Seasonally Varying Mood and Preferences:
Evidence from Bid-Ask Spreads∗
Ramon P. DeGennaro, University of Tennessee
Mark J. Kamstra, York University
Lisa A. Kramer, University of Toronto
February 2017
JEL Classification: G12, G14
Keywords: time-varying risk aversion, seasonal depression, SAD
∗
Corresponding Author: Lisa Kramer, University of Toronto, 105 St. George St., Toronto Ontario Canada M5S
3E6; Email: [email protected]; Tel: 416-978-2496; Fax: 416-971-3048. We thank Sabrina Buti, David
Goldreich, Esther Eiling, Clifton Green, Larry Harris, Paul Irvine, Tim McCormick, Andreas Park, Lukasz Pomorski,
and particularly Ingrid Werner for helpful conversations. We are grateful to seminar participants at the Copenhagen
Business School, Kent State University, Laval University, the University of Alberta, and the University of Tennessee,
and to conference participants at the Northern Finance Association meetings. This research was funded in part by
a grant from the Scholarly Research Grant Program of the College of Business and a Finance Department Summer
Faculty Research Award at the University of Tennessee. Funding was also generously provided by the Social Sciences
and Humanities Research Council of Canada. This work was begun while Kamstra was an Associate Policy Advisor
at the Federal Reserve Bank of Atlanta. We gratefully acknowledge research support from the Finance Group there.
The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta
or the Federal Reserve System. Any errors are our own.
Seasonally Varying Mood and Preferences:
Evidence from Bid-Ask Spreads∗
Abstract
We document previously undiscovered seasonal patterns in bid-ask spreads which are economically and statistically significant. After controlling for well-known conditional effects on spreads,
such as risk, liquidity, and asymmetric information effects, we find that percentage effective spreads
from both the NYSE and NASDAQ exchanges are 5-10 basis points wider during the fall and winter relative to their unconditional averages around 100 basis points in recent years. Results based
on individual dealer quotes are smaller but still strongly statistically significant. Further, inside
spreads are narrower during these periods when spreads based on individual quotes and effective
spreads are wider. We argue that these patterns are consistent with increased risk aversion among
market makers. The effects are stronger during periods of high market volatility. We hypothesize
that the seasonal patterns in spreads are the logical outcome of seasonally changing risk aversion
among market makers and investors. Independent of the cause, researchers studying spreads need
to be mindful of the strong seasonal patterns in individual dealer spreads, effective spreads, and
inside spreads, as does any market participant who has discretion in the timing of his trades.
Does Risk Aversion Vary During the Year?
Evidence from Bid-Ask Spreads
Bid-ask spreads constitute a significant cost of transacting on equity markets. Past research
suggests spreads vary across the seasons. Examples include work by Fortin, Grube, and Joy (1989)
and Clark, McConnell, and Singh (1992) who find that average spreads are wider in the second
half of the year than they are at the start of the year. Additionally, Hong and Yu (2009) find
average spreads are wider during summer, perhaps due to increased adverse selection during a
period when many people take their annual vacations. To date, there have been no studies that
investigate whether seasonal variations in spreads may arise due to seasonal changes in the risk
aversion of market makers, a question we consider here, motivated in part by the empirical evidence
that seasonality in risk aversion affects other aspects of financial markets and in part by theory
underlying the market microstructure literature.
We study the bid-ask spreads of NYSE and NASDAQ firms and find new seasonal regularities
in spreads consistent with seasonally varying risk aversion on the part of market makers, even
after controlling for known determinants of spreads including liquidity, order processing costs, and
asymmetric information effects. (See Stoll, 1978a, 1978b.) The seasonal patterns depend on the
type of spread we consider, which we will argue supports the view that market makers exhibit
seasonal variation in their taste for risk. Specifically, quoted spreads and effective spreads are wider
in the fall and winter than at other times of the year. In contrast, inside spreads (based on the best
bid and offer prices) exhibit an opposite pattern relative to quoted and effective spreads: they are
significantly narrower in the fall and winter.
These seasonal patterns in spreads intensify during high volatility periods, even after controlling
in a variety of ways for risk. Theory dating back to Stoll (1978a) and Ho and Stoll (1981, 1983)
suggests the magnitude of spreads is determined in part by the risk aversion of dealers in securities
markets. Thus, we hypothesize that the seasonality in spreads may arise due to seasonality in market
maker risk aversion. Although an increase in risk aversion is expected to increase individual dealer
quotes, we investigate what impact a change in risk aversion might have on the spread between
the best bid and best offer price (i.e. the inside or market spread). We expect that higher market
1
maker risk aversion should lead, perhaps counterintuitively, to narrower inside spreads. Indeed this
is what we find in the data. Inside spreads are significantly narrower during the periods when we
expect market makers experience more risk aversion. (We detail the reasoning that leads us to
expect a narrowed inside spread would correlate with increased risk aversion in Section IV.)
If indeed seasonal variation in market maker risk aversion causes these seasonal patterns in
spreads, what be the underlying cause of seasonality in risk aversion? Evidence suggests seasonality
in risk aversion arises from seasonality in depression caused by seasonal affective disorder (SAD) and
its less severe counterpart, winter blues. Kamstra, Kramer, and Levi (2003) and related papers cite
an extensive literature which indicates that as much as ten percent of the population suffers from
severe seasonal depression during the fall and winter, and that depression in turn causes higher
risk aversion. Kamstra, Kramer, and Levi (2003, 2012, 2015) present empirical evidence that
equity returns and Treasury bond returns exhibit economically large and statistically significant
seasonal patterns which appear to be consistent with market participants experiencing changes
in risk aversion associated with seasonal depression. Kamstra, Kramer, Levi, and Wang (2014)
calibrate the preference parameters of an Epstein and Zin (1989) representative agent endowment
economy to show that modest variation in aversion to risk (and time preference of consumption) can
match empirical moments of the equity and bond return data. If, like other market participants,
some dealers are heterogeneous in their propensity to suffer from seasonal depression and hence in
the degree to which they exhibit seasonally varying risk aversion, then they might systematically
widen their quoted spreads in the fall and winter. Further, many market makers cover a portfolio
of firms so that even if only ten percent of market makers suffer from SAD, well over ten percent
of all quoted and inside spreads could easily be influenced.
Employing the clinically observed timing of onset and recovery from SAD, we control for the
incidence of SAD among the population and find that spreads indeed covary with seasonally varying
risk aversion. We find, further, that seasonally varying risk aversion is able to account for the
opposing seasonal patterns observed empirically in effective spreads and quoted spreads versus
inside spreads. The remainder of the paper proceeds as follows. In Section I we introduce the
data used for this study. In Sections II we define the various types of spreads we consider, and we
2
perform formal regression analysis to investigate seasonality in quoted, inside, and effective spreads
in Sections III, IV, and V respectively. Section VI contains a discussion of some common concerns
and questions related to SAD, including whether financial professionals such as market makers are
less likely to suffer from seasonal depression than the general population. Section VII summarizes
a range of robustness checks, and Section VIII concludes.
I
Bid-Ask Spreads Data and Models
We employ daily NYSE and NASDAQ data from CRSP, spanning January 4, 1993 (the first date
on which NYSE bid and ask closing prices are reliably available through CRSP) through December
31, 2014. The primary series of interest are the end-of-day quoted bid-ask spreads (i.e., individual
dealer quotes), end-of-day inside spreads (the best bid and best ask quoted across all dealers), and
effective spreads (calculated using the algorithm of Corwin and Schultz, 2012, to whom we are
grateful for making their code available).1
Each NYSE firm is covered by a single specialist, and that specialist provides at all times a bid
and ask price at which s/he is willing to trade some quantity of stock. Note that there may be
market participants willing to trade at a higher bid or lower ask than posted by the specialist, but
since we are interested, in part, in the time-series behavior of an individual’s quoted spread, we
consider the quoted spread of the specialist.
The spreads we observe for each firm from the NASDAQ exchange include the inside spread and
the Corwin and Schultz (2012) effective spread. Inside spreads are the result of best bid and offer
prices across multiple market makers; more than one market maker typically covers each NASDAQ
firm. In the extreme, individual firms have been covered by well over 50 market makers, though in
practice some of the registered market makers covering a given issue may not be active at a given
point in time. The best bid and best ask prices across all market makers’ quotes define an inside
1
One might question whether our use of closing bid-ask spreads is complicated by the closing auction which takes
place on the NYSE or the closing cross which takes place on NASDAQ. While some NYSE market participants submit
limit-on-close or market-on-close orders to participate in the NYSE closing auction, continuous trading remains active
during this period. Similarly, for NASDAQ the closing cross occurs in parallel with continuous trading, with active
continuous trading taking place even while the closing cross is in progress. For both NYSE and NASDAQ stocks, the
closing bid and ask prices provided by CRSP come from the continuous trading session and are not based on quotes
that arise from the NYSE closing auction or the NASDAQ closing cross. We thank Ingrid Werner for clarifying these
issues.
3
spread. We exclude financial services firms (SIC codes 6011-6799) from our analysis; robustness
checks indicate that the findings are insensitive to whether or not we do so.
While intraday bid and ask prices are available for both the NYSE and NASDAQ markets,
we restrict our study to the end-of-day bid-ask spread and the effective spread. The statistical
properties of several of the variables we seek to study (spreads, turnover, volume, etc.), follow
distinct intraday patterns, which may complicate the analysis of intraday data. By choosing a fixed
point in the trading day, we bypass those confounding influences.2 A feature of our use of data
at market close is that the effect of risk aversion on spreads may be most evident at that point in
the trading day. That is, the desire of a seasonally risk averse market maker to reduce his long or
short position may be particularly acute at market close, when he faces the prospect of holding risk
overnight. We find in robustness checks, however, that our findings of seasonal patterns in spreads
are not limited to data from market close.
In cases where the closing bid and ask prices are not representative, in the sense that they do
not appear to be typical of the day’s trading activity, CRSP assumes that the quote was posted
by a dealer who was required to post a quote but was not interested in trading. (As described by
Harris, 2001, page 511, any NASDAQ market maker who stops making a market in a given stock
is required to wait 20 days before he resumes trading. Thus a market maker who needs to step
away from the market temporarily tends to post an exceptionally wide spread in order to avoid
trading while technically remaining in the market.) CRSP sets the closing bid and closing ask
prices to zero in such cases, pending further review. In order to ensure that we work with valid
quotes, we exclude observations where the bid price or ask price is zero. Conversations with CRSP
representatives revealed that roughly 5% of all bid and ask data points were set to zero in the
initial release, with over one-third of these cases occurring between October 2000 and May 2001.
Our analysis found that these zero bid and ask prices (which we then set to missing) were most
2
In order to compare inside quotes to individual dealer quotes on the same firms, we considered several approaches
utilizing the TAQ database, but significant problems with each of these compromise their use. We considered
comparing NASDAQ BBO quotes on NYSE-listed firms to NYSE quotes on those same firms, but NYSE quotes on
NYSE-listed firms have been, over much of the sample we consider, the most meaningful quotes, as other exchanges
post wide quotes not reflecting subsequent trade execution. We also considered intraday Nastraq data. Based on
the Nastraq database, seasonal patterns in end-of-day quotes are qualitatively similar to though statistically less
significant than the seasonal patterns in NYSE quotes on NYSE-listed firms. The slightly reduced significance may
arise from the shorter sample period. (Nastraq data is available starting only in 1999.)
4
concentrated in 72 trading days, February 6, 2011 through May 18, 2001, inclusive. Our findings
are qualitatively identical if we exclude dates with sparsely available closing bid and ask quotes.
Note that this anomaly is isolated to the NYSE data only and does not affect the NASDAQ results
we report below.
Using the firm-level data for common-share equities, we form equal-weighted daily deciles based
on the previous day’s average market capitalization. In our primary results we exclude firms for
which the previous day’s closing price was less than $5 or over $1,000. We also exclude financial
services firms and observations with spreads of over $5. We confirm in robustness checks that our
findings are unchanged when those firms are included. We analyze deciles data instead of individual
stocks for several reasons. First, the idiosyncratic noise in bid-ask spreads is high for individual
stocks, and second, the correlation between volume and turnover is high for individual stocks since
shares outstanding changes only rarely for many firms. Finally, it is infeasible to do a systems
estimation that exploits cross-firm correlation between spreads for thousands of stocks in the cross
section (which is a familiar problem in the asset pricing literature).
II
Spreads
Our study begins with a comprehensive analysis of the NYSE and NASDAQ bid-ask spreads and related data required for our conditional analysis, including volume, volatility, and turnover measures.
The three types of spreads we employ are calculated as follows.
The percentage quoted spread, QSpread, for a particular NYSE stock, k, on a particular trading
day, t, is defined as:
QSpreadk,t =
(P askk,t − P bidk,t )
(P askk,t + P bidk,t )/2
(1)
where P askk,t and P bidk,t are the closing bid and ask prices.
We calculate the inside spread for a given NASDAQ stock, k, using the best bid and ask prices at
market closing on day t, P bestbidk,t and P bestaskk,t . Then the percentage inside spread, ISpread,
is defined as:
ISpreadk,t =
(P bestaskk,t − P bestbidk,t )
.
(P bestaskk,t + P bestbidk,t )/2
5
(2)
By using inside spreads at close, we benefit from the point raised by Wahal (1997) that closing
quotes more likely reflect competitive conditions since NASDAQ spreads are at their narrowest
point of the trading day at close (an intraday regularity shown by Chan, Christie, and Schultz,
1995).
Finally, we estimate the effective spread, ESpread, for NYSE and NASDAQ firms following the
approach developed by Corwin and Schultz (2012), which employs daily high and low stock prices
over one-day and two-day intervals. Please see Corwin and Schultz (2012) for details.
Table 1 contains summary statistics for the daily firm-level data pooled across all firms and
over all trading days from 1993 to 2014. Panel A corresponds to the NYSE data, for which there
are over seven million observations. The average quoted spread is about 1 percent, with a minimum
of less than a hundredth of a percent and a maximum of over 50 percent. The average effective
spread is roughly 0.75. The bid and ask prices both have an average around $31. The mean daily
percentage return is 0.079 percent, with a minimum around -80 percent and a maximum over 360
percent. The mean dollar volume is about $33 million. Average daily turnover is about two thirds
of a percent. Most of these series are highly skewed and kurtotic.
Panel B corresponds to the NASDAQ data, for which there are over nine million observations.
