What Drives Momentum and Reversal? Evidence from Day and Night Signals

Abstract

News mostly drive overnight returns, whereas investors' trading mostly drives intraday returns. We use this fact to test theories of momentum and reversal with a sample of intraday and overnight returns spanning 1926 to 2019. Portfolios formed on past intra-day returns display short-term reversal and momentum without long-term reversal. In contrast, portfolios formed on past overnight returns display only long-term reversal. These results are consistent with underreaction theories of momentum, where investors underreact to the information conveyed by the trades of other investors.

Full text

What Drives Momentum and Reversal?
Evidence from Day and Night Signals∗
Yashar H. Barardehi
Argyros School of Business & Economics
Chapman University
barardehi@chapman.edu
Vincent Bogousslavsky
Carroll School of Management
Boston College
vincent.bogousslavsky@bc.edu
Dmitriy Muravyev
Eli Broad College of Business
Michigan State University
muravyev@msu.edu
November 18, 2022
Abstract
News mostly drive overnight returns, whereas investors’ trading mostly drives intraday
returns. We use this fact to test theories of momentum and reversal with a sample of
intraday and overnight returns spanning 1926 to 2019. Portfolios formed on past intra-
day returns display short-term reversal and momentum without long-term reversal. In
contrast, portfolios formed on past overnight returns display only long-term reversal.
These results are consistent with underreaction theories of momentum, where investors
underreact to the information conveyed by the trades of other investors.
∗We thank Zhi Da, Zhongjin Lu, Steven Malliaris, Jeff Pontiff, David Solomon, and Mitch Warachka for
helpful comments. Muravyev is also Associate Fellow at the Canadian Derivatives Institute. Barardehi is
also at the U.S. Securities and Exchange Commission. The Securities and Exchange Commission disclaims
responsibility for any private publication or statement of any SEC employee or Commissioner. This article
expresses the authors’ views and does not necessarily reflect those of the Commission, the Commissioners,
or other members of the staff.

1 Introduction
Cross-sectional return predictability patterns based on past-return predictors are key styl-
ized facts in asset pricing. Stocks that outperform (underperform) other stocks over the past
twelve months tend to yield higher (lower) future return over the next month but under-
perform at longer horizons; i.e., momentum and long-term reversal (Jegadeesh and Titman
(1993); De Bondt and Thaler (1985)). In contrast, stocks that perform well (poorly) this
month, tend to underperform (outperform) next month; i.e., short-term reversal (Jegadeesh
(1990); Lehmann (1990)). These effects hold internationally, but their interpretation is chal-
lenging as evidenced by the numerous competing theories proposed to explain them.1
Theories of momentum differ on the role of private and public information. In the model
of Daniel, Hirshleifer, and Subrahmanyam (1998), investors’ reaction to public news plays
a key role in generating momentum as overconfident investors overweight confirming public
news after receiving a private signal. In the model of Hong and Stein (1999), investors’
private information gradually diffuses into prices because some investors do not learn from
prices. According to their model, traders’ response to public news differs from their response
to private news (Hong and Stein,1999, p.2166). Furthermore, a large literature documents
underreaction to various public news (e.g., Chan (2003)). To which extent does this form of
underreaction translate into momentum?
This paper examines the role of private and public information for theories of momen-
tum and reversal. This requires proxies for price moves induced by news and for price moves
induced by private information. Intuitively, public news should drive overnight returns to a
larger extent than intraday returns since intraday returns should primarily reflect the impact
of investors’ trading. Indeed, return volatility is higher intraday than overnight (e.g., Fama
(1965)). French and Roll (1986) show that this excess volatility is driven by private informa-
tion and mispricing and not by public information. More recently, Boudoukh et al. (2019)
1See Subrahmanyam (2018) for a recent review of momentum theories and the lack of consensus about
what drives momentum.
1

show that fundamental information from public news accounts for about 50% of idiosyn-
cratic overnight volatility but for only about 12% of intraday idiosyncratic volatility. Past
day and night returns can therefore help us test news-based and trading-based explanations
of momentum and reversal. To leverage this idea, we use a unique dataset of U.S. stocks’
intraday and overnight returns that spans almost a century: from 1926 to 2019.2
Figure 1shows our main results for the 1963-2019 sample. We discuss the pre-1963
evidence separately below. This figure plots monthly three-factor alpha of strategies that
invest in past winners and short past losers with portfolio formation horizons from one month
to five years. The grey bars show the well-known cross-sectional evidence of short-term
reversal, medium-term momentum, and long-term reversal. Figure 1’s main innovation is to
split the past-return signal into two components: past intraday return, and past overnight
return (orange and navy bars) and then predict regular (close-to-close) monthly returns.
That is, the holding period is standard—one month—but the anomaly signal is decomposed
into intraday and overnight parts.
Momentum strategies based on past intraday returns work well. Momentum strategies
based on past overnight returns fail to predict future returns.3Figure 1further shows that a
reversal strategy based on past-month intraday returns is substantially more profitable than
a reversal strategy based on past-month overnight returns. However, this pattern is reversed
for strategies based on past 3- or 5-year returns. There is long-term return reversal with
portfolios sorted on past overnight returns but not with portfolios sorted on past intraday
returns.
The results in Figure 1are robust. They are not driven by liquidity effects at the open
(e.g., Berkman et al. (2012); Bogousslavsky (2021)), stale overnight returns, or by the bid-
ask bounce between day and night returns. Indeed, the results hold if intraday and overnight
2We elaborate in Section 2on potential issues associated with our interpretation. For example, after-hours
trading contributes to the revelation of private information overnight (Barclay and Hendershott (2003)) but
was introduced only in 1992 (Barardehi et al. (2021)).
3As discussed below, Lou, Polk, and Skouras (2019) show that momentum returns are realized primarily
overnight over 1993-2013. We focus on variations in the strategy’s signal rather than on when the return is
realized.
2

Figure 1. Monthly alpha of momentum and reversal long-short strategies constructed
from past close-to-close, intraday, and overnight returns. This figure presents point esti-
mates and 95% confidence intervals for three-factor alphas. Stocks are allocated into decile port-
folios based on NYSE breakpoints of past returns. 7-month and 12-month strategies exclude the
most recent month. 36-month, 48-month, and 60-month strategies exclude the most recent year.
For a stock-month to be included in the sample, we require return observations over the preceding
36 months. The sample includes all U.S. common stocks over 1963 to 2019.
−1.5 −1 −.5 0 .5 1 1.5
Three−factor alpha (%)
1−month 7−month 12−month 36−month 48−month 60−month
Portfolio formation horizon
24−hour Intraday Overnight
returns are constructed from quote midpoints taken 15 minutes after the open and at the
close over 1985 to 2015. The results also hold over the 1926-1962 period, for which open
prices are available for a subset of NYSE-listed stocks. The results are robust to using return-
weighted portfolio returns (Asparouhova, Bessembinder, and Kalcheva (2013)) and excluding
the bottom 20% of stocks by market capitalization and days around earnings announcements.
The momentum results are robust to using value-weighted portfolio returns. For short-term
and long-term reversal strategies, value-weighting produces qualitatively similar patterns but
renders all the strategies statistically insignificant.
Figure 1provides novel insights about the sources of momentum and reversal. The
short-term reversal result supports theories of imperfect liquidity in which return reversal
compensates liquidity providers for absorbing inventory shocks (e.g., Campbell, Grossman,
and Wang (1993); Hendershott and Menkveld (2014)). Intraday returns are the outcome of
3

investors’ continuous trading and are therefore much more likely to be driven by this kind
of “price pressure” than overnight returns. This intuitive result supports our interpretation
of past day and night return signals.
News-based theories of momentum are hard to reconcile with the lack of profitability
associated with the overnight signal. If investors underreact to public news, which causes
momentum, then past overnight returns should predict future returns. For example, consider
news that would increase a stock price by 5% in a fully rational market. When the news
arrives overnight, investors underreact and the price increases by only 3%. The price will
later increase by 2% as more information arrives. In this scenario, past night returns predict
future returns, inconsistent with the data.
The results are more supportive of theories where investors underreact to the information
conveyed by the trades of other investors. Investors may be overconfident in the precision of
their own signals relative to the precision of other investors’ signals or a subset of investors
may not learn from prices at all (Eyster, Rabin, and Vayanos (2019)). Over time, prices
eventually incorporate this trade-based information, leading to momentum. The intraday
return signal predicts future returns because investors’ private information will mostly be
reflected in intraday returns rather than overnight returns (French and Roll (1986)). At
short horizons, liquidity effects dominate, leading to reversal.
If investors underreact to private information contained in trading, then momentum
should be mostly associated with low-volume days. Intuitively, information flows into prices
more quickly when volume is high. In addition, the disclosure of public information is
followed by high volume empirically. To test this explanation, we split each stock-month
into below-median and above-median volume days and then compute past-return signals
using either days with below-median volume or days with above-median volume. Hence,
we decompose past returns into high and low volume parts for a given stock.4Looking at
past 7 or 12 month return, the “low volume return” predicts future return more strongly
4This differs from Lee and Swaminathan (2000), who show that momentum is stronger among high volume
stocks.
4

than the “high volume return.” In contrast, one-month return reversal is stronger when
conditioned on high volume days, which further supports the price pressure interpretation:
the higher the volume, the higher the reversal (Campbell et al. (1993)). In summary, the
fact that momentum is stronger when the momentum signal is conditioned on low volume
days supports an underreaction channel.
Price overshooting due to delayed overreaction is therefore unlikely to explain past intra-
day returns’ predictive power. Long-term reversal based on past intraday returns is absent
in Figure 1. Past long-term overnight return predicts future return negatively, whereas past
long-term intraday return predicts future return positively. This result dissociates momen-
tum from long-term reversal, consistent with George and Hwang (2004) and Yao (2012).
Evidence from analyst target price data is supportive of underreaction to information
conveyed through trading. We examine how analyst target prices incorporate information in
past trading and non-trading returns. We show a strong positive association between past
intraday returns and analyst forecast error. A one percentage point increase in past intraday
returns is associated with a 0.34 percentage point increase in analyst forecast error, with a
t-statistic of about 17. In contrast, a change in past overnight returns is not associated with
a statistically significant change in forecast error.
This paper adds new evidence to the large literature on the sources of momentum and
reversal. Pioneering behavioral models of momentum and reversal are proposed by Barberis,
Shleifer, and Vishny (1998); Daniel et al. (1998); Hong and Stein (1999).5Our results are
more consistent with Hong and Stein (1999) but for the absence of long-term reversal. That
said, our results only suggest that underreaction to trading information is an important
component of momentum. Controlling for intraday returns, past overnight returns generate
some—albeit weak—momentum. This could be consistent with theories of momentum based
on public information.
5Rational explanations of momentum and (long-term) reversal have also been proposed; e.g., Johnson
(2002); Andrei and Cujean (2017). We focus on the leading explanations of momentum, which are mostly
behavioral.
5

