The Rodney L. White Center for Financial Research

Adverse Selection and Competitive Market Making:Empirical Evidence from a Pure Limit Order Market

Patrik Sandas

014-99

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Adverse Selection and Competitive Market Making:

Empirical Evidence from a Pure Limit Order Market

Patrik Sand�as�

The Wharton School

University of Pennsylvania

Philadelphia, PA 19104-6367

e-mail: sandas@wharton.upenn.edu

This Version: July 1999

Abstract

In this paper, I estimate and test a model of liquidity provision in a pure limit

order market based on Glosten (1994). The estimation strategy is directly based on

restrictions on quotes and depths in the limit order book implied by the theoretical

model. I �nd strong evidence of insu�cient depth in the limit order books relative to

the model predictions. For most stocks, an extended version of the model with a state

dependent market order distribution predicts depths and price revisions that are closer

to the empirically observed ones.

�This paper is based on my dissertation at Carnegie Mellon University. I am indebted to my committeemembers{Burton Holli�eld, Robert Miller, and Duane Seppi. The paper has improved thanks to detailedcomments and suggestions by an anonymous referee and Larry Glosten. I thank Ulf Axelson, Bruno Biais,Thierry Foucault, Rick Green, Jonas Niemeyer, Christine Parlour, Matthew Rhodes-Kropf, Chester Spatt,Per Str�omberg, and seminar participants at the 1999 AFA meetings, Carnegie Mellon, Columbia, LBS, MIT,NYU, UBC, UNC-Chapel Hill, and Wharton for many helpful comments. I thank the Stockholm StockExchange, Stockholms Fondb�ors Jubileumsfond, and Dextel Findata AB for providing the data set, and theA.P. Sloan and the W.L. Mellon Foundations for generous �nancial support.

1

A market for a security is liquid if investors can buy or sell large amounts of the security

at a low transaction cost. Liquidity is a valuable characteristic of a security because it

allows investors to realize more of the gains from optimal risk sharing through dynamic

trading.1 In many markets liquidity is supplied by market makers, who are willing, for a

premium (discount) relative to the current fundamental value, to take the other side of a

trade. Traders, who are willing to pay this premium (discount) in order to execute a trade

immediately, demand liquidity.2 A trader may desire to transact immediately because she

has some private information about the future value of the security or because she wants

to optimally re-balance her portfolio. The presence of traders with private information

exposes the market makers to adverse selection risk and therefore a�ects the prices that

they quote. The potential market power of the market maker(s) and the speci�c trading

rules of the exchange also a�ect the premium (discount) that the trader pays for liquidity.

The interaction between trader preferences and information, the market power of the market

makers, and the trading rules is at the heart of many policy questions. In order to answer

questions about whether high trading costs are due to strategic market makers, adverse

selection costs, or frictions due to the trading mechanism, it is necessary to consider models

that can identify these e�ects in the data.

In this paper, I analyze how adverse selection, competition, and market frictions a�ect

the cost of liquidity in a pure limit order market. In this type of market, the cost of liquidity

at any point in time is determined by the \limit order book", which contains all outstanding

limit orders. A market maker's decision to submit a limit order to the order book involves a

trade-o� between the \expected pro�t" on the limit order and the value of the \free-trading

option" that the limit order provides to the market. The probability of the order being

executed and the di�erence between the order price and the value conditional on execution

determine the expected pro�t. The value of the free trading option depends on how likely

1See Amihud and Mendelson (1986) and Brennan and Subrahmanyam (1996) for estimates of the com-pensation for illiquidity in asset returns.

2An investor could either demand or supply liquidity at various points in time and thus the terms \marketmaker" and \trader" do not necessarily refer to a �xed group of individuals.

2

it is that the execution of the limit order is triggered by new information, which causes the

fundamental value to move against the limit order submitter. At any order price, competitive

market makers submit new limit orders as long as it is pro�table to do so. Due to the time

priority rule the marginal pro�ts for additional limit orders are decreasing at every price

level. Therefore the competitive limit order book is characterized by a break-even condition

for the marginal limit orders at every price. The basic idea of this paper is to use this

break-even condition to estimate the adverse selection costs and test the theoretical model.

A pure limit order market is a trading mechanism that o�ers investors two main order

choices, market and limit orders.3 A limit buy (sell) order is an order, which speci�es a �xed

order price and quantity and is not assured execution. When a limit order is submitted it

typically enters the centralized order book where it is stored until execution or cancellation.

A trade occurs when a market order arrives and is \executed" against the outstanding limit

orders. Limit orders are ranked according to a set of \priority" rules, which determine the

order in which they are to be executed. Two common priority rules are order price and order

submission time.4

In pure limit order markets, e.g., the Paris Bourse and the Stockholm Stock Exchange,

there are no designated market makers. All liquidity is supplied by limit orders. The

member �rms are \broker-dealers". They can act in their broker capacity as agents for

outside investors or in their dealer capacity as investors trading on their own account. The

potentially large number of limit order submitters and the transparent trading rules suggest

that competition would eliminate any pro�t opportunities in the limit order book. Given

the simplicity of the trading rules, the limit order market o�ers an excellent laboratory

for studying the interaction between the adverse selection risk and the trading rules in a

competitive market.

3Domowitz (1993) documents that approximately 35 �nancial markets in 16 di�erent countries use tradingsystems based on some form of limit order book. The \pure" pre�x means that in this market mechanismthere is only a limit order book in contrast to \hybrid" market mechanisms such as the NYSE where thereis both a designated market maker and a limit order book.

4A higher (lower) price buy (sell) orders have priority. At the same price, orders with an earlier submissiontime have priority.

3

Trading in this market is modeled, following Glosten (1994), as a game between market

makers, who supply liquidity, and traders, who demand liquidity.5 The model is a version

of Glosten (1994) with price discreteness and the time priority rule, similar to the pure

limit order market version of Seppi (1997). Market makers submit limit orders to a limit

order book in order to exploit pro�t opportunities. A potentially informed trader with

unobservable preferences for liquidity enters and submits an optimal market order given

the existing order book. Her reservation price for the security may di�er from the current

fundamental value due to private information or a portfolio re-balancing need. A trader

facing a price schedule, which is weakly increasing in the order quantity will submit a larger

order quantity the higher her reservation price is. In order to protect themselves against

the adverse selection risk, the market makers charge a higher price for larger orders. A

limit order is executed whenever a su�ciently large market order arrives. The expected

payo� on a limit order is given by the execution probability times the di�erence between

the current fundamental value and the order price and an adjustment term, which re ects

the expected adverse selection cost. The execution probability is weakly decreasing and the

adverse selection cost is increasing in the quantity and thus the competitive quantity is the

solution to a break-even condition.

I use the break-even conditions imposed by the model to estimate and test the model. The

estimated parameters characterize the distribution of market orders and the order processing

and adverse selection costs. The adverse selection cost is measured by the change in the

expected value of the security conditional on a trade. It is assumed that this cost is a

function of the market order size so that larger orders tend to result in larger revisions in the

expected value and thus in a higher adverse selection cost. The change in the expected asset

value conditional on a trade or the \price impact" of a trade is parameterized as a function

of the market order size.

The empirical analysis in this paper di�ers from most empirical market microstructure

5Rock (1996) and Seppi (1997) use a similar setup for the limit order book in theoretical models of theinteraction between the specialist and the limit order book at the NYSE. Parlour and Seppi (1997) extendthis framework to analyze inter-exchange competition for order ow.

4

studies by explicitly testing restrictions on both the price and quantity dimensions of liquid-

ity.6 There is a unique price and quantity pair that satis�es the break-even condition given

the market order distribution, the order processing costs, and the adverse selection costs

(the price impact function). Therefore the break-even conditions allow us to identify the

model parameters from a series of observations of limit order books and market orders. An

additional set of restrictions is obtained by considering how the price impact function relates

to the price dynamics. By de�nition, the price impact function predicts the revision in the

expected asset value conditional on a given market order quantity. If updating is rational,

this prediction should, of course, on average be correct. By combining the updating restric-

tions and the break-even conditions we obtain a set of internally consistent restrictions on

both the \shape" of the limit order book at a point in time and on the order book revisions

in response to trades.

The main empirical �ndings of the paper are:

1. The price schedules o�ered by the limit order books appears to be too steep, i.e., the

book o�ers insu�cient depth, to be internally consistent with a single price impact

function that can explain the shape of the order book at one point in time and its time

series properties.

2. I �nd evidence consistent with an endogenous market order ow distribution. Large

(small) market orders are more likely when the order book depth is large (small).

3. The deviations from the model predictions are inversely correlated with the time

elapsed since the last transaction. This is consistent with a market where it takes

time for the order book to be replenished.

4. For most stocks, a state dependent market order distribution and price impact function

predict order book depths and price revisions that are close to the observed ones. In

particular, the market order ow distribution is strongly dependent on state variables

6Exceptions include papers by Lee, Mucklow, and Ready (1993) and Kavajecz (1997).

5

such as the past trading volume. For some stocks, I also �nd evidence consistent with

more (less) severe adverse selection risk when stock speci�c volatility (market wide)

volatility is high, everything else equal.

Several theoretical and empirical studies of order placement strategies in �nancial markets

have appeared during the last few years. Foucault (1999) and Parlour (1998) develop dynamic

models, which show how the trade-o� between execution price, execution probability, and

the winner's curse risk determines the investors' optimal order placement strategies in limit

order markets. In this paper, I analyze the empirical implications of a di�erent trade-o�.

Namely, the trade-o� between the value of the \free trading option" and the surplus accruing

to liquidity providers in a competitive market.7

Biais, Hillion, and Spatt (1995) study the order ow dynamics at the Paris Bourse and

�nd evidence consistent with information asymmetries and competition. This paper builds

on their paper by directly testing a theoretical model that predicts the shape of the order

book in competitive market with asymmetric information. De Jong, Nijman, and R�oell

(1996) test a version of Glosten (1994) on data from the Paris Bourse. This paper di�ers

from theirs by providing a direct test of the model restrictions and by incorporating the

e�ects of price discreteness and the time priority rule and utilizing information both in the

limit order books as well as in the trades.8

The next section provides a brief description of the market institution and the data set.

The theoretical model is presented in Section 2. Section 3 develops the empirical strategy,

and Section 4 presents the empirical results. Concluding comments are in the �nal section.

7Other related work, include theoretical papers by Byrne (1993), Chakravarty and Holden (1995), Cohen,Maier, Schwartz, and Whitcomb (1981), Harris (1998), Kumar and Seppi (1993), Parlour and Seppi (1997),and empirical papers by Greene (1995), Handa and Schwartz (1996), Harris and Hasbrouck (1996), Kavajecz(1997), and Lo, MacKinlay, and Zhang (1997).

