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The Rodney L. White Center for Financial Research
Adverse Selection and Competitive Market Making:
Empirical Evidence from a Pure Limit Order Market
Patrik Sandas
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Adverse Selection and Competitive Market Making:
Empirical Evidence from a Pure Limit Order Market
Patrik Sandas
The Wharton School
University of Pennsylvania
Philadelphia, PA 19104-6367
e-mail: [email protected]
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 insucient 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 committee
members{Burton Hollield, Robert Miller, and Duane Seppi. The paper has improved thanks to detailed
comments 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 Stromberg, 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 Stock
Exchange, Stockholms Fondbors Jubileumsfond, and Dextel Findata AB for providing the data set, and the
A.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 aects the prices that
they quote. The potential market power of the market maker(s) and the specic trading
rules of the exchange also aect 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 eects in the data.
In this paper, I analyze how adverse selection, competition, and market frictions aect
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 prot" 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 dierence between the order price and the value conditional on execution
determine the expected prot. The value of the free trading option depends on how likely
1 See Amihud and Mendelson (1986) and Brennan and Subrahmanyam (1996) for estimates of the compensation for illiquidity in asset returns.
2 An investor could either demand or supply liquidity at various points in time and thus the terms \market
maker" 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 protable to do so. Due to the time
priority rule the marginal prots 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 oers investors two main order
choices, market and limit orders.3 A limit buy (sell) order is an order, which species 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 prot opportunities in the limit order book. Given
the simplicity of the trading rules, the limit order market oers an excellent laboratory
for studying the interaction between the adverse selection risk and the trading rules in a
competitive market.
3 Domowitz (1993) documents that approximately 35 nancial markets in 16 dierent countries use trading
systems based on some form of limit order book. The \pure" prex means that in this market mechanism
there is only a limit order book in contrast to \hybrid" market mechanisms such as the NYSE where there
is both a designated market maker and a limit order book.
4 A higher (lower) price buy (sell) orders have priority. At the same price, orders with an earlier submission
time 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 prot 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 dier 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 suciently large market order arrives. The expected
payo on a limit order is given by the execution probability times the dierence between
the current fundamental value and the order price and an adjustment term, which reects
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 diers from most empirical market microstructure
5 Rock (1996) and Seppi (1997) use a similar setup for the limit order book in theoretical models of the
interaction between the specialist and the limit order book at the NYSE. Parlour and Seppi (1997) extend
this framework to analyze inter-exchange competition for order ow.
4
studies by explicitly testing restrictions on both the price and quantity dimensions of liquidity.6 There is a unique price and quantity pair that satises 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 denition, 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 restrictions 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 oered by the limit order books appears to be too steep, i.e., the
book oers insucient 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
6 Exceptions 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 specic 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 dierent 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 Roell
(1996) test a version of Glosten (1994) on data from the Paris Bourse. This paper diers
from theirs by providing a direct test of the model restrictions and by incorporating the
eects 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.
7 Other 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).
8 Other studies of limit order markets include Coppejans and Domowitz (1998), De Jong, Nijman, and
Roell (1995), Frino and McCorry (1995), Hollield, Miller, and Sandas (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
9 See Domowitz (1993) for extensive documentation and analysis of dierent 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 oces 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 identication codes for the brokerdealers submitting the limit orders. The transparency and rich information disseminated to
all market participants facilitate detection of prot 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 dierent 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 signicant 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
10 The full sample included all shares of companies that had a least one class of shares included in the
OMX-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 securities
on the Stockholm Stock Exchange. The index is updated semiannually. A smaller sub-sample of 10 stocks
was randomly selected using two additional criteria. Firstly, only stocks with at least 500 trades during the
59-day sample period were considered. Secondly, for a company with dual-class shares no more than one
class was included in the nal sample.
8
hours these trades must be reported to the market. While other trading venues account
for a signicant 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 dierences in activity are reected
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 dierent 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 oered directly. In Table 3 the average spreads oered 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 oered
at the best and second best price levels is between 1.8 and 2.4 times the quantities oered at
11 In 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 oers 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" reects the information content of the order Qt. The expected change in the fundamental value conditional
on the order submitted at time t is E [Xt+1 , Xt jQt] = a + Qt c, 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 + It b + Qt c, where Xt denotes the fundamental value at time t: Taking the rst
dierence we get
pt+1 , pt = p = a + Ib + Qt+1 c + ";
(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 signicant 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 dierent 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 dened 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 dierent 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
coecients from the price impact regressions are \too at" to match either the average or
the median schedule oered by the order book. The dierence tends to grow as we move
further out in the limit order book. Is this evidence of prot 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 prots 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
dierences between the empirical and the implied price schedules in Figures 1 and 2?
2 Model
The model presented here follows Glosten (1994), modied 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 participants 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 prot 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 ;
where t+1 represents a random innovation in the value and is the expected change.
