Trade Size and the Adverse Selection Component of. the Spread: Which Trades Are "Big"?

Transcription

1 Trade Size and the Adverse Selection Component of the Spread: Which Trades Are "Big"? Frank Heflin Krannert Graduate School of Management Purdue University West Lafayette, IN USA Kenneth W. Shaw Robert H. Smith School of Business University of Maryland College Park, MD USA Frank Heflin gratefully acknowledges financial support from the Yale School of Management and the Krannert Graduate School of Management. The authors thank Rick Antle, Sugato Chakravarty, William Kross, and workshop participants at Yale, Purdue, and Syracuse Universities for helpful comments and suggestions. A previous version of this paper was titled Informed Trading and the Role of Depth."

2 Trade size and the adverse selection component of the spread: Which trades are "big"? ABSTRACT Existing research suggests adverse selection spread components are positively related to trade size, consistent with informed traders trading in larger sizes. However, if market makers use quoted depth to limit losses to informed traders, the size of a trade relative to the depth quoted at the time of the trade is potentially a better indicator of informed trading than is trade size alone. We show that much of the variation in adverse selection estimates can be explained by variation in the ratio of trade size to depth. Further, the previously documented relation between adverse selection spread component estimates and raw trade size largely disappears when the ratio of trade size relative to depth is held constant. Finally, we find that the previously documented intraday pattern in adverse selection estimates does not exist for trade sizes that approach or exceed the quoted depth. That is, for large (relative to depth) trades, the time of day at which a trade occurs provides virtually no incremental information about the probability the trade is informed. In sum, our results indicate whether a trade is large, and therefore indicative of informed trading, depends on the depth in effect when the trade occurs.

3 Recent empirical research suggests the informed trading (i.e. adverse selection) component of the spread is increasing in trade size [Huang and Stoll (1997), Lin, Sanger, and Booth (1995), and Glosten and Harris (1988)]. This evidence is consistent with the notions that informed traders wish to trade larger amounts, as modeled by Easley and O'Hara (1987), and that larger trades have a greater impact on prices set by market makers, as modeled by Kyle (1985), Admati and Pfleiderer (1988), and others. 1 However, in quote driven markets, such as the New York (NYSE) and American (AMEX) stock exchanges, market orders for quantities above the posted depth are not guaranteed execution at the posted price. 2, 3 A market maker concerned about trading with an informed trader can limit potential losses by quoting depth that would likely bind the executed size of an informed trader s order. 4 However, depth quotes set too low likely also constrain the liquidity trades from which market makers earn profits. Thus, under current institutional rules, when quoting depth, market makers potentially trade off constraining profits from liquidity traders against constraining losses to informed traders. If, as existing theoretical and empirical research suggests, informed traders submit larger orders than do liquidity traders, depth quotes should constrain informed trade sizes more than liquidity trade sizes. As such, a rational market maker could infer the presence of informed trading by the proximity of an order size to the currently quoted depth. 5 Thus, while existing research on spread decomposition focuses on raw trade sizes, we 1 Other theoretical work in this line includes Subrahmanyam (1991), Foster and Viswanathan (1993), and Kim and Verrecchia (1994). 2 Since market makers quote prices before receiving orders, this rule protects them from catastrophic losses to informed traders. Market makers offering to trade at price $P or better must be allowed to limit the number of shares they can be required to trade at $P, or an informed trader could place an order for all outstanding shares. 3 Market makers sometimes execute orders at a price better than the posted price. If order sizes above the quoted depth are subject to execution at prices inferior to currently quoted prices, traders might be reluctant to place such orders. Baciadore, Batallio, and Jennings (2000) find that only 16 percent of all trade sizes are greater than the quoted depth at the time of the trade. Further, they find that about 70 percent of those trades are executed at prices equivalent to, or better than, the current quoted price, consistent with the notion that at least some of these orders are submitted with assurances they are not informationally motivated, as suggested by Seppi (1990). Madhavan and Cheng (1997) provide similar evidence consistent regarding these assurances in block trades originating on the upstairs versus downstairs markets. 4 Evidence that depth quotes are smaller in situations, and at times, where informed trading is more likely can be found in Lee, Mucklow, and Ready (1993), Kavajecz (1999), Heflin and Shaw (2000a), and Heflin and Shaw (2000b). Theoretical support is in Dupont (1999). Thus, when informed trading is likely, not only are trade sizes likely larger, but depth quotes are also likely smaller. 5 While they do not study adverse selection components nor the relative incremental impacts of raw trade size and trade size relative to depth on adverse selection estimates, Knez and Ready (1996) and Ready (1999) show that the probability of an 1