The average NASDAQ bid and ask prices are around $20, the average inside spread is about 1.8
percent and the average effective spread is about 1.6 percent. Relative to the NYSE data in
Panel A, the NASDAQ prices are lower on average. On a percentage basis, NASDAQ spreads
are much wider than spreads from the NYSE, consistent with the relatively smaller, risker nature
of most NASDAQ firms. The average daily return for the NASDAQ firms in our sample is 0.14
percent. The average daily volume of about $20 million is lower than for NYSE firms. The average
daily turnover of NASDAQ firms is higher than that of NYSE firms, at about 1.0 percent and 0.7
percent respectively, and also much more variable with a standard deviation almost triple that of
NYSE firms. The average NASDAQ firm has about 25 market makers, ranging from a low of zero
to a high of 120.3
3
In the very small number of cases where we had valid bid and ask data but the recorded number of market
makers was zero, we kept the observation in our sample. Our results are not sensitive to the inclusion of these
observations. In practice, the vast majority of firms have at least two market makers. NASDAQ Marketplace Rule
4310(c)(1) indicates “For initial inclusion, the issue shall have three registered and active market makers, and for
6
Table 2 contains summary statistics for the NYSE (Panel A) and NASDAQ (Panel B) data.
For each decile, we present summary statistics over the entire sample on spreads, returns, market
capitalization, volume, turnover, and return variance (detrended). In each of Panels A and B, as
expected, Decile 1 (the decile of the smallest firms) has the widest average quoted spreads. The
average spreads narrow monotonically through the deciles. Decile 1 also contains firms with the
lowest volume, which rises monotonically through the deciles. Although this fact is not apparent
from summary statistics on percentage spreads, we note that at no point in either sample do
average spreads for the size-sorted deciles approach the minimum tick size; naturally, individual
firms’ spreads do occasionally reach the minimum. The average detrended variance values for each
decile are close to 1 by construction.
We do not present summary statistics by year, but we can report that, generally speaking,
NYSE tick size was stable over the 1990s, varying between 2 and 3 percent, and then after the 2001
minimum tick size reduction there was an abrupt decline in tick size which stabilized by 2004, not
rising again substantially until the financial crisis of 2008. The NASDAQ tick size declined much
more steadily than seen with the NYSE, stabilizing in 2006 and 2007 around 90 basis points before
rising sharply with the financial crisis. Effective spreads have been much more stable over time
than either quoted or inside spreads.
In Figure 1 we provide plots of the monthly average spreads for five quintiles and averaged across
all stocks for various types of spreads for both NYSE and NASDAQ stocks, starting with the month
of July in each plot. (Although spreads have declined in magnitude over our sample period, plots
covering 2002-2015 are similar to the ones shown here, as are plots of detrended spreads.) These
monthly average spreads are smoothed with a 3-month centered moving average. Panel A depicts
dealer quoted spreads for NYSE stocks, Panel B depicts data on inside quotes for NASDAQ stocks,
Panel C depicts data on effective quotes for NYSE stocks, and Panel D depicts data on effective
quotes for NASDAQ stocks. In each panel, the average monthly spread across all stocks on the
exchange is denoted with a starred line. The quintile associated with the smallest-capitalization
continued inclusion, the issue shall have two registered and active market makers, one of which may be a market
maker entering a stabilizing bid.” If a firm has less than two market makers for a period of 10 days or more, the
firm risks having its shares delisted.
7
stocks is denoted with the thinnest dotted line, and the dotted lines get progressively thicker as
market capitalization increases across the quintiles.
In visually evaluating the seasonal patterns in spreads shown in Figure 1, we must keep in mind
the fact that the plots depict unconditional patterns in the data; later we consider seasonal patterns
in spreads that emerge from models that control for important factors known to affect spreads. In
Panel A of 1, we see that quoted NYSE dealer spreads are relatively larger in the fall and winter,
with a fairly smooth transition in spreads from one season to another. In Panel B, the pattern
in NASDAQ inside spreads is muted with no strong seasonal pattern. Panels C and D present
monthly effective spreads, which The monthly effective spreads peak in the fall or winter, similar
to the NYSE quoted spreads. The pattern is more muted for NASDAQ stocks. As we shall see in
formal regression analysis, the seasonal patterns suggested by Panels A, C, and D are present and
significant when controlling for volume, turnover, and volatility. Additionally, an opposite seasonal
pattern in NASDAQ inside spread data emerges when controlling for these important factors.
A
Known Determinants of Spreads
Most previous studies of spreads consider effective spreads or inside spreads, not the quotes of
individual market makers. We show below that most variables known to influence effective and
inside spreads have a similar effect on quoted spreads. Following Stoll (1978a, 1978b), Ho and Stoll
(1981, 1983), and others, when modelling spreads we control for adverse selection (using turnover),
holding period (using volume), and risk (using return variance). Additionally, to control for the
effects of competition, when working with NASDAQ data we control for the number of market
makers covering a given firm.
Over the period we consider, 1993 to 2014, there have been a number of regulatory changes
affecting minimum tick size. with the NYSE and NASDAQ exchanges having reduced the minimum tick size twice. There was a shift to quote prices in sixteenths of a dollar instead of eighths in
1997, taking effect on June 2 for NASDAQ and June 24 for the NYSE, and then in 2001 decimalization took effect, with prices quoted in increments of $.01 by January 29 for the NYSE and on
April 9 for NASDAQ. We use dummy variables to capture the effect of these changes on spreads:
8
SixteenthsN AS is set to equal one for dates starting June 2, 1997, SixteenthsN Y is set to equal
one for dates starting June 24, 1997, DecimalsN Y equals one for dates starting April 9, 2001, and
DecimalsN AS equals one for dates starting January 29, 2001. All four variables equal zero otherwise. For more details on these changes, see Huang and Stoll (2001), Gibson, Sing, and Yerramilli
(2003), Chung, Chuwonganant, and McCormick (2004), and Wu, Krehbiel, and Borsen (2011).4
Before presenting detailed regression results, we report where we obtain and how we calculate
some of the variables the prior literature suggests we should control for when modeling spreads,
including volume, variance, and turnover. We obtain from the CRSP Daily Stock database daily
observations for the variables for each stock: trading volume, shares outstanding, and closing price.
Based on this set of information, we are able to calculate turnover and variance for each stock.
We calculate the lagged logarithm of dollar volume for each decile as follows: We take each
stock’s closing price for the previous day times its share volume for the previous day (in millions),
calculate the average across all stocks in the decile, then take the natural logarithm of that figure.
We calculate lagged variance for each decile on each trading day as follows. First, for the stocks
which are in a particular decile on a given day, we calculate the previous month’s average daily
variance using squared returns. This yields a daily raw variance series. Next, we detrend the decile’s
raw variance series by dividing by the decile’s raw variance averaged over the past 252 trading days.
Finally, we multiply by 100. The result is our (detrended) lagged variance series, in percentage
terms.5 We calculate lagged turnover for each decile by taking a given stock’s share volume for the
previous day divided by its number of shares outstanding for the previous day, averaging across all
stocks in the decile, and multiplying by 100.
4
Another noteworthy event was the observation by Christie and Schultz (1994) that NASDAQ market makers
avoided odd-eighths for many large stocks, and Christie, Harris, and Schultz’s (1994) observation that they stopped
doing so immediately after the media reported on the finding. Since this practice was observed in a fraction of the
full universe of NASDAQ stocks and it ended early in our sample period, we don’t include a dummy variable for this
particular event. Results are qualitatively identical if instead we include such a dummy variable.
5
This method of specifying variance takes account of the fact that volatility models and risk are highly persistent.
Our findings with respect to the seasonal patterns in bid-ask spreads, presented below, are virtually identical if we
use an alternate measure of risk, such as GARCH volatility or realized volatility.
9
B
Measuring Seasonality
Researchers in medicine and psychology have established that as much as ten percent of the population suffers from a severe form of seasonal depression known as seasonal affective disorder, or SAD.6
Additional numbers suffer from a milder form of the condition which is often called winter blues;
indeed most people may experience seasonal mood changes to some extent (see Harmatz et al.,
2000, and Kramer and Weber, 2012). Onset of seasonal depression is typically in the fall, recovery
is typically in the spring, and it is well accepted by medical professionals that the primary cause of
the seasonal variation is a reduction in hours daylight, as opposed to other environmental variables
such as rainfall or cloud cover. Importantly, seasonal depression is associated with an increase
in financial risk aversion; see Kramer and Weber (2012). Building on the conjecture that some
market participants experience seasonally varying risk aversion, we explore whether seasonality in
depression and risk aversion might help explain the seasonal behavior of bid-ask spreads.7
To model the seasonal pattern in (some) market makers’ risk aversion, we follow Kamstra,
Kramer, and Levi (2012, 2015) and adopt a measure of seasonality which is based directly on
the clinical incidence of SAD. Young et al. (1997) and Lam (1998) studied hundreds of North
Americans who suffer from SAD, documenting the precise point in the late summer or fall when
each individual’s SAD symptoms first arose as well as the point in winter or spring when symptoms
resolved. We use the Lam data to create a proxy for the timing of seasonal changes in risk aversion
among those who are affected by seasonal depression.8 First, we take the proportion of SAD-suffers
in the Lam sample who are actively experiencing SAD symptoms in a given month. Next, we
calculate the cumulative proportion of subjects who experienced the onset of their SAD symptoms
(cumulated starting in late summer, the earliest point at which any subjects are first diagnosed with
onset) and then deduct the cumulative proportion of subjects who experienced full recovery from
SAD. We then produce a daily measure of SAD incidence by smoothly interpolating the monthly
6
SAD was first documented by Rosenthal et al. (1984). See Kamstra, Kramer, and Levi (2012) for extensive
citations to the medical literature and for a discussion of variation in the prevalence by location.
7
Our work is also motivated by the findings of Goetzmann and Zhu (2005), who show that NYSE specialists’
quotes vary with New York City weather.
8
We adopt the Lam data here because that study provides the timing of both onset of and recovery from seasonal
depression symptoms, whereas Young et al. consider onset only. Our findings are unchanged if we use combine the
Lam and Young et al. data.
10
incidence of SAD to daily frequency using a spline function. The result is a daily measure of
SAD incidence, taking on values between zero percent in summer and close to 100 percent in winter
(indicating that close to 100 percent of the people who suffer from SAD have succumbed by winter).9
Because this proxy measures the true incidence of SAD with error, we use an instrumented version
of the measure to avoid an errors-in-variables bias; see Levi (1973).10 Results are very similar if we
use the SAD incidence proxy directly, without instrumenting.
III
Seasonal Patterns in Quoted Spreads
The first regression we consider is with quoted spreads as the dependent variable. Recall that we
can calculate quotes for NYSE stocks only, and the quoted spread is based on an actual bid and
ask pair posted by the specialist during end-of-day continuous trading. (See footnote 1 for more
details.) Following Stoll (1978a) and Ho and Stoll (1981, 1983), we expect to find wider spreads
during periods when specialists exhibit heightened risk aversion.
We use GMM to jointly estimate a set of regressions for the NYSE quoted spread deciles (10
equations in all), where decile 1 contains the smallest capitalization firms and decile 10 is the largest:
2
QSpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
+ µDecimalsN Y Di,t
+ it .
9
(3)
In studying security return and capital flow seasonalities, Kamstra, Kramer, and Levi (2012, 2015) and Kamstra,
Kramer, Levi, and Wermers (2016) use the change in SAD incidence (which they call SAD onset/recovery) instead of
the level of SAD incidence. In this paper we use the level, not the change, for the following reason. Security returns
and capital flows, which are income flows, should respond to the flow of SAD-affected investors (i.e., the change in
SAD incidence). In contrast, spreads, being a stock concept instead of a flow concept, should respond to the stock
of actively suffering SAD-affected investors (i.e., the level of SAD incidence).
10
To produce the instrumented version of SAD incidence, first we run a logistic regression of the daily incidence
on our chosen instrument, the length of day, i.e., the time between sunrise and sunset. (The nonlinear model is
1/(1 + eα+βdayt ), where dayt is the length of day t in hours in New York and t ranges from 1 to 365. This particular
functional form is used to ensure that the fitted values lie on the range zero to 100 percent. The β̂ coefficient estimate
is 1.24 with a standard error of 0.0255, and the regression R2 is 95.7 percent.) The fitted value from this regression
is the instrumented measure of incidence. Employing additional instruments, such as change in the length of the
day, makes no substantial difference to the fit of the regression or the subsequent results using this fitted value. For
the purpose of illustrating the annual seasonal cycle in the incidence variable, here are the mean monthly values of
the daily instrumented incidence variable we use in our regressions (starting with January): 0.94, 0.82, 0.48, 0.15,
0.04, 0.02, 0.03, 0.11, 0.38, 0.76, 0.93, 0.96. Of course, we use the daily values rather than the monthly means in our
regressions.
11
The dependent variable in each equation is the average daily percentage quoted spread, QSpreadi,t ,
where the average is formed equally weighted based on firms in decile i on day t. V olumei,t−1 is
the log of decile i’s average dollar volume (in millions) at t − 1. T urnoveri,t−1 is the average at
2
t − 1, across all firms in decile i, of share volume divided by shares outstanding, times 100. σi,t−1
is the variance of returns, detrended by dividing by average variance over the past 365 calendar
days. DtSixteenthsN Y is a dummy variable set to equal one for dates on or after June 24, 1997, and
DtDecimalsN Y is a dummy that equals one for dates on or after January 29, 2001. T rendi,t−1 is the
mean spread over the last year, and controls for the downward trend in spreads. We test for the
influence of seasonal changes in risk aversion on the width of quotes using an explanatory variable
which is a function of the clinically observed incidence of SAD over time, Incidencet . Specifically, for
2
and by
each decile we multiply Incidencet by the square root of decile i’s detrended variance σi,t−1
decile i’s one year average of spreads, T rendi,t−1 .11 We control for autocorrelation in the dependent
variable through our use of heteroskedasticity and autocorrelation consistent (HAC) robust t-tests
based on Newey and West (1994) standard errors, though our findings are qualitatively unchanged
if instead we use MacKinnon and White (1985) heteroskedasticity-consistent standard errors and
include as regressors sufficient lags of the dependent variable to produce white-noise residuals.
We now consider results from estimating this model. The top section of Table 3 contains
parameter estimates and standard errors. In all tables, one, two, and three asterisks denote significance at the ten, five, and one percent level of significance respectively, based on two-sided tests.
The lower portion of Table 3 contains R-squared, a χ2 test for the presence of up to 10 lags of
autocorrelation, and a χ2 test for the presence of up to 10 lags of ARCH. There remains significant
evidence of autocorrelation and ARCH in all cases, reinforcing our choice of HAC standard errors.