12
(2)
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+k g 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+k g. The buy side variables are denoted analogously with a negative
k and
index. Thus, any limit order book is characterized by a sequence of prices fpig+i=,
k
+
k
order quantities fQigi=,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
8
>
< fmb (mjz) = 21(z) e, mz if m 0 (market buy);
f (mjz) = >
: fms(mjz) = 21(z) e mz 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+1 jXt; 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 non14
decreasing function of the market order quantity.12 The price impact function is summarized
by a function h() dened 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 suciently 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 oer innite 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 dierence between this price level and the
fundamental value is large enough to make it protable to supply a positive quantity at this
price level. Consider the market makers problem of deciding how many units to oer at this
price level. The expected prot on the last unit, q; is determined by
Em~ [(p1 , (z) , E [Xt+1 jm;
~ z]) I (m~ q) jz];
(6)
where p1 , E [Xt+1 jm;
~ z] is the dierence between the price the market maker receives
and the expected fundamental value conditional on the market order m, I (m~ q) is an
12 Many specications of this model, including the normal-exponential version, yield a non-decreasing price
impact 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
u
(p1 , Xt , (z) , h(ujz)u) 21(z) e, z du
q
q
= 21 (p1 , Xt , (z) , (z)(q + (z))) e, z :
( )
( )
(7)
The market makers submitting limit orders to the order book are assumed to behave
competitively. Therefore the quantity oered 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
t , (z )
Q1 = p1 , X
, (z)
(8)
(z)
Given that the quantity Q1 is oered at p1 we can now ask what the quantity is that
the market makers will oer at the next price level, p2 = p1 + ; where is the tick size.
Using similar arguments we nd that the quantity oered at the second best price level, Q2 ,
is given by
Q2 = p1 + ,(Xz)t , (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" prots opportunities in the limit order book. A large number of market makers
would guarantee that the limit order book would oer prot opportunities only for brief time
intervals. The distinction between the timing of limit order submissions and market order
arrivals is of course articial.
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
briey 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
k
X
pk , Xt , (z) , (z)( Qi + (z)) = 0
i=1
k 2 f+1; +2g;
(10)
where Pki=1 Qi is the aggregate quantity oered 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)(
k
X
i=,1
Qi + (z)) = 0
k 2 f,1; ,2g;
(11)
k Q is the aggregate quantity oered at or above the -kth price level on the
where P,i=,1
i
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 f1; 2g. The error terms capture the fact that
17
at any point in time the order book may oer prot 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 prot opportunities. We will assume that on average the limit order books do not
exhibit exposed limit orders or unexploited prot 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
"
and
#
k
X
E pk , Xt , (z) , (z)( Qi + (z)) = 0;
k 2 f+1; +2g
(12)
2
3
k
X
E 4Xt , pk , (z) , (z)( Qi + (z))5 = 0:
k 2 f,1; ,2g
(13)
i=1
i=,1
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 dierence 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
k
,k
X
X
E [ek (yt; ')] = E [pk , 2 (z) , (z)( Qi + (z) + Qi + (z))] = 0;
i=1
i=,1
(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 dierence 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 information set at time t. A vector of moment conditions denoted gt (yt;') is obtained by stacking
all the moment conditions as
3
2
66 e1 (yt; ') Zt 77
66 e2 (yt; ') Zt 77
77 :
gt(yt;') = 66
...
77
66
5
4
ek (yt; ') Zt
A GMM estimator of the parameter vector ' can then be dened as:
8" T
#0 " X
#9
T
< 1X
1 g (y ') = ;
'T = arg min
g
(
y
'
)
W
t
t;
T
t t; ;
'2 : T
T
t=1
t=1
(16)
(17)
where WT is a positive denite weighting matrix. Hansen (1982) proves that the GMM estimator of ' is consistent and asymptotically normally distributed. The variance-covariance
t (') ] and S Ps=+1 E [g g 0 ].
matrix of 'T is given by T' = [D00 S0,1D0 ],1, where D0 E [ @g@'
0
t t,s
s=,1
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 , Xt jmt ] = + h(mt jzt ) = + 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
"
#
k
X
E pk;t , Xt , (z) , (z)( Qi;t + (z)) = 0
k 2 f+1; +2g;
i=1
and
2
3
k
X
E 4pk ;t+1 , Xt+1 , (z) , (z)( Qi;t+1 + (z))5 = 0
0
0
i=1
k0 2 f+1; +2g;
(19)
(20)
where the set of prices pk;t; k 2 f+1; +2g and pk ;t+1 ; k0 2 f+1; +2g need not be the
same. By subtracting Equation (19) from Equation (20) we obtain the following expression
0
k
X
k
X
i=1
i=1
0
E [Xt+1 , Xt jIt+1] = E [pk ;t+1 , pk;t , (z)( Qi;t+1 ,
0
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 , Xt jmt]: Using that fact we
can combine Equations (18) and (21) to obtain
2
0
13
k
k
X
X
E 4 + (z)mt , @pk ;t+1 , pk;t , (z)( Qi;t+1 , Qi;t ))A5 = 0:
0
0
i=1
i=1
(22)
Equation (22) states that on average we expect the limit order book to be revised in a
manner that reects 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 identied 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 dierent 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 coecients 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 reect rational behavior in a setting dierent 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 prot.
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 coecient, , that determines the price impact of market orders is
positive for all stock. It is interesting to note that the magnitude of this coecient much
larger than the coecient inferred from the standard price impact regression (c in Table
5). Note that the data used to obtain the coecients reported in Table 6 only include
21
\snapshots" of the limit order book at dierent points in time as well as the market order
series. On average the slope coecient obtained with the break-even conditions is about
nine times greater than the corresponding slope coecient 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 coecients is consistent with the large dierences
between the price schedules presented in Figures 1 and 2. Using the updating restrictions,
which use time series information, we would expect a smaller dierence.