8Other studies of limit order markets include Coppejans and Domowitz (1998), De Jong, Nijman, andR�oell (1995), Frino and McCorry (1995), Holli�eld, Miller, and Sand�as (1996), Lehmann and Modest (1994),Hamao and Hasbrouck (1995), and Hedvall, Niemeyer, and Rosenqvist (1997).

6

1 Description of the Market and the Dataset

The empirical analysis in this paper is based on data from the Stockholm Stock Exchange, a

pure limit order market. This section provides a brief description of the market institution

at the Stockholm Stock Exchange and the data set used in this study.

1.1 The Market Institution

A new limit order trading system was introduced in 1990 on the Stockholm Stock Exchange.9

This trading system is similar to the systems used on the Toronto and Paris Stock Exchanges.

In this system, a new limit order enters a computerized \limit order book" where it is

stored until execution or cancellation. A trade occurs when a market order arrives and is

\executed" against the outstanding limit orders in the order book. Orders in the book are

executed giving priority �rst according to price and secondly according to time of submission.

Investors submitting limit orders have the option to \hide" part of the limit order quantity,

that is, allowing only a part of the limit order quantity to be displayed in the book. The

hidden part of any limit order has lower priority than all the visible order quantities at

a given price level. Any hidden order quantity automatically becomes \visible" when the

initially visible part of the order has been executed. The data used in this paper does not

distinguish between the hidden and visible order quantities.

All order prices are restricted to be multiples of a �xed minimum price unit, the \tick

size". During the sample period, most stocks were traded either below 100 crowns with a

tick size of 1/2 crown or above 100 crowns with a tick size of 1 crown. During this period

a US dollar was equal to roughly 6.5 Swedish crowns. The tick size at the Stockholm Stock

Exchange is relatively coarse in comparison to other markets. The exchange has decreased

the tick size twice since the data set of this study was collected.

There are no designated market makers in this market. All liquidity is supplied by limit

9See Domowitz (1993) for extensive documentation and analysis of di�erent trading systems.

7

orders. The exchange members are \broker-dealers". Thus, limit orders are either submitted

by outside investors using brokers or by member �rms trading on their own account.

The limit order market is very transparent; all information on the status of the limit order

book is instantaneously transmitted to the computer screens in the o�ces of the exchange

members. The information disseminated includes the �ve best bid and ask quote levels

with the corresponding buy and sell quantities as well as identi�cation codes for the broker-

dealers submitting the limit orders. The transparency and rich information disseminated to

all market participants facilitate detection of pro�t opportunities. The computerized trading

system and the lack of institutional barriers allow market participants to quickly shift market

making resources from one stock to another. The relatively simple trading rules and the rich

information set available makes this market an excellent laboratory for analyzing how the

order strategies followed by di�erent agents interact with the trading rules to determine the

observed orders and prices.

1.2 The Data Set

The data set contains histories for all trades and orders that were submitted to the electronic

trading system during normal trading hours for a sample of 10 actively traded stocks at the

Stockholm Stock Exchange.10 The sample period consists of the 59 trading days between

December 3, 1991, and March 2, 1992. A list of the companies included in the sample is

presented in Table 1. The sample is representative of the electronic trading but it does not

include all trading in these stocks. Transactions made in London on the SEAQ International

and in the U.S. through the NASDAQ system accounted for a signi�cant fraction of the

turnover in some Swedish companies during this time period. In addition, block trades

can be settled outside the electronic system and if this occurs during the normal trading

10The full sample included all shares of companies that had a least one class of shares included in theOMX-index during the sample period. Using this criterium a total of 60 stocks were included in the sample.The OMX-index is a trading volume weighted index, which includes the 30 most actively traded securitieson the Stockholm Stock Exchange. The index is updated semiannually. A smaller sub-sample of 10 stockswas randomly selected using two additional criteria. Firstly, only stocks with at least 500 trades during the59-day sample period were considered. Secondly, for a company with dual-class shares no more than oneclass was included in the �nal sample.

8

hours these trades must be reported to the market. While other trading venues account

for a signi�cant proportion of the total trading volume most transactions are routed to the

electronic trading system. The analysis in this paper focuses on the trading in the electronic

system.

Summary statistics characterizing the trading activity in the sample stocks are reported

in Table 2. The total number of transactions over the 59-day sample period range from a

low of 667 for PROC to a high of 8102 for LME. Similar di�erences in activity are re ected

in the total number of limit orders submitted where the low is 1463 and the high is 12646.

The third column of Table 2 reports the average transaction price for the stocks. Most of the

stocks in the sample are traded with a tick size of 1 crown (prices above 100 crowns). The

exception is SEB and for part of the sample LME and INVE, which traded in the 1/2 crown

tick size range (prices below 100 crowns). The daily trading volume for the stocks in the

sample is quite dispersed with a minimum of 1.1 million crowns (PROC) and a maximum

of 35.2 million crowns (LME). The last column of Table 2 reports the sample period returns

for the stocks.

The model considered in this paper provides a way of decomposing the price schedule

provided by the limit order book into its di�erent components. The decomposition depends

on the order book itself as well as the distribution of market orders. I characterize the price

schedules and the market order distributions here to provide a benchmark for the empirical

analysis that follows. One way to characterize the price schedule is to look at the prices and

the quantities o�ered directly. In Table 3 the average spreads o�ered in the book and the

corresponding order quantities are reported for the two best levels at the bid and the ask

side of the book.11 Notice that for some stocks, e.g., SEB, the average spreads are very close

to 1/2 crowns, which is the minimum tick size for that stock. For other stocks, e.g., ASEA,

PROC, and STOR, spreads of several ticks are common. The total order quantities o�ered

at the best and second best price levels is between 1.8 and 2.4 times the quantities o�ered at

11In the data it is rare to observe orders that \walk up the book" beyond the two best price levels. Thus,the trade-o� we are interested in this paper are most relevant for the two best price levels.

9

the best price levels. For all stocks, the spreads between the best bid (ask) quotes and the

second best bid (ask) quotes are greater than the bid-ask spreads. In order to determine how

the trading costs are related to the quoted order books we need to know the distribution of

market order quantities.

In order to focus on the most relevant ranges of the limit order books the distributions

for market buy and sell orders were calculated for each stock. Market orders submitted in a

sequence within 30 seconds by the same broker-dealer on the same side of the market (e.g., a

market buy order by dealer A directly followed by another market buy order from dealer A

within 30 seconds) were aggregated into one market order observation. Table 4 reports the

10th, 30th, 50th, 70th, and 90th percentiles for the market buy and sell order quantities as

well as the average size for each stock. Note that for most of the stocks the median market

order size is much smaller than the average market order size.

The limit order book o�ers a price schedule for immediate liquidity demand. Suppose

that the costs of providing liquidity consists of a �xed order processing cost, b, and a variable

adverse selection cost (price impact), c, which depends on the market order size, Qt; (Qt > 0

for market buy and Qt < 0 for market sell orders). The \price impact" re ects the infor-

mation content of the order Qt. The expected change in the fundamental value conditional

on the order submitted at time t is E[Xt+1 � XtjQt] = a + Qtc, where a is the expected

change in the value. Let It be an indicator, which takes value +1 and �1 for market buy

and sell orders respectively. The competitive transaction price for a market order at time t

is pt = Xt + Itb + Qtc, where Xt denotes the fundamental value at time t: Taking the �rst

di�erence we get

pt+1 � pt = �p = a +�Ib+Qt+1c+ "; (1)

where " is a random innovation in the value between time t and time t+1 and �I = It+1�It.

This price impact regression equation was estimated for each of the ten sample stocks. Table

5 reports the results. The parameter estimates with the corresponding t-values are reported

in the �rst three columns followed by the R2 and the F-values. The �xed order processing

cost, b, and the adverse selection cost, c, are both positive and statistically signi�cant at the

10

1% level for all of the stocks. What can we say about the observed limit order books based

on the estimated price impact regressions?

Figures 1 and 2 plot the empirical price schedule as well as an implied price schedule based

on the above regression results. Each of the panels in Figures 1 and 2 were constructed

as follows. The marginal prices for \hypothetical" market orders of di�erent sizes were

computed for each limit order book observed before a transaction in the sample. For each

stock, the market order sizes were chosen to match the respective quantities reported in

Table 4. The marginal price is de�ned as the price paid for the last unit given the orders

on the book. The marginal prices were normalized by the corresponding mid-quote. The

average and the median are plotted for each stock. In order to assess whether these price

schedules are relatively \steep" or not the following benchmark was computed. Suppose

the marginal price for a market buy order for Q units is simply determined by the sum of

the order processing cost b and the price impact cQ. The marginal price schedule based

on this type of calculation is plotted for the di�erent market order size categories in each

panel in Figures 1 and 2. It is clear from the graphs that the price schedules implied by the

coe�cients from the price impact regressions are \too at" to match either the average or

the median schedule o�ered by the order book. The di�erence tends to grow as we move

further out in the limit order book. Is this evidence of pro�t opportunities in the limit order

book? This is not necessarily the case.

One aspect of liquidity provision in a limit order market is that liquidity providers can not

condition on the size of the market order that will trigger the execution of a limit order. This

implies that liquidity providers need to compute so-called \upper (lower) tail expectations",

see Rock (1996) and Glosten (1994). The tick size and the time priority can also potentially

explain some part of the discrepancy between the actual and the implied price schedules.

Price discreteness combined with a time priority rule allows non-marginal limit order to earn

positive pro�ts in the order book.

In the next section, I present a model of the liquidity provision process in a limit order

market. The model captures important institutional features of this market including the

11

transparency of the trading system, price discreteness, and the time priority rule. The model

imposes testable restrictions on the data that allow us to address questions like: Are the

observed limit order books consistent with competitive liquidity provision? What fractions

of the trading costs are due to adverse selection? Can a more detailed model account for the

di�erences between the empirical and the implied price schedules in Figures 1 and 2?

2 Model

The model presented here follows Glosten (1994), modi�ed for discrete prices and the time

priority rule as in the pure limit order market version of Seppi (1997). The theoretical results

are reviewed here to provide a framework for the empirical analysis.

2.1 General Features of the Model

There are two types of agents in the market. Agents who supply liquidity by submitting limit

orders are referred to as market makers. They can be thought of as patient market partici-

pants who choose to supply liquidity. The other class of agents is traders. Traders, who may

have private information, demand liquidity by submitting market orders. Correspondingly

they can be thought of as impatient market participants who do not want to postpone their

trading. There are a large number of both types of agents. The market makers are risk

neutral and pro�t maximizing.

