evidence on the speed of convergence to market efficiency by

Evidence on the Speed of Convergence to Market Efficiency

by

Tarun Chordia, Richard Roll, and Avanidhar Subrahmanyam

September 25, 2001

Abstract

Daily returns for large and mid-cap stocks listed on the New York Exchange are not

serially dependent. In contrast, order imbalances on the same stocks are highly persistent

from day to day. These two empirical facts can be reconciled if sophisticated investors

react to order imbalances within the trading day by engaging in countervailing trades

sufficient to remove serial dependence over the daily horizon. How long does this

actually take? The pattern of intra-day serial dependence, over intervals ranging from

five minutes to one hour, reveals traces of efficiency-creating actions. For the stocks in

our sample, it takes longer than five minutes for astute investors to begin such activities.

By thirty minutes, they are well along on their daily quest.

Contacts

Chordia Roll Subrahmanyam Voice: 1-404-727-1620 1-310-825-6118 1-310-825-5355

Fax: 1-404-727-5238 1-310-206-8404 1-310-206-5455 E-mail: [email protected] [email protected] [email protected]

Address: Goizueta Business School Emory University

Atlanta, GA 30322

Anderson School UCLA

Los Angeles, CA 90095-1481

Anderson School UCLA

Los Angeles, CA 90095-1481

We are grateful to Michael Brennan, Eugene Fama, Laura Frieder, William Goetzmann, Andrew Karolyi, Francis Longstaff, Stephen Ross, and Ross Valkanov for valuable comments and suggestions.

Convergence to Efficiency, September 25, 2001 1


Abstract

Daily returns for large and mid-cap stocks listed on the New York Exchange are not

serially dependent. In contrast, order imbalances on the same stocks are highly persistent

from day to day. These two empirical facts can be reconciled if sophisticated investors

react to order imbalances within the trading day by engaging in countervailing trades

sufficient to remove serial dependence over the daily horizon. How long does this

actually take? The pattern of intra-day serial dependence, over intervals ranging from

five minutes to one hour, reveals traces of efficiency-creating actions. For the stocks in

our sample, it takes longer than five minutes for astute investors to begin such activities.

By thirty minutes, they are well along on their daily quest.



I. The Issue.

For most of its scientific life, the field of finance has debated the question of market

efficiency. Despite a long list of empirical anomalies and extensive indications of

psychological quirks among investors, most financial economists and professionals still

profess that asset prices are difficult to predict. Schwert (2001) reviews a number of

well-documented anomalies and finds that some of them have disappeared, perhaps

revealing ephemeral market inefficiencies. But he argues also that other anomalies

appear to have been “discovered” even though they did not exist.

There is a growing literature about the irrationalities of individual investors. Odean

(1999), for instance, finds that small investors have a perverse ability to forecast future

returns; their stock purchases perform worse than their sales. Barber and Odean (2000)

find that the more individuals trade, the worse their returns. Benartzi and Thaler (2001)

document bizarre portfolio choices among individuals allocating pension assets to various

classes.

Despite their reluctance to forecast prices, most scholars admit also that some individuals

behave foolishly all the time and all individuals behave foolishly some of the time. When

reconciling these conflicting views, we usually resort to flurry of hand waving and invoke

the mantra of aggregation. Somehow, from within the blizzard of behavioral proclivities,

the “market” becomes efficient, or, at least efficient enough that professors and money

managers have a very difficult time beating passive investment strategies. But exactly

how does this happen and how long does it take?

The concepts of market efficiency as defined by Fama (1970) in his seminal review,

weak, semi-strong, or strong form efficiency, represent a road map for statistical tests.

They offer little insight about market processes that might deliver the hypothesized


phenomena. Clearly, efficiency does not just congeal from spontaneous combustion. It

depends, somehow, on individual actions.

This idea was formalized by Grossman (1976) and Grossman and Stiglitz (1980) who

proved that the market price cannot fully incorporate all knowable information. Someone

must be able to make (infra-marginal) returns from exploiting deviation of prices from

fundamental values. But whom, and how? Cornell and Roll (1981) borrowed a model

from evolutionary biology to show that efficient markets must be inhabited by both

passive investors, who take prices as correct forecasts of future value, and by active

investors who expend resources in an effort to detect errors in prices. Market efficiency

is the state in which neither the marginal active nor the marginal passive investor has an

incentive to alter his or her respective approach. Infra-marginal active investors pay to

become better informed and somehow move prices enough that passive investors can

enjoy a free ride without sacrificing much return (indeed, any return at the margin.)

Many investors still follow technical trading strategies that appear to generate little

revenue and much cost; these strategies have long been the subject of much critique by

finance professors. Recently, we uncovered a seemingly related and surprising

phenomenon during a study of market-wide order imbalances on the New York Stock

Exchange.1 Market order imbalance, defined as the aggregated daily market purchase

orders less sell orders for stocks in the S&P500 index, is highly predictable from day to

day. A day with a high imbalance on the buy side will likely be followed by several

additional days of aggregate buy side imbalance; and similarly for an imbalance on the

sell side. This implies that investors continue buying or selling for quite a long time,

either because they are herding or because they are splitting large orders across days, or

both. More than fifty percent of tomorrow’s imbalance among S&P500 stocks can be

forecast by past returns and past imbalances.

Yet the S&P500 index is virtually a random walk over a horizon of one day. During the

1988-98 sample period, it had a first order autocorrelation coefficient of 0.005 (p-

1 See Chordia, Roll, and Subrahmanyam (2001).


value=0.78) and insignificant autocorrelations at all longer daily lags. This suggests, of

course, that some astute investors must be correctly forecasting continuing price pressure

from order imbalances and conducting countervailing trades within the very first day,

trades sufficient to remove all serial dependence in returns which would otherwise be

induced by the continuing procession of order imbalances.

There are at least two puzzles here: First, why do some naïve investors persist in their

orders for days on end when it does them no good (because there is no inter-day return

dependence)? Second, how long within the day does pressure from order imbalances

continue to move prices? When thinking about this second and more imporatant

question, it seems rather obvious that some finite time period, albeit perhaps quite a short

period, is required for sophisticated investors to counteract a sudden and unexpected

preponderance of orders on the same side of the market.

It simply cannot be true that returns are independent from trade to trade or even from

minute to minute. It must take at least some time for astute investors to figure out what is

happening to orders, to ascertain whether there is new pertinent information about values,

and to expunge any serial dependence remaining after prices adjust to their new

equilibrium levels. The horizon over which this activity takes place is the object of our

study. We propose to investigate how long it takes the market to achieve weak-form

efficiency; i.e., how long it takes to remove return dependence.

Other researchers have investigated questions similar to the one we address, but in very

specific contexts. In early work, Patell and Wolfson (1984) show that dividend and

earnings announcements “interrupt” the usual pattern of return serial dependence for at

least fifteen minutes and that prices do not revert completely to their normal serial

correlation pattern for up to ninety minutes. Although they make no explicit statement

about how this happens, they clearly have in mind the activities of arbitrageurs who

offset the impulsive reactions to company announcements of naïve investors.