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 identied 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 coecients reported in Table
7 are smaller in magnitude than the corresponding coecients reported in Table 6. There
is large variation between the stocks but on average the value of the price impact coecient
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 dierence is much smaller. The main dierence between the standard price impact
regression and the updating conditions used here is that the eects 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 specication
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 reect 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 dierent 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 overidentied 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 signicant for all of the stocks. The
estimated coecients tend to be smaller than the ones reported in Table 6. The dierence
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 specication 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 dier from the ones implied by the model estimates. Figures 3
and 4 show the dierence 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 dierence 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 coecients, ,
as well as the order ow parameters, and , which together with the slope coecient
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 eects of price discreteness as well as the order processing costs. Thus, the graphs
are designed to capture eects that are related to the market order quantities rather than
quantity invariant level eects. 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 dierence
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 coecient it seems clear
that the larger slope obtained when we combined the restrictions must produce an \overreaction" eect 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 signicant \bunching" of order that the
exponential distribution can not capture. For many of the stocks there appears to be more
13 The empirical distribution was computed for discrete bins of size 200 (i.e., two round-lots for a typical
stock). The implied distribution was \discretized" by computing the fraction of order we would observe for
each 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 aect 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 dierence
grows as we move towards larger order quantities. This suggests that the functional form
assumption may be partly responsible for the rejections. Consider the eects 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 signicantly simplies the derivation 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 optimally 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
14 For example, Hollield, Miller, and Sand
as (1999) estimate and test a model where the order ow is
endogenous 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 specic 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 aect the market order distribution through the
and parameters and the price impact function through the coecient. The interaction
is assumed to be linear and given by:
(z) = 0 + 1z1 + 2 z2 + 3z3
(z) = 0 + 1z1 + 2z2 + 3z3
(z) = 0 + 1z1 + 2z2 + 3 z3 ;
(23)
where 0, 0, 0 may be interpreted as the base case for the \average " state and the other
coecients measure the eects 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 dierent specication tests based on this extended model.
The rst column in Table 10 reports the chi-squared test of overidentifying restrictions for
the extended specications. 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 coecients that capture the impact of past trading volume,
2 and 2 , are positive and signicant for most of the stocks. Thus, this specication
potentially captures some of the endogeniety of the order ow illustrated in Figures 7 and 8
and its eects on the order book. The eects of changing mid-quote or market volatility are
more mixed. For some stocks, for example, SEB the 1 and 1 coecients are negative and
signicant, suggesting that we tend to see smaller market orders in more volatile markets.
Yet, for other stocks we nd no eect or a positive eect. 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 signicant for six of the stocks. For the remaining four
we can not reject a zero slope with this specication. The specication 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
coecient, and a positive 1 coecient suggesting that orders tend to have a greater price
impact in more volatile markets. The 2 and 3 coecients 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 coecients 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 specic
(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 prot 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
aects 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 eects of time in this setting.
Instead some indirect evidence on the possible eects of time are reported.
Table 11 reports results for two regression that were designed to capture some of these
eects. 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 \prot" 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 insucient 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 \prot 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 \prot 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 coecient on this variable is negative.
This suggests that longer time periods between trades are associated with books that exhibit
fewer or at least smaller \prot 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 dene the slope of the
price schedule oered 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 dene the shape of the order book imply a relatively high
degree of information asymmetry. The estimated price impact coecient is several orders of
magnitude larger than the corresponding coecient 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 coecients 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 oers 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 reected 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
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of International Money and Finance 12, 607-631.
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Financial Markets 2, 99-134.
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[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.
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49, 1127-1161.
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Stocks", working paper, Indiana University.
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Trading Problems", Financial Markets, Institutions, and Instruments 7, Vol 2, Issue 2.
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on Order Submission", Journal of Financial and Quantitative Analysis 31, 213-231.
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Review of Financial Studies 10, 103-150.
33
Table 1: Sample of Securities
Company Name
Asea
Astra
Electrolux
Investor
Ericsson
Procordia
S-E Banken
Skandia
Skanska
Stora
Class of Shares
A, unrestricted
A
B, unrestricted
B, unrestrcied
B, unrestricted
A
A
unrestricted
B
A
Symbol
ASEA
ASTR
ELUX
INVE
LME
PROC
SEB
SDIA
SKAB
STOR
The companies included in the sample, the class of
shares included, and the symbol used to refer to each
company in this paper.
34
Table 2: Descriptive Statistics
Number of
Average
Average Daily
Number of Limit Orders Transactions Trading Volume
Stock Transactions Submitted
Price
[Million crowns]
ASEA
1374
2423
309.75
2.983
ASTR
3654
5699
522.29
28.400
ELUX
2342
4271
242.19
13.338
INVE
784
1672
124.73
2.131
LME
8102
12646
109.35
35.151
PROC
667
1463
199.33
1.079
SDIA
956
2125
154.50
2.817
SEB
2141
3719
48.36
5.598
SKAB
1367
2302
123.59
2.727
STOR
894
1868
266.74
2.737
Sample
Period
Return
12.01%
15.44%
18.53%
22.31%
9.53%
1.51%
-15.95%
-23.58%
-23.21%
-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 trading
volume, and the return over the sample period are reported. The daily trading volume is reported
in million Swedish crowns. During the sample period one US dollar equaled roughly 6.5 Swedish
crowns. The sample period return is based on the rst and the last transaction prices.