Consider the coefficient estimates and t-statistics in the top portion of Table 3. Consistent with
the theory of Stoll (1978) and Ho and Stoll (1981, 1983) that higher dealer risk aversion should
11
In robustness checks, we also find statistically significant effects using Incidence alone (not multipled by T rend
or σ 2 ) or using Incidence multiplied only by T rend as an explanatory variable, instead of using the product of
Incidence, T rend, and σ 2 . We elected to use the product of Incidence, T rend, and σ 2 because the impact of the
Incidence variable should be greater during high-volatility periods if the effect truly arises due to seasonally varying
risk aversion. Furthermore, some of our spread measures, quoted and inside spreads in particular, have strong
downward trends over our sample period. Use of the three-way product allows us to avoid constraining the effect of
SAD on spreads to be a constant and instead allows the effect to shrink as spreads shrink.
12
lead to wider spreads quoted by dealers, we find that the SAD incidence variable is positive and
significant for all ten spread deciles. That is, spreads are wider during periods when the incidence of
seasonal depression is high. The positive coefficients suggest that, on average, market makers quote
wider spreads during periods when risk aversion tends to be higher. While the relationships between
season, mood, risk aversion, and spreads is not necessarily causative, the correlation between season
and spreads is undeniably strong.
Signs and significance of other variables in the model are as follows. Consistent with theory, we
observe mostly negative volume coefficients and uniformly positive coefficients on variance. Many
of the coefficient estimates on turnover are positive, but there is some variability, perhaps due to the
high degree of multicollinearity with volume. The far-right column of Table 3 contains joint test
statistics, testing whether a given set of coefficients is jointly zero across all ten equations. All of
the tests strongly reject the null hypothesis at conventional levels of significance, notably including
the case of Incidence.
We now consider the economic impact of the statistically significant seasonal pattern in quoted
bid-ask spreads. Given the construction of the incidence variable, the average economic impact
is the coefficient on the incidence variable times the average percentage spread over the past year
times the value of the SAD incidence measure (which varies from about 0 in July to about 1 in
December). In Figure 2 we plot the value of this economic impact measure over time by decile
(lower series) along with the average value of the NYSE effective spread over time (upper series).
Figure 3 reproduces the economic impact measure using a finer scale, to allow closer inspection.
Note decile 1 is denoted with a dotted line, decile 10 appears with a solid line, and the average
across all deciles is marked with a heavy solid line. The annual peak value of this economic impact
measure exceeded 15 basis points for about the first half of the sample; in more recent years it peaks
around 1 to 5 basis points. The current magnitude of the economic impact represents about a few
percent of the overall quoted spread across the full sample period. In analysis below, we document
even larger economic effects for inside spreads and especially effective spreads.
13
IV
Seasonal Patterns in Inside Spreads
In the previous section, we saw that seasons when specialists may experience higher risk aversion
are associated with wider quoted spreads. We now consider the case of inside spreads, where we
find higher risk aversion is associated with narrower inside spreads. We start with a demonstration,
to show why we expect inside spreads to vary with risk aversion in a manner opposite to that of
quoted spreads.
A
Hypothesis
Consider contrasting two distinct markets for an asset: Market A is composed of homogeneous
dealers quoting spreads of X dollars for the asset, and Market B is composed of a mix of dealers in
which some quote a spread of X dollars and others (the more risk averse dealers) quote a spread of
2X dollars. Note that some of the market makers in either market may hold net long inventories
and others may hold net short inventories. In Market A, with homogeneous dealers, the inside
spread will tautologically be equal to X dollars. But what about Market B, with the heterogenous
dealers? If both types of dealers in the heterogenous-dealer market quote spreads with the same
midpoint, then this market would also display an inside spread of X dollars, and the risk averse
dealers would either have to trade using the prices defined by this inside spread or leave the market.
But would the midpoint necessarily be the same for both types of dealers in market B?
To answer this question, consider a dealer who holds a net long inventory and is the type of
dealer that is more averse to risk. We note that, relative to the less risk averse dealer, the more
risk averse dealer would, of course, be expected to post a lower bid. With his higher degree of risk
aversion he would prefer to buy the risky asset at a lower price than his less risk averse colleagues, to
offset his greater distaste for risk.12 Similarly, he would also choose a lower ask price than less risk
averse dealers in order to more quickly liquidate (at least some of) his risky positions, consistent
with his preference to face less exposure to risk. Although his spread would be wider, his midpoint
should be lower than that posted by the less risk averse dealers, and most importantly, his asking
12
Of course, to remain in the market he may have to have to execute trades at less favorable prices than he would
prefer under ideal circumstances, as many dealers do.
14
price should be lower than that posted by the less risk averse dealers. His wider spread (which allows
him greater compensation for any risk he does continue to face, consistent with theory) would be
facilitated only if his bid were to fall more than his ask. This is a standard tactic employed by
dealers.13
Consider now a dealer who holds a net short inventory and is the type of dealer that is more
averse to risk. Theory dictates that a short market maker with high risk aversion would also post
wider spreads than less risk averse dealers. In contrast to the long market maker, however, the
short market maker would raise his bid and ask prices relative to the less risk averse dealer, with
the ask rising more than the bid to facilitate a wider spread. His higher bid price would indicate
that he desires to reduce his short exposure, and his higher ask price would reflect that he is willing
to increase his short position only at a relatively more attractive price. With the less risk averse
dealers posting a spread of X dollars with midpoint M (i.e., an ask price of M + X/2 and a bid
price of M − X/2), the dealers who are more risk averse and net long posting wider spreads with
ask price less than M + X/2, and the dealers who are more risk averse and net short posting wider
spreads with bid price greater than M − X/2, the inside spread in this market should be less than
X dollars. (As discussed by Stoll, 1989, and many others, the inside spread need not necessarily
be based on the bid and ask of a single dealer.) With time-varying risk aversion, markets may vary
between trading sessions with dealers exhibiting homogeneous risk aversion and trading sessions
with dealers exhibiting heterogeneous risk aversion, so that individual quoted spreads and inside
spreads widen and narrow in mirror image fashion over time, as we see in the data.
In Figure 4 we present a diagrammatic representation of bid prices, ask prices, and spreads for
a representative stock during two different periods for an exchange with market makers who are
heterogeneous in their propensity to exhibit seasonally varying risk aversion. The period “before”
anyone is suffering from seasonal depression (spring and summer) is on the left. For simplicity, we
assume all market makers have the same information and quote the same bid and ask prices during
this period, so everyone’s quoted spread is exactly the same as the inside spread. For the sake of
simplicity, we also assume the SAD-affected market makers hold long inventory positions, though
13
See Table 13-2 on page 285 of Harris (2003) for a summary of the tactics dealers use to manage their inventories,
including changing bid and ask prices to influence clients’ buy or sell decisions.
15
we show below that the implications for inside spreads are invariant to whether market makers
hold long, short, or a mix of long and short positions. The quoted spread is indicated by the solid
line in two-dimensional space, with vertical distance representing price: the ask price is above (i.e.,
exceeds) the midprice, which in turn exceeds the bid price. The vertical distance between the ask
and the bid represents the spread, both quoted and inside in this case.
The period denoted “after,” shown to the right of the dotted line, depicts the period when
some market makers are suffering from SAD and consequently feeling depressed and experiencing
higher risk aversion. As stated previously, as much as ten percent of the population suffers severely
from the medical condition of SAD, and while more may suffer from sub-clinical forms of seasonal
depression, it is probably fair to assume that many market makers do not suffer from depression due
to SAD. Abstracting from the possibility of strategic behavior (which would require some market
makers having specific knowledge about SAD impacting other market makers at particular points
in time), a market maker unaffected by SAD would likely have no reason to revise his bid or ask
price. Thus the quoted spread of the market maker unaffected by SAD is identical to his “before”
quote. (Again, this quote is also depicted with a solid vertical line.) A market maker who suffers
from SAD, however, might lower his bid and ask prices. His choice to reduce his bid price would
arise because with his now-higher degree of risk aversion, he is willing to buy the risky asset only
at a lower price. The decision to reduce his ask price means he can more quickly liquidate risky
positions. Dropping the bid more than the ask would be consistent with him now demanding more
compensation for holding risky positions and would lead to the widening of his quoted spread. His
lower bid and ask prices are shown with the diagonal-striped line. With some market makers having
left their quotes unchanged and others having adjusted their quotes to be lower and their spreads
to be wider, the end result is a narrower inside spread, as depicted by the short vertical line with
horizontal stripes. (The end result is an inside quote representing the bid of one dealer and the ask
of another. As discussed by Stoll, 1989, and many others, the inside quote need not necessarily be
composed of the bid and ask of a single dealer.)
If instead of assuming the SAD-affected market maker holds long inventory we were to assume
he held short inventory, his bid and ask price would both rise instead of falling, which would still
16
result in a narrower inside spread than would be observed in absence of SAD. Likewise, if some
SAD-affected market makers are long while others are short, some would drop their bid and ask
prices while others would raise their bid and ask prices, and the end result would still be a narrower
inside spread.
In short, we hypothesize that with the onset of SAD in autumn, some market makers change
their bid and ask prices and simultaneously widen the spreads, and then with their recovery from
SAD (at some point in the winter or spring) they revert their bid and ask prices and narrow their
spreads. The implication of this hypothesis is that we should observe opposite movements in inside
versus quoted spreads during the fall and winter periods. Above, we saw evidence of seasonal
patterns in quoted spreads consistent with this hypothesis; we turn now to inside spreads.
B
Inside Spreads Data
NYSE firms in effect have only a single market maker, the specialist. To test the SAD hypothesis
more fully, however, we need inside spreads which are defined by the interaction of the quotes of
more than one market maker. Thus, we employ inside spread data for stocks traded on NASDAQ.
As discussed by Wahal (1997), the NASDAQ National Market allows a dealer who wishes to make a
market in a security to begin doing so within a day of registration. This low barrier to entry makes
for a fairly competitive dealer market. In contrast to NASDAQ intraday quoted spreads (which are
available starting from 1999), NASDAQ end-of-day inside spreads are available as far back as the
1980s, allowing us to match the sample period we consider in studying NYSE quoted spreads. We
now turn to the conditional evidence of seasonality in inside spreads.
C
Regression Analysis, NASDAQ Inside Spreads
We use GMM to jointly estimate a set of regressions for the inside NASDAQ spread deciles (10
equations in all):
17
2
ISpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
+ µM M Cnt M M Cnti,t−1
SixteenthsN AS
DecimalsN AS
+ µT rendi T rendi,t−1 + µSixteenthsN AS Di,t
+ µDecimalsN AS Di,t
+ it .
(4)
The dependent variable in each equation is the average daily percentage inside spread, ISpreadi,t ,
formed by equally weighting inside spreads for firms in decile i on day t. The Incidencet , V olumet−1 ,
2
, and T rendt−1 variables are defined analogously to the variables used in estiT urnovert−1 , σt−1
mating Equation (3), but based on the NASDAQ inside spread decile data where appropriate.
DtSixteenthsN AS is a dummy set to equal 1 for dates on or after June 2, 1997, and DtDecimalsN AS is a
dummy that equals 1 for dates on or after April 9, 2001. M M Cnti,t−1 is equal-weighted average
the market maker count for firms in decile i at t − 1. As before, we control for autocorrelation in
the dependent variable through our use of HAC robust t-tests based on Newey and West (1994)
standard errors, and our findings are qualitatively unchanged if instead we use MacKinnon-White
heteroskedasticity-consistent standard errors and include as regressors sufficient lags of the dependent variable to produce white-noise residuals.
We now consider results from estimating this model. Consider the coefficient estimate on the
SAD incidence variable of Table 4. In contrast to our earlier results based on quoted spreads, for
inside spreads we expect to find a negative coefficient on the incidence coefficient. This is precisely
what we find. The estimate is negative and significant for all deciles. That is, while quoted spreads
are wider during periods when individuals suffer from SAD, inside spreads are narrower.
As theory predicts, the coefficients on variance are everywhere positive, those on volume are
everywhere negative, and those on turnover are everywhere positive, in each case strongly significantly so. The last column of Table 4 contains several joint test statistics, testing whether a given
set of coefficients is jointly zero across all ten equations. All of the tests strongly reject the null
hypothesis at conventional levels of significance.
We plot the economic magnitude of the effect arising due to Incidencet in Figures 5 and 6. The
economic magnitude of this effect amounts to a narrowing of inside spreads by several basis points.
18
The fact that quoted and inside spreads move in opposite directions during specific seasons of
the year is perhaps a counter-intuitive result. The finding can be explained, however, by seasonally
changing risk aversion among market makers. Figure 4 shows that if, during the autumn, some
SAD-affected long-inventory market makers progressively drop their bid and ask prices while simultaneously widening the spreads, and then at some point before the end of winter they progressively
raise their bid and ask prices while narrowing their spreads, the implication is opposite movements
in inside versus effective spreads during the fall and winter periods. Similar implications hold for
effective spreads and inside spreads if market makers hold short inventories or a mix of long and
short inventories.
V
Seasonal Patterns in Effective Spreads
A feature of quoted spreads is that we cannot evaluate them for NASDAQ stocks, and similarly,
inside spreads are not defined for NYSE stocks. An advantage of effective spreads is that we can
calculate them for both NASDAQ and NYSE stocks.14 Considering effective spreads therefore helps
us examine whether the opposing results we obtained for quoted versus inside spreads are an artifact
of differences across the exchanges.
We expect the seasonal behavior of effective spreads to be somewhat similar to quoted spreads,
in the sense that they ought to be wider during periods of higher risk aversion. Thus if exchange
differences are not the underlying reason for the opposite patterns we have seen for quoted versus
inside spreads, then we ought to observe effective spreads widening during periods of heightened
risk aversion, for both the NASDAQ and NYSE effective spreads data. Indeed, that is what we
find.
A
NYSE Effective Spreads
We use GMM to jointly estimate a set of regressions for the NYSE effective spread deciles (10
equations in all):
14
Another attractive feature of effective spreads is that they are based on prices from actual trades and so reflect
the intentions of investors, while the quoted and inside spreads are best described as reflecting the intetions of market
makers and specialists. Evidence from effective spreads therefore constitute a kind of out-of-sample evidence relative
to quoted and inside spreads.
19
2
ESpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
+ µDecimalsN Y Di,t
+ it .
(5)
The dependent variable in each equation is the average daily percentage effective spread, ESpreadi,t ,
where the average is formed equally weighted based on NYSE firms in decile i on day t. The
explanatory variables are analogous to those employed above, defined based on the NYSE data.
And again we employ HAC robust t-tests based on Newey and West (1994) standard errors, and
our findings are qualitatively unchanged if instead we use MacKinnon-White heteroskedasticityconsistent standard errors and include as regressors sufficient lags of the dependent variable to
produce white-noise residuals.
We now consider results from estimating this model. The top section of Table 5 contains
parameter estimates and standard errors for the case of NYSE effective spreads. As we saw for
quoted spreads, the coefficient estimate on Incidencet is positive and significant in all cases. These
results are again consistent with theory: spreads are wider during periods when the incidence of
seasonal depression is high.
