finance, 30(3), 1023-1039.

21

Osborne, M.F.M., 1962, Periodic Structure in the Brownian Motion of Stock Prices, Operations Research,

10, 345-379.

Saar, G. (2001). Price impact asymmetry of block trades: An institutional trading explanation. Review

of Financial Studies, 14(4), 1153-1181.

Tversky, A., & Kahneman, D. (1991). Loss aversion in riskless choice: A reference-

dependentmodel. The Quarterly Journal of Economics, 106(4), 1039-1061.

Wyckoff, P., 1963, Psychology of Stock Market Timing. Prentice-Hall, Englewood Cliffs, NJ.

22

Table 1

Descriptive Statistics

This table provides summary statistics for our sample of firms. We observe 2,626 sample firms. The information is segmented into firms listed on the NYSE/Amex (1,243 firms) and NASDAQ (1,383 firms) exchanges. The sample consists of executed orders on the NASDAQ exchange for the period January 1, 2012 to December 31, 2012. Market value is constructed for each firm as the closing price times the number of shares outstanding, in $billions. Price is the average daily closing price obtained from CRSP. Number of Orders is calculated for each firm as the average number of limit orders executed daily on the NASDAQ exchange. Buy (Sell) orders is the average number of trades executed daily on the NASDAQ exchange for buy (sell) orders for each firm. Executed shares is the average number of shares traded daily on the NASDAQ exchange by firm. Buy (Sell) shares is the average daily number of shares executed on the NASDAQ exchange for buy (sell) initiated orders for each firm. Volume is average daily trading volume for each firm obtained from CRSP. Return volatility is calculated as the standard deviation of daily returns for each firm over the sample period. Spread is the percentage bid/ask spread calculated as the difference between the closing Bid-Ask prices obtained from CRSP, divided by the midpoint.

Mean Median Std. Dev.

Panel A: All Stocks

Market Value 6.0748 1.0838 22.9354

Price 33.26 22.57 58.97

Number ofOrders 1,706.14 503.00 3,249.70

Buy Orders 852.53 250.00 1,631.66

Sell Orders 853.83 247.00 1,635.73

Executed Shares 236,429.83 47,829.00 735,321.25

Buy Shares 118,226.33 23,778.00 367,565.03

Sell Shares 118,225.27 23,453.00 370,700.77

Volume 1,505,333.46 310,807.00 6,149,121.29

Return Volatility 0.0220 0.0203 0.0095

Spread% 0.0016 0.0007 0.0033

Panel B: NYSE & AMEX listed Stocks

Market Value 9.0478 2.2235 25.5455

Price 40.01 29.45 73.30

Number of Orders 2,018.79 707.00 3409.65

Buy Orders 1,008.07 351.00 1,710.12

Sell Orders 1,011.14 349.00 1,716.63

Executed Shares 277,679.57 65,722.00 817,838.78

Buy Shares 138,869.62 32,804.00 407,974.40

Sell Shares 138,848.66 32,379.00 412,529.78

Volume 2117074.30 585700.00 8028596.76

Return Volatility 0.0202 0.0189 0.0087

Spread% 0.0008 0.0004 0.0013

23

Panel C: NASDAQ listed stocks

Market Value 3.2406 0.5718 19.7215

Price 26.83 17.76 39.83

Number of Orders 1,408.07 388.00 3,059.91

Buy Orders 704.30 193.00 1,538.65

Sell Orders 703.81 189.00 1,539.76

Executed Shares 197,105.41 36,627.00 644475.52

Buy Shares 98,553.41 18,153.00 323,180.21

Sell Shares 98,557.61 17,814.00 324,628.36

Volume 922,149.34 159,580.00 3,422,296.78

Return Volatility 0.0237 0.0219 0.0098

Spread% 0.0024 0.0012 0.0044

24

Table 2

Clustering frequency

Mean clustering frequencies for buy and sell prices in limit order data from the NASDAQ exchange for the period January 1, 2012 to December 31, 2012. We separate the statistics by exchange listing into NYSE/AMEX and NASDAQ listed stocks. Clustering percentage is constructed for each firm by week, as the total number of prices (buy/sell) that rounded on $0.05 increments divided by the total number of trades. Price is the weekly average buy (sell) price for each firm. Simple pooled test-statistic are constructed to assess the difference of means between buy and sell clustering percentages.