35
Table 3: Limit Order Book Statistics
Stock
ASEA
ASTR
ELUX
INVE
LME
PROC
SDIA
SEB
SKAB
STOR
Spreads Between Order Prices Cumulative Order Quantities Oered
in the Limit Order Book
Oered in the Limit Order Book
[crowns]
[100 stocks]
Ask2, Ask1,
Bid1 ,
AskQ2 +
BidQ2+
Ask1 Bid1
Bid2
AskQ1 AskQ1 BidQ1 BidQ1
3.28
1.72
1.87
23.34 11.51 9.85
21.72
6.26
1.04
1.99
16.00 11.61 10.17 17.00
1.48
1.40
1.58
40.59 18.90 23.55 51.69
1.21
0.82
1.99
32.10 19.29 28.64 44.60
1.42
1.16
1.61
79.75 38.50 48.90 115.36
1.25
0.49
2.85
57.64 37.87 45.83 76.93
1.55
1.22
1.52
80.74 40.18 35.35 77.57
1.20
0.56
1.19
54.12 40.06 31.14 55.33
0.93
0.91
0.91
470.71 200.25 199.91 446.85
0.23
0.20
0.22
293.99 193.79 178.22 319.53
2.27
1.69
1.90
20.48 11.13 17.22 38.00
2.02
1.09
2.18
17.22 13.70 18.69 28.88
1.65
1.46
1.76
43.40 21.67 34.75 75.82
1.17
0.98
1.98
31.43 19.61 30.20 46.23
0.59
0.53
0.57
393.98 196.08 160.18 326.06
0.31
0.13
0.23
249.58 187.15 176.92 246.97
1.48
1.24
1.54
61.96 29.70 44.64 88.34
0.96
0.67
1.35
57.44 34.48 45.46 68.89
3.48
1.82
2.09
39.67 18.75 12.60 26.72
4.40
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 second
row for each stock respectively) of the dierence between the second best and the best ask price, the
best ask and best bid price, and the best and second best bid price are reported in the rst three
columns for each stock. The mean and standard deviation for the quantities oered in the limit
order book for the two best ask and two best bid price levels are reported in the last four columns
for each stock. The quantities at the second best price levels are cumulative, that is, they include
the quantities at the respective best price level (bid or ask). The spreads are reported in Swedish
crowns 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
378
Buy 50 100 200 500 1000
435
ASTR Sell
50 100 300 750 2000
759
Buy 100 200 500 1000 2500
1035
ELUX Sell 100 200 500 1100 3600
1299
Buy 100 300 800 1600 4000
1481
INVE Sell 200 400 800 1400 3600
1388
Buy 200 400 800 1000 3000
1208
LME Sell 100 400 1000 2450 7000
2757
Buy 100 300 600 1500 5000
1997
PROC Sell 100 102 200 300 700
378
Buy 100 200 315 600 1825
654
SDIA Sell 100 200 500 1000 2500
1009
Buy 200 400 900 1368 3000
1305
SEB Sell 200 800 1400 3400 9000
3309
Buy 200 600 1000 2200 8200
3155
SKAB Sell 100 200 500 1000 2300
1002
Buy 100 200 400 1000 2100
889
STOR Sell 100 200 300 600 1500
639
Buy 100 200 300 600 1800
717
The 10th, 30th, 50th, 70th, and 90th percentiles for the market buy and sell
order quantities are reported. The nal column reports the average order
size.
37
Table 5: Price Impact Regressions
Stock
a
ASEA 0.016
0.547
ASTR 0.021
1.380
ELUX 0.018
1.325
INVE 0.012
0.388
LME 0.001
0.204
PROC 0.006
0.133
SDIA -0.011
-0.337
SEB -0.006
-1.426
SKAB -0.011
-0.714
STOR -0.012
-0.31)
b
0.351
14.890
0.265
21.636
0.216
19.692
0.254
9.689
0.198
56.828
0.294
7.699
0.269
9.457
0.114
29.377
0.292
23.539
0.362
10.949
c
0.019
3.876
0.008
7.511
0.004
6.327
0.008
4.754
0.001
6.863
0.033
5.326
0.010
4.928
0.000
5.302
0.005
4.999
0.012
3.695
R2
0.20
0.17
0.21
0.19
0.32
0.18
0.16
0.33
0.36
0.17
F-Value
170.99
377.21
308.01
88.42
1878.67
72.21
91.22
529.31
376.58
91.67
Estimated coecients for the standard price impact regression for
each stock with the corresponding R2 and F-values. The t-statistics
are reported on the second row for each stock. The price impact
regression is given by
pt+1 , pt = a + Ib + Qt+1 c + ;
where I is the change in the trade indicator and Qt+1 is the signed
order quantity.