The agents trade a risk-free asset with a return normalized to zero and a risky security

whose value in period t is given by Xt. This value is the fundamental value of the security

conditional on all publicly available information. This value changes as new information

arrives. Next period's fundamental value Xt+1 is given by

Xt+1 = Xt + �+�t+1; (2)

where �t+1 represents a random innovation in the value and � is the expected change.

12

Trading occurs sequentially over a discrete number of periods indexed by t. In each

period t, there are three stages. Shortly before t market makers submit new limit orders.

Market makers are randomly given the opportunity to submit limit orders at this stage. This

process is repeated as long as someone wishes to place a new order. At time t; when the new

limit order book has been established, a trader arrives and submits a market order. After

the trade, the new fundamental value is announced and the game starts over.

The agents trade by submitting market and limit orders to a limit order book. For any

order book I will use the following notation for the order prices and quantities. Let the prices

fp+1; p+2; : : : ; p+kg denote the ask prices in the book ordered from the best ask price, p+1, to

the kth best ask price, p+k. The order quantities associated with each price level are denoted

by fQ+1; Q+2; : : : ; Q+kg. The buy side variables are denoted analogously with a negative

index. Thus, any limit order book is characterized by a sequence of prices fpig+ki=�k and

order quantities fQig+ki=�k. Let the market order quantity submitted at time t be denoted by

mt. Market buy orders correspond to positive quantities, m > 0, and market sell orders to

negative quantities, m < 0.

The market makers submit their limit orders based on the publicly available information

at time t. The public information set includes the sequence of orders and trades up to time t.

The market order submitted by the trader at time t is correlated with the innovation in the

security value. The market makers information set only includes the probability distribution

of market order quantities.

2.2 The Liquidity Demand

The trader arriving in a given period is independently drawn from a population of traders.

In principle, the optimal market order quantity submitted by the trader will depend on

factors such as the trader's current position in the security and her information about the

likely future value of the security. In order to simplify the analysis and concentrate on the

decision problem of the market makers we will use a reduced form representation for the

13

trader's demand for liquidity. It is assumed that a trader will be a buyer or a seller with

equal probability and that the desired market order quantity is exponentially distributed.

The distribution of market order quantities may depend on market conditions, which are

characterized by a state variable z. The distribution of market order quantities m can be

summarized by the following density function f(mjz), where

f(mjz) =

8><>:

fmb(mjz) =1

2�(z)e�

m�(z) if m � 0 (market buy);

fms(mjz) =1

2�(z)e

m�(z) if m � 0 (market sell).

(3)

The expected market buy and sell order quantities are �(z) and �(z) respectively in state

z. Given our parameterization of the liquidity demand we can now solve the market maker's

decision problem.

2.3 The Market Makers' Decision Problem and the Order Book

The traders who demand liquidity may be informed about the innovation in the value of

the security, �t+1: The market order is therefore informative about the likely value of this

innovation. This may occur because the traders are privately informed about the value of

the innovation. An alternative explanation, which in this case will lead to the same limit

order books, is that some traders observe public information regarding the value of the

innovation and \pick o�" limit orders that are exposed. In both cases we will �nd that

the market order quantity submitted is correlated with the value of the security in the next

period. I will summarize the information link between the market order quantity m and the

fundamental value Xt in a reduced form function, h(mjz); which relates the market order

quantity m to the next period's fundamental value as follows

E[Xt+1jXt; m; z] = Xt + �+ h(mjz) + �t; (4)

where �t represents an innovation in the value of the asset that is not associated with the

trade. The function h(mjz), which is referred to as the price impact function, is a non-

14

decreasing function of the market order quantity.12 The price impact function is summarized

by a function h() de�ned as:

h(mjz) = �(z)m; (5)

where �(z) is the potentially state dependent per unit price impact of market orders.

Given our assumptions we would expect � to be positive. For a given distribution of market

orders, a larger � parameter implies a more severe adverse selection problem.

Consider an arbitrary limit order book and an arbitrary price level p. A limit order,

which is the Qth best sell order unit in the limit order book, is executed whenever a market

order that is su�ciently large arrives, i.e., mt � Q. Multiple limit orders at the same price

level do not \split the surplus". Instead, limit orders are executed according to strict time

priority. This means that all limit order units with a higher time priority must be executed

�rst.

Market makers executing are assumed to incur a quantity invariant order processing cost,

which we will denote by (z): I assume that the order processing cost is equal for buy and

sell orders. In the absence of adverse selection costs (i.e., � = 0) the order processing costs

would determine the bid-ask spread and the order book would o�er in�nite depth at the best

bid and ask quotes.

Suppose the price level p was the lowest price, which is above the current fundamental

value, Xt. I assume, for simplicity, that the di�erence between this price level and the

fundamental value is large enough to make it pro�table to supply a positive quantity at this

price level. Consider the market makers problem of deciding how many units to o�er at this

price level. The expected pro�t on the last unit, q; is determined by

E ~m[(p1 � (z)� E[Xt+1j ~m; z])� I( ~m � q) jz]; (6)

where p1 � E[Xt+1j ~m; z] is the di�erence between the price the market maker receives

and the expected fundamental value conditional on the market order m, I( ~m � q) is an

12Many speci�cations of this model, including the normal-exponential version, yield a non-decreasing priceimpact function. See Glosten (1994), page 1137, for a counter example.

15

indicator, which is one if the market order submitted is larger than q, and (z) is the order

processing cost. We can rewrite Equation (6) as follows using Equations (3), (4), and (5),

Z1

q(p1 �Xt � (z)� h(ujz)u)

1

2�(z)e�

u�(z)du

=1

2(p1 �Xt � (z)� �(z)(q + �(z))) e�

q�(z) : (7)

The market makers submitting limit orders to the order book are assumed to behave

competitively. Therefore the quantity o�ered at the price level p must be such that the last

unit breaks even. This implies that the quantity, Q1, submitted at the best sell price level

price level, p1; is given by

Q1 =p1 �Xt � (z)

�(z)� �(z) (8)

Given that the quantity Q1 is o�ered at p1 we can now ask what the quantity is that

the market makers will o�er at the next price level, p2 = p1 + �; where � is the tick size.

Using similar arguments we �nd that the quantity o�ered at the second best price level, Q2,

is given by

Q2 =p1 + � �Xt � (z)

�(z)�Q1 � �(z): (9)

By recursively following the procedure described above we can construct the whole limit

order book at the sell side. By analogy we can construct the order book on the buy side.

The model presented above is static and therefore we can not make predictions about

the timing of limit order submissions. One way to think about the liquidity provision is

to allow one market maker to enter and replensih the limit order book until all price levels

satisfy the conditions presented above. Alternatively, several market makers could enter and

\pick o�" pro�ts opportunities in the limit order book. A large number of market makers

would guarantee that the limit order book would o�er pro�t opportunities only for brief time

intervals. The distinction between the timing of limit order submissions and market order

arrivals is of course arti�cial.

16

3 Estimation Strategy

The theoretical model imposes restrictions on the price and quantities in the limit order

book as well as on the joint dynamics of the fundamental value and the order books. I will

brie y summarize and contrast the two types of implications of the model. The empirical

strategy, which is subsequently outlined, builds directly upon these restrictions.

3.1 Break-Even Conditions

Given the model framework presented in Section 2 we know that the limit order book at time t

is characterized by a set of break-even conditions. We will concentrate on the restrictions that

apply to the two best bid and ask quotes. The limit order book on the sell side is described

by a pair of order prices fp+1; p+2g with corresponding order quantities fQ+1; Q+2g: The

break-even conditions impose the following set of restrictions on these prices and quantities

pk �Xt � (z)� �(z)(kX

i=1

Qi + �(z)) = 0 k 2 f+1;+2g; (10)

wherePk

i=1Qi is the aggregate quantity o�ered at the kth price level and all price levels

below the kth level on the sell side. Correspondingly, we have restrictions on the limit buy

side, which relates the prices fp�1; p�2g and the quantities fQ�1; Q�2g:

Xt � pk � (z)� �(z)(kX

i=�1

Qi + �(z)) = 0 k 2 f�1;�2g; (11)

whereP�ki=�1Qi is the aggregate quantity o�ered at or above the -kth price level on the

buy side.

The theoretical model makes a sharp distinction between the time of limit and market

order arrivals. In reality there is no such distinction. In order to bridge this gap between the

model and data we assume that the break-even conditions hold approximately by introducing

an error term in the Equations (10) and (11). Let the errors terms corresponding to the

break-even condition be denoted by �k, k 2 f�1;�2g. The error terms capture the fact that

17

at any point in time the order book may o�er pro�t opportunities or exposed limit orders.

Market makers are assumed to frequently enter the market and update their outstanding

limit orders in the order book and potentially add new limit orders when the order book

exhibits pro�t opportunities. We will assume that on average the limit order books do not

exhibit exposed limit orders or unexploited pro�t opportunities for liquidity suppliers. This

implies that the limit order books observed in the data are assumed to be characterized by

the following set of equations for the limit sell and buy side

E

"pk �Xt � (z)� �(z)(

kXi=1

Qi + �(z))

#= 0; k 2 f+1;+2g (12)

and

E

24Xt � pk � (z)� �(z)(

kXi=�1

Qi + �(z))

35 = 0: k 2 f�1;�2g (13)

In particular, we will assume that Equations (12) and (13) hold for the limit order

books preceding a transaction in the data. We could now construct moment conditions and

overidentifying restrictions by using variables in the information set as instruments. In order

to eliminate the unobserved fundamental value, Xt, pairs of equations can be combined

to di�erence out the fundamental value. To simplify the notation let yt denote the data

corresponding to the tth observation, yt = f(pk; Qk); k = 1; 2; ztg and let ' denote the

vector of parameters of interest, ' = f�; ; �; �g. The sum of the kth and �kth break-even

conditions, Equations (12) and (13), is then equal to

E[ek(yt; ')] = E[�pk � 2 (z)� �(z)(kX

i=1

Qi + �(z) +�kX

i=�1

Qi + �(z))] = 0; (14)

where �pk = p+k � p�k. Using the two best bid and ask quotes we can construct two

moment conditions based on Equation (14).