Garbade and Lieber (1977) formulate a model of independent changes in equilibrium

price coupled with random orders to buy or to sell at quoted ask and bid prices. They use

data on two stocks for a single month and find that this model does not describe price

moves for short time intervals (a few minutes) while it is consistent with price moves

over longer horizons.2 In concluding, they recognize that “…investors who monitor the

market continually during the day…” might be instrumental in bringing about the

observed pattern.

Epps (1979) studies price adjustments for a group of firms in the same industry

(automobiles). He finds rapid but not instantaneous adjustments across firms to common

news relevant for all industry firms. Correlations among the returns increase with the

time interval, which suggests cross-firm variation in the speed of adjustment to new

information. Epps’ overall conclusion is that “…the predictive value of a price change in

one stock endures not much more than one hour…” but “…the average lag in the

response of prices [to new information] is more then 10 minutes” (p. 298).

Related theoretical models were developed by Copeland (1976) and Hillmer and Yu

(1979). Copeland’s model predicts a positive correlation between trading volume and

absolute price change and positive skewness in volume. However, it does not include a

provision for the activities of arbitrageurs. Hillmer and Yu note that the incorporation of

information into prices “cannot be completed instantaneously” because “…in practice an

investor will not react…unless he is convinced that it is economically advantageous.” (p.

321.) They develop various alternative statistical models involving price, volume, and

volatility, all inspired by the idea that investor/arbitrageurs would be watching the market

closely and reacting occasionally. Their tests, however, involve only a handful of

anecdotal events.

Much later, Chakrabarti and Roll (1999) formulate a model populated by Bayesian

traders/arbitrageurs who attempt, through observing the trading of others, to deduce the

2 Unlike us, Garbade and Lieber (1977) do not have access bid-ask quote mid-points and hence are unable to separate bid-ask bounce in transaction prices from true serial correlation.


quality of their information. Simulations of the model show that the market usually

converges more rapidly to an equilibrium price and that it is a better predictor of true

value when arbitrageurs react to one another as opposed to trading solely on their own

information.

Section II below describes the data. Section III presents our analysis of how quickly

prices of highly liquid stocks become efficient. Section IV concludes and suggests

further investigations.

II. The Data.

Since we already know that serial dependence in returns is close to zero for active stocks

over a daily horizon, our investigation of the efficiency-creating process must focus on

intra-day trading. We would like to measure the timing of efficiency creation as

precisely as possible, so it seems sensible to examine frequently-traded stocks for which

very short term serial dependence can actually be observed. This suggests that small

stocks should be excluded until further statistical developments make it possible to

measure serial dependence even when trading is infrequent.

Because transactions data are so voluminous, this initial study uses only a limited sample

of stocks and time. Our calculations here cover twenty large and twenty mid-cap stocks

listed on the New York Exchange for two recent years, 1996 and 1998. These years were

chosen because (a) transactions data are available from the TAQ (Trade and Automated

Quotations) database recorded by the Exchange, and (b) they bracket a significant change

in the minimum tick size, which was reduced from $1/8 to $1/16 during 1997. We hoped

to discern the impact, if any, of that event. Future investigations should extend the

investigation to smaller firms, and other years, exchanges, and countries.

The forty sample firms are listed in Table 1. The first twenty were the largest listed firms

at the beginning of 1996. Their market capitalizations at that time ranged from $120.3


billion (General Electric) to $38.8 billion (DuPont). The mid-cap group consists of firms

ranked 101 to 120 by market capitalization at the beginning of 1996. The market cap

range was $9.70 billion (Duke Power) to $8.44 billion (Chubb). By 1998, one of the

large firms and one mid-cap firm had disappeared in a merger and acquisition

respectively. Two mid-cap firms were substantially restructured and we thought it

prudent to drop them also from the 1998 sample. Hence, only 19 large firms and 17 mid-

cap firms were included in calculations for 1998.

The TAQ data base provides not only trade prices, but also bid and ask quotes associated

with each transaction. This allows us to use the Lee/Ready (1991) trade assignment

algorithm to estimate whether a particular trade was buyer- or seller-initiated.3 Order

imbalance for each stock over any time interval can then be calculated variously as the

number of buyer- less the number of seller-initiated trades (OIB#), the number of buyer-

initiated shares purchased less the number of seller-initiated shares sold (OIBSh), or the

dollars paid by buyer-initiators less the dollars received by seller-initiators (OIB$).

The first of these order imbalance measures disregards the size of the trade, counting

small orders equally with large orders. The second and third measures weight large

orders more heavily. The distinction is important here because we hope to shed light on

how arbitrageurs make the market more efficient over very short horizons and presume

that arbitrageurs tend to undertake larger trades as compared to naïve investors in order to

quickly exploit deviations of prices from fundamentals.

III. The Evidence.

III.A. Evidence of efficiency at a daily horizon.

Using CRSP returns data,4 we first set out to ascertain whether our sample of stocks

conformed to semistrong-form efficiency over a daily horizon; i.e., whether future returns

could be predicted by either past returns or past order imbalances. Table 2 documents the 3 The Lee/Ready algorithm classifies a trade is as buyer- (seller-) initiated if it is closer to the ask (bid) of the prevailing quote. The quote must be at least five seconds old. If the trade is exactly at the mid-point of the quote, a “tick test” is used whereby the trade is classified as buyer- (seller-) initiated if the last price change prior to the trade is positive (negative.)


daily return serial correlations and shows that the average first-order daily autocorrelation

coefficient for the largest 20 stocks during 1996 was 0.010; the t-statistic, 0.56, was

calculated from the cross-section of sample autocorrelation coefficients assuming

independence.5

This positive (though insignificant) coefficient is somewhat surprising because negative

first-order autocorrelation in trade-to-trade returns is known to be induced by the bid-ask

bounce. During 1998, the large stocks did exhibit such negative autocorrelation as did

the mid-cap stocks for 1996 (though the coefficient is insignificant.) There are two

possibilities to explain the evident weakness of the bid-ask bounce; first, for these

relatively liquid stocks, spreads might be too narrow to induce a pronounced bounce and

second, there is actually positive dependence in bid-ask bounce-free returns that is more

or less offset by the bounce, depending on the sample period.

To avoid contamination of return serial correlations by bid-ask bounce, we compute

returns from quote mid-points as well as from transaction prices. So, for each transaction

during every day, the quotes existing at least five seconds before the trade were used to

compute a bid-ask midpoint. Returns were then computed from these midpoints. For

example, the daily midpoint returns in Table 2 are computed from the bid and ask quotes

just prior to the last transaction of the day. The daily autocorrelations in these midpoint

returns are small and insignificant for both the large and the mid-cap stocks in both years.

Thus, it appears that the first explanation above about the weakness of the bid-ask bounce

is probably the correct one; bid-ask spreads are small, and there is no evidence of positive

serial dependence in true returns over a horizon of one day.