4 posit estimates of the adverse selection component of the effective spread are increasing in the proximity of the trade size to the currently quoted depth, and not simply unconditional trade size. 6 This is the main message of this paper and the focus of our empirical analyses. Like prior research we investigate the relation between the estimated adverse selection component of the spread and trade size, but we posit that the market maker s definition of a large trade is conditioned on the currently quoted depth. We document that adverse selection spread component estimates [estimated by methods from either Lin, Sanger, and Booth (1995) or Huang and Stoll (1997)] increase steadily with trade size relative to (i.e. divided by) depth, peak at trade sizes equal to quoted depth, and are flat at trade sizes above quoted depth. When we pit unconditional trade size against trade size relative to depth in a 'horse race' to explain adverse selection component estimates, the clear winner is trade size relative to depth. Holding trade size relative to depth constant, we find very little relation between adverse selection spread component estimates and trade size. However, we find a strong positive relation between adverse selection spread component estimates and trade size relative to depth while holding trade size constant. These results are robust to several sensitivity analyses, including controls for autocorrelation, outliers, and the level of quoted depth, the use of non-parametric tests, and estimations where the quoted spread is not bound by the minimum tick size. We then examine intraday patterns in adverse selection spread component estimates in the context of trade size relative to depth. Prior research [Lin, Sanger, and Booth (1995) and Madhavan, Richardson, and Roomans (1997)] documents declining adverse selection spread component estimates across the trading day, consistent with the hypothesis that informed trading is highest early in the day and declines thereafter, as modeled in Admati and Pfleiderer (1988). We first document that trade size relative to depth is highest in the first half-hour of trading, declines steadily until about mid-day, then remains fairly constant until closing, consistent with the joint hypothesis that informed trading is highest early in the day, and high trade size relative to depth is indicative of informed trading. order receiving price improvement is declining in that order's size relative to both the depth at the time of the trade and the firm's average quoted depth for the year. 6 Throughout the paper we refer to the proximity of trade size to the currently quoted depth simply as trade size relative to depth. We also use the terms trade size and order size interchangeably. 2

5 We then examine the intraday pattern in adverse selection estimates conditional on trade size relative to depth, and find the pattern weakens as trade size relative to depth increases. When trade sizes equal (or exceed) the quoted depth, there is no intraday pattern in adverse selection spread components. These results suggest market makers rely less on time of day to infer informed trading as trade size relative to depth increases, and time of day appears irrelevant in assessing the probability of informed trading when trade size is very near or exceeds the quoted depth. We conclude from our analyses that transaction size and informed trading are, indeed, positively related, but not in the way generally thought. In terms of making inferences about informed trading, "big" trades are better defined by their proximity to quoted depth than by the raw number of shares traded. Further, while both the proximity of the trade to quoted depth and the time of day the trade occurs are associated with the probability the trade is informationally motivated, the importance of time of day declines as trade size approaches quoted depth. When trade size equals the quoted depth, the probability the trade is informationally motivated is so high that time of day becomes irrelevant. Our analysis proceeds as follows. Section I describes the sample selection procedures and the data. Section II reports empirical results on the relation between the adverse selection spread component and both trade size and trade size relative to depth. Section III investigates intraday variation in adverse selection spread component estimates conditional on trade size relative to depth. Section IV concludes. I. Sample and data The sample is drawn from 349 firms listed in the Financial Analysts Federation reports on corporate disclosure quality. The firms vary considerably in market capitalization, trade size, trading volume, spreads, and depths, and are not concentrated in any particular industry. 7 A total of 314 of these firms have sufficient quote and trade data available from the 1988 ISSM database. 8 7 The firms were selected first for use in another project. We later compare our sample to those used in prior studies and to the population of NYSE firms. 8 Some studies [e.g. Kavajecz (1999), Kavajecz and Odders-White (1999)] use the TORQ database to examine the interaction between the specialist and orders on the limit order book. Our focus is on simply documenting the importance of depth, regardless of whether provided by the specialist or the limit orders, in defining trade size. 3

6 Following Lin, Sanger, and Booth (1995), we estimate adverse selection spread components from ordinary least squares estimation of the following firm-specific regression, ( Pi, t 1 M i, t 1 ) et M = ASC + i, t i, (1) where is the first difference operator, M i,t is the log quoted spread midpoint at time t for firm i, M i,t-1 is the log quoted spread midpoint at time t-1 for firm i, and P i,t-1 is the log transaction price prior to the quoted spread at time t for firm i. We truncate observations entering these regressions at the 99 th and 1 st percentiles of their distributions to minimize the influence of extreme values. The term in parentheses is the effective spread and 9, 10 the parameter estimate, ASC i, is the estimated adverse selection component of the spread. Table I provides descriptive statistics on average share price (bid-ask midpoint), daily share volume (number of shares traded), daily dollar trading volume (daily share volume times transaction price), firm size (share price times number of outstanding common shares), trade size (number of shares traded in a single buy or sell transaction), dollar trade size (trade size times transaction price), number of trades per day, dollar spreads (ask price minus bid price), and relative spreads (dollar spread divided by share price). We first average the variables by firm across all trading days in 1988, and then compute averages across the firm-specific means to yield the descriptive statistics in Table I. The average sample firm has a share price of $39, daily share volume of 155,179 shares, daily dollar trading volume of $6.4 million, and total market capitalization of $3.6 billion. All of these variables are considerably skewed, as their respective medians are $32.80, 94,737, $3.42 million, and $1.82 billion. Average trade size is 2,063 shares, average dollar trade size is $72,460, and the firms trade on average almost 72 times per day. Raw spreads average $0.25 per share. The last row of Table I reports that the mean (median) adverse selection component is about 37.6 (38.1) percent of the effective spread, and ASC i ranges from 1.7 to 71.6 percent of the effective spread. 9 Later, we discuss sensitivity analyses employing the Huang and Stoll (1997) method. 10 Since Lee and Ready (1991) show that prevailing quotes may sometimes be recorded ahead of trades, we define the prevailing quote for each transaction as the quote in effect five seconds before the transaction. Like other studies, we exclude the opening transaction of each day since it is conducted in a call market. Like Lin, Sanger and Booth (1995), when ISSM reports a series of trades with identical price, time, and volume, we keep only the first trade in the series. 4