An additional set of moment conditions is obtained by forming the di�erence between

the expected values of the buy and sell market order quantities and the means of the market

18

order distributions given by �(z) and �(z) respectively. These restrictions are given by

E[(mt � �(z))I(mt > 0)] = 0;

E[(mt + �(z))I(mt < 0)] = 0:(15)

Additional moment conditions can be constructed by using variables in the public infor-

mation set at time t. A vector of moment conditions denoted gt(yt;') is obtained by stacking

all the moment conditions as

gt(yt;') =

2666666664

e1(yt; ') Zt

e2(yt; ') Zt

...

ek(yt; ') Zt

3777777775: (16)

A GMM estimator of the parameter vector ' can then be de�ned as:

'T = argmin'2

8<:"1

T

TXt=1

gt(yt;')

#0WT

"1

T

TXt=1

gt(yt;')

#9=; ; (17)

whereWT is a positive de�nite weighting matrix. Hansen (1982) proves that the GMM es-

timator of ' is consistent and asymptotically normally distributed. The variance-covariance

matrix of 'T is given by T' = [D0

0S�10 D0]

�1, whereD0 � E[@gt(')@'

] and S0 �Ps=+1

s=�1E[gtg0

t�s].

A consistent �rst round estimate is obtained by using the identity matrix as the weighting

matrix. In the second round, a heteroscedasticity and autocorrelation consistent estimate

of the variance-covariance matrix S0 is obtained by following the Newey-West procedure.

Hansen (1982) proves that T times the minimized value of Equation (17) is asymptotically

distributed as chi-squared with the number of degrees of freedom equal to the number of

orthogonality restrictions minus the number of parameters estimated.

3.2 Updating Restrictions and Order Book Revisions

The law of motion for the fundamental value given in Equation (2) together with the price

impact function h(�jz) given in Equation (5) provides a link between the changes in the

19

limit order books from period to period and the market order ow. This link will be used

to generate another set of restrictions on the data that complement the information in the

break-even conditions.

Consider the expected change in the fundamental value given a market order quantity

mt submitted in period t. It is given by

E[Xt+1 �Xtjmt] = �+ h(mtjzt) = �+ �mt: (18)

We also know that the limit order books on the sell side at time t + 1 and time t are

characterized by the following set of equations

E

"pk;t �Xt � (z)� �(z)(

kXi=1

Qi;t + �(z))

#= 0 k 2 f+1;+2g; (19)

and

E

24pk0;t+1 �Xt+1 � (z)� �(z)(

k0Xi=1

Qi;t+1 + �(z))

35 = 0 k0 2 f+1;+2g; (20)

where the set of prices pk;t; k 2 f+1;+2g and pk0;t+1 ; k0 2 f+1;+2g need not be the

same. By subtracting Equation (19) from Equation (20) we obtain the following expression

E [Xt+1 �XtjIt+1] = E[pk0;t+1 � pk;t � �(z)(k0Xi=1

Qi;t+1 �kX

i=1

Qi;t))]: (21)

Furthermore, given that the market order submitted at time t is in the information set

at time t + 1 we have that E [[Xt+1 �XtjIt+1] jmt] = E[Xt+1 � Xtjmt]: Using that fact we

can combine Equations (18) and (21) to obtain

E

24�+ �(z)mt �

0@pk0;t+1 � pk;t � �(z)(

k0Xi=1

Qi;t+1 �kX

i=1

Qi;t))

1A35 = 0: (22)

Equation (22) states that on average we expect the limit order book to be revised in a

manner that re ects the \price impact" of the previous trade. Based on Equation (22) we

can construct four moment conditions, one for each of the two best price levels, and a GMM

20

estimator of the parameter vector as in the previous section. Note that the order processing

cost parameter, (z), is not identi�ed from Equation (22). In the empirical analysis I use

the break-even and the updating restrictions both separately and jointly.

4 Empirical Results

4.1 Test of Main Model Predictions

The previous section discussed how we can estimate di�erent model parameters using the

two main sets of restrictions of the model. The results based on applying the break-even

and updating conditions separately are presented �rst as a starting point for the subsequent

analysis.

Table 6 presents the GMM estimates of the model using only the break-even conditions

[Equation (14)] and the mean equations for the market order distributions [Equation (15)].

The coe�cients � and �, which characterize the distribution of market order quantities, are

in general very close to the empirical averages of the market buy and sell order quantities

reported in Table 4. I will compare the distribution implied by this parametrization with the

actual distribution later in this section. The estimated order processing cost, , is negative

for all of the stock. It is, of course, unlikely that the true order processing cost is negative.

On the other hand, this could re ect rational behavior in a setting di�erent from the one

considered in this model. For example, if the relevant decision was whether to submit a

limit order or a market order, then, one could observe a negative expected trading pro�t.

In that case, one would expect this cost to be traded o� against the premium (above the

fundamental value) paid if a market order was used.

The estimated slope coe�cient, �, that determines the price impact of market orders is

positive for all stock. It is interesting to note that the magnitude of this coe�cient much

larger than the coe�cient inferred from the standard price impact regression (c in Table

5). Note that the data used to obtain the � coe�cients reported in Table 6 only include

21

\snapshots" of the limit order book at di�erent points in time as well as the market order

series. On average the slope coe�cient obtained with the break-even conditions is about

nine times greater than the corresponding slope coe�cient obtained using the standard price

impact regression. Based on these estimates the adverse selection cost for the marginal unit

of an average size market order is between 10% and 59% of the quoted bid-ask spread. The

large discrepancy between two sets of slope coe�cients is consistent with the large di�erences

between the price schedules presented in Figures 1 and 2. Using the updating restrictions,

which use time series information, we would expect a smaller di�erence.

Table 7 reports the results from estimating Equation (22) using the two best bid and ask

quotes and depths. The order processing cost, , is not identi�ed using these restrictions

alone. The drift parameter, �, is close to zero for most of the stocks. The slope of the price

impact function, �, is positive for all stocks. Note that the � coe�cients reported in Table

7 are smaller in magnitude than the corresponding coe�cients reported in Table 6. There

is large variation between the stocks but on average the value of the price impact coe�cient

obtained using Equation (22) is only about 28% of the values reported in Table 6. The

magnitude of the �'s reported in Table 7 also exceeds the estimates reported in Table 5

but this di�erence is much smaller. The main di�erence between the standard price impact

regression and the updating conditions used here is that the e�ects of price discreteness and

time priority as well as the \upper (lower) tail expectations" are appropriately accounted

for in the latter.

I use a constant and the market order ow as instruments in estimating (22) applied to

the two best bid and ask levels. The total number of orthogonality conditions is therefore

eight. The tests of overidentifying restrictions, reported in Table 7, reject this speci�cation

for LME, PROC, and SEB at the 1% level. One reason why we may reject this set of

restrictions is that order book changes may not be aligned. This could simply re ect the

fact that not enough time has elapsed for the order book to replenish to the \equilibrium"

levels. This possibility will be revisited later in this section. So far the results indicate that

the break-even and updating conditions lead us to make di�erent inference about the slope

22

of the price impact function.

Table 8 reports the results for the combined restrictions. By applying the break-even

conditions in Equation (14), the updating restrictions in Equation (22), and the moment

conditions for the market order ow in Equation (15) to the two best bid and ask levels eight

moment conditions are obtained. No instruments were used in this estimation. The system

is overidenti�ed since only �ve parameters are estimated.

The � and � estimates are comparable to the estimates reported before in Tables 6 and

7. The estimated order processing costs is negative and signi�cant for all of the stocks. The

estimated � coe�cients tend to be smaller than the ones reported in Table 6. The di�erence

between the estimated price impact in Table 6 and in Table 8 is roughly 10% on average.

The tests of over-identifying restrictions strongly reject this model speci�cation for all

stocks. The key common parameter is � and it is therefore clear that the discrepancy

between the � estimates based on the break-even and updating conditions is at least partially

responsible for the rejections. Basically, the order books together with the market order

distribution suggest that quotes should be revised by a certain amount following trades.

The actual revisions in quotes and depths are in fact much smaller. Thus, based on the

observed revisions we would expect to see more depth in the books. Various explanations

for this result are considered below.

To better understand how the model fails to match the data, it is useful to consider how

the price schedules we observe di�er from the ones implied by the model estimates. Figures 3

and 4 show the di�erence between the actual price schedules and two implied price schedules

based on the estimation results reported in Tables 6 and 7 respectively. The units on the

horizontal axis are 100 shares and the on the vertical axis we have the di�erence in price

impact measured in crowns. Thus, a \perfect" price schedule would yield a horizontal line

at zero.

The implied price schedules are computed based on the estimated slope coe�cients, �,

as well as the order ow parameters, � and �, which together with the slope coe�cient

23

determine the upper and lower tail expectations. The implied price schedules are computed

for the market order quantities that match the deciles reported in Table 4. We ignore

the e�ects of price discreteness as well as the order processing costs. Thus, the graphs

are designed to capture e�ects that are related to the market order quantities rather than

quantity invariant level e�ects. The solid line, which corresponds to the estimates obtained

using break-even conditions, is in general closer to zero. For most of the stocks the di�erence

is more pronounced for the price schedule implied by the updating restrictions (the dashed

line) than for the schedule implied by the break-even conditions (solid line).

One could, of course, also consider a similar comparison for the updating equations.

However, since the updating is only driven by the estimated slope coe�cient it seems clear

that the larger slope obtained when we combined the restrictions must produce an \over-

reaction" e�ect in the updating. There are, of course, several potential explanations for these

�ndings.

4.2 Market Order Distribution

One potential explanation is that the parametrization used for the market order distribution

provides a poor approximation of the empirical distribution. A related question is whether

the assumption of an exogenous market order distribution is a good one.

Figures 5 and 6 present a comparison of the empirical and the implied distributions. The

empirical distribution of market order quantities is plotted (solid line) against the distribution

implied by the exponential function and the parameter estimates for � and � (dashed line)

in the graphs on the left.13 For some of the stocks, for example, ASEA, PROC, and STOR,

the exponential distribution seems to match the empirical distribution pretty well. For

other stocks, for example, LME there seems to be signi�cant \bunching" of order that the

exponential distribution can not capture. For many of the stocks there appears to be more

13The empirical distribution was computed for discrete bins of size 200 (i.e., two round-lots for a typicalstock). The implied distribution was \discretized" by computing the fraction of order we would observe foreach bin if the market order ow was exponentially distributed with parameters � and �.

24

orders close to zero and far away from zero than we would expect if the true distribution

was an exponential with a parameter equal to the average order size. This is potentially

a problem as the adverse selection cost in the model is based on \upper" and \lower tail

expectations" that may be sensitive to these types of deviations.