Da, Gurun, and Warachka (2014) propose a “frog in the pan” hypothesis where in-
vestors are less attentive to information that arrives continuously in small amounts than to
information that arrives infrequently but in large amounts. They construct a measure of in-
formation discreteness and show that it explains momentum returns. Da et al. (2014) argue
that how information is partitioned matters, whereas we argue that its origin is equally im-
portant by showing that investors underreact to trading by other investors. Thus, our results
are complementary. Specifically, we show that the overnight momentum strategy remains
unprofitable after conditioning on continuous information, while the intraday momentum
strategy remains highly profitable after conditioning on discrete information.
Several papers provide important stylized facts about momentum profits. Cooper, Gutier-
rez, and Hameed (2004) show that momentum profits depend on the state of the market.
Daniel and Moskowitz (2016) show that momentum strategies can crash. Lou et al. (2019)
find that momentum returns are realized primarily overnight (intraday) during 1993-2013
(1926-1962), which they attribute to investor clienteles.6Instead, we focus on variations
in the strategy’s signal rather than on when the return is realized; i.e., how the mispricing
accumulates rather than on how it is resolved. Furthermore, our momentum result holds in
both sample periods, which indicates that the two sets of findings reflect different effects. In
other related work, Xu (2017) shows variation in the profitability of momentum and reversal
strategies constructed using morning and afternoon returns. Our focus is instead on intraday
and overnight returns.
Our paper also relates to the literature on stock price reactions to news. Most related,
Chan (2003) compares returns for stocks with and without major news headlines and shows
that news (no-news) stocks with large price moves in the previous month tend to exhibit re-
turn continuation (reversal) over 1980-2000. More recently, Jiang, Li, and Wang (2021) show
underreaction to news in a post 2000 sample. These authors focus on return predictability
6Other recent studies show that information extracted from day and night returns can help shed light
on what drives cross-sectional variation in stock returns. See Cliff, Cooper, and Gulen (2008); Bollerslev,
Li, and Todorov (2016); Hendershott, Livdan, and R¨osch (2020); Bogousslavsky (2021); Akbas, Boehmer,
Jiang, and Koch (2021); Lu, Malliaris, and Qin (2022), among others.
6

over a few days. We discuss how our results relate to evidence of underreaction to news
in Section 4.6. One of our key contributions relative to prior work is that our proxy for
news-driven price movements goes back to 1926.
Last, a large literature in accounting and finance studies how sell-side analysts’ recom-
mendations and price targets relate to past and future stock returns (e.g., Jegadeesh, Kim,
Krische, and Lee (2004); Engelberg, McLean, and Pontiff (2020) among others). We show
that analyst expectations—measured by price targets—relate to past trading information in
a different way than to past non-trading information.
This paper is organized as follows. Section 2motivates our empirical analysis. Section 3
describes our data and methodology. Section 4presents our main empirical results. Section 5
examines analyst price targets. Section 6concludes.
2 Motivation and Hypotheses Development
In this section, we explain why intraday returns and overnight returns can be used to proxy
for price moves induced by public information and private information. In a nutshell, we
expect news events to drive overnight returns to a larger extent than intraday returns. We
then lay out the main predictions to be tested in the empirical analysis.
It has long been pointed out that return volatility per unit of time is higher intraday than
overnight (e.g., Fama (1965)). French and Roll (1986) show that this excess volatility cannot
be driven by public information. Indeed, they show that the two-day return variance around
exchange holidays that are regular business days is only slightly larger than that of a normal
day. Instead, French and Roll (1986) argue that excess trading volatility is driven by private
information and mispricing. Hence, intraday returns should primarily reflect the impact of
investors’ trading. In a more recent sample, Boudoukh et al. (2019, BFKR) confirm these
findings by showing that fundamental information accounts for about 50% of idiosyncratic
overnight volatility but for only about 12% of intraday idiosyncratic volatility.
7

Importantly, we do not assume that news are irrelevant for intraday return volatility. A
substantial number of news are disclosed during trading hours. Instead, we rely on the fact
that public news events should drive overnight price moves to a larger extent than intraday
price moves. Therefore, any predictive power of intraday returns that is not observed for
overnight returns can be attributed to investors’ trading. In essence, our argument is that
public news should be relatively more important for overnight volatility than for intraday
volatility, as empirically documented by BFKR.
As detailed in Section 3, a key benefit of using intraday and overnight returns as proxies
for private and public information price moves is that they can be computed back to 1926,
whereas many news datasets are restricted to post 2000. Hence, we are able to leverage
almost a century of data to examine how private information and public information relate
to momentum and reversal strategies.
After-hours trading could be a concern for our basic assumption that overnight returns
reflect news rather than investors’ trading. However, after-hours trading was introduced in
1992 and is therefore non-existent for the majority of our sample. Even in recent years, after-
hours volume remains modest compared to regular-hours volume. Furthermore, Barclay,
Litzenberger, and Warner (1990) find that the trading of U.S. stocks listed on the Tokyo
exchange is negligible and does not generate additional volatility. They conclude that a large
volume of trading is required for private information to be incorporated into prices
One potential issue for our tests is that the type of news disclosed overnight could differ
from the type of news disclosed intraday. Related to this point, BFKR classify news into
18 categories. According to data in their Table 4, the correlation between the fraction of
news in each category disclosed intraday and that disclosed overnight is about 0.7, which
suggests considerable similarity. Moreover, our long sample period allows us to test whether
the findings are robust to shifts in the timing of firm’s information disclosure. For example,
the vast majority of earnings announcements are made outside of trading hours in recent
years, but that may not have been the case historically (Patell and Wolfson (1984)).
8

Market participants could react differently to overnight public news than to intraday
public news. On a relative basis, this does not appear to be the case: BFKR show that
the relative importance of each category of news is highly similar intraday versus overnight,
with a correlation of about 0.9. In their words, “the most important sources of news remain
consistent.” Hence, any difference in reaction must apply uniformly across news categories.
Since BFKR find that the impact of news tends to be higher overnight, we come back to
this issue in Section 4.5.
We next turn to our hypotheses. To formalize intuition, a simple model based on Eyster
et al. (2019) is presented in the Appendix. In this framework, investors observe a common
public signal about the asset fundamental value. Investors then receive private signals and
trade on these signals. The initial period can be interpreted as overnight. This period’s return
is driven by investors’ interpretation of public news. The second period can be interpreted
as intraday. This period’s return is driven by investors’ trading; that is, how much they
trade on their private information and how they weight the information conveyed through
prices. Finally, at some later date, the fundamental value is revealed.
We are interested in return autocorrelation and volume patterns. If investors’ underesti-
mate the precision of the public signal, the overnight return incompletely reflects the public
signal. This leads to momentum based on past overnight return signals. In contrast, even if
investors are overconfident about the precision of their own signal, there is positive momen-
tum based on past intraday return signals as long as investors’ sufficiently underweight the
precision of others’ signals. In this case, investors learn from prices but do not put a large
enough weight on the price signal.
While the above two predictions are intuitive, predictions that relate trading volume to
momentum are more difficult to lay out. Indeed, since the model is static, it does not allow
us to easily compare periods with low and high volume. Instead, we perform comparative
statics relative to a specific variable and assume that this variable varies across periods. One
natural candidate is the number of active investors on a given day. As long as investors learn
9

from prices to some extent, they know that, as the number of other investors’ increases,
prices become more informative. In general, this leads to weaker momentum and higher
volume.
Endowment shocks and liquidity shocks are also likely to be key drivers of trading volume,
but it is unclear how they would change the main predictions about momentum. That said,
as we discuss below, such shocks are likely to play an important role in explaining short-term
return reversal. One could add to the model an intermediate period as in Grossman and
Miller (1988) to capture reversal induced by liquidity shocks.
3 Data and Methodology
In this section, we explain how we compute intraday and overnight returns that are required
to decompose the momentum signal into the day and night parts. The data and methodology
are standard except that we use several datasets to construct reliable open prices. These
data eliminate concerns that open prices are disproportionately affected by microstructure
noise.
Our main sample covers U.S. common stocks (CRSP share code 10 and 11) listed on
NYSE, AMEX, and NASDAQ, running from January 1, 1963 through December 31, 2019.
We obtain daily price, volume, and dividend distribution information from Daily CRSP
files. We obtain monthly returns (the main dependent variable) and shares outstanding from
Monthly CRSP. From Compustat, we extract earnings announcement dates and construct
book value measures as the sum of stockholders’ equity and deferred taxes, at the end of
each fiscal year.7We obtain the time-series of monthly T-Bill rates and Fama-French factor
returns from WRDS. Finally, analyst target price data are from IBES, which we describe
further in Section 5.
To decompose the momentum signal into day and night parts, we need open and close
7The “linktable” provided by Wharton Research Data Service allows matching stocks across CRSP,
Compustat, and GFD. As such, in our regression analyses where we use information from all three sources,
a stock with no such links is excluded from the sample.
10