38
Table 6: Break-Even Conditions
Stock
ASEA
ASTR
ELUX
INVE
LME
PROC
SDIA
SEB
SKAB
STOR
4.362
36.622
10.348
58.193
14.809
48.756
12.023
29.340
19.968
51.705
6.541
32.136
13.063
42.193
31.571
27.371
8.902
35.702
7.174
19.148
3.776
40.097
7.605
42.058
12.993
40.262
13.881
34.194
27.573
76.827
3.779
20.885
10.093
26.502
33.092
40.648
10.018
32.643
6.380
28.549
-2.349
-10.187
-1.155
-18.332
-1.041
-14.150
-1.270
-11.001
-0.341
-32.391
-1.838
-9.772
-1.430
-12.299
-0.406
-17.545
-1.230
-11.006
-2.659
-8.621
0.217
13.722
0.061
28.803
0.028
20.664
0.037
13.920
0.004
65.923
0.139
13.820
0.054
18.111
0.003
26.818
0.040
15.748
0.159
11.773
Summary of estimation results obtained by imposing
the break-even conditions on the two best price levels.
The coecients characterizing the market order ow (,
), the order processing cost ( ), and the slope of the
price impact function () are reported in the rst four
columns with corresponding t-statistics on the second
row for each stock.
39
Table 7: Updating Conditions
Stock
ASEA 0.021
0.737
ASTR 0.023
1.557
ELUX 0.016
1.342
INVE 0.008
0.310
LME 0.000
0.052
PROC 0.006
0.158
SDIA 0.001
0.036
SEB -0.009
-2.164
SKAB 0.014
0.977
STOR -0.028
-0.732
0.006
1.189
0.003
2.293
0.013
18.304
0.011
7.197
0.002
26.844
0.040
3.889
0.021
8.577
0.001
10.571
0.013
9.370
0.037
5.612
2(6)
2.543
0.864
1.054
0.983
7.518
0.276
3.314
0.766
40.035
0.000
18.893
0.004
5.122
0.528
16.473
0.011
9.540
0.145
3.063
0.801
Summary of estimation results obtained using the updating restrictions applied to each of the two best bid
and ask price levels in the order books. In this case,
the drift term and the price impact coecient are
identied. The estimated coecients and the t-statistics
are reported on the rst and second row for each stock
respectively. A constant and the signed market order
quantity were used as instruments in this estimation.
Given the four moment conditions we have a total of
total of eight orthogonality conditions. A chi-squared
statistic for a test of overidentifying restrictions (six degrees of freedom) is reported in the last column with
p-values directly below.
40
Table 8: Break-Even and Updating Conditions
Stock
ASEA
ASTR
ELUX
INVE
LME
PROC
SDIA
SEB
SKAB
STOR
4.120
39.825
9.944
62.212
13.800
51.499
11.944
31.598
20.131
55.486
6.185
35.695
12.167
47.539
30.665
33.506
9.203
40.000
5.924
29.506
3.766
45.162
7.057
44.352
12.244
44.399
12.521
44.822
26.836
78.368
3.498
27.425
9.452
27.044
34.952
45.739
9.699
37.477
6.220
31.642
-1.882
-12.596
-1.076
-18.673
-0.852
-14.298
-1.057
-12.772
-0.312
-32.077
-1.588
-10.950
-1.266
-12.368
-0.318
-19.867
-0.973
-12.058
-1.831
-9.555
0.1836
17.1498
0.0597
30.1126
0.0255
22.8040
0.0339
16.1759
0.0035
67.8169
0.1272
15.6923
0.0516
19.2479
0.0028
30.7691
0.0331
18.0610
0.1327
14.8480
-0.0710
-1.7013
0.0015
0.0744
0.0101
0.7073
-0.0832
-3.2524
-0.0033
-1.0585
-0.0012
-0.0301
0.0564
1.6747
-0.0090
-1.7532
0.0471
2.6456
0.0304
0.5814
2(3)
39.6340
0.000
52.4118
0.000
70.4384
0.000
42.2795
0.000
144.1210
0.000
22.8218
0.000
39.0322
0.000
78.9999
0.000
54.6851
0.000
36.2041
0.000
Summary of estimation results obtained by jointly imposing the break-even and
updating conditions. The rst ve columns report the parameter estimates; ,
the market buy order mean, , the market sell order parameter, , the order
processing cost, , the price impact slope coecient, and , the drift term. Tstatistics are reported on the second row for each stock. The last column reports
the chi-squared statistic (three degrees of freedom) for a test of overidentifying
restrictions.