To further study how the assumed functional form may a�ect our results, I computed

the upper and lower tail expectations implied by the estimated distribution. In Figures 5

and 6 the graphs on the right show the implied tail expectations for the market order ow,

for example, E[mjm > 500], based on the estimated distribution (dashed line). The same

expectations based on the sample information is indicated with a solid line. In most cases,

the estimates produce upper and lower tail expectations that are too small. This di�erence

grows as we move towards larger order quantities. This suggests that the functional form

assumption may be partly responsible for the rejections. Consider the e�ects of using a

linear price impact function but a distribution of market order quantities, which does a

better job in matching the tail expectations. This would allow us to capture more of the

\excess" slope in the order books without violating the updating restrictions. Basically, we

could \generate" a steeper price schedule without causing too much \over-shooting" in the

dynamic updating.

Assuming an exogenous market order ow distribution signi�cantly simpli�es the deriva-

tion of the equilibrium limit order book. However, it is plausible that in reality the market

order ow is a function of the limit order book. For example, it is likely that traders opti-

mally choose between market and limit orders or follow dynamic order submission strategies

that involves order splitting.14 It is beyond the scope of this paper to formally consider

a model with endogenous market order ow. In Figures 7 and 8, I provide some evidence

on the possible endogeneity of the market orders. Each subgraph in Figure 7 and 8 shows

the empirical distribution of the market order quantities and an adjusted distribution of the

market orders. The adjustment works as follows. Each market buy (sell) order quantity is

14For example, Holli�eld, Miller, and Sand�as (1999) estimate and test a model where the order ow isendogenous because traders optimally choose between market and limit orders.

25

multiplied by the ask (bid) depth in the book and divided by the average ask (bid) depth.

Thus, if the market buy (sell) order quantities are independent of the ask (bid) depths we

should obtain a distribution that is close to the original. The results suggest that this is not

the case. Instead the adjusted distribution is more concentrated around zero. This suggest

that the depths and the market order quantities are positively correlated. Whether or not

this phenomenon can explain the rejections above is unclear. It depends on how the market

order distribution is related to the information asymmetry.

The above �ndings suggest two sets of empirical facts that could explain the rejections of

the main model restrictions. First, a exible distribution for market orders should do a better

job in matching the empirical distribution of market orders and thereby better approximate

the upper and lower tail expectations. Second, a more complicated model that allows for

an endogenous order ow would better match the empirical facts. Whether or not that will

help explain the results obtained here is unclear. One feature that one would hope to achieve

by endogenizing the order ow is to match some of the time variation in the order books.

The summary statistics reported in Table 4 suggest that there is a lot of variation in the

order books. Without changing the current model framework we can ask whether changes

in market conditions can explain some of the variation in the books.

4.3 Changing Market Conditions

It seems plausible that the relative intensity of informed and liquidity trading changes over

time. Changes in the arrival rate of earnings and macro news could generate this type of

behavior. Thus, we would expect that both the distribution of market orders as well as the

price impact of market orders change with market conditions. Without a formal model it is,

of course, hard to know what a relevant state variable is. With that limitation in mind the

following analysis focuses on a small set of state variables that would seem like natural state

variables in a formal model.

The variables used are the volatility of the mid-quote over the last 60 minutes (z1), the

26

trading volume over the last 60 minutes (z2), and the volatility of the market index over the

last 60 minutes (z3). The state variables are normalized by subtracting and dividing by the

average value of the respective state variables. The �rst and the third state variables are

selected to capture changes in the arrival rate of stock speci�c and market wide information.

The trading volume variable should capture changes in trading activity and allow us to

separate high volatility periods with thin trading from periods of high volatility with heavy

trading. The state variables are allowed to a�ect the market order distribution through the

� and � parameters and the price impact function through the � coe�cient. The interaction

is assumed to be linear and given by:

�(z) = �0 + �1z1 + �2z2 + �3z3

�(z) = �0 + �1z1 + �2z2 + �3z3

�(z) = �0 + �1z1 + �2z2 + �3z3; (23)

where �0, �0, �0 may be interpreted as the base case for the \average " state and the other

coe�cients measure the e�ects of deviations from this state. The order processing cost and

the drift term are assumed to be constant. Table 9 presents the estimation results based on

using the break-even, updating, and market order mean equations together with a vector of

instruments, which includes a constant and the three state variables.

Table 10 reports the results for di�erent speci�cation tests based on this extended model.

The �rst column in Table 10 reports the chi-squared test of overidentifying restrictions for

the extended speci�cations. The test rejects only for LME and ASTR at the 5% level

suggesting that allowing for changes in the order distribution and the slope of the price

impact function captures some of the variations in the order ow and order books. The

subsequent columns report the chi-squared statistics for tests of further restrictions on the

model. The �rst restriction imposes no dependence on market conditions for both the market

order distribution and the price impact function. This restriction is rejected for all stocks

at the 1% level. In order to determine whether the market conditions matter more for the

order ow or the price impact two further tests were performed. A test of no dependence of

27

the market order distributions on market conditions is reported in the third column. This

set of restrictions are strongly rejected as well. Finally, a test of no state dependence in the

price impact function is reported. This test rejects for seven of the stocks at the 5% level and

for �ve stocks at the 1% level. These results suggest that accounting for market conditions

appears to be most important for the market order distribution.

In Table 9 we note that the coe�cients that capture the impact of past trading volume,

�2 and �2, are positive and signi�cant for most of the stocks. Thus, this speci�cation

potentially captures some of the endogeniety of the order ow illustrated in Figures 7 and 8

and its e�ects on the order book. The e�ects of changing mid-quote or market volatility are

more mixed. For some stocks, for example, SEB the �1 and �1 coe�cients are negative and

signi�cant, suggesting that we tend to see smaller market orders in more volatile markets.

Yet, for other stocks we �nd no e�ect or a positive e�ect. It is also interesting to note that

the order processing cost is positive for all stocks in this case. The estimated slope for the

price impact function is positive and signi�cant for six of the stocks. For the remaining four

we can not reject a zero slope with this speci�cation. The speci�cation tests discussed above

indicate that in general changing market conditions do not appear to be important for all

of the stocks. There are exceptions however. One is ASTR, where we have a positive �0

coe�cient, and a positive �1 coe�cient suggesting that orders tend to have a greater price

impact in more volatile markets. The �2 and �3 coe�cients suggest that the price impact

for a given order size is smaller in markets with relatively heavy trading or high overall

market volatility. These results are consistent with a market where periods of heavy trading

are associated with smaller information asymmetries, everything else being equal. The signs

of the coe�cients for mid-quote and market volatility are consistent with a market where

the adverse selection risk faced by market makers is more (less) severe if the stock speci�c

(market wide) volatility is high.

28

4.4 Time to Replenish the Book

One aspect of liquidity provision that does not enter the model explicitly is time. It takes

time for the market participants to learn about pro�t opportunities or exposed limit orders

in the order book. Thus, it is possible that some of the observations we use in the estimation

represent order book that are not at their \equilibrium" levels. It is also possible that time

a�ects the shape of the \equilibrium" order book along the lines of Easley and O'Hara (1992)

so that periods of no-trade may signal that information based trades are less likely to arrive.

It is beyond the scope of this study to formally model the e�ects of time in this setting.

Instead some indirect evidence on the possible e�ects of time are reported.

Table 11 reports results for two regression that were designed to capture some of these

e�ects. The residual from the break-even conditions given by Equation (14) were computed

for the best bid and and ask quotes based on the parameter estimates reported in Table 8. In

order to focus on situation where there are \pro�t" opportunities in the book the maximum

of zero and this residual was used as the dependent variable. Note that if the residual is

positive it suggests that according to the model there is insu�cient depth at the quotes or

the bid (ask) quote is too low (high). The �rst two columns of Table 11 report the results

for a regression of the measure of \pro�t opportunities" in the order book on a constant and

an indicator that takes on value one if the time elapsed from the last transaction is less than

the median time between transactions. The results suggest that in general we see larger

deviation or \pro�t opportunities" when a relatively short period of time has elapsed since

the last transaction. This supports the idea that it takes time to replenish the limit order

book. The second regression takes the average time elapsed between the last three trades

as an independent variable. This measure captures changes in the arrival rate of market

orders. For most of the stock we observe that the coe�cient on this variable is negative.

This suggests that longer time periods between trades are associated with books that exhibit

fewer or at least smaller \pro�t opportunities". This is consistent with the idea of time to

replenish the order book. It is also consistent with the idea that periods of no-trade are

informative as in Easley and O'Hara (1992).

29

5 Conclusions

I this paper, I estimate and test a model of liquidity provision in a limit order market. The

model is based on models by Glosten (1994), Rock (1996), and Seppi (1997). The empirical

strategy is directly based on two types of restrictions that the model imposes on the data.

Break-even conditions for marginal orders in the limit order book de�ne the slope of the

price schedule o�ered at a given point in time. Updating restrictions determine how the

order book or price schedules respond to the information content in the market orders.

The break-even conditions that de�ne the shape of the order book imply a relatively high

degree of information asymmetry. The estimated price impact coe�cient is several orders of

magnitude larger than the corresponding coe�cient obtained from a standard price impact

regression. The updating restrictions that relate the changes in the order books over time

to the information content of the market order ow generate coe�cients that are much

closer to estimates using the standard method. Given these results it is not surprising

that we reject the combined set of restrictions implied by the break-even and the updating

conditions. These �ndings imply that the limit order books o�ers too little depth or imply

price schedules that are too steep relative to the order book changes in response to trades.

Several explanations for this �nding are explored in the paper: the functional form of the

market order distribution, endogenous order ow, time-varying market conditions, and time.

I �nd evidence suggesting that the market order ow distribution changes with market

conditions. In particular, the variance of the market order ow distribution is positively

correlated with the past trading volume. This dependence is also re ected in the order

books and I tend to �nd less evidence against this model when I use a state dependent

market order ow distribution.

30

References

[1] Amihud, Y., and Mendelson, H., 1986, \Asset Pricing and the Bid-Ask Spread", Journal

of Financial Economics 17, 223-250.

[2] Biais, B., Hillion, P., and Spatt C., 1995, \An Empirical Analysis of the Limit Order

Book and the Order Flow in the Paris Bourse", Journal of Finance 50, 1655-1689.

[3] Brennan, M., and A. Subrahmanyam, 1996, \Market Micro-structure and Asset Pricing:

On the Compensation for Illiquidity in Stock Returns", Journal of Financial Economics

41, 441-460.

[4] Byrne, R., 1993, \Discrete Prices in Securities Markets", working paper, Carnegie Mel-

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[5] Chakravarty, S., and C. Holden, 1995, \An Integrated Model of Market and Limit

Orders", Journal of Financial Intermediation 4, 213-241.