stock prices. While close prices from CRSP are widely used, open prices from CRSP are
less commonly used and are only available starting in 1991. We use three additional data
sources that provide alternative open prices and extend our sample—daily data from Global
Financial Data (GFD) and intraday data from the Institute for the Study of Security Mar-
kets (ISSM) database pre-1993 and NYSE Trade and Quote database (TAQ) 1993-onward.
Specifically, for stock-days with open price observations only reported by GFD, we match
daily observations across CRSP and GFD using unique combinations of security identi-
fiers PERMNO, PERMCO, and CIK.8Hence, from 29,225,292 unique daily observations in
CRSP, we can match 25,468,167 observations with GFD.9Barardehi et al. (2021) document
that the attributes of the matched GFD-CRSP set of stocks are similar those of the entire
population of stocks covered by CRSP, i.e., the matched subsample is representative.10 In
robustness analyses, (1) we rely on open prices of NYSE-listed common shares reported in
Daily CRSP files from 1926 through 1962; and (2) we use quote midpoints taken 15 minutes
after the open, instead of actual open prices, in the 1985-2015 period using ISSM/TAQ data
as described in Bogousslavsky (2021). In the pre-1963 CRSP records, 79% of stock-day ob-
servations feature both open and close prices. In the post-1963 GFD-CRSP matched data,
this ratio is over 87%.
We construct measures of intraday and overnight returns over various horizons. On each
trading day, the intraday return measures the percentage change in price from open to close
and the overnight return measures the percentage change of today’s open price relative to
previous day’s close, accounting for overnight adjustments to returns, e.g., dividend distri-
butions or stock splits, obtained from CRSP as in Lou et al. (2019).
We compound intraday and overnight returns over all trading days in a given horizon;
8Open prices reported by GFD reflect transaction prices when there is a transaction at open; the mid-point
of best bid and ask at open is reported when a transaction price is not available.
9To control for potential data errors in CRSP or GFD, we use similarity in closing prices reported by
CRSP and GFD, dropping a matched stock-day observation if its CRSP closing price deviates from its GFD
closing price by more than 0.1%.
10The ratio of means for the GFD-CRSP sample to the CRSP population are all very close to one for key
stock characteristics at the beginning of each year: 1.009 for market capitalization, 1.046 for share price,
and 1.002 for volatility.
11

e.g., one month, to construct intraday and overnight signals, denoted IDRHand ON RH
respectively, that are then used to predict future monthly returns. To predict month m
return, Rm, intraday and overnight signals are constructed over horizons H∈ {m−1,(m−
7, m −2),(m−12, m −2),(m−36, m −13),(m−48, m −13),(m−60, m −13)}. We
also construct the standard return signals using daily close-to-close returns, RHover the
analogous horizons. For a stock-month to be included in the sample, we require return
observations over the preceding 36 months. As such, some stocks have missing observations
on return signals that span beyond a 3-year horizon.
Table 1presents summary statistics for day and night return signals for short-term re-
versal, momentum, and long-term reversal. Monthly compounded intraday returns are more
volatile than monthly compounded overnight returns; i.e., 0.14 vs 0.11 volatility. However,
monthly overnight returns are still highly volatile. Similar patterns are reported in Table A.1
in the Appendix, which presents summary statistics for the midpoint sample. Two impor-
tant points are worth emphasizing. First, even though we winsorize returns in the table,
intraday and overnight returns signals can take extreme values when compounded over long
horizons. These extreme values do not affect our results since our asset pricing tests mostly
focus on portfolio sorts. When we use regressions, we convert the return signals to percentile
statistics, as explained below. Second, intraday and overnight returns are negatively corre-
lated. This is apparent from the fact that the product of average intraday and overnight
gross returns does not equal average monthly gross return, which is not entirely explained by
the winsorization. This negative correlation increases with the horizon over which returns
are compounded. We go back to this point in Section 4.4.
[Insert Table 1about here.]
12

4 News Signals, Trading Signals, and Future Returns
In this section, we study how signals that proxy for public news (past cumulative overnight
return) and for investors’ trading (past cumulative intraday return) predict future monthly
returns.
4.1 Main Results
We start our empirical analysis with portfolio sorts. We estimate three-factor alphas of
trading strategies that buy past winners and sell past losers based on past return signals
that reflect close-to-close returns, intraday returns, and overnight returns. Following Hou,
Xue, and Zhang (2020), in each month, we first assign each stock to one of the ten portfolios
whose breakpoints represent deciles of a given signal for the cross-section of NYSE-listed firms
in that month. We then calculate equally-weighted month mreturns in each portfolio to
construct ten time-series of portfolio returns. We regress the excess return of each portfolio,
in excess of the one-month T-Bill rate, as well as the return difference between in top and
bottom portfolios on market, size, and value factor returns to estimate the corresponding
three-factor alpha; i.e., intercept. To account for autocorrelated error terms, we correct
standard errors of estimates using the Newey-West approach based on three lags.
[Insert Table 2about here.]
Table 2reports portfolios’ average monthly returns and associated t-statistics for different
signal horizons. Panel A uses the previous-month return as signal. The first row of the
table uses every hour of the day to form the return signal (24-hour) and confirms the well-
known evidence of short-term reversal in the cross-section of stock returns (Jegadeesh (1990);
Lehmann (1990)). A portfolio that shorts previous-month winners and goes long previous-
month losers earns a monthly alpha of 1.41 with a t-statistic of 7.14. The second and third
rows reveal a striking difference. A reversal strategy based on past-month intraday return
is substantially more profitable (monthly alpha of 1.22% with a t-statistic of 7.13) than a
13

reversal strategy based on past-month overnight return (monthly alpha of 0.37% with a t-
statistic of 3.35). Furthermore, alpha declines almost monotonically as one moves from the
portfolio of past intraday losers to the portfolio of past intraday winners. The pattern is
much less clear for portfolios formed on past overnight returns.
The above result supports theories of imperfect liquidity in which return reversal com-
pensates liquidity providers to absorb inventory shocks. Intraday returns are the outcome
of investors’ continuous trading and are therefore more likely to be driven by “price pres-
sure” than overnight returns. Overnight returns are more affected by news and, as a result,
should be less subject to price pressure than intraday returns. Since the price pressure
effect has been confirmed by many studies (e.g., Campbell et al. (1993); Hendershott and
Menkveld (2014); Nagel (2012)), the above intuitive result supports the view that a day-
night decomposition of return signals has economic content. Furthermore, this validates our
interpretation of intraday returns and overnight returns as reflecting investors’ trading and
news, respectively.
Panel B of Table 2considers momentum strategies based on return signals over the past
12 months, excluding the last month. The intraday momentum strategy yields a monthly
alpha of 1.16% with a t-statistic of 6.87. This is close to the regular momentum strategy,
which yields an alpha of 1.26% with a t-statistic of 5.75. The overnight momentum strategy
does not produce any discernible abnormal return pattern. Figure 1plots Table 2long-short
portfolios’ alphas with additional horizons. The results do not depend on the specific horizon
of the momentum or reversal strategy (e.g., past 7-month vs past 12-month). In sum, when
the momentum signal is decomposed into day and night components, the day signal appears
to account for the bulk of momentum alpha.
The lack of profitability of the strategy based on past overnight returns seems difficult to
reconcile with news-based explanations of momentum. If the primary cause of momentum is
investors’ underreaction to news, then past overnight returns should predict future returns.
For example, consider a positive news that would increase a stock price by 5% in a fully
14

rational market. Investors underreact to the news and the stock price increases by 3%
overnight when the news is disclosed. Over time, the stock price increases by 2% as more
information arrives. In this scenario, past overnight returns predict future returns. This
channel can only explain the evidence if most price-relevant news are disclosed intraday,
which seems at odd with the evidence discussed in Section 2.
The results appear more consistent with investors’ underreaction to the information con-
veyed by the trades of other investors. Investors may be overconfident in the precision of
their own signals relative to that of others (Banerjee (2011)). Alternatively, a subset of in-
vestors may not learn from prices at all (Eyster et al. (2019)). Over time, prices incorporate
this trading-based information, leading to momentum. Since investors’ private information
will mostly be reflected in day returns rather than night returns (French and Roll (1986)),
it is the intraday return signal that predicts future returns. This result is not inconsistent
with the reversal result in Panel A: At short horizons, liquidity effects dominate, leading to
reversal.
Momentum could be also driven by investors’ continued overreaction to past price move-
ments. In that case, it is possible that the intraday signal reflects the price pressure of
investors, which is continued in the future. Hence, the intraday signal generates momentum
profit, whereas the overnight signal does not. Delayed overreaction predicts that momen-
tum profits should be realized intraday as investors keep trading in the same direction as
past price pressure. This is inconsistent with the results of Lou et al. (2019) in a sample of
post 1993 U.S. stock returns. These authors find that momentum returns accrue primarily
overnight. Overreaction also predicts long-term reversal as mispricing is corrected, which we
examine next.
Panel C of Table 2examines long-term reversal strategies; i.e., strategies based on returns
over the previous 5 years, excluding the last year. The pattern observed for momentum
and short-term reversal strategies is reversed for long-term reversal strategies. A long-term
reversal strategy based on past intraday returns is not profitable with a monthly alpha of -
15