Clustering Frequency

Price level subsample

Orders Buy Sell Difference Sell-

Buy Avg. Buy

Price Avg. Sell

Price

Panel A. NYSE, AMEX, and NASDAQ listed stocks

All stocks 25.26% 24.82% 25.69% 0.87%*** $33.08 $33.11

Over $200 33.16 32.99 33.33 0.33 514.70 514.89

$40-$200 23.89 23.58 24.20 0.63*** 63.77 63.82

$20-$40 24.61 24.20 25.01 0.81*** 28.90 28.92

$10-$20 26.13 25.62 26.65 1.03*** 14.65 14.66

Under $10 26.52 25.99 27.04 1.05*** 7.36 7.37

Panel B. NYSE & AMEX listed stocks

All stocks 23.70 23.46 23.93 0.48*** 40.01 40.04

Over $200 33.38 33.22 33.54 0.32 541.98 542.64

$40-$200 22.93 22.68 23.17 0.50*** 64.54 64.60

$20-$40 23.40 23.18 23.62 0.44*** 29.58 29.60

$10-$20 24.42 24.18 24.66 0.48*** 14.82 14.82

Under $10 24.50 24.24 24.76 0.52*** 7.43 7.43

Panel C. NASDAQ listed stocks

All stocks 26.73 26.11 27.35 1.23*** 26.54 26.56

Over $200 32.77 32.60 32.94 0.34 465.30 464.78

$40-$200 25.72 25.28 26.16 0.87*** 62.32 62.34

$20-$40 25.96 25.35 26.58 1.23*** 28.14 28.16

$10-$20 27.22 26.53 27.91 1.38*** 14.54 14.55

Under $10 27.50 26.85 28.16 1.31*** 7.34 7.35

***,**, * denote statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 3

Clustering Frequency Advancing vs. Declining

Mean clustering frequencies for executed limit orders separated by buys and sells. Clustering percentage is constructed weekly for each, and is the total number of prices that rounded on $0.05 increments divided by the number of executed orders. We categorize advancing and declining markets in three separate ways: First in Panel A, we report the results under extreme market conditions which we define as follows. The advancing market period is when the NASDAQ index increased by an unprecedented 17.78% (471 points), which occurred between January 3, 2012 and April 01, 2012. Conversely, the declining market is defined as when the NASDAQ index experienced a significant decrease of 11.93% between the period April 2, 2012 to June 1, 2012. Second in Panel B, we report the results from defining advancing markets as weeks when the average return on the S&P 500 is positive, and declining markets as weeks when the average return on the S&P 500 is negative. Lastly in Panel C, for the firm-specific measure, we define a firm decline when the average weekly returns for a particular stock is negative, and the opposite holds true for firm advances. Weekly averaged buy/sell executed shares are also reported. Simple pooled test-statistics are measured to assess the difference of means between the buy and sell clustering percentages across market conditions, which are reported in parentheses.

All Stocks NYSE/AMEX NASDAQ

Advance Decline Difference Advance Decline Difference Advance Decline Difference

Panel A. Extreme Market Advances vs. Declines

Orders 25.55% 25.05% 0.49%***

23.83% 23.44% 0.39%***

27.16% 26.56% 0.60%***

(9.21) (7.46) (6.65)

Buy 25.02 25.29 -0.27***

23.51 23.60 -0.09

26.44 26.88 -0.44***

(-3.71) (-1.25) (-3.58)

Sell 26.07 24.81 1.26***

24.16 23.29 0.87***

27.87 26.25 1.63***

(16.11) (11.61) (12.42)

Buy Shares 620,604.34 636,743.00 742,229.68 764,959.52 506,179.48 516,562.52

Sell Shares 624,933.70 627,211.53 744,122.66 755,854.69 512,861.79 506,576.80

Panel B. S&P 500 Declining vs. Advancing Weeks

Orders 25.27% 25.24% 0.04%

23.75% 23.62% 0.13%***

26.71% 26.76% -0.05

(1.06) (3.45) (-0.87)

Buy 24.59 25.16 -0.57***

23.32 23.65 -0.34***

25.79 26.57 -0.79***

(-12.17) (-6.45) (-10.46)

Sell 25.96 25.31 0.65***

24.18 23.58 0.60***

27.63 26.95 0.69***

(12.18) (10.60) (8.00)

Buy Shares 549,603.18 552,900.49 649,874.75 654,399.44 454,850.61 457,310.36

Sell Shares 555,438.85 544,825.45 656,642.50 645,300.03 459,911.11 450,043.95

2

Panel C. Firm-Specific Advancing vs. Declining Weeks

Orders 25.32% 25.18% 0.14%***

23.72% 23.67% 0.05%

26.84% 26.60% 0.25%***

(3.86) (1.24) (4.35)

Buy 24.63 25.06 -0.43***

23.37 23.57 -0.19***

25.83 26.45 -0.62***

(-9.21) (-3.76) (-8.29)

Sell 26.01 25.31 0.70***

24.06 23.78 0.29***

27.85 26.74 1.11***

(13.27) (5.12) (13.03)