[6] Cohen, K., S. Maier, R. Schwartz, and D. Whitcomb, 1981, \Transactions Costs, Order

Placement Strategies, and the Existence of the Bid-Ask Spread", Journal of Political

Economy 89, 287-305.

[7] Coppejans, M., and I. Domowitz, 1998, Stock and Order Flow Information as inputs to

Limit Order Book Trading Activity, working paper, Duke University.

[8] Domowitz, I., 1993, \A Taxonomy of Automated Trade Execution Systems", Journal

of International Money and Finance 12, 607-631.

[9] Easley, D., and M. O'Hara, 1992, Time and the Process of Security Price Adjustment,

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[10] Foucault, T., 1999, \Price Formation in a Dynamic Limit Order Market", Journal of

Financial Markets 2, 99-134.

31

[11] Frino, A., and M. McCorry, 1995, \Why are Spreads Tighter on the Australian Stock

Exchange than on the NYSE? An Electronic Limit Order Book versus the Specialist

Structure", working paper, University of Sydney.

[12] Glosten, L., 1994, \Is the Electronic Limit Order Book Inevitable ?", Journal of Finance

49, 1127-1161.

[13] Greene, J., 1995, \The Impact of Limit Order Executions on the Trading Costs in NYSE

Stocks", working paper, Indiana University.

[14] Hamao, Y., and J. Hasbrouck, 1995, \Securities Trading in the Absence of Dealers:

Trades and Quotes on the Tokyo Stock Exchange", Review of Financial Studies 8, 849-

878.

[15] Handa, V., and R. Schwartz, 1996, \Limit Order Trading", Journal of Finance 51,

1835-1861.

[16] Hansen, L.P., 1982, \Large Sample Properties of Generalized Method of Moments Es-

timators", Econometrica 50, 1024-1054.

[17] Harris, L., 1998, \Optimal Dynamic Order Submission Strategies in Some Stylized

Trading Problems", Financial Markets, Institutions, and Instruments 7, Vol 2, Issue 2.

[18] Harris, L., and J. Hasbrouck, 1996, \Market vs. Limit Orders: The SuperDot Evidence

on Order Submission", Journal of Financial and Quantitative Analysis 31, 213-231.

[19] Hedvall, K., J. Niemeyer, and G. Rosenqvist, 1997, \Do Buyers and Sellers Behave

Similarly in a Limit Order Book? A High Frequency Data Examination of the Finnish

Stock Exchange", Journal of Empirical Finance 4, 279-293.

[20] Holli�eld, B., R.A. Miller, and P. Sand�as, 1999, \An Empirical Analysis of a Pure Limit

Order Market", working paper, Carnegie Mellon University.

32

[21] De Jong, F., T. Nijman, T., and A. R�oell, 1995, \A Comparison of the Cost of Trading

French shares on the Paris Bourse and on SEAQ International", European Economic

Review 39, 1277-1301.

[22] De Jong, F., T. Nijman, T., and A. R�oell, 1996, \Price E�ects of Trading and Com-

ponents of the Bid-Ask Spread on the Paris Bourse", Journal of Empirical Finance 3,

1-35.

[23] Kavajecz, K., 1999, \A Specialist's Quoted Depth and the Limit Order Book", Journal

of Finance 54, 747-771.

[24] Kumar, P., and D. Seppi, 1993, \Limit Orders and Market Orders with Optimizing

Traders", working paper, Carnegie-Mellon University.

[25] Lee, C., B. Mucklow, and M. Ready, 1993, \Spreads, Depths, and the Impact of Earnings

Information: An Intraday Analysis", Review of Financial Studies 6, 345-374.

[26] Lehmann, B., and D. Modest, 1994, \Trading and Liquidity on the Tokyo Stock Ex-

change: A Bird's Eye View", Journal of Finance 49, 951-984.

[27] Lo, A.W., A.C. MacKinlay, and J. Zhang, 1997, \Econometric Models of Limit-Order

Executions", NBER working paper No. 6257.

[28] Parlour, C., 1998, \Price Dynamics in Limit Order Markets", Review of Financial Stud-

ies 11, 789-816.

[29] Parlour, C., and D. Seppi, 1997, \Liquidity-Based Competition for Order Flow", work-

ing paper, Carnegie Mellon University.

[30] Rock, K., 1996, \The Specialist's Order Book and Price Anomalies", Review of Financial

Studies, forthcoming.

[31] Seppi, D., 1997, \Liquidity Provision with Limit Orders and a Strategic Specialist",

Review of Financial Studies 10, 103-150.

33

Table 1: Sample of Securities

Company Name Class of Shares Symbol

Asea A, unrestricted ASEAAstra A ASTRElectrolux B, unrestricted ELUXInvestor B, unrestrcied INVEEricsson B, unrestricted LMEProcordia A PROCS-E Banken A SEBSkandia unrestricted SDIASkanska B SKABStora A STOR

The companies included in the sample, the class ofshares included, and the symbol used to refer to eachcompany in this paper.

34

Table 2: Descriptive Statistics

Number of Average Average Daily Sample

Number of Limit Orders Transactions Trading Volume Period

Stock Transactions Submitted Price [Million crowns] Return

ASEA 1374 2423 309.75 2.983 12.01%ASTR 3654 5699 522.29 28.400 15.44%ELUX 2342 4271 242.19 13.338 18.53%INVE 784 1672 124.73 2.131 22.31%LME 8102 12646 109.35 35.151 9.53%PROC 667 1463 199.33 1.079 1.51%SDIA 956 2125 154.50 2.817 -15.95%SEB 2141 3719 48.36 5.598 -23.58%SKAB 1367 2302 123.59 2.727 -23.21%STOR 894 1868 266.74 2.737 -3.85%

Descriptive statistics for all stocks over the 59-day sample period. The total number of transactions,the total number of limit orders submitted, the average transaction price, the average daily tradingvolume, and the return over the sample period are reported. The daily trading volume is reportedin million Swedish crowns. During the sample period one US dollar equaled roughly 6.5 Swedishcrowns. The sample period return is based on the �rst and the last transaction prices.

35

Table 3: Limit Order Book Statistics

Spreads Between Order Prices Cumulative Order Quantities O�eredin the Limit Order Book O�ered in the Limit Order Book

[crowns] [100 stocks]Ask2� Ask1� Bid1� AskQ2+ BidQ2+

Stock Ask1 Bid1 Bid2 AskQ1 AskQ1 BidQ1 BidQ1

ASEA 3.28 1.72 1.87 23.34 11.51 9.85 21.726.26 1.04 1.99 16.00 11.61 10.17 17.00

ASTR 1.48 1.40 1.58 40.59 18.90 23.55 51.691.21 0.82 1.99 32.10 19.29 28.64 44.60

ELUX 1.42 1.16 1.61 79.75 38.50 48.90 115.361.25 0.49 2.85 57.64 37.87 45.83 76.93

INVE 1.55 1.22 1.52 80.74 40.18 35.35 77.571.20 0.56 1.19 54.12 40.06 31.14 55.33

LME 0.93 0.91 0.91 470.71 200.25 199.91 446.850.23 0.20 0.22 293.99 193.79 178.22 319.53

PROC 2.27 1.69 1.90 20.48 11.13 17.22 38.002.02 1.09 2.18 17.22 13.70 18.69 28.88

SDIA 1.65 1.46 1.76 43.40 21.67 34.75 75.821.17 0.98 1.98 31.43 19.61 30.20 46.23

SEB 0.59 0.53 0.57 393.98 196.08 160.18 326.060.31 0.13 0.23 249.58 187.15 176.92 246.97

SKAB 1.48 1.24 1.54 61.96 29.70 44.64 88.340.96 0.67 1.35 57.44 34.48 45.46 68.89

STOR 3.48 1.82 2.09 39.67 18.75 12.60 26.724.40 1.28 1.79 30.69 21.04 13.90 23.24

Summary statistics for the limit order books. The mean and standard deviation (�rst and secondrow for each stock respectively) of the di�erence between the second best and the best ask price, thebest ask and best bid price, and the best and second best bid price are reported in the �rst threecolumns for each stock. The mean and standard deviation for the quantities o�ered in the limitorder book for the two best ask and two best bid price levels are reported in the last four columnsfor each stock. The quantities at the second best price levels are cumulative, that is, they includethe quantities at the respective best price level (bid or ask). The spreads are reported in Swedishcrowns and the quantities in 100 stocks.

36

Table 4: Market Order Flow Statistics

Market Order Flow Percentiles Average

Stock Type 10th 30th 50th 70th 90th Order Size

ASEA Sell 50 100 200 400 1000 378Buy 50 100 200 500 1000 435

ASTR Sell 50 100 300 750 2000 759Buy 100 200 500 1000 2500 1035

ELUX Sell 100 200 500 1100 3600 1299Buy 100 300 800 1600 4000 1481

INVE Sell 200 400 800 1400 3600 1388Buy 200 400 800 1000 3000 1208

LME Sell 100 400 1000 2450 7000 2757Buy 100 300 600 1500 5000 1997

PROC Sell 100 102 200 300 700 378Buy 100 200 315 600 1825 654

SDIA Sell 100 200 500 1000 2500 1009Buy 200 400 900 1368 3000 1305

SEB Sell 200 800 1400 3400 9000 3309Buy 200 600 1000 2200 8200 3155

SKAB Sell 100 200 500 1000 2300 1002Buy 100 200 400 1000 2100 889

STOR Sell 100 200 300 600 1500 639Buy 100 200 300 600 1800 717

The 10th, 30th, 50th, 70th, and 90th percentiles for the market buy and sellorder quantities are reported. The �nal column reports the average ordersize.

37

Table 5: Price Impact Regressions

Stock a b c R2 F-Value

ASEA 0.016 0.351 0.019 0.20 170.990.547 14.890 3.876

ASTR 0.021 0.265 0.008 0.17 377.211.380 21.636 7.511

ELUX 0.018 0.216 0.004 0.21 308.011.325 19.692 6.327

INVE 0.012 0.254 0.008 0.19 88.420.388 9.689 4.754

LME 0.001 0.198 0.001 0.32 1878.670.204 56.828 6.863

PROC 0.006 0.294 0.033 0.18 72.210.133 7.699 5.326

SDIA -0.011 0.269 0.010 0.16 91.22-0.337 9.457 4.928

SEB -0.006 0.114 0.000 0.33 529.31-1.426 29.377 5.302

SKAB -0.011 0.292 0.005 0.36 376.58-0.714 23.539 4.999

STOR -0.012 0.362 0.012 0.17 91.67-0.31) 10.949 3.695

Estimated coe�cients for the standard price impact regression foreach stock with the corresponding R2 and F-values. The t-statisticsare reported on the second row for each stock. The price impactregression is given by

pt+1 � pt = a+�Ib+Qt+1c+ �;

where �I is the change in the trade indicator and Qt+1 is the signedorder quantity.