0.12% (t-statistic of -0.96). In contrast, a long-term reversal strategy based on past overnight
returns yields a monthly alpha of 0.45% with a t-statistic of 4.16. In fact, the overnight
strategy performs better than the standard long-term reversal strategy, which only yields an
alpha of 0.27% with a t-statistic of 2.00. Alpha decreases almost monotonically as one moves
from the decile of past losers to the decile of past winners. Hence, the above results dissociate
momentum from long-term reversal, which is inconsistent with overreaction theories. We go
back to this result in Section 4.6.
Next, we confirm the robustness of the results and alleviate several data concerns.
The results are robust to excluding stocks in the bottom 20% of market capitalization
and to dropping the 3-day windows around earning announcements before constructing re-
turn signals as reported in Table A.2 in the Appendix. Tables A.3 and A.4 report return-
weighted portfolio returns, as suggested by Asparouhova et al. (2013), and value-weighted
portfolio returns. Return-weighting produces results that are highly similar to the results in
Table 2. With value-weighting, the momentum results are robust. The value-weighted intra-
day momentum strategy generates a monthly alpha of 1.06% with a t-statistic of 5.32. The
value-weighted overnight momentum strategy generates a monthly alpha of 0.27%, which is
statistically insignificant. Both short-term reversal and long-term reversal are statistically
insignificant with value-weighted returns. For short-term reversal, this result is consistent
with liquidity theories as small stocks tend to be less liquid than large stocks.
Overnight returns can be noisy due to stale opening prices or other related issues. As
detailed in Section 3, the data is screened for such errors. Furthermore, the profitability of
long-term reversal based on past overnight returns suggests that overnight returns contain
information. If past overnight returns were pure noise, we would not expect to see a pattern
in average returns. Bogousslavsky (2021) shows that the choice of the opening price affects
the intraday and overnight average returns of various investment strategies. To assess the
robustness of our results, we replace the opening price with a quote midpoint taken a few
minutes after the open. Following Bogousslavsky (2021), we use a sample of intraday and
16

overnight returns constructed using quote midpoints fifteen minutes after the open (i.e., at
9:45am). Fifteen minutes is enough time for quotes to stabilize but avoids that a substantial
portion of the intraday period is attributed to the overnight period. In addition, the use
of quote midpoints eliminates any noise induced by the bid-ask spread. The availability of
intraday data limits this sample to 1985-2015.
[Insert Table 3about here.]
Table 3reports the portfolio sorts analysis for this sample. In this more recent sample,
the standard one-month reversal strategy yields an alpha of only 0.12% per month, which is
not statistically significant. However, the striking contrast between intraday and overnight
reversal strategies remains. The intraday reversal strategy yields an alpha of 0.38% per
month (t-statistic of 1.84), more than three times higher than the unconditional strategy.
While a reversal strategy based on past-month overnight return yields an alpha of -0.40%
(t-statistic of -2.73). Panel B of Table 3shows that the previous momentum results are
robust. A momentum strategy based on past overnight returns is unprofitable, whereas a
momentum strategy based on past intraday returns is profitable. Panel C shows that while
there is no evidence of long-term reversal in this more recent sample, a long-term reversal
strategy based on past overnight returns yields a positive albeit insignificant monthly alpha.
Open prices are available in CRSP pre-1963 for NYSE-listed stocks. We next examine
whether the results hold in this earlier period. There may be additional issues with the
quality of opening prices in this period. Nevertheless, any noise in opening prices should
attenuate our findings since it translates into noise in the intraday and overnight signals.
Figure 2reports the average monthly alpha of close-to-close, intraday, and overnight strate-
gies with different portfolio formation horizons. The patterns in Figure 1are mostly similar.
The one key difference is that there is evidence of reversal at all horizons for strategies based
on past overnight returns, which we discuss further in Section 4.6. In contrast, intraday
strategies display a stable pattern across all sample periods. Hence, our key finding is robust
in this sample period. This robustness is worth emphasizing since Lou et al. (2019) show
17

that the return of a regular momentum strategy tends to be realized intraday pre 1963 and
overnight post 1993. They attribute this switch to changes in investor clienteles as there
were only few institutional investors in the pre 1963 period. Figure 2indicates that what we
find is distinct and cannot be attributed to clientele effects.
[Insert Figure 2about here.]
In a nutshell, the patterns in Figure 1and Table 2are robust. They do not appear
to be driven by liquidity effects at the open (e.g., Berkman et al. (2012); Bogousslavsky
(2021)), stale overnight returns, or by some form of bid-ask bounce between intraday and
overnight returns. Hence, we argue that these patterns provide new economic insights about
the sources of momentum and reversal strategies’ profitability.
4.2 Prior-Week Return
Gutierrez and Kelley (2008) argue that a weekly horizon provides a better horizon to identify
the news that underlies the return. These authors document long-lasting momentum in
weekly returns. In particular, they show that the results of Chan (2003) about the relation
between return continuation and news do not extend to weekly returns. To assess the
robustness of our results, we replicate their approach. We form decile portfolios based on
prior-week intraday and overnight returns and then examine the cumulative profits over time
of the strategy that is long past winners and short past losers. Figure 3plots the strategy’s
cumulative profit. The figure confirms our previous tests. There is no evidence of momentum
based on prior-week overnight returns. If anything, the long-short overnight return portfolio
performs poorly over the following week and this poor performance appears to be permanent.
In contrast, after an initial period of reversal there is strong momentum based on prior-week
intraday returns.
[Insert Figure 3about here.]
18

4.3 Volume-Based Return Signal
In this section, we condition the momentum and reversal strategies’ signals on past volume
to further test our proposed explanation. Standard price pressure theories predict that short-
term reversal should increase with trading volume (Campbell et al. (1993)). According to the
model discussed in Section 2, if investors underweight the precision of others’ signals but still
learn from prices, volume and momentum are in general inversely associated. Intuitively,
when volume is high, information is better incorporated into prices. On days with high
volume, investors’ recognize that prices are likely more informative, which results in lower
price drift. Another possibility is that the disclosure of public information generates high
volume and resolves information asymmetry, which is well documented empirically (e.g.,
Chae (2005)).
For each stock, we split each month into below-median and above-median volume days.
We then compute past-return signals using either days with below-median volume or days
with above-median volume. Table 4reports alphas and associated t-statistics for these low
and high volume portfolios formed with different horizons. The detailed results across deciles
are reported in Tables A.5 and A.6 in the Appendix. Figure 4plots the alphas.
[Insert Figure 4about here.]
[Insert Table 4about here.]
Table 4shows that one-month return reversal is stronger when the strategy’s signal is
conditioned on high volume days. Indeed, monthly alpha is 1.14% (t-statistic 7.37) vs 0.36%
(t-statistic 3.3) when conditioned on low volume days. This result further supports the
price pressure interpretation: the higher the volume, the higher the reversal (Campbell et al.
(1993)). Conditioning the strategies on low volume days, the difference in alpha between
intraday reversal and overnight reversal strategies is 0.15%. This difference jumps to 0.85%
when conditioning the strategies on high volume days.
19

The pattern is reversed when we look at past 7 and 12 month return signals. A momen-
tum strategy conditioned on low volume days outperforms a momentum strategy conditioned
on high volume days by about 0.37% per month. This result is consistent with the above in-
tuition. When volume is high, information is incorporated into prices more quickly. We note,
however, that the intraday momentum strategy remains profitable even when conditioned on
high volume days. This last result seems inconsistent with a pure investor attention theory.
The long-term reversal results in Table 4provide additional evidence that supports the
underreaction channel. A five-year reversal strategy conditioned on low-volume days is not
profitable. Even more striking, a five-year reversal strategy based on intraday returns associ-
ated with low-volume days generates a negative monthly alpha of 0.26%, which is statistically
significant (t-statistic of 2.26). In other words, there is statistically significant continuation
of past long-term intraday returns associated with low-volume days. This further suggests
that the profitability of the intraday momentum strategy is not due to overreaction. In con-
trast, the five-year reversal strategy based on past intraday returns is weakly profitable when
the signal is further conditioned on high volume days (Panel B). This potentially suggests
some overreaction associated with high volume.
As a comparison, past overnight return strategies do not exhibit sizable differences
whether conditioned on low-volume or high-volume. If investors underreact to public in-
formation, one would expect stronger momentum based on “low attention days” returns.
If volume is taken as a proxy for investor attention, this is inconsistent with the results in
Table 4.
4.4 Interaction Between Intraday and Overnight Returns
In this section, we examine how interactions between intraday and overnight returns affect
the momentum results.
We start with double portfolio sorts. First, we sort stocks into quintiles based on past
overnight (intraday) returns over the prior 2 to 12 months. Second, within each overnight
20