Table 2 also reports simple correlations between returns and the three measures of order

imbalance, both contemporaneous correlations and correlations with OIB lagged by one

day. As could be expected, there is a very strong positive contemporaneous correlation 4 From the Center for Research in Securities Prices (CRSP) of the University of Chicago. 5 It seems likely that the assumption of cross-sectional independence actually results in an overstatement of statistical significance because returns, and hence sample correlation coefficients, are mostly positively


between either measure of return (trade or midpoint) and any of the OIB measures. Not

surprising also, the share and dollar measures, OIBSh and OIB$, are considerably more

highly correlated with contemporaneous returns for the large firms.

The correlations between daily returns and lagged (by one day) order imbalances are

completely insignificant in all cases for the share and dollar imbalances. However,

lagged OIB# is significantly correlated with returns during 1996 though not during 1998.

The magnitude of the correlation is 0.06 or less, so the economic value of the implied

prediction would be relatively small. We are not sure whether this represents a statistical

aberration or something truly material such as a small improvement in market efficiency

perhaps brought about by the minimum tick size reduction.

Notice that the order imbalance measures themselves are strongly and positively

autocorrelated from day to day, a feature particularly striking for OIB# (which weights all

trades equally regardless of size). For the large stock group, its autocorrelation

coefficient exceeds 0.5 in both 1996 and 1998. In an earlier paper, Chordia, Roll, and

Subrahmanyam (2001) show that even aggregate market order imbalances persist for

several days.

III.B. Evidence about efficiency over short horizons with the trading day.

We computed short-horizon returns from prices closest to the end of various time

intervals within the trading day. For example, ten-minute returns are computed for each

stock by finding the transaction closest to 9:40 a.m., 9:50 a.m., etc.6. Since some

calculations involve lagged values, the first interval of each trading day is discarded

because it would have been correlated with a lagged interval from the previous trading

day.7 Throughout this sub-section, all the reported correlations were first computed

within the trading day for each stock, then averaged across all trading days and stocks.

correlated. This implies that the estimated standard error of the sample mean is too small since it omits the mostly positive covariance terms that would be in the true standard error. 6 During 1996 and 1998, New York Stock Exchange trading hours were 9:30 a.m. to 4:00 p.m. 7 Intervals of sixty minutes were set backward from the end of the trading day. For example, each day has five one-hour intervals (11-12, 12-1,…,3-4) included in the calculations; the interval from 10 to 11 a.m. provides lagged observations only and data from 9:30 to 10 a.m. are not used at all.


There is admittedly some sloppiness involved in computing very short-term returns

because trades do not necessarily occur at the exact ending time of each interval. If the

closest price to the end of an interval was more than 150 seconds away, either before or

after, the return for that interval was not used in our calculations. Within the large stock

sample, the average difference between the transaction time and the end of a five-minute

interval was 25 seconds. Over intervals longer than five minutes, this problem obviously

becomes progressively less material.

Order imbalances were computed over all trades within each time interval. For example,

contemporaneous OIB# during the ten-minutes ending at 9:50 a.m. consists of the

number of buyer-initiated trades less the number of seller-initiated trades between

9:40:01 a.m. and 9:50:00 a.m. The lagged ten-minute OIB# is the corresponding

accumulation between 9:30:01 a.m. and 9:40:00 a.m.

The contemporaneous correlation between trade-based returns and midpoint-based

returns is, as one would expect, quite large, positive, and significant. However, it is not

perfect, particularly for the very short time intervals. The correlation is only 0.622 over

five-minute intervals on average for large stocks during 1996.8 For the same group/year,

the correlation grows steadily as the interval lengthens; it is 0.749 at 10 minutes, 0.802 at

15 minutes, 0.868 at 30 minutes, and 0.882 at 60 minutes. During 1998, all these

correlations were considerably higher regardless of interval9 but the same pattern

prevailed; they increased with interval length. The mid-cap stocks displayed somewhat

lower correlations than the large stocks, undoubtedly because they do not trade as

frequently; again, however, the same pattern of increase with interval length is evident.

Table 3 reports intra-day autocorrelations for returns and order imbalances over horizons

ranging from five minutes to one hour. The microstructure issue of bid-ask bounce can

be easily discerned by comparing the sizes of autocorrelation coefficients from trade

8 These numbers are not reported in a Table. 9 A possible explanation for this is the reduced tick size in 1998, which could lead to a stronger correlation between prices and trading activity.


returns as opposed to midpoint returns. In every case, they are larger (more negative) for

trade returns. For instance, over five minutes intervals for large stocks in 1996 the

autocorrelation using trade returns is -.203 while it is only -.043 with midpoint returns.

The relative difference declines steadily as the interval lengthens, but some difference

remains even at 60 minutes. During 1998, the five-minute interval shows about the same

relative difference, -.094 versus -.026 and the same change for longer horizons. The

mid-cap sample conforms closely to the large stock sample in both years.

For both large and mid-cap stocks, autocorrelations fell in absolute magnitude from 1996

to 1998, the reduction being particularly prominent at the shorter intervals. Perhaps the

June 24, 1997 reduction in the minimum tick size reduced the cost of arbitrage and

increased its reaction speed. It seems likely that many highly liquid firms had quoted

spreads equal to the minimum tick size; consequently, they experienced a fifty percent

reduction in quoted spreads between 1996 and 1998. Ball and Chordia (2001) confirm

that the average quoted spread declined from 21.3 cents to 11.9 cents between February

and November 1997 in a sample of seven large firms.

As Table 3 shows, order imbalances are highly positively autocorrelated over a five-

minute interval. For example, OIB# has an autocorrelation coefficient of 0.126 (t-

statistic 32.0) for the large stock group in 1996. Share and dollar order imbalance

measures have autocorrelations only about half as large, but they remain highly

significant and positive. There is a similar pattern in 1998 for the large stocks and for the

mid-cap stocks in both years. We propose that the autocorrelation is higher for the OIB#

because it is more likely to pick up the actions of naïve traders (e.g., retail investors), who

might follow unsophisticated herding strategies.

By ten minutes, autocorrelation in order imbalances has been attenuated, but is still

significantly positive. The autocorrelation is negative at 60 minutes; (for OIBSH and

OIB$ this happens at 30 minutes.) For reasons to be discussed shortly, however, we do

not assert that these negative autocorrelations are truly significant despite their large

computed t-values.


III.C. Bias in estimating the autocorrelation coefficient.

It has been long known that the autocorrelation coefficient is downward biased, rather

severely so in small samples (Cf. Kendall, 1954, and Marriott and Pope 1954).10 The

number of observations per day decreases sharply with interval length. While there are

78 five-minute intervals each day, there are only six sixty-minute intervals.

Bias is undoubtedly responsible for some of the systematic decline in all autocorrelation

coefficients in Table 3 as the interval grows. For example, the midpoint return

autocorrelation for large stocks in 1996 falls from -.043 at five minutes to -.148 at sixty

minutes. A similar pattern can be observed in autocorrelation coefficients for all the

variables; even the OIB measures, which are strongly positively autocorrelated over five-

minute intervals, become negatively autocorrelated at sixty-minute intervals. Those

negative long-interval autocorrelations are possibly spurious and the true autocorrelation

could even be positive.