38

Table 6: Break-Even Conditions

Stock � � �

ASEA 4.362 3.776 -2.349 0.21736.622 40.097 -10.187 13.722

ASTR 10.348 7.605 -1.155 0.06158.193 42.058 -18.332 28.803

ELUX 14.809 12.993 -1.041 0.02848.756 40.262 -14.150 20.664

INVE 12.023 13.881 -1.270 0.03729.340 34.194 -11.001 13.920

LME 19.968 27.573 -0.341 0.00451.705 76.827 -32.391 65.923

PROC 6.541 3.779 -1.838 0.13932.136 20.885 -9.772 13.820

SDIA 13.063 10.093 -1.430 0.05442.193 26.502 -12.299 18.111

SEB 31.571 33.092 -0.406 0.00327.371 40.648 -17.545 26.818

SKAB 8.902 10.018 -1.230 0.04035.702 32.643 -11.006 15.748

STOR 7.174 6.380 -2.659 0.15919.148 28.549 -8.621 11.773

Summary of estimation results obtained by imposingthe break-even conditions on the two best price levels.The coe�cients characterizing the market order ow (�,�), the order processing cost ( ), and the slope of theprice impact function (�) are reported in the �rst fourcolumns with corresponding t-statistics on the secondrow for each stock.

39

Table 7: Updating Conditions

Stock � � �2(6)

ASEA 0.021 0.006 2.5430.737 1.189 0.864

ASTR 0.023 0.003 1.0541.557 2.293 0.983

ELUX 0.016 0.013 7.5181.342 18.304 0.276

INVE 0.008 0.011 3.3140.310 7.197 0.766

LME 0.000 0.002 40.0350.052 26.844 0.000

PROC 0.006 0.040 18.8930.158 3.889 0.004

SDIA 0.001 0.021 5.1220.036 8.577 0.528

SEB -0.009 0.001 16.473-2.164 10.571 0.011

SKAB 0.014 0.013 9.5400.977 9.370 0.145

STOR -0.028 0.037 3.063-0.732 5.612 0.801

Summary of estimation results obtained using the up-dating restrictions applied to each of the two best bidand ask price levels in the order books. In this case,the drift term � and the price impact coe�cient � areidenti�ed. The estimated coe�cients and the t-statisticsare reported on the �rst and second row for each stockrespectively. A constant and the signed market orderquantity were used as instruments in this estimation.Given the four moment conditions we have a total oftotal of eight orthogonality conditions. A chi-squaredstatistic for a test of overidentifying restrictions (six de-grees of freedom) is reported in the last column withp-values directly below.

40

Table 8: Break-Even and Updating Conditions

Stock � � � � �2(3)

ASEA 4.120 3.766 -1.882 0.1836 -0.0710 39.634039.825 45.162 -12.596 17.1498 -1.7013 0.000

ASTR 9.944 7.057 -1.076 0.0597 0.0015 52.411862.212 44.352 -18.673 30.1126 0.0744 0.000

ELUX 13.800 12.244 -0.852 0.0255 0.0101 70.438451.499 44.399 -14.298 22.8040 0.7073 0.000

INVE 11.944 12.521 -1.057 0.0339 -0.0832 42.279531.598 44.822 -12.772 16.1759 -3.2524 0.000

LME 20.131 26.836 -0.312 0.0035 -0.0033 144.121055.486 78.368 -32.077 67.8169 -1.0585 0.000

PROC 6.185 3.498 -1.588 0.1272 -0.0012 22.821835.695 27.425 -10.950 15.6923 -0.0301 0.000

SDIA 12.167 9.452 -1.266 0.0516 0.0564 39.032247.539 27.044 -12.368 19.2479 1.6747 0.000

SEB 30.665 34.952 -0.318 0.0028 -0.0090 78.999933.506 45.739 -19.867 30.7691 -1.7532 0.000

SKAB 9.203 9.699 -0.973 0.0331 0.0471 54.685140.000 37.477 -12.058 18.0610 2.6456 0.000

STOR 5.924 6.220 -1.831 0.1327 0.0304 36.204129.506 31.642 -9.555 14.8480 0.5814 0.000

Summary of estimation results obtained by jointly imposing the break-even andupdating conditions. The �rst �ve columns report the parameter estimates; �,the market buy order mean, �, the market sell order parameter, , the orderprocessing cost, �, the price impact slope coe�cient, and �, the drift term. T-statistics are reported on the second row for each stock. The last column reportsthe chi-squared statistic (three degrees of freedom) for a test of overidentifyingrestrictions.

41

Table 9: Changing Market Conditions

Stock �0 �1 �2 �3 �0 �1 �2 �3

ASEA 4.3528 -0.0030 0.6457 -0.0340 3.9429 -0.1875 0.8106 -0.052638.926 -0.036 5.583 -0.369 45.717 -2.685 7.294 -0.577

ASTR 10.5308 1.9687 1.6866 -1.2203 7.3604 -0.4964 1.2743 -0.268761.863 6.267 8.296 -6.146 42.356 -1.677 4.905 -1.184

ELUX 14.7798 -3.7257 3.1298 0.3445 13.3618 0.3389 2.4503 1.948052.073 -7.574 7.806 1.064 45.094 0.753 5.632 5.448

INVE 12.0081 0.5682 0.1033 -2.2610 13.6766 0.2569 2.3933 2.179730.474 1.852 0.344 -6.720 38.077 0.685 5.091 5.105

LME 20.5211 1.0671 1.7485 -0.3906 28.2658 -4.9270 1.2998 1.382953.722 1.246 2.518 -1.146 79.165 -5.386 2.076 3.482

PROC 6.4565 -0.2124 0.2015 -1.1848 3.6917 0.1300 0.4095 -0.306833.956 -3.057 1.680 -4.800 24.093 1.451 2.322 -3.354

SDIA 13.2319 0.7423 1.3062 -0.0911 10.1546 0.8399 0.6441 1.045245.123 3.815 3.183 -0.380 28.305 3.166 1.660 2.203

SEB 30.6114 -3.4108 3.9848 3.5050 33.5435 -3.3892 5.7436 -1.980828.170 -2.488 2.002 4.533 43.112 -3.426 9.583 -3.106

SKAB 9.1896 0.2393 0.4817 0.1462 9.9447 -1.3747 0.2764 0.616438.360 1.262 2.033 0.532 33.817 -5.479 0.987 2.105

STOR 6.3875 -0.1615 0.7487 0.9650 5.6863 0.3509 0.3657 -3.484919.299 -1.046 0.993 3.343 26.439 2.625 2.371 -10.282

Summary of estimation results using both the updating and the break-even conditions and allowingthe �, �, and � parameters to depend on the vector of state variables (mid-quote volatility, tradingvolume, market index volatility), as stated in Equation (23). Estimated � and � coe�cients arereported on the �rst row and t-statistics on the second row for each stock.

42

Table 9 continued: Changing Market Conditions

Stock �0 �1 �2 �3 �

ASEA 0.82456685 0.07651970 0.40833491 -0.16574544 -0.08523460 0.044339.592 2.048 3.261 -2.939 -1.728 1.6968

ASTR 0.67170879 0.07755265 0.60103024 -0.09598911 -0.07839473 0.018970.442 3.257 6.024 -3.098 -2.067 1.3085

ELUX 0.56479102 0.00333083 0.00714961 -0.00491233 0.01265179 0.014093.582 2.360 2.165 -2.943 3.278 1.1017

INVE 0.56266861 0.00000003 0.00000204 -0.00000327 0.00000074 0.017649.129 0.039 2.222 -1.831 0.683 0.8629

LME 0.46692563 0.00033028 0.00071262 -0.00043867 0.00111226 0.0008194.472 1.388 1.613 -1.276 1.485 0.2338

PROC 0.78489008 0.00093371 0.00278645 -0.00030745 -0.00049924 0.011626.838 2.827 5.311 -1.446 -1.571 0.3218

SDIA 0.67197040 0.05605111 0.13781222 -0.02202459 0.04831057 -0.015936.098 2.826 3.836 -1.063 1.717 -0.6199

SEB 0.25604658 -0.00000013 -0.00000071 0.00000092 -0.00000039 -0.0078205.410 -1.121 -1.395 1.432 -2.223 -1.8258

SKAB 0.59439120 -0.00054724 -0.00046846 0.00134248 -0.00288881 -0.024355.114 -0.355 -0.445 0.803 -0.590 -1.6829

STOR 0.85649054 0.01625957 -0.00226596 -0.01085169 0.08383445 0.006129.563 3.073 -0.576 -3.144 3.840 0.1561

Summary of estimation results using both the updating and the break-even conditions and allowing the�, �, and � parameters to depend on the vector of state variables (mid-quote volatility, trading volume,market index volatility), as stated in Equation (23). Estimated , �, and � coe�cients on the �rst rowand t-statistics on the second row for each stock.

43

Table 10: Speci�cation Tests

Test of No State Dependence forJ-test of Order Distr. Order Pricefull model and Price Imp. Distribution Impact

Stock �2(18) �2(9) �2(6) �2(3)

ASEA 25.4168 97.4775 80.4596 11.05120.1139 0.0001 0.0001 0.0115

ASTR 33.3838 189.5236 154.7074 36.92380.0150 0.0001 0.0001 0.0010

ELUX 18.3616 153.9619 152.0689 16.44120.4321 0.0001 0.0001 0.0009

INVE 24.6170 142.3539 134.1297 4.98580.1358 0.0001 0.0001 0.1728

LME 113.8042 46.6465 42.1270 4.91040.0001 0.0001 0.0001 0.1785

PROC 18.4870 90.9706 76.4995 29.64850.4240 0.0001 0.0001 0.0001

SDIA 25.7446 78.1550 62.7280 21.85490.1057 0.0001 0.0001 0.0001

SEB 22.0791 158.5346 149.0737 9.12000.2285 0.0001 0.0001 0.0278

SKAB 23.2414 40.0508 37.9900 0.93600.1815 0.0001 0.0001 0.8167

STOR 26.9146 225.0759 124.4510 21.52290.0806 0.0001 0.0001 0.0001

Summary of speci�cations tests. The �rst column reports a chi-squared testof overidentifying restrictions for the extended model. The correspondingp-values are reported immediately below the chi-squared test statistic. Thefollowing three columns report speci�cation tests of no state dependenceat all, no state dependence in the market order distribution, and no statedependence in the price impact function respectively. The p-values for eachof the tests are reported below the chi-squared statistics.