Coefficient estimates on variance, volume, and turnover are as expected. And all of the joint
tests strongly reject the null hypothesis at conventional levels of significance.
B
Regression Analysis, NASDAQ Effective Spreads
Consider next the results for NASDAQ effective spreads. Just like the previous regression analysis,
we use GMM to jointly estimate a set of regressions, now for the NASDAQ effective spread deciles
(10 equations in all):
2
ESpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
+ µM M Cnt M M Cnti,t−1
SixteenthsN AS
DecimalsN AS
+ µT rendi T rendi,t−1 + µSixteenthsN AS Di,t
+ µDecimalsN AS Di,t
+ it .
(6)
The dependent variable in each equation is the average daily percentage effective spread, ESpreadi,t ,
where the average is formed equally weighted based on NASDAQ firms in decile i on day t. The
20
explanatory variables are defined analogously to those used in the regression model for NYSE
effective spreads, Equation (5), but now based on the NASDAQ data. Additionally, we include
M M Cnti,t−1 , the average the market maker count formed equally weighted across firms in decile i
at t−1. As before, we control for autocorrelation in the dependent variable through our use of HAC
robust t-tests based on Newey and West (1994) standard errors, and our findings are qualitatively
unchanged if instead we use MacKinnon-White heteroskedasticity-consistent standard errors and
include as regressors sufficient lags of the dependent variable to produce white-noise residuals.
We now consider results from estimating this model. The format of the table mirrors that of
other regression result tables shown above. There remains significant evidence of autocorrelation
and ARCH in all cases, reinforcing the choice to use HAC standard errors. Consider the coefficient
estimates and t-statistics in the top portion of Table 6. Again, consistent with theory that suggests
higher dealer risk aversion should lead to wider spreads quoted by dealers, we find that the Incidence
variable is positive and significant for all ten spread deciles. That is, spreads are wider during periods
when the incidence of seasonal depression is high. The positive coefficients suggest that, on average,
market makers quote wider spreads during periods when more individuals are experiencing higher
risk aversion. As theory predicts, the coefficients on variance are almost everywhere positive, those
on volume are almost everywhere negative, and those on turnover are almost everywhere positive,
in each case strongly significantly so. The statistics in the last column of Table 6 strongly reject
the null hypothesis that each set of coefficients is jointly zero at conventional levels of significance.
We now consider the economic impact of the statistically significant seasonal pattern in bid-ask
spreads. Figure 7 and 8 correspond to the NYSE effective spreads, and Figures 9 and 10 reflect
the NASDAQ data. The annual peak value of the economic impact measure has been fairly stable
at about 10 basis points for the NYSE. For the NASDAQ, the peak has stabilized around 5 basis
points in recent years. These are economically large effects given the unconditional average effective
spread is about 75 basis points for the NYSE and about 160 basis points for NASDAQ.
How can we reconcile these economically large effects found for the effective spreads data with
the medical fact that only ten percent of the population suffers from the clinical condition of
SAD? First, it is very likely that the individuals who suffer from milder seasonal mood fluctuations
21
influence markets in a similar fashion. Second, market makers are operating in an asymmetric
information environment. Thus it is plausible that when a SAD-affected market maker widens the
spread between her bid and ask quotes, other market makers are inclined to widen their spreads as
well, as they must set their own prices without benefit of the knowledge of who is widening their
spreads for information-related reasons and who is widening their spreads due to heightened risk
aversion. Further, many market makers cover a portfolio of firms. In cases where a SAD-affected
market maker widens her spreads, we expect she would do so for all the stocks in her portfolio. The
implication is that even if only ten percent of market makers suffer from SAD, well over ten percent
of all stocks could easily be influenced.
VI
Discussion
Having documented some previously unknown findings about the seasonal behavior of spreads, we
turn now to a discussion of reasonable questions and concerns, including issues relating to whether
individuals who work indoors should be immune to the effects of SAD, whether market makers may
be less prone to SAD perhaps due to their relatively high education and income levels, whether
SAD-affected market makers would be relatively less likely to survive (financially) as market makers,
and other general questions about the plausibility of the risk aversion hypothesis we consider.
A valid question is whether people who work indoors, such as NASDAQ market makers, are
immune to the effects of SAD. The medical condition of Seasonal Affective Disorder arises due to
reduced exposure to daylight, which is independent of weather.15 Daylight is literally a function
of the earth’s position relative to the sun, and hours of light are calculated as the time between
sunrise and sunset, independent of weather conditions. During the fall and winter seasons, when
hours of daylight are diminished, people in general have fewer opportunities to be exposed to direct
light than they do in the spring and summer. According to the medical literature, for people prone
to SAD who work indoors, the impact of the reduced daylight through the fall and winter is at
least equivalent to that for SAD sufferers who work outdoors, and may even be more severe. (See
15
See Kamstra, Kramer, and Levi (2003, 2012) for citations to papers in the medical literature which establish
that SAD arises due to seasonal changes in daylight, as opposed to seasonal changes in weather conditions such as
cloudiness or precipitation.
22
Wirz-Justice et al., 1992, and Magnusson and Stefansson, 1993, for clinical evidence.) Thus, the
fact that market makers work indoors would not seem sufficient to make them invulnerable to the
effects of SAD.
An additional question is whether people who hold professional jobs, people who are relatively
wealthy, or people who have relatively high incomes are perhaps less likely to suffer from seasonal
depression than others. The evidence suggests that if anything, higher socioeconomic status is
associated with a greater risk of seasonal depression. Medical research does suggest that low socioeconomic status may be associated with a higher disposition to non-seasonal depression; see Lynch
et al. (1997) for example. This association does not seem to apply to seasonal depression, however.
For instance, Blazer et al. (1998) show that people in high-income families are much more likely to
suffer from SAD than those in low-income families, and a study conducted in Finland by Saarijarvi
et al. (1999) finds that higher levels of education are associated with a higher likelihood to suffer
from SAD. In light of these findings, market makers would seem to be at least as likely as the rest
of the population to experience the sort of seasonally varying risk aversion we investigate here.
Still another possible challenge to our findings is whether a dealer with SAD would be eventually identified and exploited by other market participants due to the seasonal predictability of his
actions.16 A SAD-affected market maker may change his bid and ask prices during seasons when
he is more risk averse, but in practice market makers change their prices for any number of reasons.
There is no reason to believe that other market participants would conclude seasonally varying risk
aversion is behind any particular dealer’s quote change at any given point in time. A long-inventory
SAD-affected market maker’s lower bid price would discourage clients from selling to him, and his
lower ask price would encourage clients to buy from him, but this is precisely the long-inventory
market maker’s goal when he experiences an increase in his risk aversion: he wishes to reduce his
inventory. In addition, as he reduces his inventory, he approaches the point at which the risk level
of his new portfolio is exactly consistent with the new price level. Similarly, buyers of his shares
initially take the lower price as a favorable shift in the tradeoff between expected return and risk.
However, as these buyers purchase more shares (and consequently add risk to their portfolios), they
16
Note that financial theory certainly allows for the simultaneous existence of market makers who are heterogeneous
in their tolerance for risk. See Stoll (1978a), for instance.
23
approach the point at which the higher expected return exactly compensates them for the added
risk. When the SAD-influenced market maker has reduced his inventory sufficiently and the rest of
the market has increased its holdings, SAD-driven trading subsides, at least until the market maker
experiences another shift in his risk aversion, or until the quote of another SAD-affected market
maker arrives.
How likely is the arrival of the quote of another SAD-affected market maker, given the average
NASDAQ firm has only 25 market makers? The medical literature suggests that as much as ten
percent of the population suffers from SAD, with additional numbers suffering from the milder
condition of winter blues. Further, many market makers cover a portfolio of stocks. Thus it seems
reasonable to expect that the majority of NASDAQ stocks have at least one or two market makers
who might experience seasonal depression and that many stocks in each decile will be affected
accordingly.
Another possible issue is that during periods when SAD is evident, a SAD-affected market maker
might leave the market entirely. As mentioned above, a NASDAQ market maker who stops making
a market in a given stock is required to wait 20 days before he resumes trading (see Harris ,2001,
page 511). Thus a SAD-affected market maker who is uninterested in trading would likely widen his
spread to reduce the likelihood of attracting a trading counterparty rather than discontinue making
a market entirely.
VII
Robustness Checks
We conducted a wide range of (untabulated) robustness checks, all of which are available from the
authors on request. Results suggest that our findings are robust to a wide variety of changes. For
example, we considered quintiles instead of deciles, we modified the model to include dummies for
day of the week or effects from lagged returns, we included dummy variables for each of the third
and fourth quarters (and alternately a single dummy for the second half of the year, or a dummy for
the third quarter, or a dummy for the second quarter, and/or a dummy for the month of January),
and we included firms with SIC codes 6011-6799.
Additionally, inference is unaffected by monotonic transformations of the way we define many
24
of the explanatory variables, including volume, turnover, and variance. Our findings are unchanged
if we control directly for autocorrelation by including lags of the dependent variable as regressors
(a sufficient number to produce white noise residuals) instead of using Newey West standard errors,
and our findings are also unchanged if we increase the number of Newey West HAC lags (from 2 to
6) used to calculate standard errors in our GMM estimation.
NASDAQ closing bid-ask spreads are available back to the early-to-mid 1980s.17 Our findings
with respect to the SAD effect are the same if we employ a longer sample of NASDAQ closing
inside spreads data dating back to the 1980s; for the sake of symmetry with the NYSE sample
period, we report results for the shorter period only. Finally, our conclusions are unchanged by
considering value-weighted deciles instead of equal-weighted, considering median or mean intraday
quoted spreads over various one-hour intervals during the day (first hour of trading, last hour of
trading, etc.) instead of closing spreads, estimating the 10 inside spread deciles and 10 quoted
spread deciles together as a system of 20 equations instead of separately as two independent sets
of 10 equations, and restricting our study to stocks trading at prices greater than or equal to $5
instead of including all stocks. Results based on value-weighted deciles and based on excluding
stocks with prices less than $5 are most comparable to the results we presented above for deciles of
larger-sized firms.
Because dollar as well as percentage spreads have narrowed over the sample period we investigate,
we also incorporated year dummies for fixed year effects instead of detrending spreads by the
previous year’s average spread. Again, the statistical and economic significance of the results
remain.
VIII
Conclusions
Seasonal variation in bid-ask spreads, as well as variation conditional on inventory cost changes,
adverse selection events, and competition among market makers, has been extensively documented
in past studies. Studies by Kamstra, Kramer and Levi (2003, 2012, 2015) detect strong seasonal
patterns in equity and Treasury returns, perhaps as a consequence of time-varying risk aversion
17
CRSP provides NASDAQ data starting as early as 1982, but the bid and ask price data are sparse until the
beginning of 1984.
25
due to seasonal depression. We consider whether there is evidence of time-varying risk aversion
influencing bid-ask spreads and to do so we consider dealer quoted spreads on the NYSE, inside
quotes from the NASDAQ, and effective spreads from both. We find that while effective spreads and
individual NYSE dealer quotes (quoted spreads) widen during the fall and winter seasons, NASDAQ
inside spreads narrow during that period. These findings are consistent with some market makers
experiencing seasonal changes in risk aversion that align with the timing of seasonal depression
associated with SAD. The impact of SAD on spreads, including effective, inside, and quoted spreads,
appears greater for smaller firms, with the magnitude of the effect varying monotonically across
deciles. The economic impact of this seasonal effect is such that effective spreads widen by about
10 basis points during the period of maximum possible impact (the period when daylight is most
diminished) and inside spreads narrow by about 2 basis points, relative to the unconditional averages
of 100 basis points. The finding of opposing seasonal patterns in individual dealer spreads versus
inside spreads has clear implications for the timing of trades by liquidity traders and discretionary
traders. We rule out several alternative explanations for and interpretations of our results, and our
conclusions are insensitive to a variety of robustness checks.
26
References
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29
Table 1: Summary Statistics for Pooled Data
We provide summary statistics on daily firm-level data pooled across common-share equities over
the January 4, 1993 to December 31, 2014 time period. Data are obtained from CRSP. Panel A
contains NYSE data, and Panel B contains NASDAQ data. Percentage spreads are calculated
according to Equations (1) and (2), using closing bid and ask prices (for which we provide summary
statistics as well). Daily returns are calculated using closing prices (which are not reported). We
also report the effective spread, dollar volume, and turnover. In Panel B, the number of market
makers is the reported number of market makers covering a particular stock on a given day. Note
that these summary statistics are based on the following restrictions: We require that all firms in
the sample have data available for bid and ask, which we use to compute the quoted spread. We
exclude firms for which the previous day’s closing price was less than $5 or over $1,000. We also
exclude observations with spreads of over $5 and financial firms.
Panel A: NYSE Data 1993-2014
Variable
Quoted Spread(%)
Effective Spread(%)
Bid ($)
Ask ($)
Daily Return (%)
Turnover
Volume ($ millions)
N
7,116,695
7,456,508
7,144,968
7,144,968
7,456,777
7,458,131
7,458,131
Mean
1.084
0.743
31.251
31.477
0.079
0.655
33.629
Std. Dev.
1.38
1.08
31.86
31.91
2.72
1.22
105.95
Min
<0.01
<0.01
4.00
4.47
-80.20
<0.01
<0.01
Max
51.66
67.34
1005.0
1006.0
360.49
540.97
28581
Skew
1.908
3.120
10.152
10.163
2.272
52.781
15.074
Kurt
6.66
24.03
194.73
194.99
149.72
12608
1138.3
Std. Dev.
2.47
2.10
22.57
22.59
4.33
3.11
157.92
17.31
Min
<0.01
<0.01
0.01
0.50
-81.90
<0.01
<0.01
<0.01
Max
93.88
68.80
997.51
1000.0
1277.8
1422.8
31682
118.00
Skew
2.893
2.702
10.720
10.712
8.397
94.756
42.568
0.923
Kurt
16.43
15.46
241.01
240.75
1246.9
22056
3420.3
0.31
Panel B: NASDAQ Data 1993-2014
Variable
Inside Spread (%)
Effective Spread (%)
Bid ($)
Ask ($)
Daily Return(%)
Turnover (%)
Volume ($ millions)
Market Maker Count
N
9,026,404
9,199,422
9,204,565
9,204,563
9,199,947
9,204,947
9,204,947
9,204,902
Mean
1.817
1.592
19.893
20.123
0.144
0.993
19.560
25.260
30
Table 2: Summary Statistics for Deciles
We provide summary statistics based on daily data for the NYSE and NASDAQ stocks we
consider, with data obtained from CRSP. For each market, we form deciles on the basis of each
stock’s lagged average daily market capitalization. The series run from January 3, 1993 through
to December 31, 2014. The summary statistics we present for each decile, by market, include the
mean, standard deviation, minimum, maximum, skewness, and excess kurtosis. Return variance is
detrended by dividing by the mean of the (rolling) last year’s return variance. Panel A contains
the NYSE deciles’ summary statistics. Panel B contains the NASDAQ summary statistics. The
number of market makers for each decile is defined as the number of market makers covering a stock
on a given day, averaged across all stocks in the decile.