To investigate this phenomenon, we decided to engage the bootstrap using a subsample,

the large stock group for 1996 and midpoint returns. For each of the twenty stocks and

for each trading interval, the bootstrap method resamples from the original returns in

random order (with replacement). Consequently, the true autocorrelation coefficient

should be approximately zero because resampled pairs of observations are almost

invariably far apart in true calendar time. The sample variance, however, remains the

same. Moreover, each individual stock’s estimated autocorrelation coefficient from the

resampled data should be almost completely independent of that computed from data for

any other stock.

The results are shown in Figure 1 for the twenty large stocks. The sampling distributions

of the bootstrapped autocorrelation coefficients are depicted by plotting the mean, the 5th

percentile, and the 95th percentile. The autocorrelation estimates from the original data

are also plotted. Notice that the negative bias is clearly apparent. While the bootstrap

10 Cross-correlation coefficients do not suffer from this problem.


mean is only about -.01 for five-minute intervals, it declines to around -.25 for sixty-

minute intervals.

Comparing the bootstrapped fractiles to our original point estimates, it seems apparent

that autocorrelations in midpoint returns really are significantly negative for five-, ten-,

and fifteen-minute intervals. In these cases, most of our point estimates fall well below

the bootstrap 5th percentile. However, the point estimates for the thirty-minute interval

are scattered within the extreme percentiles of the bootstrap distribution and more of

them are actually near the 95th percentile. At sixty minutes, virtually all are above the

95th percentile. This implies, of course, that the true autocorrelation at sixty minutes is

actually significantly positive even though the point estimate is negative; the same

conclusion, albeit with lesser confidence appears possible at thirty minutes as well.

We did not bootstrap the OIB variables or the trade returns, but the similarity in patterns

seems to indicate clearly that the same phenomenon is at work. In the case of the OIB

measures, they are likely positively autocorrelated at all intervals.

III.D. Conclusions about autocorrelations.

Our sample autocorrelation coefficients confound three distinct effects. First, there is the

true autocorrelation within a sample interval; second, there is the small sample negative

bias; and third, there is a positive bias induced by a shifting mean over the time interval

in which the autocorrelation is measured. Since we computed the autocorrelation

coefficients in Table 3 within each trading day, and then averaged them across trading

days, the sample mean return for each trading day served as the implicit conditional

expected return for the autocorrelation computed on that day. This conditional (sample)

mean is, of course, highly variable across time. Consequently, if time variation in the

conditional mean is large enough, it could mask intra-day negative serial dependence.

The results for midpoint returns confirm that negative serial dependence is not just a

spurious microstructure phenomenon, at least for very short intervals of five and ten

minutes. By sixty minutes, however, after correcting for small sample bias, midpoint


autocorrelation becomes positive. This seems likely to be caused by a weakening of the

true negative autocorrelation as the interval lengthens; in addition, over longer intervals,

the shifting mean effect becomes dominant.

The striking negative autocorrelation at very short intervals and its weakening over

longer intervals is consistent with (1) specialists temporarily changing price quotes away

from fundamentals in order to manage their inventory, and (2) arbitrageurs engaging in

countervailing trades after they have witnessed short-term price moves. Both actions

could, of course, be taking place. This seems all the more likely in that order imbalances

are very strongly positively autocorrelated. If arbitrageurs were not taking offsetting

actions, positive serial dependence in order imbalances would induce the same thing in

returns.

III.E. Multiple regressions.

Our explanation of how the market converges to weak-form efficiency has been

supported to this point by an examination of simple autocorrelation coefficients. The

stylized facts are these: (1) very short term returns are negatively autocorrelated;11 (2) As

the return interval lengthens, from five minutes up to sixty minutes, the negative

correlation disappears;12 (3) order imbalances are strongly positively autocorrelated.

We have interpreted these results to reveal the actions of three distinct groups. Order

imbalances in the first instance arise from traders who believe themselves to be in the

possession of pertinent information. Order imbalances are positively autocorrelated,

which suggests that naïve traders are jumping on the bandwagon or spreading their orders

out over time (or both). Second, NYSE specialists react to initial order imbalances by

altering quotes away from fundamental value in an effort to control inventory. Finally,

astute traders intervene with countervailing trades in the direction opposite to the initial

11 This is not merely a bid/ask bounce effect because midpoint returns display the same phenomenon, though the magnitude is smaller than for trade returns. 12 The sample autocorrelation remains negative as the interval lengthens, but the bootstrap results reveal that the small sample bias is so severe that the correct inference (for sixty minutes) is that the true autocorrelation is significantly positive.


order imbalances. Arbitrage takes at least some time, which explains why the

autocorrelation in returns changes sign as the time interval grows.

To help elucidate this interpretation, Table 4 presents a series of multiple regressions with

the same variables.13 In all regressions, the dependent variable is the midpoint return14

for an individual stock while explanatory variables include the lagged midpoint return

and contemporaneous and lagged measures of order imbalance for that stock.15 Since all

intra-day observations for an entire year are used in the same regression, there is no small

sample bias of the sort that affected the autocorrelations in the previous subsection.

Individual regressions are condensed by averaging the coefficients.. Two t-statistic

estimates are also provided. The first is calculated from the cross-sectional array of

estimated individual coefficients, assuming independence. The second is simply the

average individual coefficient’s t-statistic. For a given intra-day return interval, all

returns over the entire year are included in the regression except for the first interval

return on each trading day. (It appears only as a lagged value.)

The table reports two different regressions for each return interval. All include the

lagged return as a regressor. The two regressions differ by the measure of order

imbalance employed, OIB# for the number of trades and OIB$ for the dollar amount

traded.

Focusing first on large stocks in 1996, the lagged returns have significant negative

coefficients in all regressions for five-minute intervals and, confirming our earlier

findings, they become mostly positive or insignificant at the longer intervals.

13 To conserve space, we consider only the OIB# and OIB$ measures of order imbalance. The OIBSh measure yields results similar to those for OIB$. 14 Similar regressions were also estimated for trade returns but are not reported in the interest of brevity. The main difference involves the bid-ask bounce, which impacts the trade returns and is absent from the midpoint returns. This results in the coefficients for lagged trade returns being algebraically smaller and, for the shorter intervals, more significantly negative, than the corresponding coefficients for lagged midpoint returns. 15 While there is clear multicollinearity induced by the inclusion of both contemporaneous and lagged imbalance, this should attenuate standard errors and reduce significance. Thus, multicollinearity does not detract from the significant coefficients on which we focus.


Contemporaneous order imbalances, whatever the measure or interval, are positive and

highly significant.

In all of the regressions, the coefficient for OIB$t-1 is larger and more significant than the

coefficient for OIB#t-1 in explaining the return at time t. OIB#t-1 has a positive, though

insignificant coefficient at five minutes. By ten minutes, it becomes negative and

significant while OIB$t-1 remains positive. This pattern persists out to sixty minutes but

OIB$t-1 is insignificant beyond fifteen minutes. Both coefficients decline monotonically

as the trading interval lengthens from five to sixty minutes. Notice that the

contemporaneous OIB coefficients do not decline very much with interval length; indeed,

in the case of OIB$, they do not decline at all.