7 In panel B of Table I, we provide descriptive information at the transaction level. The effective spread is twice the absolute value of the difference between the trade price and the quote midpoint at the time of the trade. Trade size is the number of shares traded in the transaction, depth is the number of shares quoted at the time the trade occurred, on the side (bid or ask) on which the trade occurred, and trade size relative to depth is trade size divided by depth. 11 From panel B of Table I, effective spreads average 15.7 cents per share, and the transaction level mean trade size is just over 2,100 shares. The median trade size is 500 shares, and is less than 1,700 shares for over 75 percent of our observations. Quoted depth at the time of the trade, on the side of the market on which the trade occurs, averages just over 74 round lots (7,400 shares), with a median of 45. The mean trade size relative to depth is 0.895; however, the median is only 0.20, and 75 percent of the transactions occur in sizes less than 70 percent of the quoted depth. 12 At both the firm and transaction level, our sample appears very similar to those of other microstructure studies [e.g. Lin, Sanger, and Booth (1995), Lee, Mucklow, and Ready (1993), and Krinsky and Lee (1996)] sampling different NYSE firms over similar time periods. II. Results A. Univariate analyses We first replicate previous research on adverse selection spread component estimates and trade size. To facilitate comparison to prior research [Lin, Sanger, and Booth (1995)], we partition each firm's trades into six 11 Trades are classified as buys or sells by applying the Lee and Ready (1991) algorithm as follows. Trades at or nearer the quoted ask are classified as buys. Trades at or nearer the quoted bid are classified as sells. Trades at the quote midpoint are classified using the tick test, as described in Lee and Ready (1991). If the change in price is zero the tick test is applied to the first non-zero-price-change transaction in the preceding five transactions. Remaining unclassified transactions are excluded. Lee and Radhakrishna (2000) find the Lee and Ready (1991) algorithm accurately classifies 92 percent of the trades classifiable in the TORQ database, with 96 percent accuracy on trades executed at the bid or ask price, 70 percent accuracy on trades executed inside the spread, but not at the quote midpoint, and 65 percent accuracy on trades executed at the quote midpoint. 71 percent of our trades were executed at the bid or ask price, 10 percent were executed inside the spread but not at the quote midpoint, and 19 percent were executed at the quote midpoint. Odders-White (2000) and Ellis, Michaely, and O'Hara (2000) document that the Lee and Ready algorithm correctly classifies over 83 percent of the transactions in their respective samples, and that trades executed inside the spread are more likely to be misclassified. To the extent this algorithm fails to accurately classify trades, our analyses are biased away from finding significant relations between adverse selection estimates and trade size relative to depth. 12 Some orders may be batched together and executed as a single trade. Lee and Radhakrishna (2000) report that 76 percent of all NYSE trades for their sample firms are not batched. Further, batching is more likely for larger trade sizes. They find over 86 percent of trades of under 1,900 shares are not batched. In our sample, 71 (60) percent of trades within percent of quoted depth are for 2,000 (1900) shares or less. Thus, although some of the trades at the quoted depth might result from batched orders, most likely do not. 5

8 trade size percentile-categories. To construct the categories, we first rank each firm's trades from smallest to largest (for that firm) in terms of number of shares traded. Then, for each firm, we place each trade into percentiles, and then into one of six percentile-categories. The first category contains the trades falling into the first through the 24th percentiles. The second category contains trades falling into the 25th through 49th percentiles. Remaining trades are placed into groups comprising the 50 th through 74 th percentiles, the 75 th through 89 th percentiles, the 90 th through 94 th percentiles, or the 95 th through 100 th percentiles. We estimate equation (1) within each firm-percentile-category having at least 30 observations, and compute the sample-wide average adverse selection estimate within each of the 6 percentile categories. Results are presented in panel A of Table II. Clearly, mean adverse selection spread component estimates increase as trade size increases, averaging 21.5 percent of the effective spread in the smallest trade size percentile-category and increasing monotonically to 62.5 percent in the largest percentile category. The average adverse selection spread component estimate in each trade size group is significantly greater (1 percent level) than that in the next smaller trade size category. Like prior research, evidence in Table II suggests the risk of informed trading is increasing in trade size. Panel B of Table II presents statistical tests of the relation between adverse selection spread component estimates and trade size relative to depth, using the same percentiles as in panel A. Clearly, average adverse selection estimates increase with trade size relative to depth. In the smallest trade size to depth category, adverse selection component estimates average about 10.7 percent of the total effective spread. This percentage steadily increases throughout the remaining trade size to depth categories, and is over 60 percent of the total effective spread once trade size is greater than or equal to 75 percent of the quoted depth. T-tests indicate that the mean adverse selection estimate in each trade size relative to depth category is significantly greater than the mean adverse selection estimate in the preceding trade size relative to depth category at the 1 percent level or greater. B. Multivariate analyses Our evidence thus far suggests adverse selection estimates are increasing in both trade size and in trade size relative to depth. We next examine the relation between adverse selection estimates and the two transaction 6