Panel A: NYSE Data
Decile
1
2
3
4
5
Variable
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Mean
2.074
0.984
0.193
84.355
0.701
0.413
1.018
1.498
0.920
0.116
234.58
1.661
0.569
1.019
1.277
0.849
0.085
430.65
3.544
0.663
1.019
1.098
0.793
0.071
693.95
6.518
0.760
1.025
0.966
0.737
0.063
1057.0
10.269
0.800
1.025
(Table 2, Panel A is continued on the next page)
31
Std
1.39
0.34
1.01
51.16
2.62
0.26
0.56
1.39
0.40
1.37
129.79
1.60
0.33
0.44
1.23
0.39
1.37
227.60
3.24
0.39
0.45
1.10
0.38
1.35
361.13
5.80
0.44
0.48
0.99
0.36
1.31
546.22
8.86
0.47
0.48
Min
0.43
0.25
-8.91
26.77
0.03
0.08
0.32
0.10
0.05
-11.56
65.95
0.09
0.08
0.26
0.07
0.08
-11.04
119.63
0.14
0.09
0.29
0.05
0.03
-11.02
207.70
0.27
0.09
0.25
0.04
0.06
-11.03
342.53
0.37
0.09
0.22
Max
6.23
4.11
9.22
948.42
138.09
5.69
8.99
6.83
6.02
9.96
541.96
13.01
2.98
8.12
4.02
5.23
10.30
950.80
23.59
4.21
5.59
6.15
5.15
10.76
1521.6
35.19
3.33
5.18
6.90
5.21
11.74
2305.9
52.82
3.38
4.64
Skew
0.578
1.930
0.102
2.337
33.922
5.994
6.241
0.489
3.603
-0.076
0.569
1.221
1.272
2.866
0.465
3.533
-0.150
0.427
0.948
1.112
2.733
0.476
3.626
-0.187
0.409
0.790
0.918
2.843
0.572
3.469
-0.232
0.493
0.683
1.019
2.874
Kurt
-1.21
9.66
11.20
20.35
1561.2
76.17
60.37
-1.44
22.37
9.21
-0.85
1.72
2.49
19.51
-1.49
21.31
7.21
-0.96
0.56
2.48
11.86
-1.41
22.98
7.89
-1.03
-0.12
0.96
12.48
-0.86
21.76
8.07
-0.90
-0.50
1.21
13.04
Table 2 (continued)
Panel A: NYSE Data
Decile
6
7
8
9
10
Variable
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Quoted Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
Mean
0.876
0.700
0.062
1566.7
15.614
0.825
1.023
0.794
0.670
0.056
2365.1
23.066
0.828
1.028
0.703
0.653
0.050
3848.4
36.535
0.826
1.046
0.586
0.634
0.050
7830.0
67.982
0.772
1.032
0.464
0.610
0.042
40466
203.49
0.562
1.034
(Table 2 is continued on the next page)
32
Std
0.91
0.35
1.25
780.44
13.14
0.46
0.50
0.82
0.33
1.18
1122.8
18.76
0.45
0.57
0.75
0.33
1.15
1765.1
28.31
0.43
0.78
0.64
0.33
1.13
3480.4
50.53
0.38
0.54
0.52
0.32
1.04
15330
134.30
0.26
0.54
Min
0.03
0.04
-10.70
577.84
0.64
0.09
0.24
0.03
0.01
-9.55
940.76
1.03
0.09
0.22
0.02
0.02
-9.67
1562.4
1.30
0.07
0.21
0.02
0.01
-10.46
3007.5
3.20
0.10
0.21
0.02
0.01
-8.93
13373
12.07
0.09
0.21
Max
6.42
5.06
11.86
3377.3
64.86
3.28
5.01
4.48
4.62
12.64
5117.9
91.14
3.37
14.66
4.83
4.80
11.11
8503.3
188.55
3.24
15.42
3.86
5.03
12.94
16477
279.22
3.33
6.00
2.71
4.56
11.84
75073
871.23
2.42
5.79
Skew
0.763
3.583
-0.297
0.538
0.635
0.839
2.889
0.680
3.250
-0.153
0.614
0.677
0.882
6.628
0.916
3.241
-0.335
0.680
0.679
0.935
8.516
1.200
3.261
-0.168
0.447
0.573
0.934
3.467
1.442
3.049
-0.077
-0.163
0.616
1.107
3.339
Kurt
0.37
23.93
8.34
-0.80
-0.74
0.55
13.93
-0.57
20.14
9.24
-0.59
-0.66
0.72
108.72
0.41
20.19
8.98
-0.37
-0.40
1.01
112.10
1.77
21.39
11.68
-0.72
-0.70
1.28
20.99
2.57
19.54
10.27
-0.55
-0.10
1.91
20.20
Table 2 (continued)
Panel B: NASDAQ Data
Decile
1
2
3
4
5
Variable
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Mean
3.615
2.045
0.454
40.677
0.658
0.715
1.014
14.266
2.439
1.812
0.237
84.727
0.506
0.606
1.002
18.445
1.912
1.702
0.176
132.83
0.908
0.680
1.006
21.665
1.624
1.657
0.124
189.35
1.534
0.803
1.009
24.120
1.396
1.568
0.100
264.60
2.458
0.902
1.011
26.338
Std. Deviation
2.17
1.05
0.88
26.66
2.64
0.67
0.69
6.63
2.01
0.88
1.11
37.36
0.41
0.34
0.48
8.73
1.82
0.77
1.33
62.32
0.62
0.28
0.72
10.33
1.67
0.70
1.44
90.00
0.96
0.30
0.67
11.25
1.53
0.64
1.47
131.12
1.59
0.30
0.58
12.28
(Table 2, Panel B is continued on the next page)
33
Min
0.88
0.63
-6.73
14.93
0.03
0.10
0.26
6.62
0.35
0.53
-9.63
27.96
0.05
0.11
0.31
7.90
0.19
0.31
-12.61
40.60
0.08
0.13
0.31
8.50
0.15
0.20
-11.51
56.81
0.12
0.16
0.28
8.55
0.11
0.20
-10.12
76.20
0.15
0.18
0.22
8.75
Max
14.32
5.08
8.42
1494.7
172.25
11.78
12.57
29.19
9.46
5.01
9.86
154.65
8.92
7.71
14.78
35.95
8.23
5.54
10.42
256.00
9.99
5.57
17.56
41.67
5.80
4.91
10.61
381.49
11.43
5.50
16.61
44.47
5.41
5.39
9.45
576.41
24.61
3.80
23.33
48.16
Skew
0.802
0.889
0.473
30.810
50.144
5.951
5.494
0.545
0.908
0.912
0.114
-0.030
5.824
6.413
8.419
0.443
0.959
0.869
-0.119
0.041
3.679
4.765
13.141
0.312
1.010
0.747
-0.079
0.034
1.653
3.921
11.363
0.139
1.054
0.708
-0.156
0.199
1.747
2.243
13.607
0.078
Kurt
-0.71
-0.51
8.60
1617.1
3203.4
56.37
47.78
-1.22
-0.65
-0.36
9.56
-1.45
71.33
82.80
177.71
-1.35
-0.57
-0.09
8.84
-1.34
35.81
46.87
242.97
-1.42
-0.49
0.03
6.22
-1.27
9.57
34.17
202.61
-1.46
-0.40
0.27
4.53
-0.98
12.25
11.40
425.75
-1.44
Table 2 (continued)
Panel B: NASDAQ Data
Decile
6
7
8
9
10
Variable
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Inside Spread (%)
Effective Spread (%)
Daily Return (%)
Market Cap ($ millions)
Volume ($ millions)
Turnover (%)
Detrended Return Variance
# of Market Makers
Mean
1.196
1.456
0.079
371.42
3.913
1.020
1.007
28.458
1.018
1.353
0.054
527.03
6.176
1.129
1.008
30.538
0.837
1.245
0.057
799.50
10.601
1.258
1.005
33.145
0.645
1.137
0.043
1490.8
22.055
1.422
1.017
36.935
0.378
1.008
0.052
13355
172.28
1.537
1.016
48.419
Std. Deviation
1.36
0.58
1.52
194.90
2.54
0.33
0.42
13.04
1.19
0.55
1.54
284.01
3.91
0.37
0.49
13.68
1.01
0.52
1.57
443.81
6.75
0.39
0.42
14.53
0.79
0.50
1.65
933.79
14.29
0.44
0.50
15.30
0.48
0.55
1.74
8700.3
109.16
0.48
0.61
16.25
34
Min
0.09
0.12
-11.23
100.65
0.22
0.19
0.28
9.19
0.06
0.06
-10.96
134.67
0.30
0.17
0.32
9.55
0.05
0.06
-11.33
190.97
0.47
0.17
0.28
10.20
0.04
0.04
-12.76
303.75
0.90
0.24
0.23
11.55
0.02
0.03
-12.07
1539.1
4.17
0.25
0.24
18.15
Max
4.67
4.92
10.71
843.63
25.58
4.55
8.55
49.79
4.14
5.18
10.94
1257.0
38.42
3.60
19.92
51.72
3.45
5.31
12.00
1905.2
47.37
3.84
4.73
55.16
2.74
5.68
11.14
3911.1
103.67
4.18
7.23
57.85
1.89
7.00
15.09
37801
631.40
4.09
11.04
66.14
Skew
1.110
0.657
-0.153
0.395
1.235
1.771
3.630
-0.031
1.126
0.744
-0.132
0.485
0.828
1.217
11.674
-0.147
1.173
0.964
-0.069
0.544
0.565
0.734
2.109
-0.233
1.231
1.516
-0.077
0.785
0.483
0.566
3.180
-0.421
1.365
2.216
0.119
0.575
0.267
0.773
6.557
-0.743
Kurt
-0.28
0.49
4.25
-0.71
4.31
7.78
40.18
-1.46
-0.28
1.39
4.18
-0.52
2.20
3.79
400.48
-1.46
-0.19
3.09
4.17
-0.44
0.34
1.87
7.93
-1.43
-0.04
6.15
5.05
-0.18
0.00
0.93
21.74
-1.43
0.39
9.73
6.58
-0.17
-0.55
0.96
78.41
-1.21
Table 3: Estimation Results for NYSE Quoted Spread Deciles
We report regression results from jointly estimating the following model for the 10 NYSE quoted spread deciles in a GMM framework:
2
QSpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
+ µDecimalsN Y Di,t
+ it .
(3)
QSpreadi,t is the equal-weighted average quoted spread for NASDAQ firms in decile i on day t. V olumei,t−1 is the log of decile i’s average dollar volume
2
(in millions) at t − 1. T urnoveri,t−1 is the average at t − 1, across all firms in decile i, of share volume divided by shares outstanding, times 100. σi,t−1
is the variance of returns, detrended by dividing by average variance over the past 365 calendar days. T rendi,t−1 is the mean spread over the last year,
and controls for the downward trend in spreads. DtSixteenthsN Y is a dummy variable set to equal 1 for dates on or after June 24, 1997, DtDecimalsN Y is
a dummy that equals 1 for dates on or after January 29, 2001. Incidencet is formed for each decile by taking the clinically observed incidence of SAD
and multiplying by the square root of decile i’s detrended variance σi,t−1 and decile i’s one year average of spreads, T rendi,t−1 . The one year average of
spreads T rendi,t−1 is lagged by one day to avoid overlapping the dependent variable in time. The top portion of the table contains parameter estimates
and HAC robust t-tests in parentheses. At the bottom the table, we present the value of R2 for each equation, a χ2 test for the presence of up to 10 lags
of autocorrelation with 10 degrees of freedom, and a χ2 test for the presence of up to 10 lags of ARCH with 10 degrees of freedom. We also present Wald
χ2 test statistic to test whether coefficient estimates are jointly statistically different from 0 across the quoted spread series. The sample period is January
4 1993 through December 31 2014.