quintile, we sort stocks based on past intraday (overnight) returns over the prior 2 to 12
months. Table 5shows the results. Panel A shows that, within every overnight return
quintile, sorting on past intraday return generates strong momentum profits. Most notably,
among stocks in the bottom quintile of overnight returns, stocks with high past intraday
returns earn a positive alpha of 0.26% per month with a t-statistic of 2.81. In Panel B, the
results are weaker. Among stocks with low past intraday returns, a long-short strategy based
on past overnight returns generates a near-zero and insignificant monthly alpha. Among
other intraday return quintiles, the overnight returns’ strategy long-short alpha is statistically
significant but always smaller than that of the corresponding intraday returns’ strategy. In
summary, past intraday returns generate consistently sizable and significant momentum
profits whereas past overnight returns do not.
[Insert Table 5about here.]
Next, we use Fama and MacBeth (1973) regressions to estimate the predictive powers
of past return signals for future month mexcess returns while include multiple predictors
simultaneously. We regress stock j’s month mreturn on percentile statistics of past return
signals and those of several control variables.11 The set of control variables include (1)
market-capitalization, defined as that product of closing price and the number of shares
outstanding at the end of month m−12; (2) book-to-market ratio, defined as the most
recent book value of equity divided by market value of shares outstanding at the end of
month m−1; (3) dividend yield, defined at the sum of cash dividends over the 12 months
ending in month m−2 divided by share price at the end on month m−2; and (4) the
open-to-close version of Amihud’s measure of stock liquidity, as proposed by Barardehi et al.
(2021), from the previous calendar year. We estimate
Rj
m−RFm=C0+C>
1SH+C>
2Controls + uj
m(1)
11Percentile statistics of each variable are calculated for each month.
21

where Rj
mis stock j’s return in month m;RFmis T-Bill rate in month m;SHis the matrix
of percentile statistics of return signals that includes, Rj
H,IDRj
H,ON Rj
H, or both I DRj
H
and ON Rj
H, with signal horizon H∈ {m−1,(m−12, m −2),(m−36, m −12)}; and
Controls is the matrix of percentile statistics control variables described above. To account
for autocorrelated error terms, we adopt Newey-West-corrected standard errors based on
three lags.
The results are reported in Table 6. Controlling for past intraday returns, past overnight
returns predict future returns in the cross-section, but this effect is significantly weaker than
the effect of past intraday returns. This is consistent with the results in Table 5.
[Insert Table 6about here.]
4.5 Information Discreteness and Investor Attention
As discussed in Section 2, one potential issue for the interpretation of our results is that
market participants could react differently to overnight public news than to intraday public
news. It is, however, unclear in which direction such effects would go. A natural possibility is
that investors are less attentive to overnight information than to intraday information since
the market is closed overnight. This channel predicts that there should be greater underreac-
tion to news disclosed overnight than to news disclosed intraday. We should therefore expect
stronger momentum based on past overnight returns, which cannot explain our results.
Another possibility is that public news disclosed overnight appears more salient to in-
vestors than public news disclosed intraday. Related to this point, Da et al. (2014) hy-
pothesize that investors are less attentive to information that arrives continuously in small
amounts than to information that arrives infrequently but in large amounts. To test this
hypothesis, they construct a measure of information discreteness based on past returns. For
example, over a five-day period, a return of 0% on the first four days followed by a return of
5% on the last day is more “discrete” than a return of 1% on every day, which is interpreted
as more “continuous.” Fixing the total return signal, Da et al. (2014) show that momentum
22

is much stronger when the return signal is based on continuous past returns than when based
on discrete past returns. If intraday returns are more continuous than overnight returns, this
can potentially explain why the intraday momentum strategy is more profitable than the
overnight momentum strategy.
To shed light on this possibility, Table 7reports the information discreteness of past
intraday and overnight return signals. Average information discreteness is highly similar
across intraday and overnight return signals. We also perform a similar analysis as Da et al.
(2014) but with the intraday and overnight return signals instead of the total return signal.
The top panel of Table 7shows that the profitability of intraday momentum decreases with
information discreteness from a monthly alpha of 1.61% with a t-statistic of 9.28 to an alpha
of 0.42% with a t-statistic of 3.32. This pattern is consistent with Da et al. (2014) and the
underreaction channel. Importantly, intraday momentum remains strongly profitable even
when conditioning on discrete information. Hence, the profitability of intraday momentum
cannot be simply ascribed to continuous information.
The bottom panel of Table 7shows a reversed pattern for the overnight momentum strat-
egy. This strategy tends to become more profitable when conditioned on discrete returns.
This result supports our methodology because it suggests that the overnight return signal
reflects fundamentally different economic forces than the intraday return signal. Indeed, a
profitable overnight momentum strategy when conditioned on continuous returns would sug-
gest that underreaction to “small” news plays a role in explaining our result. In summary,
our results are consistent with Da et al. (2014) but indicate that investors are more likely to
underreact to trading by other investors than to regular news.
[Insert Table 7about here.]
4.6 Public News and Future Returns
We next discuss how our results compare to previous work that documents underreaction to
public news.
23

Chan (2003) compares returns for stocks with and without major news headlines and
shows that news (no-news) stocks with large price moves in the previous month tend to
exhibit return continuation (reversal) over 1980-2000. Reversal of no-news driven price moves
accords well with our intraday reversal result under our interpretation of intraday returns
as primarily driven by trading. Continuation of news-driven price movement is consistent
with the fact that, when we focus on our 1985-2015 midpoint return sample, stocks with
low prior-month overnight return tend to do poorly in the following month. In Panel A of
Table 3, past-month overnight losers earn an alpha of -0.34% with a t-statistic of -3.58 in the
following month, but there is no evidence of return continuation for past-month overnight
winners. This result is consistent with Chan (2003) who finds stronger drift after bad news
and further validates our interpretation of overnight returns as news-driven. Since Chan
(2003) focuses on a one-month formation period, we cannot directly compare momentum
and long-term reversal results.
Jiang et al. (2021) show underreaction to intraday news in a post 2000 sample of U.S.
stocks. These authors focus on a five-day period after news are reported. Since our momen-
tum signals exclude the previous month return, this short-term drift is unlikely to explain
the strong profitability of the intraday momentum strategy reported in Tables 2and 3. To be
clear, our results do not go against underreaction to news in general, but they suggest that
it is not the dominant channel to explain the profitability of standard momentum strategies.
5 Underreaction to Trading Signals and Analyst Tar-
get Prices
We postulate that investors’ underreaction to information conveyed through trading helps
explain the results in Section 4. Simply put, investors’ beliefs should respond differently to
past trading and non-trading returns. Accordingly, this section examines how analyst target
prices incorporate information in past trading and non-trading returns. Analyst price targets
24

provide a useful measure of price expectations to test our explanation for several reasons.
As discussed in Bradshaw (2011), a large literature argues that analysts are an important
group of economic agents. Hence, an investigation of how they form their price targets is
informative. Moreover, data on target prices is available for a broad cross-section of stocks
from 1999.
Our data collection process follows Palley, Steffen, and Zhang (2019). We use IBES
consensus target price data to obtain the 12-month mean analyst target price, which is
available on a stock-month basis. We exclude observations for which the mean target price
differs from the median target price by more than 50%. We compute the mean analyst
predicted 12-month return by dividing the target price by the current stock price when
consensus information is calculated (Palley et al. (2019)) and taking the logarithm. We
winsorize the analyst predicted 12-month return at 1% and 99%. The predicted return data
is joined to our sample of midquote intraday and overnight returns (Section 3), which we
aggregate to the monthly frequency. In this section, all returns refer to logarithmic returns.
We use log returns to attenuate the high skewness of intraday and overnight returns that
are compounded over long horizons. Summary statistic for the main variables are reported
in Table A.7 in the Appendix.
How do analyst predicted 12-month return and forecast error vary with past intraday
and overnight return? To answer this question, we use panel regressions with month fixed
effects and cluster standard errors by month. Double clustering standard errors by stock
and month yields similar results, as does using Fama-MacBeth regressions and Newey-West
adjusted standard errors with up to 12 lags. We first regress future returns on past intraday
and overnight returns:
ri,t+1:t+12 =at+bovrov
i,t−12:t−1+binrin
t−12,t−1+ei,t,(2)
where ri,t+1:t+12 is stock i’s return over the next 12 months, rov
t−12:t−1is the cumulative
25

overnight return over the previous 12 months, and rin
t−12:t−1is the cumulative intraday return
over the previous 12 months. The results are reported in Column (1) of Table 8. In line
with our prior results, past intraday returns are strongly positively associated with future
returns, whereas overnight returns are not. Equation (2) focuses on returns over the next 12
months to be consistent with the horizon of analyst forecasts that we use.
Our key specification is similar to Equation (2) but with analyst predicted return over
the next 12 months (pret) as dependent variable. Hence, we estimate
preti,t+1:t+12 =at+βovrov
i,t−12:t−1+βinrin
t−12:t−1+ei,t.(3)
This specification can be mapped to the framework in Section 2. We can think of the
analyst as an agent who observes a price increase of 1% either overnight or intraday. With
fully rational expectations, this increase conveys information about the future fundamental
one-for-one and therefore the predicted future return remains constant; i.e., βin =βov = 0.
If the analyst underweights the precision of other traders’ signals or is overconfident in the
precision of his own signal, however, then βin < βov = 0. The analyst correctly incorporates
the public signal but gives less weight to the price signal than to his private signal.
Column (2) of Table 8reports the results. First, past intraday and overnight returns
account for about 19% of the within-month variation in analyst predicted return. Hence, they
are correlated with key information driving analyst forecasts. Second, both coefficients on
past intraday and overnight returns are negative. This is consistent with Brav, Lehavy, and
Michaely (2005), who find that the prior 11-month return is negatively associated with one-
year analyst expected returns over 1975-2001. Finally, βin is more than twice lower than βov,
as expected from underreaction to intraday return information relative to overnight return
information. The difference between the two coefficients is strongly statistically significant,
as reported in the last row.
Equation (3) does not inform on absolute underreaction to information. Column (3)
26