Buy Shares 534,498.53 570,767.45 631,893.93 675,890.15 441,653.04 472,774.70

Sell Shares 550,464.76 551,718.35 646,211.91 658,914.12 459,459.18 451,435.81

***,**, * denote statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 4 Cross-section Regressions

This table reports the results from estimating the following equation using cross-sectional intra-day limit order data from January to December 2012:

𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡 = 𝛼𝑖 + 𝛿𝑡 + 𝛽1𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽2𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 + 𝛽31/√𝑁𝑇𝑖,𝑡 + 𝛽4𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡+𝛽5𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽6𝑆𝑒𝑙𝑙𝑖,𝑡+ 𝛽7𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡 + 𝛽8𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 + 𝛽9𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 + 𝛽10𝑆𝑒𝑙𝑙 × 𝐷𝑒𝑐𝑖,𝑗,𝑡 + 𝜀𝑖,𝑗,𝑡

The dependent variable𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡 is the observed frequency of transaction prices that fall on $0.05 increments for

firm i in week t. We include the following as independent variables: 𝑆𝑖𝑧𝑒𝑖,𝑡 is the market capitalization, or the daily

closing price multiplied by the number of shares outstanding, in $billions. 𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 is the average daily closing price

obtained from CRSP. 1/√𝑁𝑇𝑖,𝑡 is the inverse of the square root of the number of transactions for each firm by week.

𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡 is the difference between the CRSP closing bid and ask prices divided by the midpoint. 𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡 is the average trade size, and is constructed by dividing the number of executed shares by the total number of transactions.

𝑆𝑒𝑙𝑙𝑖,𝑡 is an indicator variable labeled 1 if it is a sell order and 0 for a buy order. We construct indicator variables which

take values of 1 for each of the three-defined declining periods:Extreme Decline (𝐸𝑥𝐷𝑒𝑐𝑖,𝑡), Market Decline (𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡), and Firm Decline (𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡). 𝑆𝑒𝑙𝑙 × 𝐷𝑒𝑐𝑖,𝑗,𝑡 is the independent interaction between the 𝑆𝑒𝑙𝑙𝑖,𝑡 indicator variable and each of the three specified declining period dummy variables. Intra-day limit order data are used to measure daily values for all executed orders that we average across weeks to calculate clustering percentage

(𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡), number of trades, and executed shares. We report t-statistics in parentheses after controlling for firm and time fixed effects.

(1) (2) (3) (4) (5) (6) (7)

𝑆𝑖𝑧𝑒𝑖,𝑡 0.0001 0.0001 0.0001 0.0002 0.0001 0.0001*** 0.0002 (1.26) (1.27) (1.28) (1.05) (1.26) (1.27) (1.05)

𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 0.0002*** 0.0002*** 0.0002*** 0.0001* 0.0002*** 0.0002*** 0.0001*

(9.99) (10.02) (10.03) (1.93) (10.01) (10.03) (1.92)

1/√𝑁𝑇𝑖,𝑡 0.0432*** 0.0427*** 0.0428*** -0.1071*** 0.0425*** 0.0430*** -0.1064***

(5.91) (5.84) (5.85) (-8.40) (5.81) (5.88) (-8.36)

𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡 0.0204*** 0.0203*** 0.0203*** 0.0040*** 0.0204*** 0.0204*** 0.0040***

(23.64) (23.51) (23.52) (2.97) (23.67) (23.65) (3.00)

𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡 0.0003*** 0.0003*** 0.0003*** 0.0004*** 0.0003*** 0.0003*** 0.0004***

(56.56) (56.41) (56.49) (39.79) (56.47) (56.58) (39.81)

𝑆𝑒𝑙𝑙𝑖,𝑡 0.0087*** 0.0133*** 0.0136*** 0.0104***

(28.64) (32.64) (34.54) (18.49)

𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡 -0.0006* 0.0046*** (-1.93) (10.63)

𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 -0.0004 0.0056*** (-1.41) (12.86)

𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 -0.0045*** 0.0030*** (-10.20) (4.83)

𝑆𝑒𝑙𝑙 × 𝐷𝑒𝑐𝑖,𝑗,𝑡 -0.0103*** -0.0120*** -0.0151*** (-17.01) (-19.59) (-17.20)

Observations 258,520 258,520 258,520 108,076 258,520 258,520 108,076

Adj R2 0.2598 0.2574 0.2574 0.3415 0.2606 0.2609 0.3440

***,**, * denote statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 5

Cross-section Regressions Partitioned Across Buy and Sell Orders

This table reports the results from partitioning the data into buy and sell trades and estimating the following equation:

𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡 = 𝛼𝑖 + 𝛿𝑡 + 𝛽1𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽2𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 + 𝛽31/√𝑁𝑇𝑖,𝑡 + 𝛽4𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡+𝛽5𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡+ 𝛽6𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡 + 𝛽7𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 + 𝛽8𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 + 𝜀𝑖,𝑗,𝑡

The dependent variable 𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡 is the observed frequency of transaction prices that fall on $0.05 increments for firm i in week t. The independent variables have been defined previously. We partition the sample by limit-sell and limit-buy orders. We report t-statistics in parentheses after controlling for firm and time fixed effects.