41
Table 9: Changing Market Conditions
Stock
ASEA
ASTR
ELUX
INVE
LME
PROC
SDIA
SEB
SKAB
STOR
0
4.3528
38.926
10.5308
61.863
14.7798
52.073
12.0081
30.474
20.5211
53.722
6.4565
33.956
13.2319
45.123
30.6114
28.170
9.1896
38.360
6.3875
19.299
1
-0.0030
-0.036
1.9687
6.267
-3.7257
-7.574
0.5682
1.852
1.0671
1.246
-0.2124
-3.057
0.7423
3.815
-3.4108
-2.488
0.2393
1.262
-0.1615
-1.046
2
0.6457
5.583
1.6866
8.296
3.1298
7.806
0.1033
0.344
1.7485
2.518
0.2015
1.680
1.3062
3.183
3.9848
2.002
0.4817
2.033
0.7487
0.993
3
-0.0340
-0.369
-1.2203
-6.146
0.3445
1.064
-2.2610
-6.720
-0.3906
-1.146
-1.1848
-4.800
-0.0911
-0.380
3.5050
4.533
0.1462
0.532
0.9650
3.343
0
3.9429
45.717
7.3604
42.356
13.3618
45.094
13.6766
38.077
28.2658
79.165
3.6917
24.093
10.1546
28.305
33.5435
43.112
9.9447
33.817
5.6863
26.439
1
-0.1875
-2.685
-0.4964
-1.677
0.3389
0.753
0.2569
0.685
-4.9270
-5.386
0.1300
1.451
0.8399
3.166
-3.3892
-3.426
-1.3747
-5.479
0.3509
2.625
2
0.8106
7.294
1.2743
4.905
2.4503
5.632
2.3933
5.091
1.2998
2.076
0.4095
2.322
0.6441
1.660
5.7436
9.583
0.2764
0.987
0.3657
2.371
3
-0.0526
-0.577
-0.2687
-1.184
1.9480
5.448
2.1797
5.105
1.3829
3.482
-0.3068
-3.354
1.0452
2.203
-1.9808
-3.106
0.6164
2.105
-3.4849
-10.282
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 coecients are
reported on the rst row and t-statistics on the second row for each stock.
42
Table 9 continued: Changing Market Conditions
Stock
ASEA 0.82456685
39.592
ASTR 0.67170879
70.442
ELUX 0.56479102
93.582
INVE 0.56266861
49.129
LME 0.46692563
194.472
PROC 0.78489008
26.838
SDIA 0.67197040
36.098
SEB 0.25604658
205.410
SKAB 0.59439120
55.114
STOR 0.85649054
29.563
0
0.07651970
2.048
0.07755265
3.257
0.00333083
2.360
0.00000003
0.039
0.00033028
1.388
0.00093371
2.827
0.05605111
2.826
-0.00000013
-1.121
-0.00054724
-0.355
0.01625957
3.073
1
0.40833491
3.261
0.60103024
6.024
0.00714961
2.165
0.00000204
2.222
0.00071262
1.613
0.00278645
5.311
0.13781222
3.836
-0.00000071
-1.395
-0.00046846
-0.445
-0.00226596
-0.576
2
-0.16574544
-2.939
-0.09598911
-3.098
-0.00491233
-2.943
-0.00000327
-1.831
-0.00043867
-1.276
-0.00030745
-1.446
-0.02202459
-1.063
0.00000092
1.432
0.00134248
0.803
-0.01085169
-3.144
3
-0.08523460
-1.728
-0.07839473
-2.067
0.01265179
3.278
0.00000074
0.683
0.00111226
1.485
-0.00049924
-1.571
0.04831057
1.717
-0.00000039
-2.223
-0.00288881
-0.590
0.08383445
3.840
0.0443
1.6968
0.0189
1.3085
0.0140
1.1017
0.0176
0.8629
0.0008
0.2338
0.0116
0.3218
-0.0159
-0.6199
-0.0078
-1.8258
-0.0243
-1.6829
0.0061
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 coecients on the rst row
and t-statistics on the second row for each stock.
43
Table 10: Specication Tests
Stock
ASEA
ASTR
ELUX
INVE
LME
PROC
SDIA
SEB
SKAB
STOR
Test of No State Dependence for
J-test of Order Distr.
Order
Price
full model and Price Imp. Distribution Impact
2 (18)
2 (9)
2(6)
2 (3)
25.4168
97.4775
80.4596 11.0512
0.1139
0.0001
0.0001
0.0115
33.3838
189.5236
154.7074 36.9238
0.0150
0.0001
0.0001
0.0010
18.3616
153.9619
152.0689 16.4412
0.4321
0.0001
0.0001
0.0009
24.6170
142.3539
134.1297 4.9858
0.1358
0.0001
0.0001
0.1728
113.8042
46.6465
42.1270
4.9104
0.0001
0.0001
0.0001
0.1785
18.4870
90.9706
76.4995 29.6485
0.4240
0.0001
0.0001
0.0001
25.7446
78.1550
62.7280 21.8549
0.1057
0.0001
0.0001
0.0001
22.0791
158.5346
149.0737 9.1200
0.2285
0.0001
0.0001
0.0278
23.2414
40.0508
37.9900
0.9360
0.1815
0.0001
0.0001
0.8167
26.9146
225.0759
124.4510 21.5229
0.0806
0.0001
0.0001
0.0001
Summary of specications tests. The rst column reports a chi-squared test
of overidentifying restrictions for the extended model. The corresponding
p-values are reported immediately below the chi-squared test statistic. The
following three columns report specication tests of no state dependence
at all, no state dependence in the market order distribution, and no state
dependence in the price impact function respectively. The p-values for each
of the tests are reported below the chi-squared statistics.