of Table 8combines Columns (1) and (2) to show a strong positive association between
past intraday returns and analyst forecast error. A one percentage point increase in past
intraday returns is associated with a 0.34 percentage point increase in forecast error, with
at-statistic of about 17. In contrast, a change in past overnight returns is not associated
with a statistically significant change in forecast error. Therefore, the results in Table 8
suggest analyst’s underreaction to trading information relative to non-trading information,
in line with the discussion in Section 2. Differences in the volatility of past intraday and
overnight returns do not drive the results in Table 8. Past intraday returns are slightly
more volatile than past overnight returns (Table A.7). Hence, scaling past returns by their
standard deviation in Table 8would increase the difference between intraday and overnight
coefficients.
Evidence from analyst target price data is supportive of our proposed underreaction
channel. That said, extrapolating the results beyond analysts would require analysts’ ex-
pectations to proxy well for investors’ expectations in general. As pointed out in Bradshaw,
Brown, and Huang (2013), analyst price targets tend to be overoptimistic and display sub-
stantial noise. It is unclear, however, how such biases could translate into the differential
associations with past day and night returns that we uncover.
6 Conclusion
In this paper, we implement a novel test of momentum and reversal theories that distin-
guishes between public and private information flows. In essence, we examine the impor-
tance of news-related and trading-related price moves for reversal and momentum. Com-
peting theories put different emphasis on the importance of such components. For example,
if momentum is fundamentally about underreaction to news, then a proxy for news, such
as overnight returns, should generate strong momentum profits. In contrast, if momentum
is fundamentally about investors’ underreaction to private information, then a proxy for
27

trading-related price moves, such as intraday returns, should generate strong momentum
profits.
To implement our test, we proxy for news by using past cumulative overnight returns
and for trading-related price moves by using past cumulative intraday returns. We vary the
horizon over which we consider these signals and use them to predict future monthly returns.
Portfolios formed on past intraday returns display short-term reversal and momentum with-
out long-term reversal. In contrast, portfolios formed on past overnight returns display only
long-term reversal.
The reversal results support price pressure explanations of short-term reversal. Investors’
trading generate price pressure intraday, and this intraday return subsequently partially
reverts. The momentum results support underreaction theories of momentum. In particular,
they are more consistent with investors’ underreaction to the information conveyed by the
trades of other investors’ rather than underreaction to regular news. The absence of long-
term reversal is inconsistent with delayed overreaction and over-extrapolation. We also show
that momentum is stronger when the momentum signal is conditioned on low volume days,
which supports an inattention channel.
Evidence from analyst target price data is supportive of underreaction to information
conveyed through trading. We show a strong positive association between past intraday
returns and analyst forecast error, but a change in past overnight returns is not associated
with a statistically significant change in forecast error.
The proposed test can be used in other settings to separate the effects of information
contained in news from that contained in investors’ trades. A significant advantage of using
intraday and overnight U.S. stock returns is that they can be computed back to 1926. A
better understanding of how investors learn from the trades of others is an interesting avenue
for future research.
28

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32

Figure 2. Monthly alpha of momentum and reversal long-short strategies constructed
from past close-to-close, intraday, and overnight returns - 1929-1962. This figure point
presents estimates and 95% confidence intervals for three-factor alphas. Stocks are allocated into
decile portfolios based on NYSE breakpoints of past returns. 7-month and 12-month strategies
exclude the most recent month. 36-month, 48-month, and 60-month strategies exclude the most
recent year. The sample includes all U.S. common stocks from 1929–1962.
1929–1962
−2 −1.5 −1 −.5 0 .5 1
Three−factor alpha (%)
1−month 7−month 12−month 36−month 48−month 60−month
Portfolio formation horizon
24−hour Intraday Overnight
33

Figure 3. Weekly Cumulative Profits to Momentum Strategies Based on Intraday and
Overnight Return Signals. This figure compares cumulative profits to a trading strategy that
buys past winners and sells past losers based on prior week’s returns (Gutierrez and Kelley,2008).
Stocks are sorted into ten equally-sized portfolios of past week’s intraday returns (the orange line)
and past week’s overnight returns (the grey line). Raw profits associated with a trading strategy
that is long on week-zero highest-return decile of stocks and is short on week-zero lowest return
decile of stocks are calculated for the following 52 weeks. Cumulative profits associated momentum
trading strategies based on week-zero intraday and overnight are plotted against event week. The
sample includes all U.S. common stocks over 1963 to 2019.
−2 −1 0 1 2 3 4 5
Cumulative profit (%)
0 4 8 12 16 20 24 28 32 36 40 44 48 52
Event week
Intraday Overnight
34

Figure 4. Monthly alpha of momentum and reversal long-short strategies constructed
from past close-to-close, intraday, and overnight returns - low- vs. high-volume signals.
This figure point presents estimates and 95% confidence intervals for three-factor alphas. Stocks
are allocated into decile portfolios based on NYSE breakpoints of past returns. Each stock’s past
daily returns are decomposed into low- vs. high-volume days according to the respective month’s
median daily trading volume. 7-month and 12-month strategies exclude the most recent month.
36-month, 48-month, and 60-month strategies exclude the most recent year. The sample includes
all U.S. common stocks over 1966 to 2019.
Panel A: Low-volume day signals
−1.5 −1 −.5 0 .5 1 1.5
Three−factor alpha (%)
1−month 7−month 12−month 36−month 48−month 60−month
Portfolio formation horizon
24−hour Intraday Overnight
Panel B: High-volume day signals
−1.5 −1 −.5 0 .5 1 1.5
Three−factor alpha (%)
1−month 7−month 12−month 36−month 48−month 60−month
Portfolio formation horizon
24−hour Intraday Overnight
35

Table 1. Summary statistics of past return signals. This table provides sample statistics of past intraday and overnight return
signals, denoted IDR, and ONR, respectively, constructed at horizons H∈ {m−1,(m−12, m −2),(m−60, m −13)}and winsorized
at the 1st and the 99th percentiles for common shares listed on NYSE, AMEX, and NASDAQ over the period of 1966-2019.
Return Signal Count Mean St Dev. Skewness 1st Pctile 5th Pctile Median 95th Pctile 99th Pctile
R−11,754,100 0.009 0.14 0.61 -0.37 -0.20 0.00 0.24 0.52
IDR−11,754,100 0.010 0.14 0.56 -0.42 -0.22 0.00 0.25 0.57
ON R−11,754,100 0.004 0.11 0.90 -0.33 -0.16 0.00 0.17 0.48
R(−12,−2) 1,754,100 0.12 0.53 1.57 -0.81 -0.59 0.06 1.06 2.48
IDR(−12,−2) 1,754,100 0.27 1.09 4.33 -0.94 -0.69 0.06 1.67 7.58
ON R(−12,−2) 1,754,091 0.16 0.96 4.80 -0.91 -0.63 0.01 1.24 6.83
R(−60,−12) 1,450,998 0.69 1.57 2.52 -0.96 -0.82 0.30 3.60 8.67
IDR(−60,−12) 1,450,978 7.65 46.09 8.09 -1.00 -0.94 0.32 13.48 419.09
ON R(−60,−12) 1,450,942 3.97 23.99 7.64 -1.00 -0.95 0.03 7.80 211.03
36

Table 2. Three-Factor Alphas of Portfolios Based on Past Return Signals. This table
reports three-factor alphas of trading strategies that buy past winners and sell past losers according
to past return signals reflecting close-to-close, intraday, and overnight returns. A stock’s signals
are constructed at horizons H∈ {m−1,(m−12, m −2),(m−60, m −13)}. Each month, stocks are
assigned to one of the ten portfolios of past return signals constructed based on NYSE breakpoints.
The time-series of equally-weighted month mportfolio returns, in excess of 1-month T-Bill rates, are
regressed on market, size, and value factor returns to estimate the corresponding three-factor alpha,
i.e., the intercept. The sample includes common shares listed on NYSE, AMEX, and NASDAQ
from 1966 to 2019. Standard errors of estimates are constructed using the Newey-West approach
based on three lags with t-statistics reported in brackets.
Panel A: Portfolios of last month’s return (H=m−1)
Signal type 1 2 3 4 5 6 7 8 9 10 10−1
24-hour 0.61 0.14 0.1 0.11 0.11 0.022 0.015 −0.14 −0.27 −0.79 −1.41
[3.79] [1.73] [1.61] [1.93] [1.91] [0.42] [0.28] [−2.49] [−4.57] [−7.63] [−7.14]
Intraday 0.46 0.27 0.19 0.18 0.11 0.12 0.015 −0.039 −0.2 −0.75 −1.22
[2.88] [4.02] [3.38] [3.19] [2.11] [2.3] [0.3] [−0.82] [−3.59] [−8.28] [−7.13]
Overnight 0.099 0.012 0.048 0.081 0.077 −0.013 0.02 0.04 0.054 −0.27 −0.37
[0.99] [0.23] [0.84] [1.59] [1.62] [−0.27] [0.42] [0.83] [0.98] [−2.23] [−3.35]
Panel B: Portfolios of months −12 to −2 return (H= (m−12, m −2))
Signal type 1 2 3 4 5 6 7 8 9 10 10−1
24-hour −0.71 −0.25 −0.11 −0.005 0.13 0.2 0.24 0.33 0.37 0.55 1.26
[−3.92] [−2.38] [−1.55] [−0.08] [2.54] [4.14] [4.79] [5.96] [6.35] [5.35] [5.75]
Intraday −0.79 −0.17 −0.002 0.045 0.069 0.16 0.15 0.23 0.29 0.38 1.17
[−4.68] [−1.96] [−0.04] [0.75] [1.37] [3.22] [2.93] [4.81] [5.28] [5.14] [6.87]
Overnight −0.005 −0.041 −0.058 0.048 0.053 0.12 0.16 0.042 0.12 −0.051 −0.046
[−0.05] [−0.65] [−1.00] [0.98] [1.11] [2.51] [3.58] [0.82] [2.1] [−0.38] [−0.31]
Panel C : Portfolios of months −60 to −13 return (H= (m−60, m −13))
Signal type 1 2 3 4 5 6 7 8 9 10 10−1
24-hour 0.096 0.062 0.11 0.1 0.1 0.074 0.12 0.022 0.026 −0.17 −0.27
[0.59] [0.86] [2.00] [2.09] [2.03] [1.29] [1.84] [0.32] [0.36] [−4.26] [−2.00]
Intraday −0.18 0.006 0.07 0.15 0.039 0.071 0.11 0.13 0.13 −0.061 0.12
[−1.17] [0.08] [1.19] [2.6] [0.68] [1.31] [1.96] [2.09] [2.09] [−0.86] [0.96]
Overnight 0.23 0.1 0.09 0.065 0.073 0.026 0.011 −0.007 −0.082 −0.21 −0.45
[2.48] [1.6] [1.58] [1.14] [1.22] [0.45] [0.2] [−0.12] [−1.1] [−2.26] [−4.16]
37

