Skewness preference and measurement of abnormal returns, a comparative evaluation of current vs proposed event study paradigm

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Skewness Preference and Measurement of Abnormal Returns: A Comparative
Evaluation of Current Vs Proposed Event Study Paradigm

by

Suchismita Mishra

A DISSERTATION

Presented to the Faculty of
The Graduate College at the University o f Nebraska
In Partial Fulfillment o f Requirements
For the Degree o f Doctor o f Philosophy

Major: Interdepartmental Area o f Business (Finance)

Under the Supervision o f Professor John M. Geppert

Lincoln, Nebraska

December, 2002

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UMI Number: 3074090

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DISSERTATION TITLE
Skewnes Preference and Measurement of Abnormal Returns:

A Comparative

Fvalnainn of Curraftt--v-s-gg°Posed Event Study-Earadigm
BY
Suchismita Mishra

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Dr. John Geppert
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Dr. Gordon V.. KarelsTyped Name

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Dr. Manferd 0. Peterson
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r. James Schmidt
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GRADUATE
COLLEGE


Skewness Preference and Measurement of Abnormal Returns: A Comparative
Evaluation of Current Vs Proposed Event Study Paradigm

Suchismita Mishra, Ph.D.
University o f Nebraska, 2002

Advisor: John M. Geppert

If asset returns have systematic skewness, expected returns should include
rewards for accepting this risk. Many recent empirical studies such as by Kraus and
Litzenberger (1976), Sears and Wei (1988), Harvey and Sddique (2000) etc. have shown
that the pricing o f assets can be better explained by the three-moment capital asset pricing
model that accounts for systematic skewness, rather than the traditional two-moment
CAPM. Event studies in finance are concerned with abnormal returns after removing an

estimate o f the portion o f total return that represents the premium for bearing risk. Till
date event studies have used the return generating models that are consistent with some
form o f the traditional capital asset pricing model. Typically the market model (linear
characteristic line model) or a variant o f it is used as the underlying return generating
process in the computation of event specific abnormal returns. We investigate the
possible implications o f recognizing skewness preference for event study by using the
quadratic characteristic lines model (QCL), which is the return generating model
consistent with the three moment CAPM. We replicate the pioneer event study on stock
split by Fama, Fisher, Jensen and Roll (FFJR) (1969) on a new data set using their
methodology as well as other methodologies used by other event studies. First we test the

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conditions when QCL will be the appropriate return generating model and be applicable
to the given data set. With the market model we obtain the same intertemporal trend in
the abnormal return as reported by FFJR. Using QCL the same trend is maintained but
the level o f the values o f the abnormal returns are statistically significantly different than
that obtained using the market model.

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ACKNOWLEDGEMENTS
I would like to begin by thanking my elder brother Dr. Subhendu K. Mishra and
my husband Mr. Rohit Singh. My elder brother is the reason I am here today with my
Ph.D. I honor my husband’s support and sacrifice for my success in the graduate school.
He has done everything he could to take care o f me during the grueling years of my
doctoral study. Rohit, I love you.
I would like to thank my brother Dr. Sandip K. Mishra, who was there for me

through out my education in India. I must admit both my brothers are one of the most
wonderful gifts I have.
To my Mom (Mrs, Kumudini Mishra) and Dad (Mr. Nanda Kishore Mishra), a
thanks is never enough to express my feeling o f gratitude towards you. Your sacrifices
made me successful and my Ph.D. belongs to you.
I would like to thank my dissertation committee Dr. John Geppert, Dr. Gordon
Karels, Dr. Manferd Peterson, Dr. James Schmidt and Dr Richard Defusco. To Dr.
Geppert my committee chair, thank you for your patience, guidance and support. Dr.
Karels, you showed me how the world o f academics looks like. You taught me how to
become better scholar, by thinking critically and accepting challenges. Dr. Peterson you
are the one responsible for my joining the finance Ph. D. program. I can never thank you
enough for that. Dr. Defusco, thank you for your guidance and I look forward to working
with you. Dr. Schmidt, your in-depth teaching o f econometrics has enabled Ph.D.
students like me to be able to conduct research. In addition to the members of my
committee, I would like to thank all the faculty members in the finance department at
UNL for your guidance and willingness to provide support in various issues concerning
research. I have immensely enjoyed my time at UNL and I must admit whenever I go
back there, I feel like I am going back to a family.
Looking back on my graduate school experience I realize any achievement in a
life is not solely due to the personal effort o f the individual but also because of the effort
and support o f the people who stand by the person. To all o f you, thanks for caring and
thanks for helping me to achieve my goal.

