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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 014-99 The Rodney L. White Center for Financial Research The Wharton School University of Pennsylvania 3254 Steinberg Hall-Dietrich Hall 3620 Locust Walk Philadelphia, PA 19104-6367 (215) 898-7616 (215) 573-8084 Fax http://finance.wharton.upenn.edu/~rlwctr The Rodney L. White Center for Financial Research is one of the oldest financial research centers in the country. It was founded in 1969 through a grant from Oppenheimer & Company in honor of its late partner, Rodney L. White. The Center receives support from its endowment and from annual contributions from its Members. The Center sponsors a wide range of financial research. It publishes a working paper series and a reprint series. It holds an annual seminar, which for the last several years has focused on household financial decision making. The Members of the Center gain the opportunity to participate in innovative research to break new ground in the field of finance. Through their membership, they also gain access to the Wharton School’s faculty and enjoy other special benefits. Members of the Center 1999 – 2000 Directing Members Ford Motor Company Fund Geewax, Terker & Company Miller, Anderson & Sherrerd The New York Stock Exchange, Inc. Twin Capital Management, Inc. Members Aronson + Partners Credit Suisse Asset Management EXXON Goldman, Sachs & Co. Merck & Co., Inc. The Nasdaq Stock Market Educational Foundation, Inc. Spear, Leeds & Kellogg Founding Members Ford Motor Company Fund Merrill Lynch, Pierce, Fenner & Smith, Inc. Oppenheimer & Company Philadelphia National Bank Salomon Brothers Weiss, Peck and Greer 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. 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[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 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