Table 3. Three-Factor Alphas of Portfolios Based on Quote-Midpoint Past Return
Signals. This table reports three-factor alphas of trading strategies that buy past winners and
sell past losers according to past return signals reflecting close-to-close, intraday, and overnight
quote-midpoint returns. The quote midpoint 15 minutes after open is used instead of open price.
A stock’s signals are constructed at horizons H∈ {m−1,(m−12, m −2),(m−60, m −13)}. Each
month, stocks are assigned to one of the ten portfolios of past return signals constructed based
on NYSE breakpoints. The time-series of equally-weighted month mportfolio returns, in excess
of 1-month T-Bill rates, are regressed on market, size, and value factor returns to estimate the
corresponding three-factor alpha, i.e., the intercept. The sample includes common shares listed on
NYSE, AMEX, and NASDAQ from 1989 to 2015, excluding the stocks with smallest 20% market-
capitalizations. Standard errors of estimates are constructed using the Newey-West approach based
on three lags with t-statistics reported in brackets.
Panel A: Portfolios of last month’s return (H=m−1)
Signal type 1 2 3 4 5 6 7 8 9 10 10−1
24-hour −0.13 0.14 0.19 0.16 0.2 0.16 0.12 −0.009 −0.054 −0.25 −0.12
[−0.91] [1.19] [1.68] [1.87] [2.51] [2.04] [1.66] [−0.13] [−0.67] [−2.17] [−0.56]
Intraday −0.017 0.22 0.2 0.18 0.2 0.15 0.14 0.087 −0.14 −0.4 −0.38
[−0.12] [2.07] [2.01] [2.14] [2.33] [2.3] [1.81] [1.22] [−1.84] [−3.82] [−1.84]
Overnight −0.34 −0.075 0.073 0.18 0.12 0.12 0.14 0.13 0.16 0.052 0.4
[−3.58] [−0.8] [0.84] [2.19] [1.41] [1.52] [1.59] [1.64] [1.81] [0.53] [2.73]
Panel B: Portfolios of months −12 to −2 return (H= (m−12, m −2))
Signal type 1 2 3 4 5 6 7 8 9 10 10−1
24-hour −0.64 −0.24 −0.025 0.015 0.16 0.22 0.24 0.25 0.22 0.36 0.99
[−3.24] [−1.68] [−0.22] [0.16] [1.84] [2.81] [3.06] [3.3] [2.57] [2.57] [3.27]
Intraday −0.55 −0.054 −0.003 0.14 0.1 0.17 0.24 0.18 0.15 0.22 0.77
[−3.26] [−0.49] [−0.03] [1.56] [1.2] [2.3] [3.36] [2.71] [2.21] [2.44] [3.35]
Overnight −0.28 −0.076 0.068 0.12 0.1 0.16 0.24 0.17 0.14 −0.062 0.22
[−2.52] [−0.76] [0.84] [1.5] [1.27] [2.19] [3.02] [2.33] [1.71] [−0.59] [1.38]
Panel C : Portfolios of months −60 to −13 return (H= (m−48, m −13))
Signal type 1 2 3 4 5 6 7 8 9 10 10−1
24-hour −0.047 0.052 0.17 0.13 0.11 0.074 0.082 0.065 0.049 −0.16 −0.11
[−0.59] [0.74] [2.23] [1.83] [1.57] [0.94] [0.98] [0.7] [0.53] [−1.24] [−0.72]
Intraday −0.07 −0.003 0.1 0.12 0.16 0.094 0.08 0.024 0.081 −0.048 0.022
[−0.56] [−0.02] [1.33] [1.52] [1.89] [1.13] [1.08] [0.27] [0.93] [−0.72] [0.17]
Overnight 0.026 0.043 0.11 0.12 0.14 0.067 0.16 0.019 0.039 −0.12 −0.14
[0.32] [0.54] [1.35] [1.57] [1.73] [0.86] [1.89] [0.19] [0.37] [−1.31] [−1.23]
38

Table 4. Three-Factor Alphas of Portfolios Based on Past Return Signals from Low-
vs. High-Volume Days. This table reports three-factor alphas of trading strategies that buy
past winners and sell past losers according to past return signals reflecting close-to-close, intraday,
and overnight returns. A stock’s signals are constructed at horizons H∈ {m−1,(m−12, m −
2),(m−60, m −13)}sampling daily returns from trading days with trading volumes no more than
the respective month’s median daily volume (Panel A) and from trading days with trading days
with trading volumes higher that the respective month’s median (Panel B). Each month, stocks are
assigned to one of the ten portfolios of past return signals constructed based on NYSE breakpoints.
The time-series of equally-weighted month mportfolio returns, in excess of 1-month T-Bill rates, are
regressed on market, size, and value factor returns to estimate the corresponding three-factor alpha,
i.e., the intercept. The sample includes common shares listed on NYSE, AMEX, and NASDAQ
from 1966 to 2019. Standard errors of estimates are constructed using the Newey-West approach
based on three lags with t-statistics reported in brackets.
Panel A: low-volume-day signals
Portfolio formation horizon, H
Signal type m−1 (m−7, m −2) (m−12, m −2) (m−36, m −12) (m−48, m −12) (m−60, m −12)
24-hour −0.36 0.81 0.93 0.086 0.12 0.087
[−3.3] [6.45] [6.66] [0.67] [0.98] [0.79]
Intraday −0.35 0.83 0.96 0.33 0.36 0.26
[−3.1] [6.38] [7.09] [2.73] [3.12] [2.21]
Overnight −0.2 −0.1 −0.083 −0.47 −0.42 −0.35
[−2.47] [−0.99] [−0.68] [−4.77] [−4.4] [−3.72]
Panel B: high-volume-day signals
Portfolio formation horizon, H
Signal type m−1 (m−7, m −2) (m−12, m −2) (m−36, m −12) (m−48, m −12) (m−60, m −12)
24-hour −1.14 0.38 0.41 −0.41 −0.38 −0.36
[−7.37] [2.38] [2.35] [−3.49] [−3.27] [−3.34]
Intraday −1.15 0.48 0.59 −0.17 −0.14 −0.19
[−7.89] [3.22] [3.79] [−1.39] [−1.21] [−1.83]
Overnight −0.3 0.032 −0.052 −0.48 −0.47 −0.43
[−2.91] [0.26] [−0.38] [−4.3] [−4.35] [−4.12]
39

Table 5. Three-Factor Alphas of Portfolios based on Sequential Double-Sorts of Past
Overnight and Intraday Returns. This table reports three-factor alphas of trading strategies
that buy past winners and sell past losers according to past intraday return signals after sorting each
cross-section first into quintiles of past overnight signals. A stock’s overenight and intraday return
signals are constructed at horizons H∈ {(m−12, m−2)}. In Panel A, each month, stocks are sorted
first into five portfolios (quintiles) of past overnight return signals and second into five portfolios
(quintiles) of past intraday return signals. In Panel B, each month, stocks are sorted first into five
portfolios (quintiles) of past intraday return signals and second into five portfolios (quintiles) of past
overnight return signals. Average monthly portfolio returns are calculated by quintiles of overnight
and intraday returns. Average portfolio returns as well as the return difference between past high-
and low-intraday return portfolios, given each past overnight return portfolio, are regressed on
market, size, and value factor returns to estimate the corresponding three-factor alpha, i.e., the
intercept. The sample includes common shares listed on NYSE, AMEX, and NASDAQ from 1966
to 2019. Standard errors of estimates are constructed using the Newey-West approach based on
three lags.
Panel A: sequential sort on past overnight and intraday signals
Portfolios of past intraday returns
Low 2 3 4 High High - Low
Portfolios of past overnight returns
Low −0.46 −0.14 0.020 0.12 0.26 0.72
[−2.67] [−1.40] [0.23] [1.55] [2.81] [4.62]
2−0.53 −0.025 0.026 0.17 0.37 0.90
[−4.49] [−0.32] [0.39] [2.53] [5.74] [5.82]
3−0.45 0.028 0.14 0.25 0.53 0.98
[−4.08] [0.35] [2.29] [3.62] [7.67] [6.72]
4−0.36 0.030 0.14 0.24 0.53 0.89
[−3.03] [0.43] [2.08] [3.82] [6.36] [5.26]
High −0.63 0.056 0.32 0.35 0.56 1.19
[−3.12] [0.56] [3.75] [4.16] [4.87] [5.30]
Panel B: sequential sort on past intraday and overnight signals
Portfolios of past overnight returns
Low 2 3 4 High High - Low
Portfolios of past intraday returns
Low −0.56 −0.58 −0.37 −0.42 −0.51 0.045
[-3.23] [-4.21] [-3.04] [-3.70] [-2.93] [0.27]
2−0.10 −0.028 −0.0078 0.027 0.27 0.37
[-0.93] [-0.35] [-0.10] [0.36] [3.48] [2.63]
3−0.045 0.037 0.19 0.19 0.29 0.34
[-0.48] [0.53] [2.84] [3.27] [4.03] [2.55]
4 0.0022 0.19 0.26 0.31 0.34 0.33
[0.03] [2.81] [3.89] [4.39] [4.44] [2.66]
High 0.18 0.31 0.38 0.59 0.60 0.42
[1.82] [4.86] [5.68] [6.92] [5.14] [2.62]
40