This pattern is consistent with the traces of two types of investors. Smaller traders,

whose actions are weighted equally in the OIB# measure, are presumably more likely to

be “naïve.” Their order imbalances tend to be offset by arbitrageurs and/or specialists.

This takes at least ten minutes. The relative sizes of coefficients for OIB#t and OIB#t-1

give a proximate indication of the naïve trades that are offset. At ten minutes, the initial

price impact is offset by about eleven percent (-0.943/8.94) while at 15, 30, and 60

minutes it is offset by roughly 16%, 24%, and 33%, respectively.

Turning to OIB$, which presumably weights more astute traders more heavily, we find

that its lagged coefficients are significantly positive for five, ten, and fifteen minutes (for

large stocks in 1996). They fall to insignificance at thirty minutes but remain positive.

This pattern in 1996 indicates that traders were responding on average to larger orders by

jumping on the bandwagon, placing additional orders in the same direction, rather than

conducting countervailing trades as they appeared to be doing after smaller orders. This

happened rapidly; notice the relative sizes of the contemporaneous and lagged

coefficients for OIB$. At five minutes, the lagged coefficient is about 18 percent as large

as the contemporaneous coefficient. The percentage drops to 10%, 7%, 2%, and 0.1% as

the interval lengthens from 10 to 60 minutes.


In 1998 for large stocks, there is a similar algebraic decline for OIB#t-1 as the return

interval lengthens. In contrast to 1996, however, it is negative even at five minutes. This

seems to suggest that arbitrageurs were intervening more quickly with countervailing

trades in 1998. Moreover, the coefficients for OIB$t-1 show no bandwagon effect in

1998. They too are negative after five minutes. However, the coefficients for OIB$t-1 are

much smaller (in absolute value) than the coefficients for OIB#t-1, and they also represent

smaller percentages of their corresponding contemporaneous coefficient. For instance, at

sixty minutes, about 36% of OIB# is reversed (-2.87/7.89) while only about 10% of OIB$

is reversed (-.455/4.43). Evidently, larger orders contain more accurate information and

thus offer no genuine arbitrage opportunities.

The pattern of coefficients for mid-cap stocks is similar in many respects. For example,

the coefficient of the contemporaneous order imbalance is always positive and highly

significant, regardless of the return interval or the OIB measure employed. The

magnitudes of these contemporaneous coefficients are considerably larger than for large

cap stocks, perhaps revealing that order imbalances of a given size have a greater impact

on mid-cap stocks, presumably because inventory and asymmetric information concerns

are more important in stocks that trade relatively less frequently.

There are some differences between the mid-cap and large patterns in the other

coefficients. Notice, for example, that the coefficient of the lagged return remains

significantly negative in some cases out to thirty minutes; this is a longer delay than for

large stocks. The coefficient for lagged OIB#t-1 does not become negative until thirty

(fifteen) minutes in 1996 (1998). This also is a delay relative to large cap stocks, where

the corresponding coefficient was negative at ten minutes in 1996 and five minutes in

1998. Evidently, countervailing arbitrage trading takes a bit longer for mid-cap than for

the largest stocks.

There is also a small contrast between mid-cap and large stocks in the pattern of

coefficients for OIB$t-1. The coefficient declines as the return interval lengthens but is


larger at all intervals for mid-cap stocks and is negative only after thirty minutes in 1998.

This too is consistent with a slower pace of arbitrage activity.

IV. Conclusions The long and continuing debate about financial market efficiency has been relatively

silent about the behavior of actual traders. Somehow, perhaps unwittingly, they act

collectively to push markets toward efficiency. Except in an idealized theoretical world,

this cannot happen instantaneously. There must be some time interval, albeit very short,

over which the actions of efficiency-creating traders remain incomplete. A central goal

of this paper is to present evidence about this important issue, the speed of convergence

to market efficiency.

For convenience, we study weak-form efficiency (Fama, 1970), which is concerned only

with serial dependence in returns. Of course, even weak-form efficiency cannot be

attained immediately. Using a sample of intra-day returns for large and mid-cap stocks

during calendar years 1996 and 1998, we find that weak-form efficiency does appear to

prevail over intervals of a day or longer. There is evidence, however, that some traders

cause serial dependence in prices over short intervals of a few minutes. But there is also

strong evidence that other traders become aware of price-moving order imbalances and

undertake countervailing trades.

To obtain these results, we circumvent the bid-ask bounce by using returns computed

from bid-ask quote midpoints. Yet like trade returns, midpoint returns also are negatively

serially correlated over intervals up to ten minutes for large stocks and over somewhat

longer intervals for mid-cap stocks.16 (Order imbalances themselves are highly positively

dependent over short intervals.) We argue that this is consistent with NYSE specialists

altering quotes away from fundamentals for the purpose of inventory control, while

16 Because of the bid-ask bounce, the negative dependence in trade returns is larger in absolute magnitude.


awaiting countervailing trades. By thirty to sixty minutes, depending on firm size, there

is no remaining serial dependence in returns.

Multiple regressions of midpoint returns on lagged midpoint returns plus

contemporaneous and lagged order imbalances are consistent with the gist of this story.

Order imbalances measured in number of trades are reversed as the return interval

lengthens, evidently because sophisticated investors undertake countervailing actions.

Order imbalances measured in dollars, which reflect larger orders, are not reversed as

soon, though they are attenuated to some extent with time.

There is suggestive evidence that that arbitrage activity became more effective between

1996 and 1998, perhaps as a result of the reduction in the minimum tick size from $1/8 to

$1/16 during 1997.

These results make one wonder about the existence of market anomalies and

inefficiencies in general. If there is no significant evidence of weak-form inefficiency at

intervals of thirty minutes, it is hard to understand how the market could be inefficient at

horizons of six to twelve months as in the extensive literature on much longer-term

anomalies.17 Investigation of this apparent conundrum could be a worthwhile area for

future research.

17E.g., the momentum (Jegadeesh and Titman, 1993) effect. Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998, 2001), and Hong and Stein (1999) attempt to explain momentum and other inefficiencies using models with irrational investors.