44
Table 11: Time and Price Schedule Deviations
Indicator for
Stock Constant fast trading
ASEA 1.0968
0.1846
19.9842
2.3778
ASTR 0.6936
0.1516
29.0247
4.4863
ELUX 0.5458
0.1050
23.3094
3.1721
INVE 0.6297
0.1172
13.3710
1.7637
LME
0.2606
0.1051
45.5149
12.9993
PROC 1.2381
-0.0085
15.2642
-0.0743
SDIA 0.6415
0.2496
12.0365
3.3114
SEB
0.2494
0.0791
24.0514
5.4042
SKAB 0.7301
0.2555
17.7593
4.3991
STOR 1.0788
0.2816
14.2062
2.6222
Avg. Time for
Constant last three trades
1.3098
-0.0002
23.5681
-3.0314
0.9047
-0.0005
36.6724
-7.4914
0.6855
-0.0002
29.0429
-5.1589
0.7193
0.0000
16.6490
-1.1134
0.3838
-0.0006
70.3333
-18.9753
1.2252
0.0000
15.7124
0.1647
0.9507
-0.0002
17.0287
-4.4611
0.3382
-0.0001
34.1080
-7.3081
0.9611
-0.0002
22.4054
-3.2684
1.2513
0.0000
16.4564
-0.5897
Summary of results for two regressions of a measure of price schedule errors on
variables that measure the time between arrivals. The price schedule deviation
is based on the maximum of zero and the residual from applying Equation (14)
to the best bid and ask quotes, i.e., the dierence between the actual and the
predicted bid-ask spread conditional on the estimated price impact function and
order distribution. A positive value indicates that there are \prot" opportunities in the order book. The rst two columns report the coecients and the
t-values for a regression of the price schedule deviations on a constant and an
indicator for whether time since the last trade is below the median time between
trades (i.e., trading is relatively fast). The last two columns report the results
for a regression of the price schedule deviation on a constant and the average
time elapsed between the last three trades.
45
ASEA
5
4
3
2
1
0
−1
−2
−3
−4
−5
−10
ASTR
4
3
2
1
0
−1
−2
−3
−4
−5
−1
1
5
10
−20
−10
−5
−1 1
ELUX
20
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
−6
3
2
1
0
−1
−2
−3
−25
−15−10 −5
5 10 15
25
35
−30
−20
−10 −5 −1 1
LME
5
10
20
30
PROC
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
−6
0.5
0.25
0
−0.25
−0.5
−70
10
INVE
4
−4
−35
5
−50
−30 −20 −10 −11
10
20
30
50
−7 −5 −3 −1
1
3
5
7
10
15
Figure 1: Each panel plots the average [dash] and median [dash-dot] mark-up/discount
paid/received relative to the midquote (on the vertical axis) as a function of the market
order size, i.e., the marginal price schedule. The units are round-lots [100 shares] on the
horizontal axis and Swedish crowns on the vertical axis. The implied price schedule based
on a standard price impact regression results reported in Table 5 is plotted [solid] with a 5%
condence interval [dot].
46
SDIA
SEB
6
5
5
4
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
−4
−5
−5
−6
−25
−15 −10
−5 −1 1
5
10
15
25
−80
−50
−30−20−10
SKAB
10 20 30
50
80
STOR
3
10
8
2
6
4
1
2
0
0
−2
−1
−4
−6
−2
−8
−10
−3
−20
−15
−10
−5
5
10
15
20
−15
−10
−5
−1
1
5
10
15
Figure 2: Each panel plots the average [dash] and median [dash-dot] mark-up/discount
paid/received relative to the midquote (on the vertical axis) as a function of the market
order size, i.e., the marginal price schedule. The units are round-lots [100 shares] on the
horizontal axis and Swedish crowns on the vertical axis. The implied price schedule based
on a standard price impact regression results reported in Table 5 is plotted [solid] with a 5%
condence interval [dot].
47
ASEA
ASTR
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
−10
−5
−3
−1
1
3
5
10
−20
−10
−5
−1 1
ELUX
20
INVE
2
1
0
−1
−2
−3
−25
−15−10 −5
5 10 15
25
35
−30
−20
−10 −5 −1 1
LME
5
10
20
30
PROC
5
4
3
2
1
0
−1
−2
−3
−4
−5
0.1
0.05
0
−0.05
−0.1
−70
10
5
4
3
2
1
0
−1
−2
−3
−4
−5
3
−35
5
−50
−30 −20 −10 −11
10
20
30
50
−7 −5 −3 −1
1
3
5
7
10
15
Figure 3: Each panels plots the dierence between the average price schedule based on the
observed order books and the price schedules implied by two dierent sets of parameter
estimates for the rst six sample stocks. The price impact [Swedish crowns] is a function
of the order quantities [100 shares]. The rst dierence (solid line) is based on the parameter estimates obtained when using only the break-even conditions. The second dierence
(dashed line) is based on the parameter estimates obtained when using only the updating
restrictions. The implied price schedules were computed as the appropriate \upper" and
\lower" tail expectations based on the estimated slope coecients. Note that the quantity
invariant coecient in the break-event conditions was excluded to facilitate comparison with
the second case.
48
SDIA
SEB
0.75
4
3
0.5
2
0.25
1
0
0
−1
−0.25
−2
−0.5
−3
−4
−0.75
−25
−15 −10
−5 −1 1
5
10
15
25
−80
−50
−30−20−10
SKAB
10 20 30
50
80
STOR
3
10
2
8
6
1
4
2
0
0
−2
−4
−1
−6
−8
−2
−10
−3
−20
−15
−10
−5
5
10
15
20
−15
−10
−5
−1
1
5
10
15
Figure 4: Each panels plots the dierence between the average price schedule based on the
observed order books and the price schedules implied by two dierent sets of parameter
estimates for the last four sample stocks. The price impact [Swedish crowns] is a function
of the order quantities [100 shares]. The rst dierence (solid line) is based on the parameter estimates obtained when using only the break-event conditions. The second dierence
(dashed line) is based on the parameter estimates obtained when using only the updating
restrictions. The implied price schedules were computed as the appropriate \upper" and
\lower" tail expectations based on the estimated slope coecients. Note that the quantity
invariant coecient in the break-event conditions was excluded to facilitate comparison with
the second case.