9 size measures, with each conditioned on the other. We first examine the relation between adverse selection estimates and trade size, holding trade size divided by depth constant. We partition each firm s transactions into quintiles based on the magnitude of trade size relative to the prevailing depth quote at the time of the trade. Then, within each trade size relative to depth quintile, we partition trades into the six trade size categories employed in Table II, producing 30 distinct groups of observations for each firm. We estimate equation (1) by firm within each of the 30 groups, producing up to 30 adverse selection estimates for each firm. 13 Table III reports mean (across firms, within each of the 30 groups) adverse selection estimates for each trade size group, within each trade size to depth quintile. Results in Table III show that, once trade size to depth is held constant, there is no relation between trade size and adverse selection spread component estimates. 14 None of the mean adverse selection estimates, in any of the trade size percentiles, are significantly greater than the mean adverse selection estimates in the immediately preceding (smaller) trade size percentile. 15 In the first trade size to depth quintile, the mean adverse selection estimate is in the smallest trade size group and 0.09 in the largest trade size group. In the second (third) trade size to depth quintiles, the mean adverse selection estimate falls from (0.335) to (0.204) as trade size increases from the smallest 24 percent of trades to the largest 5 percent of trades. Thus, within the second and third trade size to depth quintiles, adverse selection estimates actually decrease with increasing trade size. Adverse selection estimates show little variability within the fourth and fifth trade size to depth quintiles. In trade size to depth quintile four (five), the mean adverse selection estimate is (0.636) in the smallest 24 percent of trades, and (0.654) in the largest 5 percent of trades. In sum, Table III suggests no evidence of a positive relation between adverse selection estimates and trade size, once trade size to depth is held constant. Next we ask whether the reverse is true: are adverse selection estimates related to trade size to depth after holding trade size constant? Thus, we reverse the procedure employed in Table III, and partition each 13 Some firms did not have at least 30 observations in one or more of the 30 groups. Thus, the number of firms contributing to the mean for a particular group can be less than the total of 314 sample firms. 14 Note that we are holding trade size to depth constant, which is not the same as simply holding depth constant. Later we implement a control for the level of depth in sensitivity analyses. 15 We repeated all the tests reported in Tables III through V using medians, and obtained the same results. 7

10 firm's trades into five quintiles based on trade size. Within each trade size quintile for each firm, we further partition the trades into six groups based on the same six percentile categories (as in Tables II and III) of trade size divided by depth, and estimate equation (1) by firm, separately for each of these 30 groups. Table IV presents the average adverse selection estimate within each trade size to depth group and trade size quintile. Table IV suggests that, even after holding trade size constant, a strong positive relation exists between adverse selection spread component estimates and trade size relative to depth. In the first trade size quintile, mean adverse selection estimates are 7.5 percent of the total effective spread in the lowest trade size to depth group, increase steadily in each of the remaining trade size to depth groups, and reach a maximum of over 43 percent of the total effective spread in the highest 5 percent of trade size to depth observations. Four of the five increases in the mean adverse selection estimates across successive trade size to depth groups in this first trade size quintile are significant at the 1 percent level or better. Similar evidence is found in the second and third trade size quintiles. In particular, mean adverse selection estimates increase from 8.3 (10.4) percent of the total effective spread in the smallest trade size to depth group in trade size quintile 2 (3), to over 57 (62) percent of the total effective spread in the highest 5 percent of trade size to depth observations in trade size quintile 2 (3). In each of these two trade size quintiles, mean adverse selection estimates increase uniformly with trade size relative to depth, and all of the increases in the mean adverse selection estimates across successive trade size to depth groups are significant at the 1 percent level or better. Average adverse selection estimates increase with trade size divided by depth in the fourth and fifth trade size quintiles, though fewer of the increases in mean adverse selection estimates are statistically significant. In the fourth trade size quintile, adverse selection averages 14.5 percent of the total effective spread in the lowest trade size to depth group, and over 61 percent of the total effective spread for trades in the highest 25 percent of trade size to depth (i.e., the 75-89, 90-94, and percentile groups). Three of the five increases in mean adverse selection estimates are significant at the 1 percent level in the fourth trade size 8