References

Ball, Clifford A., and Tarun Chordia, 2001, True spreads and equilibrium prices, Journal of Finance, forthcoming. Barber, Brad M., and Terence Odean, 2000, Trading is hazardous to your health: The common stock investment performance of individual investors, Journal of Finance 55, 2 (April), 773-806. Barberis, Nicholas, Andrei Shleifer, and Robert W. Vishny, 1998, A model of investor sentiment, Journal of Financial Economics 49, 3 (1998), 307-343 Benartzi, Shlomo, and Richard Thaler, 2001, Naive diversification strategies in retirement saving plans, American Economic Review 91, 1 (March), 79-98. Chakrabarti, Rajesh, and Richard Roll, 1999, Learning from others, reacting, and market quality, Journal of Financial Markets, 2, 2 (May), 153-178. Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2001, Order Imbalance, Liquidity and Market Returns, Journal of Financial Economics, forthcoming. Copeland, Thomas E., 1976, A model of asset trading under the assumption of sequential information arrival, Journal of Finance 31, 4 (September), 1149-1168. Cornell, Bradford and Richard Roll, 1981, Strategies for pairwise competitions in markets and organizations, Bell Journal of Economics, 12, 1 (Spring), 201-213. Cramér, Harald, 1954, Mathematical Methods of Statistics, (Princeton: Princeton University Press). Daniel, Kent, David Hirshleifer, and Avanidhar Subrahmanyam, 1998, Investor psychology and security market under- and overreactions, Journal of Finance 53, 6 (December), 1839-1885. Daniel, Kent, David Hirshleifer, and Avanidhar Subrahmanyam, 2001, Overconfidence, arbitrage, and equilibrium asset pricing, Journal of Finance 56, 3 (June), 921-965. Epps, Thomas W., 1979, Comovements in stock prices in the very short run, Journal of the American Statistical Association 74, 2 (June), 291-298. Fama, Eugene F., 1970, Efficient capital markets: A review of theory and empirical work, Journal of Finance 25, 2 (May), 383-417.


Fama, Eugene F, and French, Kenneth R., 1992, The cross-section of expected stock returns, Journal of Finance 47, 2 (1992), 427-465. Garbade, Kenneth D., and Zvi Lieber, 1977, On the independence of transactions on the New York Stock Exchange, Journal of Banking and Finance 1, 2 (October), 151-172. Grossman, Sanford J., 1976, On the efficiency of competitive stock-markets where traders have diverse information, Journal of Finance 31, 2 (May), 573-585. Grossman, Sanford J. and Joseph E. Stiglitz, 1980, On the impossibility of informationally efficient markets, American Economic Review 70, 3 (June), 393-408. Hillmer, S.C., and P. L. Yu, 1979, The market speed of adjustment to new information, Journal of Financial Economics 7, 4 (December), 321-345. Hong, Harrison, and Jeremy C. Stein, 1999, A unified theory of underreaction, momentum trading, and overreaction in assets markets, Journal of Finance 54, 6 (December), 2143-2184. Jegadeesh, Narasimhan, and Sheridan Titman, Returns to buying winners and selling losers: Implications for stock market efficiency, Journal of Finance 48, 1 (March), 65-91. Kendall, Maurice G., 1954, Note on bias in the estimation of autocorrelation, Biometrika 41, 3/4 (December), 403-404. Lee, Charles, and M. Ready, 1991, Inferring trade direction from intra-day data, Journal of Finance 46, 733-747. Marriott, F. H. C., and J. A. Pope, 1954, Bias in the estimation of autocorrelations, Biometrika, 41, 3/4 (December), 390-402. Odean, Terrance, 1999, Do investors trade too much? American Economic Review 89, 1279-1298. Patell, James M., and Mark A. Wolfson, 1984, The intra-day speed of adjustment of stock prices to earnings and dividend announcements, Journal of Financial Economics 13, 2 (June), 223-252. Schwert, G. William, 2001, Anomalies and market efficiency, Chapter 17 in George Constantinides, Milton Harris, and René Stulz, eds., Handbook of the Economics of Finance, (North-Holland.)


Table 1

Firms in the Sample

Large Mid-Cap American International Group Alcoa

AT&T Archer Daniels Midland Bell South Chubb18

Bristol Myers Squib CSX Coca Cola Deere

DuPont Digital Equipment19 Exxon Duke Power

General Electric Enron General Motors First Union

GTE FPL group Hewlett Packard Gannett

IBM General Mills Johnson and Johnson Keycorp

Merck Loews Mobil20 Merrill Lynch Pepsi Phillips Petroleum Pfizer PPG industries

Philip Morris Texas Utilities Proctor and Gamble US West21

Walmart Weyerhaeuser

18 Chubb underwent a major restructuring in 1997. Because it became a substantially different firm, Chubb was not included in the 1998 sample. 19 Acquired by Compaq, not in 1998 sample. 20 Merged with Exxon, not in 1998 sample. 21 US West became Qwest after expanding into the cable television business. It was not included in the 1998 sample.


Table 2

Correlation Coefficients at a Daily Horizon for Returns and Order Imbalances

For stocks listed in Table 1, trade returns are computed from the last transaction price of each day and midpoint returns are computed from the average of the bid-ask quotes associated with the last transaction of each day. Trade returns are from CRSP. Bid-Ask quotes and order imbalances (OIB) are from the NYSE TAQ data base. OIB# is the number of buyer-initiated less the number of seller-initiated trades during the same day as the return; OIBSh is the number of buyer-initiated shares purchased less the number of seller-initiated shares sold that day; OIB$ is the total dollars paid by buyer-initiators less the total dollars received by seller-initiators that day. The product-moment correlation coefficient is reported along with a t-statistic computed from the cross-sectional distribution of correlation coefficients.

Trade Returnt

Midpoint Returnt

OIB#t OIBSht OIB$t Trade

Returnt

Midpoint Returnt

OIB#t OIBSht OIB$t

1996 1998 Large Stocks

Returnt-122 0.010

(0.56) -0.002 (-0.20) -0.051

(-2.33) -0.033 (-1.63)

OIB#t 0.208 (6.34)

0.116 (2.17) 0.132

(3.77) 0.101 (2.58)

OIB#t-1 0.061 (3.68)

0.050 (2.96)

0.505 (11.5) -0.005

(-0.26) -0.008 (-0.51)

0.515 (16.1)

OIBSht 0.551 (31.6)

0.473 (11.3)

0.217 (6.02) 0.540

(28.4) 0.502 (18.1)

0.163 (3.11)

OIBSht-1 0.003 (0.19)

-0.008 (-0.57)

-0.112 (-3.08)

0.153 (6.77) -0.031

(-1.42) -0.013 (-0.96)

-0.115 (-2.66)

0.196 (5.74)

OIB$t 0.549 (28.0)

0.476 (12.1)

0.197 (5.36)

0.986 (215.) 0.540

(31.5) 0.500 (18.5)

0.160 (3.13)

0.990 (582.)

OIB$t-1 0.000 (0.01)

-0.010 (-0.68)

-0.126 (-3.80)

0.148 (6.74)

0.155 (6.67)

-0.035 (-1.56)

-0.016 (-1.24)

-0.114 (-2.74)

0.187 (5.70)

0.192 (6.83)

Mid-Cap Stocks

Returnt-122 -0.023

(-1.35) -0.009 (-0.69) 0.012

(0.57) 0.013 (0.63)

OIB#t 0.360 (12.2)

0.330 (10.1) 0.324

(11.9) 0.295 (9.10)

OIB#t-1 0.046 (2.93)

0.053 (2.94)

0.236 (5.87) 0.015

(0.85) 0.022 (1.12)

0.349 (9.85)

OIBSht 0.387 (12.6)

0.362 (10.7)

0.296 (11.3) 0.369

(13.8) 0.348 (11.3)

0.349 (10.3)

OIBSht-1 -0.015 (-1.02)

-0.015 (-0.97)

-0.048 (-2.35)

0.121 (6.06) -0.005

(-0.32) -0.001 (-0.06)

0.054 (1.63)

0.129 (3.97)

OIB$t 0.391 (13.1)

0.365 (11.0)

0.294 (10.8)

0.994 (438.) 0.368

(13.7) 0.346 (11.1)

0.343 (11.6)

0.989 (263.)