Table 6. Reversals and Momentum Anomalies - Fama-MacBeth Regressions. This table
reports partial Fama-MacBeth regression results of anomalies in monthly returns with respect to
past return signals. A stock’s signals reflect close-to-close, intraday, and overnight returns, denoted
R,IDR, and ON R, respectively, and are constructed at horizons H∈ {m−1,(m−12, m −
2),(m−36, m −13)}. Month mreturns are regressed on percentile statistics of a set of past return
signals and controls as described in Equation (1). Control variables include market-capitalization,
book-to-market ratio, dividend yield, and Amihud liquidity measure. The sample includes common
shares listed on NYSE, AMEX, and NASDAQ from 1966 to 2019. Standard errors of estimates
are corrected according to Newey-West approach based on three lags, with t-statistics reported in
brackets. Symbols *, **, and *** reflect statistical significance at type I error levels of 10%, 5%,
and 1%, respectively.
Dependent variable: Next-month return
R−1−1.98***
[−8.75]
IDR−1−1.75*** −1.82***
[−9.70] [−9.42]
ON R−1−0.30** −0.59***
[−2.43] [−5.28]
R(−12,−2) 1.15***
[4.75]
IDR(−12,−2) 0.91*** 1.02***
[6.16] [4.39]
ON R(−12,−2) 0.29*** 0.57***
[2.78] [3.52]
R(−36,−12) −0.22
[−1.17]
IDR(−36,−12) 0.053 −0.077
[0.42] [−0.47]
ON R(−36,−12) −0.23** −0.27**
[−2.54] [−2.12]
Controls Yes Yes Yes Yes
Observations 1,702,130 1,702,130 1,702,130 1,702,130
41

Table 7. Information Continuity and Momentum: Intraday vs. Overnight Signals.
This table reports three-factor alphas of trading strategies that buy past winners and sell past losers
according to past return signals as well as Da et al. (2014)’s information discreetness measure, both
reflecting intraday and overnight returns. A stock’s signals and information discreteness measures
are constructed at horizons H∈ {(m−12, m −2)}. Each month, stocks are assigned to one of the
five portfolios of past return signals. In each monthly portfolio, stocks are sorted into quintiles of the
respective information discreteness measure. Average monthly portfolio returns are calculated by
quintiles of past return and information discreteness, for overnight and intraday signals. For a given
information discreteness quintile, the time-series of the difference between winner and loser stock
returns are regressed on market, size, and value factor returns to estimate the corresponding three-
factor alpha, i.e., the intercept. The sample includes common shares listed on NYSE, AMEX, and
NASDAQ from 1966 to 2019. Standard errors of estimates are constructed using the Newey-West
approach based on three lags.
unadjusted 3-factor alpha
Information winner loser average ID return t-stat alpha t-stat
Intraday signal
discrete 1.29 0.94 0.02 0.35 2.6 0.42 3.32
2 1.35 0.99 −0.03 0.37 2.71 0.47 3.62
3 1.46 0.64 −0.07 0.82 5.34 0.95 6.68
4 1.53 0.46 −0.10 1.07 6.89 1.26 8.76
continuous 1.52 0.14 −0.18 1.38 7.16 1.61 9.28
Overnight signal
discrete 1.13 0.96 0.02 0.16 1.44 0.22 1.95
2 1.24 1.00 −0.05 0.23 1.84 0.29 2.25
3 1.15 0.97 −0.08 0.18 1.26 0.25 1.79
4 1.08 1.03 −0.13 0.052 0.32 0.098 0.63
continuous 1.06 1.01 −0.21 0.048 0.28 0.05 0.3
42

Table 8. Analyst predicted return association with intraday and overnight returns.
This table reports estimates from panel regressions with month fixed effects. All observations are
measured at the stock-month level. rt+1,t+12 is the cumulative return over the next 12 months.
pret is the consensus analyst predicted return over the next 12 months. ferr is the forecast error
defined as rt+1,t+12-pret. rov
t−1,t−12 is the cumulative overnight return over the previous 12 months.
rin
t−1,t−12 is the cumulative intraday return over the previous 12 months. All the above returns are
logarithmic returns. The row labeled p-value ∆ in/ov reports the p-value from the test that the
coefficients on rov
t−1,t−12 and rin
t−1,t−12are equal. The sample includes common shares listed on NYSE,
AMEX, and NASDAQ over 1999 to 2015 with a price greater than $5. pret is winsorized at 1%
and 99%. Intraday and overnight returns are constructed based on quote midpoints at 9:45am and
4:00pm. Standard errors of estimates are clustered by month, with associated t-statistics reported
in brackets. Symbols *, **, and *** reflect statistical significance at type I error levels of 10%, 5%,
and 1%, respectively.
(1) (2) (3)
Dep. Variable rt+1:t+12 prettferrt
rov
t−12:t−1-0.0608* -0.1146*** 0.0538
(-1.8182) (-14.910) (1.4756)
rin
t−12:t−10.1109*** -0.2318*** 0.3427***
(5.8508) (-30.052) (16.051)
R20.0165 0.1884 0.0650
Obs. 477,179 477,179 477,179
p-value ∆ in/ov 0.0000 0.0000 0.0000
43

A Appendix: Model
The model is directly based on Eyster et al. (2019). There is an economy with four dates (t=
0,1,2,3), which is populated by Ninvestors with exponential utility and identical risk aversion.
Figure 5illustrates the timeline. There is a risk-free asset, whose return is set to zero for simplicity.
There is a single risky asset, which pays at date 3 a terminal payoff equal to
d=¯
d+dn+dp+δ. (4)
First, ¯
dis a constant equal to the common prior of investors about the final value of the asset at
time 0. For simplicity, we assume that the asset is in zero net supply. Hence, p0=¯
d. Second,
the other three components are normally distributed zero-mean shocks that are uncorrelated with
each other: dnrepresents the news component of the fundamental value, which has variance τ−1
n;
dprepresents the private information component of the fundamental value, which has variance τ−1
p;
and δis the fundamental uncertainty, which has variance τ−1
δ.
Figure 5. Model timeline.
t=0 t=1 t=2 t=3
p0=¯
dPublic news Trading Payoff d
Overnight Intraday
Right before date 1, investors observe a common public signal about dn:sn=dn+n, where
n∼N(0, τ −1
n). We assume that investors agree on the interpretation of the public signal at t= 1.
Since investors do not face endowment shocks, there is no trade at date 1. However, investors may
underreact or overreact to the public signal. More formally, they believe that the precision of the
signal snis αnτn, where αn>1 (αn<1) indicates overreaction (underreaction) to the signal.
Right before date 2, investor ireceives a private signal about dp:sp,i =dp+p,i, where
p,i ∼N(0, τ −1
p). Investors are overconfident in the precision of their own signal relative to the
precision of others’ signals; i.e., investor ibelieves that the precision of her signal is αpτpand that
the precision of others’ signals is βτp, where αp≥1 and 0 ≤β≤1.
The period between date 0 and date 1 can be interpreted as overnight. This period’s return
(labeled rov) is driven by investors’ interpretation of public news. The period between date 1 and
date 2 can be interpreted as intraday. This period’s return (labeled rin) is driven by investors’
trading; that is, how much they trade on their private information and how they weight the infor-
mation conveyed through prices. Finally, the period between date 2 and date 3 can be interpreted
as a future period over which uncertainty is resolved. We label the return over this period rfut.
We first solve for the optimal demand and price at date 2. Agent imaximizes −Ee−γxi(d−p2)
as a function of her demand xi, where γis the risk aversion and p2is the equilibrium price. We
44

focus on linear price functions and conjecture that the price at date 2 is a function of the average
private signal
p2=a+b1
N
N
X
i=1
sp,i,(5)
where aand bare constants that are determined in equilibrium. Since the price is normally
distributed, it follows that the optimal demand is given by
xi=1
γ(V ar[d|Fi
2])−1E[d|Fi
2]−p2,(6)
where the expectation and variance are relative to the information set of agent iat date 2, Fi
2, which
includes her private signal (sp,i), the price, and all information from previous dates. Using the price
conjecture, agent’s ican extract the following signal from the price: s0
p,i =1
N−1p2−a
b/N −sp,i=
dp+1
N−1PN
j6=ip,j.According to agent’s ibeliefs, 1
N−1PN
j6=ip,j is a mean-zero shock with precision
β(N−1)τp. We can then compute the agent’s expectation and variance of the final value. Denote
by ¯
d1the common expected prior across agents about the final value right before date 2 and by τ¯
d1
its precision. We then get that
E[d|Fi
2] = ¯
d1+E[dp|Fi
2] (7)
=¯
d1+αpτp
τp+ (αp+β(N−1))τp
sp,i +β(N−1)τp
τp+ (αp+</