35
Parameter
α
µIncidence
µV olume
µT urnover
µV ariance
µT rend
µSixteenthsN Y
µDecimalsN Y
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
1.201∗∗∗
(68.25)
0.015∗∗∗
(7.68)
-0.074∗∗∗
(-12.5)
0.025∗∗
(2.11)
0.079∗∗∗
(10.67)
0.559∗∗∗
(95.97)
0.472∗∗∗
(47.38)
-1.435∗∗∗
(-97.6)
0.9608
5608.46∗∗∗
2061.37∗∗∗
Decile 2
(t-test)
0.924∗∗∗
(84.20)
0.030∗∗∗
(15.14)
-0.019∗∗∗
(-4.41)
-0.016∗∗
(-2.37)
0.079∗∗∗
(25.70)
0.652∗∗∗
(141.5)
0.177∗∗∗
(22.99)
-1.070∗∗∗
(-90.0)
0.9775
5084.19∗∗∗
1991.92∗∗∗
Decile 3
(t-test)
0.690∗∗∗
(60.95)
0.062∗∗∗
(26.34)
-0.004
(-1.04)
-0.012∗∗
(-2.36)
0.094∗∗∗
(28.14)
0.687∗∗∗
(139.6)
0.045∗∗∗
(5.51)
-0.770∗∗∗
(-69.5)
0.9746
6464.59∗∗∗
1660.44∗∗∗
Decile 4
(t-test)
0.552∗∗∗
(50.97)
0.057∗∗∗
(21.06)
-0.035∗∗∗
(-8.66)
0.033∗∗∗
(7.90)
0.085∗∗∗
(30.39)
0.704∗∗∗
(145.5)
0.029∗∗∗
(4.00)
-0.596∗∗∗
(-55.7)
0.9648
6794.23∗∗∗
2841.69∗∗∗
Decile 5
(t-test)
0.535∗∗∗
(49.85)
0.066∗∗∗
(23.82)
-0.048∗∗∗
(-13.4)
0.044∗∗∗
(13.30)
0.058∗∗∗
(26.54)
0.697∗∗∗
(140.9)
-0.008
(-1.33)
-0.485∗∗∗
(-48.3)
0.9438
6416.28∗∗∗
3262.24∗∗∗
Decile 6
(t-test)
0.596∗∗∗
(57.06)
0.087∗∗∗
(30.34)
-0.037∗∗∗
(-11.4)
0.035∗∗∗
(11.24)
0.059∗∗∗
(30.88)
0.627∗∗∗
(124.7)
-0.007
(-1.24)
-0.548∗∗∗
(-58.2)
0.8882
4392.48∗∗∗
3391.17∗∗∗
Decile 7
(t-test)
0.584∗∗∗
(61.48)
0.086∗∗∗
(26.88)
-0.031∗∗∗
(-11.5)
0.033∗∗∗
(11.39)
0.044∗∗∗
(27.35)
0.599∗∗∗
(116.9)
0.045∗∗∗
(7.75)
-0.575∗∗∗
(-65.1)
0.8756
6695.12∗∗∗
4054.15∗∗∗
Decile 8
(t-test)
0.436∗∗∗
(45.10)
0.147∗∗∗
(37.73)
0.010∗∗∗
(3.90)
0.026∗∗∗
(8.12)
0.014∗∗∗
(13.28)
0.588∗∗∗
(106.8)
0.089∗∗∗
(16.03)
-0.592∗∗∗
(-72.0)
0.8411
5267.38∗∗∗
3930.02∗∗∗
Decile 9
(t-test)
0.279∗∗∗
(26.92)
0.199∗∗∗
(48.03)
0.039∗∗∗
(13.06)
-0.008∗∗
(-2.41)
0.038∗∗∗
(25.18)
0.545∗∗∗
(97.92)
0.065∗∗∗
(13.26)
-0.535∗∗∗
(-77.9)
0.7965
2592.59∗∗∗
4439.51∗∗∗
Decile 10
(t-test)
-0.061∗∗∗
(-4.54)
0.225∗∗∗
(43.90)
0.091∗∗∗
(26.95)
-0.086∗∗∗
(-17.4)
0.028∗∗∗
(19.16)
0.568∗∗∗
(94.21)
0.039∗∗∗
(8.06)
-0.447∗∗∗
(-77.4)
0.7880
4557.85∗∗∗
5047.58∗∗∗
χ2
P-value
10189.5
<0.001
2755.02
<0.001
1511.58
<0.001
750.429
<0.001
1299.93
<0.001
27944.1
<0.001
4056.11
<0.001
11439.1
<0.001
Table 4: Estimation Results for NASDAQ Inside Spread Deciles
We report regression results from jointly estimating the following model for the 10 inside spread deciles in a GMM framework:
2
ISpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µM M Cnt M M Cnti,t−1
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN AS
DecimalsN AS
+ µDecimalsN AS Di,t
+ it .
+ µT rendi T rendi,t−1 + µSixteenthsN AS Di,t
(4)
ISpreadi,t , which is the equal-weighted average inside spread for NASDAQ firms in decile i on day t. DtSixteenthsN AS is a dummy set to equal
1 for dates on or after June 2, 1997, and DtDecimalsN AS is a dummy that equals 1 for dates on or after April 9, 2001. Other variables are
defined analogously to those defined in the notes to Table 3, but but based on the NASDAQ inside spread deciles. Additionally, the model
includes M M Cnti,t−1 , which is the average the market maker count formed equally weighted across firms in decile i at t − 1. The layout of
the table, the tests for autocorrelation, ARCH, and joint significance, and the sample period mirror that of Table 3.
Parameter
36
α
µIncidence
µV olume
µT urnover
µV ariance
µM M Cnt
µT rend
µSixteenthsN AS
µDecimalsN AS
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
1.511∗∗∗
(23.42)
-0.005∗∗
(-2.27)
-0.265∗∗∗
(-17.9)
0.050∗∗∗
(5.62)
0.155∗∗∗
(22.65)
0.008∗∗∗
(5.23)
0.668∗∗∗
(87.43)
-0.995∗∗∗
(-42.4)
-0.558∗∗∗
(-21.3)
0.9303
3200.02∗∗∗
521.26∗∗∗
Decile 2
(t-test)
1.351∗∗∗
(29.29)
-0.023∗∗∗
(-11.4)
-0.249∗∗∗
(-25.4)
0.128∗∗∗
(12.86)
0.192∗∗∗
(25.89)
-0.001
(-1.26)
0.658∗∗∗
(101.7)
-0.959∗∗∗
(-48.6)
-0.453∗∗∗
(-26.4)
0.9740
2590.32∗∗∗
53.09∗∗∗
Decile 3
(t-test)
1.208∗∗∗
(32.62)
-0.007∗∗∗
(-2.96)
-0.213∗∗∗
(-24.0)
0.120∗∗∗
(13.24)
0.071∗∗∗
(23.90)
-0.001∗∗
(-2.51)
0.672∗∗∗
(102.4)
-0.811∗∗∗
(-43.0)
-0.325∗∗∗
(-23.0)
0.9772
4613.76∗∗∗
65.59∗∗∗
Decile 4
(t-test)
1.102∗∗∗
(36.87)
-0.010∗∗∗
(-4.10)
-0.292∗∗∗
(-33.1)
0.202∗∗∗
(22.17)
0.048∗∗∗
(20.59)
0.003∗∗∗
(8.75)
0.648∗∗∗
(97.84)
-0.774∗∗∗
(-48.7)
-0.304∗∗∗
(-25.0)
0.9810
4486.96∗∗∗
201.35∗∗∗
Decile 5
(t-test)
0.854∗∗∗
(34.44)
-0.009∗∗∗
(-3.57)
-0.334∗∗∗
(-45.6)
0.204∗∗∗
(28.43)
0.061∗∗∗
(25.74)
0.006∗∗∗
(20.80)
0.672∗∗∗
(113.7)
-0.619∗∗∗
(-45.1)
-0.212∗∗∗
(-20.2)
0.9814
5134.30∗∗∗
491.09∗∗∗
Decile 6
(t-test)
0.944∗∗∗
(39.39)
-0.021∗∗∗
(-8.01)
-0.258∗∗∗
(-43.4)
0.127∗∗∗
(26.09)
0.108∗∗∗
(35.98)
0.004∗∗∗
(16.04)
0.656∗∗∗
(107.5)
-0.649∗∗∗
(-49.1)
-0.194∗∗∗
(-22.5)
0.9830
6468.23∗∗∗
410.09∗∗∗
Decile 7
(t-test)
0.845∗∗∗
(42.80)
-0.020∗∗∗
(-8.42)
-0.241∗∗∗
(-51.0)
0.123∗∗∗
(35.53)
0.061∗∗∗
(31.62)
0.006∗∗∗
(26.17)
0.671∗∗∗
(117.3)
-0.541∗∗∗
(-51.3)
-0.181∗∗∗
(-24.1)
0.9859
2988.89∗∗∗
313.66∗∗∗
Decile 8
(t-test)
0.802∗∗∗
(44.93)
-0.014∗∗∗
(-5.70)
-0.184∗∗∗
(-52.1)
0.090∗∗∗
(37.81)
0.056∗∗∗
(31.20)
0.005∗∗∗
(30.10)
0.659∗∗∗
(111.0)
-0.515∗∗∗
(-51.9)
-0.173∗∗∗
(-28.4)
0.9867
4115.25∗∗∗
266.05∗∗∗
Decile 9
(t-test)
0.663∗∗∗
(41.06)
-0.018∗∗∗
(-6.48)
-0.133∗∗∗
(-50.8)
0.053∗∗∗
(33.63)
0.036∗∗∗
(27.80)
0.006∗∗∗
(38.02)
0.668∗∗∗
(100.9)
-0.422∗∗∗
(-50.1)
-0.182∗∗∗
(-33.9)
0.9844
3777.57∗∗∗
96.46∗∗∗
Decile 10
(t-test)
0.250∗∗∗
(16.98)
-0.029∗∗∗
(-9.37)
-0.022∗∗∗
(-16.5)
0.003∗∗∗
(3.03)
0.021∗∗∗
(25.57)
0.003∗∗∗
(26.11)
0.770∗∗∗
(98.86)
-0.246∗∗∗
(-46.5)
-0.062∗∗∗
(-23.4)
0.9802
6473.52∗∗∗
1108.03∗∗∗
χ2
P-value
2675.68
<0.001
293.595
<0.001
4044.18
<0.001
2105.16
<0.001
2350.93
<0.001
2478.67
<0.001
20548.6
<0.001
4015.58
<0.001
1625.84
<0.001
Table 5: Estimation Results for NYSE Effective Spread Deciles
We report regression results from jointly estimating the following model for the 10 quoted spread deciles in a Hansen (1982) GMM
framework:
2
ESpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µDecimalsN Y Di,t
+ it .
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
(5)
ESpreadi,t is the equal-weighted average effective spread for NYSE firms in decile i on day t. Other variables are defined analogously to those
defined in the notes to Table 3, based on the NYSE effective spread deciles. The layout of the table, the tests for autocorrelation, ARCH,
and joint significance, and the sample period mirror that of Table 3.
Parameter
α
37
µIncidence
µV olume
µT urnover
µV ariance
µT rend
µSixteenthsN Y
µDecimalsN Y
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
0.266∗∗∗
(8.65)
0.079∗∗∗
(12.26)
0.062∗∗∗
(10.40)
0.064∗∗∗
(3.83)
0.096∗∗∗
(14.73)
0.748∗∗∗
(51.10)
-0.039∗∗∗
(-5.87)
-0.111∗∗∗
(-12.1)
0.5143
374.09∗∗∗
1011.62∗∗∗
Decile 2
(t-test)
-0.300∗∗∗
(-13.3)
0.078∗∗∗
(10.73)
-0.132∗∗∗
(-17.8)
0.529∗∗∗
(26.24)
0.194∗∗∗
(27.83)
0.706∗∗∗
(51.08)
0.004
(0.50)
0.046∗∗∗
(5.17)
0.4487
113.89∗∗∗
906.89∗∗∗
Decile 3
(t-test)
-0.274∗∗∗
(-15.6)
0.084∗∗∗
(11.06)
-0.154∗∗∗
(-20.7)
0.454∗∗∗
(26.45)
0.206∗∗∗
(31.17)
0.706∗∗∗
(48.16)
0.049∗∗∗
(6.85)
0.078∗∗∗
(8.26)
0.4101
67.40∗∗∗
1026.45∗∗∗
Decile 4
(t-test)
-0.180∗∗∗
(-14.1)
0.091∗∗∗
(12.20)
-0.199∗∗∗
(-27.1)
0.504∗∗∗
(32.75)
0.194∗∗∗
(32.75)
0.633∗∗∗
(43.11)
0.085∗∗∗
(12.54)
0.085∗∗∗
(8.95)
0.4241
72.27∗∗∗
1205.79∗∗∗
Decile 5
(t-test)
-0.002
(-0.16)
0.081∗∗∗
(10.66)
-0.188∗∗∗
(-27.7)
0.467∗∗∗
(34.43)
0.180∗∗∗
(34.96)
0.494∗∗∗
(34.04)
0.110∗∗∗
(15.73)
0.067∗∗∗
(6.91)
0.4249
46.70∗∗∗
1169.15∗∗∗
Decile 6
(t-test)
0.036∗∗∗
(3.69)
0.116∗∗∗
(14.41)
-0.152∗∗∗
(-21.0)
0.406∗∗∗
(28.29)
0.174∗∗∗
(34.32)
0.536∗∗∗
(37.26)
0.112∗∗∗
(15.09)
-0.012
(-1.36)
0.4127
80.74∗∗∗
1096.41∗∗∗
Decile 7
(t-test)
0.136∗∗∗
(12.42)
0.109∗∗∗
(13.00)
-0.156∗∗∗
(-20.8)
0.410∗∗∗
(29.43)
0.135∗∗∗
(29.64)
0.561∗∗∗
(40.23)
0.123∗∗∗
(15.38)
-0.051∗∗∗
(-5.70)
0.4130
98.78∗∗∗
969.97∗∗∗
Decile 8
(t-test)
0.274∗∗∗
(19.20)
0.164∗∗∗
(18.25)
-0.152∗∗∗
(-19.5)
0.423∗∗∗
(28.98)
0.066∗∗∗
(15.86)
0.512∗∗∗
(34.38)
0.154∗∗∗
(17.77)
-0.092∗∗∗
(-10.4)
0.4036
222.58∗∗∗
1145.24∗∗∗
Decile 9
(t-test)
0.153∗∗∗
(8.90)
0.076∗∗∗
(8.90)
-0.115∗∗∗
(-15.8)
0.356∗∗∗
(25.00)
0.166∗∗∗
(35.18)
0.639∗∗∗
(44.44)
0.125∗∗∗
(14.54)
-0.078∗∗∗
(-9.35)
0.4439
81.08∗∗∗
966.72∗∗∗
Decile 10
(t-test)
0.026
(1.11)
0.073∗∗∗
(7.93)
-0.051∗∗∗
(-7.52)
0.320∗∗∗
(18.38)
0.179∗∗∗
(35.02)
0.739∗∗∗
(50.27)
0.086∗∗∗
(8.57)
-0.103∗∗∗
(-13.1)
0.4648
89.59∗∗∗
972.03∗∗∗
χ2
P-value
1087.64
<0.001
475.610
<0.001
1210.29
<0.001
1407.03
<0.001
1831.67
<0.001
4507.57
<0.001
575.153
<0.001
988.855
<0.001
Table 6: Estimation Results for NASDAQ Effective Spread Deciles
We report regression results from jointly estimating the following model for the 10 effective spread deciles in a GMM framework:
2
ESpreadi,t = αi + µIncidence Incidencet · T rendi,t−1 · σi,t−1
+ µV olumei V olumei,t−1
2
+ µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
+ µM M Cnt M M Cnti,t−1
SixteenthsN AS
DecimalsN AS
+ µDecimalsN AS Di,t
+ it .
+ µT rendi T rendi,t−1 + µSixteenthsN AS Di,t
(6)
ESpreadi,t is the equal-weighted average effective spread for NASDAQ firms in decile i on day t. DtSixteenthsN AS is a dummy variable set to
equal 1 for dates on or after June 2, 1997, and DtDecimalsN AS is a dummy that equals 1 for dates on or after April 9, 2001. Other variables
are defined analogously to those defined in the notes to Table 3, but based on the NASDAQ effective spread deciles. The layout of the table,
the tests for autocorrelation, ARCH, and joint significance, and the sample period mirror that of Table 3.