49
ASEA
ASEA
3000
0.4
0.3
2000
0.2
1000
0.1
0
−2500 −2000 −1500 −1000 −500
0
500
ASTR
0
−2500 −2000 −1500 −1000 −500
1000 1500 2000 2500
0.4
0
500
ASTR
1000 1500 2000 2500
6000
0.3
4000
0.2
2000
0.1
0
−4000
−3000
−2000
−1000
0
ELUX
1000
2000
3000
0
−4000
4000
−3000
−2000
−1000
0
ELUX
1000
2000
3000
4000
6000
0.2
0.15
4000
0.1
2000
0.05
0
−3000
−2000
−1000
0
INVE
1000
2000
0
−3000
3000
−2000
−1000
0
INVE
1000
2000
3000
−2000
−1000
0
LME
1000
2000
3000
6000
0.2
4000
0.1
2000
0
−3000
−2000
−1000
0
LME
1000
2000
0
−3000
3000
0.2
15000
0.15
10000
0.1
5000
0.05
0
−8000
−6000
−4000
−2000
0
2000
4000
6000
0
−8000
8000
−6000
−4000
−2000
0
2000
4000
6000
8000
Figure 5: The distribution plotted on the left represent the distribution of market order
quantities implied by the parameter estimates reported in Table 8 (dashed) and the empirical
distribution 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 quantities
appear with negative signs. The units for the market order quantities (x-axis) are 100 shares
and the interval shown matches the relevant range presented in Table 4.
50
PROC
PROC
3000
0.4
0.3
2000
0.2
1000
0.1
0
−2500 −2000 −1500 −1000 −500
0
SDIA
500
0
−2500 −2000 −1500 −1000 −500
1000 1500 2000 2500
0
SDIA
500
1000 1500 2000 2500
6000
0.2
0.15
4000
0.1
2000
0.05
0
−4000
−3000
−2000
−1000
0
SEB
1000
2000
3000
0
−4000
4000
−3000
−2000
−1000
0
SEB
1000
2000
3000
4000
0.15
10000
0.1
5000
0.05
0
−6000
−4000
−2000
0
SKAB
2000
4000
0
−6000
6000
−4000
−2000
0
SKAB
2000
4000
6000
−2000
−1000
0
STOR
1000
2000
3000
−2000
−1000
0
1000
2000
3000
5000
4000
0.2
3000
2000
0.1
1000
0
−3000
−2000
−1000
0
STOR
1000
2000
0
−3000
3000
5000
0.3
4000
0.2
3000
2000
0.1
1000
0
−3000
−2000
−1000
0
1000
2000
0
−3000
3000
Figure 6: The distribution plotted on the left represent the distribution of market order
quantities implied by the parameter estimates reported in Table 8 (dashed) and the empirical
distribution 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 quantities
appear with negative signs. The units for the market order quantities (x-axis) are 100 shares
and the interval shown matches the relevant range presented in Table 4.
51
ASEA
ASTR
0.5
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
−2500 −2000 −1500 −1000 −500
0
500
0
−4000
1000 1500 2000 2500
−3000
−2000
−1000
ELUX
0
1000
2000
3000
4000
INV
0.4
0.35
0.3
0.3
0.25
0.2
0.2
0.15
0.1
0.1
0.05
0
−3000
−2000
−1000
0
1000
2000
0
−3000
3000
−2000
−1000
LME
0
1000
2000
3000
PROC
0.35
0.6
0.3
0.5
0.25
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
−8000
−6000
−4000
−2000
0
2000
4000
6000
0
−2500 −2000 −1500 −1000 −500
8000
0
500
1000 1500 2000 2500
Figure 7: Each graph shows the empirical distribution of market order quantities (solid) and
an adjusted market order quantity distribution. The adjusted market order quantities were
computed as (Q+1;t mt =Q+1 (if m > 0), that is, each market buy (sell) order quantity is
multiplied by the ask (bid) depth on the other side and divided by the average ask (bid)
depth.
52
SDIA
SEB
0.25
0.35
0.2
0.3
0.25
0.15
0.2
0.1
0.15
0.1
0.05
0.05
0
−6000
−4000
−2000
0
2000
4000
0
−6000
6000
−4000
−2000
SKAB
0
2000
4000
6000
1000
2000
3000
STOR
0.45
0.5
0.4
0.45
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
−3000
−2000
−1000
0
1000
2000
0
−3000
3000
−2000
−1000
0
Figure 8: Each graph shows the empirical distribution of market order quantities (solid) and
an adjusted market order quantity distribution. The adjusted market order quantities were
computed as (Q+1;t mt =Q+1 (if m > 0), that is, each market buy (sell) order quantity is
multiplied by the ask (bid) depth on the other side and divided by the average ask (bid)
depth.
53