OIB$t-1 -0.015 (-1.08)

-0.014 (-0.94)

-0.050 (-2.36)

0.120 (6.05)

0.121 (6.04)

-0.006 (-0.36)

-0.002 (-0.11)

0.047 (1.72)

0.124 (4.05)

0.126 (4.05)

22 Trade (Midpoint) Returnt-1 in the Trade Returnt (Midpoint Returnt) column.


Table 3

Autocorrelation Coefficients at Intra-Day Horizons for Returns and Order Imbalances

Daily returns and order imbalances are obtained from the NYSE TAQ data base for stocks listed in Table 1. The return is computed from the actual trade price (Trade Return) or from the midpoint of the bid-ask spread (Midpoint Return) associated with the transaction nearest the end of an intra-day time interval of fixed length. The first interval of each day is excluded. OIB# is the number of buyer-initiated less the number of seller-initiated trades during the same time interval as the return; OIBSh is the number of buyer-initiated shares purchased less the number of seller-initiated shares sold during that interval; OIB$ is the total dollar amount expended by buyer-initiators less the total dollar amount received by seller-initiators during that interval. The product-moment autocorrelation coefficient is reported along with a t-statistic computed from the cross-sectional distribution of correlation coefficients.

Trade Return

Midpoint Return

OIB# OIBSh OIB$ Trade Return

Midpoint Return

OIB# OIBSh OIB$

1996 1998 Time Interval

(Minutes) Large Stocks

Five -0.203 (-11.9)

-0.043 (-6.33)

0.126 (32.0)

0.060 (26.5)

0.060 (26.5)

-0.094 (-12.6)

-0.026 (-3.36)

0.089 (19.2)

0.064 (18.8)

0.065 (18.7)

Ten -0.173 (-13.8)

-0.079 (-16.5)

0.080 (11.6)

0.025 (9.43)

0.025 (9.46)

-0.066 (-7.75)

-0.029 (-4.57)

0.070 (16.7)

0.059 (14.8)

0.059 (14.8)

Thirty -0.111 (-11.2)

-0.066 (-8.54)

0.044 (4.28)

-0.024(-5.07)

-0.024 (-5.04)

-0.079 (-9.12)

-0.063 (-8.24)

0.016 (2.63)

0.004 (0.55)

0.004 (0.56)

Sixty (from 10 am)

-0.171 (-19.6)

-0.148 (-16.5)

-0.099 (-18.9)

-0.137 (-27.9)

-0.137 (-27.6)

-0.172 (-19.3)

-0.163 (-19.1)

-0.101 (-10.2)

-0.124 (-14.7)

-0.124 (-14.6)

Mid-Cap Stocks Five -0.248

(-11.6) -0.052 (-4.86)

0.161 (25.6)

0.057 (18.5)

0.057 (18.5)

-0.104 (-5.58)

-0.010 (-0.98)

0.101 (27.4)

0.037 (9.13)

0.037 (9.12)

Ten -0.206 (-10.2)

-0.074 (-7.01)

0.103 (17.9)

0.033 (8.89)

0.033 (8.88)

-0.078 (-5.04)

-0.022 (-2.19)

0.071 (14.8)

0.021 (4.28)

0.022 (4.31)

Thirty -0.167 (-11.8)

-0.096 (-12.6)

0.011 (1.53)

-0.043 (-9.20)

-0.043 (-9.20)

-0.107 (-11.6)

-0.084 (-15.1)

-0.012 (-1.96)

-0.046 (-6.93)

-0.045 (-6.88)

Sixty (from 10 am)

-0.218 (-21.8)

-0.184 (-27.4)

-0.130 (-16.1)

-0.164 (-29.8)

-0.164 (-29.8)

-0.192 (-20.2)

-0.180 (-22.4)

-0.143 (-29.3)

-0.147 (-15.5)

-0.146 (-15.5)


Table 4

Multiple Regressions of Returns on Lagged Returns and Two Different Measures of Contemporaneous and Lagged Order Imbalance

for Return Intervals from Five to Sixty Minutes

Daily returns and order imbalances are obtained from the NYSE TAQ data base for the twenty large and twenty mid-cap stocks listed in Table 1. The return is computed from the midpoint of the bid-ask spread associated with the transaction nearest the end of an intra-day time interval of fixed length. OIB# is the number of buyer-initiated less the number of seller-initiated trades during the same time interval as the return. OIB$ is the total dollar amount expended by buyer-initiators less the total dollar amount received by seller-initiators during that interval. The first interval of each day is excluded and all other interval observations during each calendar year, (either 1996 or 1998), are included in the same regression. A separate regression is estimated for each individual stock. The first number in each cell is the cross-sectional mean of the estimated regression coefficient. The second number (the first number in parentheses) is a t-statistic computed from the cross-sectional distribution of the estimated coefficients assuming independence. The third number (also in parentheses) is the average t-statistic from the individual regressions. The R2 is the cross-sectional average adjusted R-square in percent. To adjust the units for presentation, the coefficients for OIB# have been multiplied by 105 and the coefficients for OIB$ have been multiplied by 1010.


Table 4 (continued)

Return Interval (minutes) Explanatory

Variable Five Ten Fifteen Thirty Sixty Dependent Variable is the Midpoint Returnt, Large Stocks, 1996

Midpoint Returnt-1

-0.038 (-5.42) (-3.55)

-0.066 (-7.40) (-6.52)

-0.050 (-5.55) (-3.29)

-0.095 (-15.8) (-6.44)

-0.025 (-2.44) (-1.31)

-0.083 (-9.89) (-4.51)

0.037 (3.33) (1.52)

-0.028 (-2.27) (-0.95)

0.090 (5.99) (2.26)

-0.009 (-0.80) (-0.12)

OIB#t 8.78

(7.10) (37.5)

8.94

(6.83) (30.1)

8.54

(6.81) (25.1)

7.53

(6.28) (16.9)

6.77

(6.10) (10.8)

OIB#t-1 3.51

(1.38) (0.30)

-0.943 (-3.58) (-3.89)

-1.40

(-5.24) (-4.93)

-1.79

(-5.51) (-4.21)

-2.23

(-6.05) (-3.71)

OIB$t 4.60

(20.1) (46.3)

4.96

(11.6) (38.1)

4.99

(11.2) (32.7)

4.91

(10.7) (24.9)

4.95

(10.7) (17.0)

OIB$t-1 0.829 (5.45) (6.48)

0.481 (4.44) (2.67)