Parameter
α
38
µIncidence
µV olume
µT urnover
µV ariance
µM M Cnt
µT rend
µSixteenthsN AS
µDecimalsN AS
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
1.068∗∗∗
(16.00)
0.011∗∗
(2.31)
0.029∗∗∗
(3.69)
-0.007
(-1.44)
0.090∗∗∗
(17.28)
-0.001
(-1.15)
0.697∗∗∗
(44.76)
-0.471∗∗∗
(-18.9)
-0.249∗∗∗
(-12.4)
0.9139
2273.31∗∗∗
1243.19∗∗∗
Decile 2
(t-test)
0.324∗∗∗
(6.63)
0.011∗∗
(2.40)
0.011
(1.13)
0.044∗∗∗
(4.00)
0.229∗∗∗
(24.46)
0.002∗∗∗
(3.89)
0.794∗∗∗
(62.19)
-0.289∗∗∗
(-14.8)
-0.106∗∗∗
(-6.83)
0.8674
833.92∗∗∗
931.82∗∗∗
Decile 3
(t-test)
0.206∗∗∗
(4.79)
0.031∗∗∗
(6.60)
-0.052∗∗∗
(-4.24)
0.210∗∗∗
(14.17)
0.097∗∗∗
(15.03)
0.003∗∗∗
(5.38)
0.780∗∗∗
(59.81)
-0.216∗∗∗
(-12.8)
-0.077∗∗∗
(-5.25)
0.7899
732.92∗∗∗
1056.91∗∗∗
Decile 4
(t-test)
0.053
(1.35)
0.038∗∗∗
(7.92)
-0.052∗∗∗
(-3.84)
0.243∗∗∗
(14.65)
0.094∗∗∗
(15.40)
0.003∗∗∗
(4.96)
0.812∗∗∗
(64.30)
-0.136∗∗∗
(-8.74)
-0.085∗∗∗
(-5.81)
0.7328
687.82∗∗∗
1177.51∗∗∗
Decile 5
(t-test)
-0.024
(-0.70)
0.044∗∗∗
(9.00)
-0.101∗∗∗
(-7.60)
0.271∗∗∗
(17.13)
0.145∗∗∗
(24.28)
0.003∗∗∗
(4.32)
0.790∗∗∗
(59.79)
-0.079∗∗∗
(-5.62)
-0.062∗∗∗
(-4.31)
0.6851
379.97∗∗∗
681.47∗∗∗
Decile 6
(t-test)
-0.294∗∗∗
(-9.41)
0.013∗∗
(2.46)
-0.060∗∗∗
(-4.68)
0.236∗∗∗
(16.72)
0.258∗∗∗
(32.83)
0.004∗∗∗
(6.37)
0.875∗∗∗
(70.86)
-0.065∗∗∗
(-5.02)
-0.077∗∗∗
(-5.80)
0.6552
326.82∗∗∗
647.90∗∗∗
Decile 7
(t-test)
-0.240∗∗∗
(-8.60)
-0.003
(-0.61)
-0.077∗∗∗
(-6.94)
0.265∗∗∗
(22.55)
0.205∗∗∗
(30.56)
0.005∗∗∗
(10.04)
0.860∗∗∗
(68.48)
-0.058∗∗∗
(-4.87)
-0.154∗∗∗
(-11.0)
0.6021
158.85∗∗∗
624.70∗∗∗
Decile 8
(t-test)
-0.303∗∗∗
(-12.0)
0.022∗∗∗
(3.92)
-0.045∗∗∗
(-4.14)
0.191∗∗∗
(17.26)
0.284∗∗∗
(35.46)
0.004∗∗∗
(8.01)
0.866∗∗∗
(64.40)
-0.039∗∗∗
(-3.44)
-0.173∗∗∗
(-11.4)
0.5631
232.05∗∗∗
694.54∗∗∗
Decile 9
(t-test)
-0.115∗∗∗
(-5.33)
0.034∗∗∗
(5.14)
-0.011
(-1.32)
0.202∗∗∗
(22.49)
0.182∗∗∗
(27.06)
0.003∗∗∗
(5.76)
0.751∗∗∗
(58.32)
-0.031∗∗∗
(-2.87)
-0.284∗∗∗
(-16.3)
0.5022
254.49∗∗∗
716.31∗∗∗
Decile 10
(t-test)
-0.293∗∗∗
(-11.0)
0.029∗∗∗
(3.67)
0.064∗∗∗
(9.51)
0.259∗∗∗
(28.59)
0.120∗∗∗
(17.64)
0.003∗∗∗
(4.79)
0.601∗∗∗
(44.89)
-0.076∗∗∗
(-6.39)
-0.387∗∗∗
(-17.2)
0.5326
132.86∗∗∗
643.06∗∗∗
χ2
P-value
615.941
<0.001
294.442
<0.001
285.864
<0.001
1107.28
<0.001
1888.69
<0.001
147.393
<0.001
7104.87
<0.001
515.818
<0.001
538.629
<0.001
Panel A
NYSE Quoted
Panel B
NASDAQ Inside
Panel C
NYSE Effective
Panel D
NASDAQ Effective
Figure 1: Monthly Average NYSE and NASDAQ Percentage Spreads Panels A and B present
spreads sorted into quintiles based on market capitalization. Panel A presents data on dealer quotes on
NYSE stocks, equal weighted. Panel B presents data on inside quotes on NASDAQ stocks, equal weighted.
Panel C presents data on effective quotes on NYSE stocks, equal weighted. Panel D presents data on
effective quotes on NASDAQ stocks, equal weighted. The vertical axes correspond to percentage spread,
and the horizontal axes represent months (by first letter, starting with July). The starred solid lines are the
monthly average percentage spread across stocks, equal weighted, the thinnest dashed line is the monthly
average percentage spread for stocks in the smallest-cap quintile, equal weighted, the next-thinest dashed
line is the next quintile, and so on.
39
NYSE Quoted Spread
Figure 2: Average NYSE Quoted Spread Over Time and Predicted Seasonal Variation in NYSE
Quoted Spread (by Decile).
40
NYSE Quoted Spread - Finer Scale
Figure 3: Predicted Seasonal Variation in NYSE Quoted Spread (by Decile).
41
Before: Non-SAD-affected &
SAD-affected market makers
have the same bid and ask, so
the quoted spread equals
the inside spread.
Non-SAD
market
k t
maker’s ask
SAD & non-SAD
market makers’ best ask
42
midprice
SAD & non-SAD
market makers’ best bid
After: The non-SAD-affected market maker has the same
bid and ask as before, but the SAD-affected market maker
p his bid and ask p
prices and widens his spread.
p
drops
Inside
spread
Non-SAD
market maker’s
maker s
midprice
The inside
spread is
narrower
than either
market
maker’s
quoted
spread
p
Non-SAD
market
maker’s bid
Conclusion: The SAD-affected market-maker’s
spread widens
idens when
hen he becomes more risk
averse but the inside spread narrows.
SAD market
maker’s ask
SAD market
maker’s
midprice
SAD market
maker’s bid
Figure 4: Narrowing of the Inside Spread. With multiple heterogeneous market makers, the inside spread narrows when a subset of the
market makers experiences an increase in risk aversion due to SAD. To preserve simplicity in this diagram, we assume that the SAD-affected
market makers hold a long-inventory position. though as we explain in the main body of the text, our findings do not rely on this assumption.
NASDAQ Inside Spread
Figure 5: Average NASDAQ Inside Spread Over Time and Predicted Seasonal Variation in NASDAQ
Inside Spread (by Decile).
43
NASDAQ Inside Spread - Finer Scale
Figure 6: Predicted Seasonal Variation in NASDAQ Inside Spread (by Decile).
44
NYSE Effective Spread
Figure 7: Average NYSE Effective Spread Over Time and Predicted Seasonal Variation in NYSE
Effective Spread (by Decile). The deciles are formed on the basis of lagged market capitalization.
45
NYSE Effective Spread - Finer Scale
Figure 8: Predicted Seasonal Variation in NYSE Effective Spread (by Decile). The deciles are formed
on the basis of lagged market capitalization.
46
NASDAQ Effective Spread
Figure 9: Average NASDAQ Effective Spreads Over Time and Predicted Seasonal Variation in NASDAQ Effective Spread (by Decile). The deciles are formed on the basis of lagged market capitalization.
47
NASDAQ Effective Spread - Finer Scale
Figure 10: Predicted Seasonal Variation in NASDAQ Effective Spread (by Decile). The deciles are
formed on the basis of lagged market capitalization.
48
Appendix: Baseline Regression Models, without Incidencet
For comparison with the models presented in the main text, we provide each regression model
excluding the Incidencet variable. All variables are as previously defined, and all details of the
estimation technique, standard errors, tests, and table format are unchanged. The models are as
follows for NYSE quoted spreads, NASDAQ inside spreads, NYSE effective spreads, and NASDAQ
effective spreads, respectively.
2
QSpreadi,t = αi + µV olumei V olumei,t−1 + µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
+ µDecimalsN Y Di,t
+ it
(7)
2
ISpreadi,t = αi + µV olumei V olumei,t−1 + µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
+ µM M Cnt M M Cnti,t−1 + µT rendi T rendi,t−1
SixteenthsN AS
DecimalsN AS
+ µSixteenthsN AS Di,t
+ µDecimalsN AS Di,t
+ it
(8)
2
ESpreadi,t = αi + µV olumei V olumei,t−1 + µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
+ µDecimalsN Y Di,t
+ it
(9)
2
ESpreadi,t = αi + µV olumei V olumei,t−1 µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN AS
+ +µM M Cnt M M Cnti,t−1 + µT rendi T rendi,t−1 + µSixteenthsN AS Di,t
DecimalsN AS
+ µDecimalsN AS Di,t
+ it
49
(10)
Table A1: Baseline Estimation Results for NYSE Quoted Spread Deciles (without Incidencet )
We report regression results from jointly estimating the following model for the 10 NYSE quoted spread deciles in a GMM framework:
2
QSpreadi,t = αi + µV olumei V olumei,t−1 + µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µDecimalsN Y Di,t
+ it .
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
(7)
Variables are defined identically to those defined in the notes to Table 3. The layout of the table, the tests for autocorrelation, ARCH, and
joint significance, and the sample period mirror that of Table 3.
Parameter
α
µV olume
µT urnover
50
µV ariance
µT rend
µSixteenthsN Y
µDecimalsN Y
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
1.187∗∗∗
(53.37)
-0.073∗∗∗
(-10.9)
0.020
(1.52)
0.087∗∗∗
(9.14)
0.568∗∗∗
(79.39)
0.472∗∗∗
(36.58)
-1.427∗∗∗
(-70.5)
0.9609
5618.40∗∗∗
2058.11∗∗∗
Decile 2
(t-test)
0.908∗∗∗
(59.36)
-0.020∗∗∗
(-4.00)
-0.015∗∗
(-2.12)
0.088∗∗∗
(24.69)
0.666∗∗∗
(109.9)
0.185∗∗∗
(19.14)
-1.071∗∗∗
(-63.1)
0.9773
5041.04∗∗∗
2089.78∗∗∗
Decile 3
(t-test)
0.682∗∗∗
(47.28)
-0.000
(-0.09)
-0.021∗∗∗
(-3.55)
0.109∗∗∗
(27.57)
0.715∗∗∗
(117.6)
0.055∗∗∗
(5.64)
-0.783∗∗∗
(-53.5)
0.9734
6716.63∗∗∗
1901.66∗∗∗
Decile 4
(t-test)
0.547∗∗∗
(40.06)
-0.030∗∗∗
(-6.18)
0.023∗∗∗
(4.63)
0.096∗∗∗
(28.68)
0.728∗∗∗
(124.2)
0.038∗∗∗
(4.57)
-0.610∗∗∗
(-45.2)
0.9638
7017.61∗∗∗
2990.78∗∗∗
Decile 5
(t-test)
0.529∗∗∗
(38.89)
-0.043∗∗∗
(-10.4)
0.037∗∗∗
(9.90)
0.069∗∗∗
(25.81)
0.726∗∗∗
(117.8)
0.000
(0.04)
-0.502∗∗∗
(-38.2)
0.9424
6746.40∗∗∗
3446.57∗∗∗
Decile 6
(t-test)
0.591∗∗∗
(44.64)
-0.036∗∗∗
(-9.24)
0.028∗∗∗
(8.07)
0.071∗∗∗
(29.03)
0.663∗∗∗
(103.6)
0.005
(0.82)
-0.562∗∗∗
(-46.3)
0.8870
4803.31∗∗∗
3616.62∗∗∗
Decile 7
(t-test)
0.570∗∗∗
(45.35)
-0.027∗∗∗
(-7.92)
0.027∗∗∗
(7.79)
0.052∗∗∗
(24.76)
0.639∗∗∗
(99.65)
0.053∗∗∗
(8.16)
-0.586∗∗∗
(-50.7)
0.8749
6953.46∗∗∗
4041.08∗∗∗
Decile 8
(t-test)
0.435∗∗∗
(35.75)
0.009∗∗∗
(2.84)
0.030∗∗∗
(7.31)
0.019∗∗∗
(11.38)
0.652∗∗∗
(96.52)
0.108∗∗∗
(17.83)
-0.614∗∗∗
(-58.7)
0.8375
5500.63∗∗∗
3942.49∗∗∗
Decile 9
(t-test)
0.246∗∗∗
(19.39)
0.045∗∗∗
(12.54)
-0.016∗∗∗
(-3.71)
0.059∗∗∗
(26.59)
0.636∗∗∗
(89.57)
0.079∗∗∗
(14.75)
-0.555∗∗∗
(-60.1)
0.7907
2803.69∗∗∗
4460.68∗∗∗
Decile 10
(t-test)
-0.107∗∗∗
(-6.74)
0.101∗∗∗
(25.80)
-0.098∗∗∗
(-16.7)
0.042∗∗∗
(21.26)
0.667∗∗∗
(88.20)
0.044∗∗∗
(8.27)
-0.468∗∗∗
(-60.8)
0.7801
5215.77∗∗∗
5067.14∗∗∗
χ2
P-value
5606.23
<0.001
1247.87
<0.001
664.327
<0.001
1259.71
<0.001
18566.3
<0.001
2825.20
<0.001
6172.82
<0.001
Table A2: Estimation Results for NASDAQ Inside Spread Deciles (without Incidencet )
We report regression results from jointly estimating the following model for the 10 inside spread deciles in a GMM framework:
2
+ µM M Cnt M M Cnti,t−1
ISpreadi,t = αi + µV olumei V olumei,t−1 + µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN AS
DecimalsN AS
+ µDecimalsN AS Di,t
+ it .
+ µT rendi T rendi,t−1 + µSixteenthsN AS Di,t
(8)
ISpreadi,t , which is the equal-weighted average inside spread for NASDAQ firms in decile i on day t. Other variables are defined analogously
to those defined in the notes to Table 3, but but based on the NASDAQ inside spread deciles. Additionally, the model includes M M Cnti,t−1 ,
which is the average the market maker count formed equally weighted across firms in decile i at t − 1. The layout of the table, the tests for
autocorrelation, ARCH, and joint significance, and the sample period mirror that of Table 3.