11 quintile. 16 In the fifth trade size quintile, adverse selection estimates average 25.5 percent of the total effective spread in the smallest trade size to depth percentile, 58.2 in the second largest trade size to depth group, and over 63 percent in the remaining trade size to depth groups. Two of the five increases in mean adverse selection estimates are significant at the 1 percent level in this quintile. The evidence in the fourth and five trade size quintiles suggest that estimated adverse selection components are increasing in trade size to depth, but level off at high levels of trade size relative to depth. Perhaps the latter result is due to a combination of (a) increased batching of multiple trades in these quintiles (see footnote 11), or (b) the presence of block or other large trades. Existing research [Keim and Madhavan (1996), Barclay and Warner (1993), Chakravarty (2000)] suggests the price impact of trades is non-linear in trade size, either because informed traders choose not to submit very large orders and/or because large blocks precipitate the block trader to locate more counterparties to the trade and lower its liquidity cost. Also, some large trades are shopped in the upstairs market with possible assurances they are not informationally motivated. We leave exploration of these issues for future research. Our evidence thus far suggests adverse selection spread component estimates are related to the size of a trade, but not in the way suggested by previous research. We find the size of a trade is much better defined by its proximity to quoted depth than by the raw number of shares traded. We next estimate a multiple regression designed to further examine which measure (trade size relative to depth or raw trade size) better explains adverse selection estimates. For this test we assume that, within the relevant range of the data (trades in sizes up to 100 percent of the quoted depth at the time of the trade, the risk of informed trading for firm i (ASC i ) is a linear function of firm i s trade size (TRSZ i ), trade size divided by depth (TRSZ i /DEPTH i ), and an intercept term that captures other (unknown) determinants of the risk of informed trading. 17 That is, 16 Mean trade sizes in the first four trade size quintiles in Table V suggest batching of multiple orders does not likely affect the results. Mean trade sizes range from about 107 shares in the smallest trade size to depth group in the first trade size quintile, to about 648 shares in the largest 5 percent of trades in the third trade size quintile. Evidence in Lee and Radhakrishna (2000) suggests about 90 percent of all trades in these size ranges come from a single participant on the active side. Mean trade sizes vary from 1,170 shares to 1,530 shares in the fourth trade size quintile. Lee and Radhakrishna (2000) indicate that about 75 percent of all trades in these ranges come from a single participant on the active side. 17 We restrict the observations due to non-linearity (discussed later in the paper) in the relation between adverse selection spread estimates and trade size to depth. 9

12 ASCi = α 0, i + α1, itrszi + α 2, i TRSZi. DEPTHi (2) Substituting equation (2) for ASC i in equation (1) yields, TRSZ log[m i,t ] = α 1,i Z i,t-1 + α 2,i Z i,t-1 TRSZ i,t-1 + α 3,i Z i, t 1 i,t-1 + e i,t, (3) DEPTH i, t 1 where Z i,t-1 = log[p i,t-1 ] log[m i,t-1 ]. We add an intercept and estimate equation (3) by firm using ordinary least squares. 18 The coefficients on the terms in equation (3) containing the interaction of trade size and trade size divided by depth with Z i,t-1 (i.e., α 2,i and α 3,i ) capture the extent to which firm i's trade size and/or trade size divided by depth drives its adverse selection estimate. If both trade size and trade size relative to depth are important in explaining the risk of informed trading, a preponderance of the α 2,i and α 3,i coefficients from these individual firm-specific regressions should be positive and significant. Further, the magnitudes of their respective t-statistics will indicate the relative explanatory power of trade size and trade size divided by depth for ASC i. Earlier results, presented in Tables II and IV, suggest the relation between adverse selection spread component estimates and trade size relative to depth may be non-linear. Specifically, as trade size relative to depth rises above one, the rate of increase in adverse selection estimates declines (see panel B of Table II and cells where the mean trade size divided by depth is greater than one in the third, fourth, and fifth trade size quintiles of Table IV). If so, the linearity assumption we make in estimating equation (3) is not descriptive across the entire range of our data. To examine potential non-linearity more carefully, we sort our trades into 30 quantiles based simply on the value of trade size relative to depth. We place all trades with trade size to depth less than 0.05 in the first quantile. The second quantile contains trades where trade size divided by depth is between 0.05 and 0.10, etc., up to the 19 th quantile, which contains trades where trade size divided by depth is between 0.90 and The 20 th quantile contains trades where trade size divided by depth is between 0.95 and No firm had at least 30 observations where trade size divided by depth was between 0.99 and 1.0 The 21 st 18 We also estimate equation (3) without an intercept and obtain similar (untabulated) results. 10

13 quantile contains trades where trade size divided by depth equals exactly 1.0. We estimate equation (1) for each firm-quantile, with the average for each quantile shown in Figure 1. Figure 1 shows a reasonably linear relation between adverse selection spread estimates and trade size to depth, up to the trade size to depth quantile including 1.0. Right at the trade size to depth quantile including 1.0, the slope of this relation spikes sharply upward. Beyond this quantile, there appears to be no or a slightly inverse relation between adverse selection spread estimates and trade size relative to depth. This pattern is consistent with our intuition, which predicts the market maker s adverse selection problem is worst at trade sizes approximately equal to the quoted depth. Beyond this point, there are likely to be considerably more trades for which the market maker has received assurances or otherwise determined that the trade is uninformed, and more batching of multiple orders. Because of this non-linearity, and because we expect many trades in sizes greater than quoted depth are less likely to be informed trades, we limit estimation of equation (3) to trades that are less than 100 percent of the quoted depth at the time of the trade. Table V presents descriptive information on the distribution of coefficient estimates and t-statistics from the 314 firm-specific estimations of equation (3). Table V shows that in most of the firm-specific regressions, the estimates of the coefficient on the trade size interaction (i.e. α 2,i ) are not significantly different from zero. In particular, the mean (median) of the estimates of α 2,i equals ( ), and the mean (median) of the t- statistics on α 2,i equals (0.421). The last three rows of Table V present the number of times the OLS t- statistics from the individual regressions are significant at the 1, 5, or 10 percent levels respectively, in onetailed tests. A total of 118 of the t-statistics are significant at the 10 percent level or better; 81 at the 1 percent level, 23 at the 5 percent level, and 14 at the 10 percent level. Thus, in almost two-thirds (200/314) of the firmspecific regressions, the parameter estimate on the trade size interaction is not statistically different from zero. Consistent with evidence reported in Table IV, some of the estimates of α 2,i are actually negative and statistically significant. In stark contrast are the estimates of α 3,i, the coefficient on the trade size divided by depth interaction term. Both the mean and median of the individual estimates of the trade size divided by depth interaction coefficient (α 3,i ) are positive. The mean (median) estimate of α 3,i is (0.272), and the related mean and 11