Sell Buy

(1) (2) (3) (4) (5) (6)

𝑆𝑖𝑧𝑒𝑖,𝑡 0.00002 0.00003 -0.0002 0.0002 0.0002 0.0005** (0.14) (0.22) (-0.99) (1.52) (1.44) (2.49)

𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 0.0004*** 0.0004*** 0.0004*** 0.0001* 0.0001* -0.0002**

(11.74) (11.92) (4.93) (2.10) (1.94) (-2.25)

1/√𝑁𝑇𝑖,𝑡 0.0465*** 0.0464*** -0.2547*** 0.0462*** 0.0471*** 0.0467***

(4.22) (4.22) (-13.32) (4.85) (4.95) (2.79)

𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡 0.0214*** 0.0215*** 0.0029 0.0219*** 0.0218*** 0.0064***

(16.54) (16.61) (1.44) (18.78) (18.71) (3.42)

𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡 0.0004*** 0.0004*** 0.0003*** 0.0003*** 0.0003*** 0.0004***

(44.19) (44.46) (26.32) (35.23) (35.08) (29.03)

𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡 0.0053*** 0.0043*** (-11.72) (10.45)

𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 -0.0065*** 0.0057*** (-14.46) (13.87)

𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 -0.0126*** 0.0035*** (-20.06) (5.68)

Observations 129,270 129,270 54,043 129,250 129,250 54,033

Adj R2 0.2983 0.2986 0.3989 0.2569 0.2574 0.3501

***,**, * denote statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 6

Cross-section Regressions Separated Across Buy/Sell Orders and Exchange Listings

This table reports the results from estimating the following equation:

𝐶𝑙𝑢𝑠𝑡𝑒𝑟%𝑖,𝑡 = 𝛼𝑖 + 𝛿𝑡 + 𝛽1𝑆𝑖𝑧𝑒𝑖,𝑡 + 𝛽2𝑃𝑟𝑖𝑐𝑒𝑖,𝑡 + 𝛽31/√𝑁𝑇𝑖,𝑡 + 𝛽4𝑆𝑝𝑟𝑒𝑎𝑑%𝑖,𝑡+𝛽5𝑇𝑟𝑎𝑑𝑒𝑆𝑖𝑧𝑒𝑖,𝑡+ 𝛽6𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡 + 𝛽7𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 + 𝛽8𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 + 𝜀𝑖,𝑗,𝑡

All variables have been defined previously. For robustness, we partition the sample by sell and buy orders and also by exchange listings (NYSE/AMEX, NASDAQ). We note that the sample consists of executed orders on the NASDAQ exchange only, but the exchange listings differ among stocks. We draw seemingly identical coefficients on all control variables; therefore, we report only the coefficients on the declining market indicator variables. T-statistics are reported in parentheses after controlling for firm and time fixed effects.

Panel A. NYSE/AMEX

Sell Buy

(1) (2) (3) (4) (5) (6)

𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡 -0.0023*** 0.0018***

(-4.42) (3.80)

𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 -0.0060*** 0.0034*** (-11.56) (7.23)

𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 -0.0079*** 0.0017*** (-11.84) (2.67)

Control Variables Yes Yes Yes Yes Yes Yes

FE Yes Yes Yes Yes Yes Yes

Observations 62,762 62,762 26,175 62,751 62,751 26,172

Adj R2 0.1931 0.1946 0.2661 0.1798 0.1803 0.2544

Panel B. NASDAQ

Sell Buy

(1) (2) (3) (4) (5) (6)

𝐹𝑖𝑟𝑚𝐷𝑒𝑐𝑖,𝑡 -0.0081 0.0065 (-11.16) (9.95)

𝑀𝑘𝑡𝐷𝑒𝑐𝑖,𝑡 -0.0070 0.0078 (-9.67) (11.86)

𝐸𝑥𝐷𝑒𝑐𝑖,𝑡 -0.0172 0.0052 (-16.36) (5.08)

Control Variables Yes Yes Yes Yes Yes Yes

FE Yes Yes Yes Yes Yes Yes

Observations 66,508 66,508 27,868 66,499 66,499 27,861

Adj R2 0.3068 0.3065 0.4095 0.2639 0.2644 0.3525

***,**, * denote statistical significance at the 1%, 5%, and 10% levels, respectively.