Parameter
α
µV olume
51
µT urnover
µV ariance
µM M Cnt
µT rend
µSixteenthsN AS
µDecimalsN AS
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
1.553∗∗∗
(21.23)
-0.256∗∗∗
(-15.7)
0.049∗∗∗
(5.01)
0.150∗∗∗
(16.85)
0.008∗∗∗
(4.57)
0.664∗∗∗
(75.54)
-1.012∗∗∗
(-35.6)
-0.563∗∗∗
(-19.2)
0.9303
3219.95∗∗∗
524.08∗∗∗
Decile 2
(t-test)
1.366∗∗∗
(26.27)
-0.243∗∗∗
(-22.7)
0.121∗∗∗
(11.02)
0.179∗∗∗
(21.14)
-0.001
(-1.37)
0.649∗∗∗
(84.94)
-0.961∗∗∗
(-40.4)
-0.445∗∗∗
(-23.3)
0.9738
2645.77∗∗∗
55.18∗∗∗
Decile 3
(t-test)
1.225∗∗∗
(28.90)
-0.210∗∗∗
(-21.5)
0.118∗∗∗
(11.74)
0.065∗∗∗
(15.34)
-0.001∗∗∗
(-2.61)
0.668∗∗∗
(84.32)
-0.819∗∗∗
(-36.2)
-0.323∗∗∗
(-19.7)
0.9772
4670.81∗∗∗
67.08∗∗∗
Decile 4
(t-test)
1.111∗∗∗
(32.64)
-0.285∗∗∗
(-29.5)
0.196∗∗∗
(20.09)
0.043∗∗∗
(14.98)
0.003∗∗∗
(7.73)
0.646∗∗∗
(84.82)
-0.779∗∗∗
(-41.4)
-0.301∗∗∗
(-21.6)
0.9809
4501.35∗∗∗
205.26∗∗∗
Decile 5
(t-test)
0.859∗∗∗
(31.28)
-0.322∗∗∗
(-38.4)
0.195∗∗∗
(24.16)
0.058∗∗∗
(20.19)
0.005∗∗∗
(17.64)
0.671∗∗∗
(99.96)
-0.624∗∗∗
(-38.6)
-0.205∗∗∗
(-17.5)
0.9813
5237.49∗∗∗
502.41∗∗∗
Decile 6
(t-test)
0.940∗∗∗
(34.04)
-0.246∗∗∗
(-35.9)
0.120∗∗∗
(21.63)
0.101∗∗∗
(28.32)
0.004∗∗∗
(13.17)
0.652∗∗∗
(91.12)
-0.649∗∗∗
(-41.3)
-0.185∗∗∗
(-19.1)
0.9828
6464.56∗∗∗
437.21∗∗∗
Decile 7
(t-test)
0.850∗∗∗
(36.99)
-0.234∗∗∗
(-43.6)
0.118∗∗∗
(30.81)
0.054∗∗∗
(23.77)
0.005∗∗∗
(22.46)
0.664∗∗∗
(98.86)
-0.544∗∗∗
(-42.9)
-0.176∗∗∗
(-21.0)
0.9857
2981.46∗∗∗
335.30∗∗∗
Decile 8
(t-test)
0.802∗∗∗
(38.56)
-0.180∗∗∗
(-45.0)
0.087∗∗∗
(32.73)
0.053∗∗∗
(24.55)
0.005∗∗∗
(25.84)
0.655∗∗∗
(94.33)
-0.517∗∗∗
(-44.4)
-0.168∗∗∗
(-24.4)
0.9866
4129.40∗∗∗
273.57∗∗∗
Decile 9
(t-test)
0.662∗∗∗
(34.80)
-0.131∗∗∗
(-42.5)
0.052∗∗∗
(28.42)
0.034∗∗∗
(21.32)
0.006∗∗∗
(31.76)
0.662∗∗∗
(85.93)
-0.423∗∗∗
(-42.0)
-0.177∗∗∗
(-29.5)
0.9843
3791.29∗∗∗
103.25∗∗∗
Decile 10
(t-test)
0.244∗∗∗
(13.42)
-0.021∗∗∗
(-13.6)
0.003∗∗
(2.50)
0.020∗∗∗
(21.38)
0.003∗∗∗
(20.27)
0.760∗∗∗
(78.86)
-0.243∗∗∗
(-37.1)
-0.059∗∗∗
(-19.8)
0.9799
6381.70∗∗∗
1163.50∗∗∗
χ2
P-value
2160.35
<0.001
2921.70
<0.001
1570.90
<0.001
1461.66
<0.001
1725.40
<0.001
14924.0
<0.001
2947.14
<0.001
1262.78
<0.001
Table A3: Estimation Results for NYSE Effective Spread Deciles (without Incidencet )
We report regression results from jointly estimating the following model for the 10 quoted spread deciles in a Hansen (1982) GMM framework:
2
ESpreadi,t = αi + µV olumei V olumei,t−1 + µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
SixteenthsN Y
DecimalsN Y
+ µT rendi T rendi,t−1 + µSixteenthsN Y Di,t
+ µDecimalsN Y Di,t
+ it .
(9)
ESpreadi,t is the equal-weighted average effective spread for NYSE firms in decile i on day t. Other variables are defined analogously to those defined in
the notes to Table 3, but based on the NYSE effective spread deciles. The layout of the table, the tests for autocorrelation, ARCH, and joint significance,
and the sample period mirror that of Table 3.
Parameter
α
µV olume
µT urnover
52
µV ariance
µT rend
µSixteenthsN Y
µDecimalsN Y
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
0.264∗∗∗
(7.90)
0.063∗∗∗
(9.60)
0.055∗∗∗
(3.06)
0.110∗∗∗
(13.65)
0.777∗∗∗
(48.74)
-0.036∗∗∗
(-4.87)
-0.120∗∗∗
(-11.6)
0.5037
372.83∗∗∗
1038.82∗∗∗
Decile 2
(t-test)
-0.303∗∗∗
(-12.1)
-0.129∗∗∗
(-15.9)
0.515∗∗∗
(23.44)
0.215∗∗∗
(24.41)
0.734∗∗∗
(48.48)
0.003
(0.40)
0.041∗∗∗
(4.17)
0.4420
121.13∗∗∗
894.88∗∗∗
Decile 3
(t-test)
-0.279∗∗∗
(-13.9)
-0.145∗∗∗
(-17.9)
0.426∗∗∗
(22.48)
0.229∗∗∗
(28.23)
0.744∗∗∗
(44.87)
0.045∗∗∗
(5.77)
0.071∗∗∗
(6.83)
0.4032
73.94∗∗∗
1009.28∗∗∗
Decile 4
(t-test)
-0.192∗∗∗
(-12.9)
-0.191∗∗∗
(-23.4)
0.482∗∗∗
(28.05)
0.217∗∗∗
(28.43)
0.673∗∗∗
(42.60)
0.084∗∗∗
(11.13)
0.078∗∗∗
(7.43)
0.4166
79.06∗∗∗
1193.82∗∗∗
Decile 5
(t-test)
-0.021∗
(-1.71)
-0.182∗∗∗
(-24.2)
0.448∗∗∗
(29.66)
0.200∗∗∗
(30.88)
0.539∗∗∗
(32.68)
0.109∗∗∗
(13.58)
0.062∗∗∗
(5.61)
0.4195
53.53∗∗∗
1143.98∗∗∗
Decile 6
(t-test)
0.010
(0.89)
-0.144∗∗∗
(-17.8)
0.382∗∗∗
(24.00)
0.201∗∗∗
(31.97)
0.595∗∗∗
(38.19)
0.108∗∗∗
(13.32)
-0.015
(-1.55)
0.4030
92.93∗∗∗
1098.39∗∗∗
Decile 7
(t-test)
0.117∗∗∗
(9.41)
-0.150∗∗∗
(-17.7)
0.398∗∗∗
(24.77)
0.153∗∗∗
(27.35)
0.606∗∗∗
(38.88)
0.124∗∗∗
(14.11)
-0.059∗∗∗
(-5.78)
0.4031
97.89∗∗∗
957.25∗∗∗
Decile 8
(t-test)
0.271∗∗∗
(16.89)
-0.156∗∗∗
(-17.6)
0.427∗∗∗
(24.93)
0.081∗∗∗
(15.54)
0.571∗∗∗
(34.51)
0.164∗∗∗
(17.16)
-0.103∗∗∗
(-10.4)
0.3811
230.60∗∗∗
1214.59∗∗∗
Decile 9
(t-test)
0.130∗∗∗
(6.91)
-0.111∗∗∗
(-13.9)
0.342∗∗∗
(21.57)
0.180∗∗∗
(31.89)
0.676∗∗∗
(42.55)
0.126∗∗∗
(13.31)
-0.082∗∗∗
(-9.03)
0.4396
84.80∗∗∗
961.43∗∗∗
Decile 10
(t-test)
0.011
(0.43)
-0.048∗∗∗
(-6.47)
0.306∗∗∗
(15.83)
0.190∗∗∗
(30.59)
0.765∗∗∗
(47.10)
0.089∗∗∗
(8.07)
-0.109∗∗∗
(-12.3)
0.4607
92.34∗∗∗
975.39∗∗∗
χ2
P-value
801.000
<0.001
923.658
<0.001
1056.18
<0.001
1357.69
<0.001
3976.48
<0.001
472.492
<0.001
805.612
<0.001
Table A4: Estimation Results for NASDAQ Effective Spread Deciles (without Incidencet )
We report regression results from jointly estimating the following model for the 10 effective spread deciles in a GMM framework:
2
ESpreadi,t = αi + µV olumei V olumei,t−1 µT urnoveri T urnoveri,t−1 + µV ariancei σi,t−1
+ µM M Cnt M M Cnti,t−1 + µT rendi T rendi,t−1
SixteenthsN AS
DecimalsN AS
+ µSixteenthsN AS Di,t
+ µDecimalsN AS Di,t
+ it .
(10)
ESpreadi,t is the equal-weighted average effective spread for NASDAQ firms in decile i on day t. Other variables are defined analogously to
those defined in the notes to Table 3, but based on the NASDAQ effective spread deciles. The layout of the table, the tests for autocorrelation,
ARCH, and joint significance, and the sample period mirror that of Table 3.
Parameter
α
µV olume
53
µT urnover
µV ariance
µM M Cnt
µT rend
µSixteenthsN AS
µDecimalsN AS
R2
AR(10)
ARCH(10)
Decile 1
(t-test)
1.080∗∗∗
(14.61)
0.032∗∗∗
(3.72)
-0.008
(-1.61)
0.093∗∗∗
(15.39)
-0.001
(-1.45)
0.700∗∗∗
(40.71)
-0.477∗∗∗
(-16.6)
-0.248∗∗∗
(-11.1)
0.9138
2268.74∗∗∗
1262.37∗∗∗
Decile 2
(t-test)
0.344∗∗∗
(6.39)
0.009
(0.86)
0.046∗∗∗
(3.73)
0.230∗∗∗
(21.85)
0.002∗∗∗
(3.19)
0.794∗∗∗
(56.34)
-0.301∗∗∗
(-13.4)
-0.109∗∗∗
(-6.24)
0.8673
830.35∗∗∗
963.20∗∗∗
Decile 3
(t-test)
0.221∗∗∗
(4.55)
-0.050∗∗∗
(-3.73)
0.208∗∗∗
(12.60)
0.107∗∗∗
(13.83)
0.003∗∗∗
(4.45)
0.789∗∗∗
(54.85)
-0.226∗∗∗
(-11.4)
-0.079∗∗∗
(-4.71)
0.7890
721.82∗∗∗
1070.12∗∗∗
Decile 4
(t-test)
0.070
(1.58)
-0.045∗∗∗
(-2.94)
0.231∗∗∗
(12.50)
0.104∗∗∗
(14.38)
0.003∗∗∗
(4.15)
0.826∗∗∗
(58.55)
-0.146∗∗∗
(-7.88)
-0.090∗∗∗
(-5.33)
0.7315
682.89∗∗∗
1191.27∗∗∗
Decile 5
(t-test)
-0.013
(-0.33)
-0.098∗∗∗
(-6.59)
0.266∗∗∗
(15.03)
0.152∗∗∗
(21.90)
0.002∗∗∗
(3.79)
0.805∗∗∗
(56.16)
-0.087∗∗∗
(-5.27)
-0.066∗∗∗
(-4.04)
0.6829
362.45∗∗∗
677.70∗∗∗
Decile 6
(t-test)
-0.274∗∗∗
(-7.71)
-0.062∗∗∗
(-4.44)
0.230∗∗∗
(15.05)
0.259∗∗∗
(29.58)
0.004∗∗∗
(5.66)
0.875∗∗∗
(64.58)
-0.068∗∗∗
(-4.60)
-0.080∗∗∗
(-5.18)
0.6549
326.50∗∗∗
650.14∗∗∗
Decile 7
(t-test)
-0.215∗∗∗
(-6.89)
-0.077∗∗∗
(-6.31)
0.260∗∗∗
(20.04)
0.201∗∗∗
(27.25)
0.005∗∗∗
(8.80)
0.853∗∗∗
(61.87)
-0.060∗∗∗
(-4.44)
-0.154∗∗∗
(-9.67)
0.6019
162.93∗∗∗
639.73∗∗∗
Decile 8
(t-test)
-0.288∗∗∗
(-10.4)
-0.049∗∗∗
(-4.02)
0.191∗∗∗
(16.01)
0.287∗∗∗
(32.13)
0.004∗∗∗
(7.33)
0.868∗∗∗
(59.70)
-0.039∗∗∗
(-3.06)
-0.173∗∗∗
(-10.4)
0.5626
234.42∗∗∗
693.79∗∗∗
Decile 9
(t-test)
-0.109∗∗∗
(-4.37)
-0.014
(-1.48)
0.203∗∗∗
(21.28)
0.190∗∗∗
(25.97)
0.004∗∗∗
(5.33)
0.758∗∗∗
(53.13)
-0.032∗∗
(-2.51)
-0.288∗∗∗
(-14.9)
0.5014
258.88∗∗∗
716.29∗∗∗
Decile 10
(t-test)
-0.285∗∗∗
(-9.54)
0.062∗∗∗
(8.44)
0.260∗∗∗
(26.88)
0.125∗∗∗
(17.40)
0.003∗∗∗
(4.67)
0.604∗∗∗
(42.22)
-0.075∗∗∗
(-5.60)
-0.398∗∗∗
(-16.0)
0.5320
134.06∗∗∗
637.97∗∗∗
χ2
P-value
494.871
<0.001
233.549
<0.001
944.426
<0.001
1510.28
<0.001
117.371
<0.001
5948.30
<0.001
420.695
<0.001
446.494
<0.001