14 median t-statistics are and respectively. In 311 of the 314 individual firm regressions the estimates of the trade size to depth coefficient are positive and statistically significant, and all but one of these is significant at the 1 percent level or better. Unlike the distribution of parameter estimates on the trade size interaction variable, we find no statistically significant negative coefficient estimates on the trade size divided by depth interaction variable. In sum, the evidence in Table V suggests trade size divided by depth explains much more of estimated adverse selection spread component estimates than does trade size. In fact, we must conclude from Table V that after controlling for trade size relative to depth, raw trade size has almost no explanatory power for most firms' adverse selection spread component estimates. Our evidence suggests that, with respect to information conveyed, the size of a trade is much better defined by how big it is relative to currently quoted depth than by its raw size in number of shares. C. Sensitivity analyses The sensitivity analyses outlined in this section focus on the potential effects of (a) autocorrelation, (b) the (known) inverse relation between spreads and depths, (c) alternative spread decomposition models, and (d) the potential impact of price discreteness on variation in quoted depth. Lee, Mucklow, and Ready (1993) document autocorrelation in intraday data. Thus, we also estimated equation (3) with an autoregressive error structure using a maximum of sixty lags. 19 The coefficient estimates and t-statistics from this procedure (untabulated) are very similar to those reported in Table 5 and we conclude our results are robust to autocorrelation controls. Lee, Mucklow, and Ready (1993) also find an inverse relation between spreads and depths. Since our variable of interest is trade size scaled by depth, we endeavor to determine whether our results are driven simply by the inclusion of the inverse of depth. We add the inverse of depth (1/depth) as an additional explanatory variable to equation (3) and estimate the following using ordinary least squares, 19 We used the BACKSTEP option in PROC AUTOREG in SAS. 12

15 log[m i,t ] = α 0,i + α 1,i Z i,t-1 + α 2,i Z i,t-1 TRSZ i,t-1 + α 3,i Z i,t-1 α 4,i Z i,t-1 1 DEPTH i, t - 1 TRSZ DEPTH i, t 1 + i, t 1 + e i,t. (4) Consistent with prior research we find the coefficient on the inverse depth interaction term (α 4,i ) is positive; the mean (median) parameter estimate of α 4,i equals ( ), with related mean and median t-statistics close to More importantly, our previous inferences regarding the relative importance of trade size versus trade size relative to depth are essentially unchanged. Results regarding the coefficient estimates of α 3,i remain strong; the mean (median) parameter estimate on the trade size to depth interaction equals ( ) and the corresponding mean (median) t-statistics are (11.31). In 309 (over 98 percent) of the firm-specific regressions, the parameter estimate on the trade size to depth interaction is positive and significant at the 5 percent level or better. Adding the inverse depth interaction term enhances the explanatory power of the raw trade size interaction beyond that reported in Table V. However, the results on this variable remain far weaker than those on the trade size relative to depth interaction. The mean and median parameter estimates of α 2,i are positive ( and respectively), but the t-statistics (mean = 2.17 and median = 1.93) are relatively weak. Additionally, in only 171 (55 percent) of the firm-specific regressions is the parameter estimate for the trade size variable positive and significant at the 5 percent level or better. Further, several of the estimates on this variable are actually negative and significant. Thus, we conclude that the dominance of trade size to depth over trade size in explaining adverse selection spread components is not due to a simple relation between adverse selection estimates and depth. We also examine the sensitivity of our results to an alternative spread decomposition approach. Glosten and Harris (1988) develop a spread decomposition model by relating changes in price to transaction size and an indicator for whether the trade was buyer or seller initiated. This method has been used [Glosten and Harris 20 Results from equation (4) are untabulated. Since we use 1/depth a positive coefficient is consistent with an inverse relation between spreads and depths noted in prior research. 13