44

Table 11: Time and Price Schedule Deviations

Indicator for Avg. Time forStock Constant fast trading Constant last three trades

ASEA 1.0968 0.1846 1.3098 -0.000219.9842 2.3778 23.5681 -3.0314

ASTR 0.6936 0.1516 0.9047 -0.000529.0247 4.4863 36.6724 -7.4914

ELUX 0.5458 0.1050 0.6855 -0.000223.3094 3.1721 29.0429 -5.1589

INVE 0.6297 0.1172 0.7193 0.000013.3710 1.7637 16.6490 -1.1134

LME 0.2606 0.1051 0.3838 -0.000645.5149 12.9993 70.3333 -18.9753

PROC 1.2381 -0.0085 1.2252 0.000015.2642 -0.0743 15.7124 0.1647

SDIA 0.6415 0.2496 0.9507 -0.000212.0365 3.3114 17.0287 -4.4611

SEB 0.2494 0.0791 0.3382 -0.000124.0514 5.4042 34.1080 -7.3081

SKAB 0.7301 0.2555 0.9611 -0.000217.7593 4.3991 22.4054 -3.2684

STOR 1.0788 0.2816 1.2513 0.000014.2062 2.6222 16.4564 -0.5897

Summary of results for two regressions of a measure of price schedule errors onvariables that measure the time between arrivals. The price schedule deviationis based on the maximum of zero and the residual from applying Equation (14)to the best bid and ask quotes, i.e., the di�erence between the actual and thepredicted bid-ask spread conditional on the estimated price impact function andorder distribution. A positive value indicates that there are \pro�t" opportu-nities in the order book. The �rst two columns report the coe�cients and thet-values for a regression of the price schedule deviations on a constant and anindicator for whether time since the last trade is below the median time betweentrades (i.e., trading is relatively fast). The last two columns report the resultsfor a regression of the price schedule deviation on a constant and the averagetime elapsed between the last three trades.

45

−10 −5 −1 1 5 10−5−4−3−2−1

012345

ASEA

−20 −10 −5 −1 1 5 10 20

−4

−3

−2

−1

0

1

2

3

4

ASTR

−35 −25 −15−10 −5 5 10 15 25 35−4

−3

−2

−1

0

1

2

3

4ELUX

−30 −20 −10 −5 −1 1 5 10 20 30−6−5−4−3−2−1

0123456

INVE

−70 −50 −30 −20 −10 −11 10 20 30 50

−0.5

−0.25

0

0.25

0.5

LME

−7 −5 −3 −1 1 3 5 7 10 15

−6−5−4−3−2−1

0123456

PROC

Figure 1: Each panel plots the average [dash] and median [dash-dot] mark-up/discountpaid/received relative to the midquote (on the vertical axis) as a function of the marketorder size, i.e., the marginal price schedule. The units are round-lots [100 shares] on thehorizontal axis and Swedish crowns on the vertical axis. The implied price schedule basedon a standard price impact regression results reported in Table 5 is plotted [solid] with a 5%con�dence interval [dot].

46

−25 −15 −10 −5 −1 1 5 10 15 25

−5

−4

−3

−2

−1

0

1

2

3

4

5

SDIA

−80 −50 −30−20−10 10 20 30 50 80

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

SEB

−20 −15 −10 −5 5 10 15 20

−3

−2

−1

0

1

2

3

SKAB

−15 −10 −5 −1 1 5 10 15

−10

−8

−6

−4

−2

0

2

4

6

8

10

STOR

Figure 2: Each panel plots the average [dash] and median [dash-dot] mark-up/discountpaid/received relative to the midquote (on the vertical axis) as a function of the marketorder size, i.e., the marginal price schedule. The units are round-lots [100 shares] on thehorizontal axis and Swedish crowns on the vertical axis. The implied price schedule basedon a standard price impact regression results reported in Table 5 is plotted [solid] with a 5%con�dence interval [dot].

47

−10 −5 −3 −1 1 3 5 10

−4

−3

−2

−1

0

1

2

3

4

ASEA

−20 −10 −5 −1 1 5 10 20

−3

−2

−1

0

1

2

3

ASTR

−35 −25 −15−10 −5 5 10 15 25 35

−3

−2

−1

0

1

2

3

ELUX

−30 −20 −10 −5 −1 1 5 10 20 30

−5−4−3−2−1

012345

INVE

−70 −50 −30 −20 −10 −11 10 20 30 50

−0.1

−0.05

0

0.05

0.1

LME

−7 −5 −3 −1 1 3 5 7 10 15

−5−4−3−2−1

012345

PROC

Figure 3: Each panels plots the di�erence between the average price schedule based on theobserved order books and the price schedules implied by two di�erent sets of parameterestimates for the �rst six sample stocks. The price impact [Swedish crowns] is a functionof the order quantities [100 shares]. The �rst di�erence (solid line) is based on the param-eter estimates obtained when using only the break-even conditions. The second di�erence(dashed line) is based on the parameter estimates obtained when using only the updatingrestrictions. The implied price schedules were computed as the appropriate \upper" and\lower" tail expectations based on the estimated slope coe�cients. Note that the quantityinvariant coe�cient in the break-event conditions was excluded to facilitate comparison withthe second case.

48

−25 −15 −10 −5 −1 1 5 10 15 25

−4

−3

−2

−1

0

1

2

3

4

SDIA

−80 −50 −30−20−10 10 20 30 50 80

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

SEB

−20 −15 −10 −5 5 10 15 20−3

−2

−1

0

1

2

3SKAB

−15 −10 −5 −1 1 5 10 15

−10

−8

−6

−4

−2

0

2

4

6

8

10

STOR

Figure 4: Each panels plots the di�erence between the average price schedule based on theobserved order books and the price schedules implied by two di�erent sets of parameterestimates for the last four sample stocks. The price impact [Swedish crowns] is a functionof the order quantities [100 shares]. The �rst di�erence (solid line) is based on the param-eter estimates obtained when using only the break-event conditions. The second di�erence(dashed line) is based on the parameter estimates obtained when using only the updatingrestrictions. The implied price schedules were computed as the appropriate \upper" and\lower" tail expectations based on the estimated slope coe�cients. Note that the quantityinvariant coe�cient in the break-event conditions was excluded to facilitate comparison withthe second case.

49

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

ASEA

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 25000

1000

2000

3000

ASEA

−4000 −3000 −2000 −1000 0 1000 2000 3000 40000

0.1

0.2

0.3

0.4ASTR

−4000 −3000 −2000 −1000 0 1000 2000 3000 40000

2000

4000

6000

ASTR

−3000 −2000 −1000 0 1000 2000 30000

0.05

0.1

0.15

0.2

ELUX

−3000 −2000 −1000 0 1000 2000 30000

2000

4000

6000ELUX

−3000 −2000 −1000 0 1000 2000 30000

0.1

0.2

INVE

−3000 −2000 −1000 0 1000 2000 30000

2000

4000

6000INVE

−8000 −6000 −4000 −2000 0 2000 4000 6000 80000

0.05

0.1

0.15

0.2LME

−8000 −6000 −4000 −2000 0 2000 4000 6000 80000

5000

10000

15000LME

Figure 5: The distribution plotted on the left represent the distribution of market orderquantities implied by the parameter estimates reported in Table 8 (dashed) and the empiricaldistribution of market order quantities (solid). The graphs on the right show the \upper"and \lower" tail expectations computed (i) based on the estimated market order distribution(solid) and (ii) based on the empirical distribution [dashed]. The market sell order quantitiesappear with negative signs. The units for the market order quantities (x-axis) are 100 sharesand the interval shown matches the relevant range presented in Table 4.

50

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

PROC

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 25000

1000

2000

3000

PROC

−4000 −3000 −2000 −1000 0 1000 2000 3000 40000

0.05

0.1

0.15

0.2

SDIA

−4000 −3000 −2000 −1000 0 1000 2000 3000 40000

2000

4000

6000

SDIA

−6000 −4000 −2000 0 2000 4000 60000

0.05

0.1

0.15

SEB

−6000 −4000 −2000 0 2000 4000 60000

5000

10000

SEB

−3000 −2000 −1000 0 1000 2000 30000

0.1

0.2

SKAB

−3000 −2000 −1000 0 1000 2000 30000

1000

2000

3000

4000

5000SKAB

−3000 −2000 −1000 0 1000 2000 30000

0.1

0.2

0.3

STOR

−3000 −2000 −1000 0 1000 2000 30000

1000

2000

3000

4000

5000STOR

Figure 6: The distribution plotted on the left represent the distribution of market orderquantities implied by the parameter estimates reported in Table 8 (dashed) and the empiricaldistribution of market order quantities (solid). The graphs on the right show the \upper"and \lower" tail expectations computed (i) based on the estimated market order distribution(solid) and (ii) based on the empirical distribution [dashed]. The market sell order quantitiesappear with negative signs. The units for the market order quantities (x-axis) are 100 sharesand the interval shown matches the relevant range presented in Table 4.

51

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

ASEA

−4000 −3000 −2000 −1000 0 1000 2000 3000 40000

0.1

0.2

0.3

0.4

0.5

ASTR

−3000 −2000 −1000 0 1000 2000 30000

0.1

0.2

0.3

0.4ELUX

−3000 −2000 −1000 0 1000 2000 30000

0.05

0.1

0.15

0.2

0.25

0.3

0.35INV

−8000 −6000 −4000 −2000 0 2000 4000 6000 80000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

LME

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

PROC

Figure 7: Each graph shows the empirical distribution of market order quantities (solid) andan adjusted market order quantity distribution. The adjusted market order quantities werecomputed as (Q+1;t �mt=Q+1 (if m > 0), that is, each market buy (sell) order quantity ismultiplied by the ask (bid) depth on the other side and divided by the average ask (bid)depth.

52

−6000 −4000 −2000 0 2000 4000 60000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SDIA

−6000 −4000 −2000 0 2000 4000 60000

0.05

0.1

0.15

0.2

0.25SEB

−3000 −2000 −1000 0 1000 2000 30000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

SKAB

−3000 −2000 −1000 0 1000 2000 30000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

STOR

Figure 8: Each graph shows the empirical distribution of market order quantities (solid) andan adjusted market order quantity distribution. The adjusted market order quantities werecomputed as (Q+1;t �mt=Q+1 (if m > 0), that is, each market buy (sell) order quantity ismultiplied by the ask (bid) depth on the other side and divided by the average ask (bid)depth.

53