0.349 (3.35) (1.52)

0.092 (0.97) (-0.11)

0.007 (0.05) (-0.31)

R2 15.8 19.7 19.1 24.8 19.7 26.8 19.0 30.3 18.6 32.8 Dependent Variable is the Midpoint Returnt, Large Stocks, 1998

Midpoint Returnt-1

-0.018 (-2.53) (-1.68)

-0.044 (-5.28) (-4.36)

0.00 (0.01) (0.11)

-0.043 (-5.75) (-3.11)

0.007 (0.52) (0.52)

-0.045 (-4.00) (-2.68)

0.055 (4.79) (2.37)

-0.023 (-2.55) (-0.32)

0.106 (8.83) (2.61)

-0.014 (-1.06) (-0.39)

OIB#t 11.0

(10.5) (50.9)

10.36 (9.87) (37.0)

10.01 (9.62) (30.8)

9.05

(9.85) (21.4)

7.89

(8.93) (13.2)

OIB#t-1 -0.864 (-5.30) (-4.71)

-1.47

(-7.01) (-5.75)

-1.75

(-7.70) (-5.90)

-2.12

(-9.26) (-5.57)

-2.87

(-10.2) (-4.77)

OIB$t 4.56

(16.9) (48.8)

4.65

(17.4) (37.8)

4.69

(17.2) (32.9)

4.59

(17.8) (24.7)

4.43

(17.4) (17.0)

OIB$t-1 0.023 (0.44) (0.12)

-0.167 (-2.59) (-1.22)

-0.221 (-3.15) (-1.46)

-0.423 (-4.97) (-2.10)

-0.455 (-4.30) (-1.73)

R2 21.7 19.9 22.9 22.9 23.8 25.5 24.0 28.5 22.4 30.8


Table 4 (continued)

Return Interval (minutes) Explanatory Variable Five Ten Fifteen Thirty Sixty

Dependent Variable is the Midpoint Returnt, Mid-Cap Stocks, 1996

Midpoint Returnt-1

-0.052 (-2.28) (-5.66)

-0.009 (-0.30) (-3.62)

-0.069 (-5.17) (-4.03)

-0.058 (-3.62) (-3.88)

-0.056 (-3.48) (-2.44)

-0.045 (-3.01) (-2.69)

-0.003 (-0.22) (-0.25)

-0.028 (-1.84) (-0.85)

0.024 (0.97) (0.63)

-0.042 (-1.66) (-0.59)

OIB#t 19.8

(10.4) (24.8)

24.2

(8.33) (23.2)

23.6

(8.77) (19.8)

22.02 (9.26) (14.4)

20.2

(8.37) (9.70)

OIB#t-1 6.42

(5.57) (7.94)

1.59

(1.69) (1.83)

1.52

(1.88) (0.47)

-1.32

(-2.36) (-0.78)

-2.87

(-3.46) (-1.66)

OIB$t 6.85

(10.2) (20.2)

8.15

(10.5) (17.8)

8.72

(10.91) (15.92)

9.77

(10.79) (12.67)

10.4

(7.95) (8.88)

OIB$t-1 2.43

(5.45) (6.86)

1.41

(2.77) (3.21)

1.17

(4.79) (2.01)

0.588 (2.15) (0.82)

0.716 (1.35) (0.32)

R2 17.5 9.88 23.2 11.6 24.7 13.3 26.8 18.1 28.0 21.0 Dependent Variable is the Midpoint Returnt, Mid-Cap Stocks, 1998

Midpoint Returnt-1

-0.033 (-3.54) (-3.63)

0.001 (0.09) (0.55)

-0.017 (-1.63) (-1.39)

-0.012 (-1.30) (-1.14)

-0.022 (-2.33) (-1.38)

-0.034 (-4.81) (-2.05)

0.009 (0.84) (0.52)

-0.035 (-2.62) (-1.22)

0.061 (3.56) (1.68)

-0.017 (-0.93) (-0.35)

OIB#t 23.8

(12.6) (46.2)

25.6

(11.7) (37.7)

25.90 (10.7) (32.3)

23.9

(10.3) (22.9)

21.67 (9.97) (14.9)

OIB#t-1 5.18

(5.14) (7.40)

0.225 (0.37) (-0.59)

-1.82

(-3.54) (-2.49)

-3.91

(-9.65) (-3.67)

-6.23

(-9.18) (-3.95)

OIB$t 6.28

(6.58) (25.6)

7.08

(6.61) (20.8)

7.61

(6.39) (18.2)

8.01

(6.18) (13.8)

8.42

(6.44) (10.2)

OIB$t-1 1.51

(4.93) (5.06)

0.690 (3.38) (1.68)

0.135 (1.17) (0.38)

-0.038 (-0.27) (-0.05)

-0.509 (-1.40) (-0.66)

R2 19.4 6.68 22.9 8.03 24.6 9.06 25.3 10.7 25.4 13.6

Figure 1. First-Order Mid-Point Return Autocorrelation Coefficients and Bootstrap Bands

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

GEAT&T

EXXONCOKE

MRK

PHILLIP MORRIS

P&G JNJ

WALMART

IBM

MOBILPEPSI

AMERICAN IN

TL GROUP

BRISTOL MYERS

BELLSOUTH

HEWLETT PACKARD

GTE

PFEIZER

GM

DUPONT

Aut

ocor

rela

tion

5% Mean 95% Estimate

Five-Minute Intervals

Ten-Minute Intervals

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

GEAT&T

EXXONCOKE

MRK

PHILLIP M

ORRISP&G JN

J

WALMART

IBM

MOBILPEPSI

AMERICAN IN

TL GROUP

BRISTOL M

YERS

BELLSOUTH

HEWLETT PACKARD

GTE

PFEIZER

GM

DUPONT

Aut

ocor

rela

tion

Fifteen-Minute Intervals

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

GEAT&T

EXXONCOKE

MRK

PHILLIP M

ORRISP&G JN

J

WALMART

IBM

MOBILPEPSI

AMERICAN IN

TL GROUP

BRISTOL M

YERS

BELLSOUTH

HEWLETT PACKARD

GTE

PFEIZER

GM

DUPONT

Aut

ocor

rela

tion

Thirty-Minute Intervals

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

GEAT&T

EXXONCOKE

MRK

PHILLIP M

ORRISP&G JN

J

WALMART

IBM

MOBILPEPSI

AMERICAN IN

TL GROUP

BRISTOL M

YERS

BELLSOUTH

HEWLETT PACKARD

GTE

PFEIZER

GM

DUPONT

Aut

ocor

rela

tion

Sixty-Minute Intervals

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

GEAT&T

EXXONCOKE

MRK

PHILLIP M

ORRISP&G JN

J

WALMART

IBM

MOBILPEPSI

AMERICAN IN

TL GROUP

BRISTOL M

YERS

BELLSOUTH

HEWLETT PACKARD

GTE

PFEIZER

GM

DUPONT

Aut

ocor

rela

tion

28

evidence on the speed of convergence to market efficiency by

Documents