16 (1988), Hasbrouck (1991), Brennan and Subramanyam (1995), and others], to develop estimates of the 'Kyle lambda', a measure of market liquidity based on Kyle (1985). In the Glosten and Harris (1988) method the adverse selection component as a percentage of the spread is modeled to increase with trade size. As shown in Huang and Stoll (1997), the Glosten and Harris (1988) model, and variants of it used in related papers, is a restricted version of the Huang and Stoll (1997) model using trade indicators. Estimating the Huang and Stoll model (1997) in different trade size categories provides estimates of the Kyle lambda that are allowed to vary with trade size in an unrestricted manner, while also separating potential inventory holding costs from adverse selection. We repeated our tests using the Huang and Stoll (1997) adverse selection estimates and found similar results. Finally, we consider whether our results are attributable to variation in quoted depths at different levels of the discrete quoted spreads. We repeated the tests reported in Table V on those observations where the quoted spread, at the time of the transaction, is at least 2/8 ths, and again where the quoted spread at the time of the transaction is only 1/8 th. The results were very similar in both groups and nearly identical to those reported in Table V. III. Intraday patterns in informed trading Prior research [Lin, Sanger, and Booth (1995), Madhavan, Richardson, and Roomans (1997)] documents a distinct and declining intraday pattern in adverse selection estimates. 21 If informed trading is highest early in the day and declining thereafter, and if trade size relative to depth is an indicator of informed trading, we expect to see a declining pattern in trade size relative to depth during the trading day. Figure 2 shows the median trade size relative to depth (top panel) and the proportion of trades in sizes at the depth (bottom panel) across the 13 half-hour intervals of the trading day. Consistent with the notion that trade size relative to depth is related to adverse selection risk, both the median trade size to depth and the proportion of trades at the depth exhibit intraday patterns almost identical to the intraday pattern in adverse 21 We confirmed the intraday adverse selection spread component pattern in our data. Our adverse selection spread component estimates decline from over 43 percent of the effective spread during the first half-hour of trading to about 35 percent by the middle of the afternoon, and are relatively flat thereafter. 14

17 selection spread components documented in prior research. 22 Median trade size relative to depth begins the day at about 30 percent, i.e., average trade size is about 30 percent of the quoted depth. Trade size to depth declines steadily and rapidly throughout the morning, and by midday, trade sizes average only about 20 percent of quoted depth. Trade size to depth remains at about that level until the close. The proportion of trades at the depth displays a very similar pattern. During the first half-hour of trading, over 11 percent of trades have size equal to the quoted depth. By midday, about eight percent of trade sizes are equal to quoted depth. 23 Since trade size relative to depth exhibits essentially the same intraday pattern as that for adverse selection spread component estimates, we next ask whether the intraday pattern in adverse selection estimates persists, conditional on trade size relative to depth. In other words, do market makers infer a large trade size relative to depth trade is more likely an informed trade than the same trade size relative to depth trade, if it is submitted in the morning rather than in the afternoon? Or, is a trade with large size relative to depth so likely informed that the time of day it is submitted is largely irrelevant? To provide evidence on this question, we partition each firm s observations into 6 categories based on the magnitude of trade size relative to depth (less than 0.20, between 0.21 and 0.50, between 0.51 and 0.75, between 0.75 and 0.89, between 0.90 and 1.0, and above 1.0), and then estimate equation (1) within each firm half-hour and trade size relative to depth category. This yields up to 78 (13 half-hour intervals times 6 trade size relative to depth categories) adverse selection estimates per firm, and thus up to 24,492 (78 times 314 firms) adverse selection estimates for the full sample. 24 We then compute the 78 (13 times 6) mean adverse selection estimates (ASC h,j ) for each half-hour h and trade size to depth category j. We estimate ordinary least squares regressions of the following equation for each of the six trade size to depth categories ASC h,j = β 0 + β 1 TIME h,j + β 2 TIME 2 h,j + e h,j, (5) 22 Mean trade size to depth displays the same pattern as median trade size to depth, although the level is considerably higher due to the highly skewed nature of the distribution. 23 Note that this pattern is not simply the inverse of quoted depth (inverted smile ) documented in Lee, Mucklow, and Ready (1993). 24 Firm half-hour categories with less than 30 observations are excluded. 15

18 where TIME h,j equals either 1, 2, 3,.,13, i.e., it is the number of the half-hour interval from which ASC h,j is computed. We include the square of TIME h,j to capture the curvilinear shape of intraday adverse selection spread components. Since adverse selection estimates decline over at least the first half of the day, the first derivative of equation (5) with respect to TIME h,j should be negative, i.e., β (β 2 ) (TIME h,j ) < 0, during the first half of the day. However, during at least the first half of the day, adverse selection estimates appear to decline at a decreasing rate, thus the second derivative of equation (5) with respect to TIME h,j should be positive, i.e., 2 (β 2 ) (TIME h,j ) > 0. This is equivalent to a simple test of whether β 2 > 0. Results are presented in Table VI. Despite the small number of observations (13) per regression, the adjusted R 2 s (53.45 to percent) indicate a good fit. For the four estimations of equation (5) where trade size relative to depth is less than 0.90, the coefficient on TIME h,j (TIME 2 h,j) is negative (positive), and significant at less than the 0.01 level, and the adjusted R 2 s range from percent to percent. 25 Further, the first derivative is negative and significant at the 5 percent level or better for each of at least the first seven half-hours. This suggests adverse selection spread component estimates decline, at a decreasing rate, across the trading day for trades in sizes less than 90 percent of the quoted depth. Estimation of equation (5) for the last two trade size percentile categories yields markedly different results, however. In these two trade size to depth categories, the estimated first derivative of equation (5) is not negative for any of the half hours. Thus, as trade size to depth approaches and exceeds 1.0, adverse selection estimates do not appear to decline from morning through midday. In other words, the intraday pattern in adverse selection estimates does not appear to be present for very large (relative to depth) trades. Figure 3 displays these patterns graphically. Figure 3 shows the deviation in the mean adverse selection estimate in each of the 12 half-hours from half-hour 2 through half-hour 13 from the mean adverse selection estimate in the first half-hour for 3 distinct trade size relative to depth categories (0-24, 50-74, and ). For trade size relative to depth less than 0.75 (bottom 2 graphs in Figure 3), mean adverse selection estimates are higher in the early part of the trading day, decline to midday, and remain constant thereafter. The top graph in Figure 3, displays a distinctly different pattern for trade sizes between 90 and 100 percent of quoted depth 25 All p-values in Table VI are for two-tailed hypothesis tests of whether the coefficient differs from zero. 16

