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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
Evidence on the Speed of Convergence to Market Efficiency
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.
Convergence to Efficiency, September 25, 2001 2
Evidence on the Speed of Convergence to Market Efficiency
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
Convergence to Efficiency, September 25, 2001 3
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).
Convergence to Efficiency, September 25, 2001 4
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.
Convergence to Efficiency, September 25, 2001 5
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.
Convergence to Efficiency, September 25, 2001 6
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
Convergence to Efficiency, September 25, 2001 7
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.)
Convergence to Efficiency, September 25, 2001 8
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
Convergence to Efficiency, September 25, 2001 9
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.
Convergence to Efficiency, September 25, 2001 10
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.
Convergence to Efficiency, September 25, 2001 11
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.
Convergence to Efficiency, September 25, 2001 12
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.
Convergence to Efficiency, September 25, 2001 13
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
Convergence to Efficiency, September 25, 2001 14
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.
Convergence to Efficiency, September 25, 2001 15
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$. 14 Similar regressions were also estimated for trade returns but are not reported in the interest of brevity. The main difference involves the bid-ask bounce, which impacts the trade returns and is absent from the midpoint returns. This results in the coefficients for lagged trade returns being algebraically smaller and, for the shorter intervals, more significantly negative, than the corresponding coefficients for lagged midpoint returns. 15 While there is clear multicollinearity induced by the inclusion of both contemporaneous and lagged imbalance, this should attenuate standard errors and reduce significance. Thus, multicollinearity does not detract from the significant coefficients on which we focus.
Convergence to Efficiency, September 25, 2001 16
Contemporaneous order imbalances, whatever the measure or interval, are positive and
highly significant.
In all of the regressions, the coefficient for OIB$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$. At five minutes, the lagged coefficient is about 18 percent as large
as the contemporaneous coefficient. The percentage drops to 10%, 7%, 2%, and 0.1% as
the interval lengthens from 10 to 60 minutes.
Convergence to Efficiency, September 25, 2001 17
In 1998 for large stocks, there is a similar algebraic decline for OIB#t-1 as the return
interval lengthens. In contrast to 1996, however, it is negative even at five minutes. This
seems to suggest that arbitrageurs were intervening more quickly with countervailing
trades in 1998. Moreover, the coefficients for OIB$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
Convergence to Efficiency, September 25, 2001 18
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.
Convergence to Efficiency, September 25, 2001 19
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.
Convergence to Efficiency, September 25, 2001 20
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.
Convergence to Efficiency, September 25, 2001 21
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.)
Convergence to Efficiency, September 25, 2001 22
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.
Convergence to Efficiency, September 25, 2001 23
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.
Convergence to Efficiency, September 25, 2001 24
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)
Convergence to Efficiency, September 25, 2001 25
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.
Convergence to Efficiency, September 25, 2001 26
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
Convergence to Efficiency, September 25, 2001 27
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