19 however. The deviations in mean adverse selection estimates for trade size close to quoted depth essentially hover around zero. That is, for trades in sizes very near quoted depth, adverse selection estimates do not vary across the day. In summary, we conclude that trade sizes near the quoted depth are so likely informationally motivated that no additional information about the likelihood of informed trading is obtained from the time of day the trade is submitted. IV. Conclusions Previous research [Huang and Stoll (1997), Lin, Sanger, and Booth (1995), Glosten and Harris (1988)] suggests adverse selection spread component estimates are increasing in trade size. However, quoted depth provides the maximum amount the market maker "guarantees" to trade at prevailing prices. Thus, we investigate whether adverse selection spread component estimates are related to the proximity of trade size relative to quoted depth and run a "horse race" between trade size and trade size relative to depth in explaining adverse selection estimates. Our empirical analyses reveal several new findings. We find estimates of the adverse selection component of the spread are uniformly increasing in the magnitude of trade size relative to depth. Further, we find adverse selection spread component estimates are at best weakly related to raw trade size if trade size relative to depth is held constant. However, adverse selection spread component estimates remain strongly related to trade size relative to depth even when raw trade size once held constant. We interpret these results as support for the notion that the size of a trade is related to the likelihood the trade is informationally motivated, but in a manner different from previously thought. Large trades appear better defined by the number of shares relative to the quoted depth at the time of the trade than simply by raw number of shares. We also examine the intraday pattern in adverse selection estimates in the context of trade size relative to depth. We find trade size relative to depth exhibits an intraday pattern very similar to that documented for adverse selection estimates. Trade size relative to depth is highest at the beginning of the day and declines through midday, remaining fairly constant to the end of the day. We find that the intraday pattern in adverse selection estimates persists across most trade size relative to depth levels, but, interestingly, disappears as trade 17

20 size relative to depth approaches and exceeds one. This evidence suggests that for large trades (relative to depth), the time of day at which the trade occurs contains no information about the likelihood the trade comes from an informed trader. References Admati, R. and P. Pfleiderer, 1988, A theory of intraday patterns: Volume and price variability, Review of Financial Studies 1, Barclay, M.J. and J.B. Warner, 1993, Stealth and volatility: Which trades move prices?, Journal of Financial Economics, 34, Bacidore, J., Battalio, R., and R. Jennings, May 2000, Depth improvement and adjusted price improvement on the NYSE, working paper, NYSE. Brennan, M. and A. Subrahmanyam, 1995, The determinants of average trade size, Journal of Business, 71, Chakravarty, S, 2000, Stealth trading: The next generation, Journal of Financial Economics, forthcoming. Dupont, D., 1999, Market making, prices, and quantity limits, working paper, Board of Governors of the Federal Reserve System. Easley, D. and M. O'Hara, 1987, Price, trade size, and information in securities markets, Journal of Financial Economics 19, Ellis, K., R. Michaely, and M. O'Hara, 2000, The accuracy of trade classification rules: evidence from NASDAQ, Journal of Financial and Quantitative Analysis, forthcoming. Foster, F. and S. Viswanathan, 1993, Variations in trading volume, return volatility, and trading costs: Evidence on recent price formation models, Journal of Finance, 48, Glosten, L.R. and L. Harris, 1988, Estimating the components of the bid-ask spread, Journal of Financial Economics, 21, Hasbrouck, J., 1991, Measuring the information content of stock trades, Journal of Finance, 22, Heflin, F. and K.W. Shaw, 2000a, Block ownership and market liquidity, Journal of Financial and Quantitative Analysis, forthcoming. Heflin, F. and K.W. Shaw, 2000b, Adverse selection, inventory holding costs, and depth, Journal of Financial Research, forthcoming. Huang, R. and H. Stoll, 1997, The components of the bid-ask spread: A general approach, Review of Financial Studies, 7, Kavajecz, K., 1999, A specialist's quoted depth and the limit order book, Journal of Finance, 54, and E. Odders-White, 1999, An examination of changes in specialists' posted price schedules. Working paper, University of Wisconsin. 18