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TABLE OF CONTENTS
Chapter 1

Chapter 2

Introduction and Purpose o f Study
I.

Introduction.......................................................................................... 1

II.

Theoretical Justification and Background........................................3

Literature Review............................................................................................ 9
I.

Introduction..........................................................................................9

II.

A Short Description o f the Event Study........................................ 10

HI.
IV.

Model Specification...........................................................................11
Empirical Application o f the Market Model and Specification
Error Biases....................................................................................... 14

V.

The Single Index Market Model Versus Multifactor Models:
The Issue o f The Associated Pricing Mechanism.......................... 18

VI.

Beyond the Two-Moment CAPM.................................................... 20

VII. Recent Support for the Three-Moment CAPM................................24
VIII. Purpose o f This Study....................................................................... 28
Chapter 3

Methodological Framework.........................................................................30
I.

A Brief Survey of Event Study Methodologies...............................30

II.

The FFJR Methodology.................................................................... 35

III.

Event Study with the QCL Model......................

IV.

Tests of significance..........................................................................38

V.

Comparison o f the QCL and the FFJR Market Model

38

Results............................................................................................... 40
VI.

Some Additional Graphical Comparisons....................................... 40

VII.

Suggested Preliminary Tests of the Appropriatenesso f the QCL
Model................................................................................................ 41

V m . Data..................................................................................................... 43
Chapter 4

An Analytical Evaluation of the Expected Results with the QCL
Model.............................................................................................................44

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Chapter 5

Empirical Findings........................................................................................ 51
I.

Appropriateness o f the QCL Model................................................. 51

II.

The Information Content o f Stock Splits and the Signaling
Hypothesis...................................................................................... 62

III.

Analysis o f the Residuals for the Market Model
(FFJR Technique)............................................................................ 63

IV.

Analysis of the Residuals for the Market Model (Standard
Technique)........................................................................................72

V.

Analysis o f the Residuals for the QCL Model
(FFJR Technique)............................................................................ 74

Chapter 6

VI.

Analysis o f the Residuals for the QCL Model (Standard

VI.

Technique)........................................................................................78
Standardized Abnormal Returns.......................................................83

Summary and Conclusion............................................................................ 102

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1
Chapter 1
Introduction and Purpose of Study
(i).

Introduction
No theoretical development in finance had such a profound effect on the

academics and professionals as the Capital Asset Pricing Model (CAPM) developed by
Sharpe (1964), Lintner (1965) and its attendant return generating process (Markowitz,
1959). The CAPM evolved over time with improvements advanced by Black (1972) and
Rubeinstein (1983). Furthermore, its (the CAPM’s) underlying return generating model
(commonly referred to as the market model) also became a very useful tool in studying
the abnormal performance o f securities and portfolios. In this context it is appropriate to
say that the market model and the development and use of CAPM are intertwined.
The original development of the CAPM is based on the assumption that investors
make their decisions based solely on the expected rate o f return and the variance. That is,
implicitly it is assumed that the investor either has a quadratic utility function and the rate
o f return follows a normal probability distribution (see Tobin, 1958) or the investor’s
utility function belongs to HARA (hyperbolic absolute risk version) class (Rubinstein,
1973) with cut-off point taken at the second power in the Taylor’s expansion o f the utility
function. However, the empirical findings of Fisher and Lorie (1970) and Ibbotson and
Sinquefield (1976) have shown that the rates of return distributions are skewed to the
right. Also, Friend and Blume (1975), using the data provided by the Internal Revenue
Service, empirically showed that the investor’s utility function does belong to the HARA
class. These empirical findings seem to negate not only the assumption o f normal

probability distribution but also the choice of second power as the cut-off point in the
expansion o f the utility function. These apparent discrepancies in the theory and
empirical findings prompted Arditti (1971), Jean (1973) and others to argue that some
improvement or modifications are needed in the original two moment asset pricing
paradigm.

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2
Kraus and Litzenberger (1976) were able to successfully reconcile these apparent
discrepancies in theory and empirical findings. By incorporating the investor’s preference
for positive skewness they developed the three moment CAPM. Furthermore, they
empirically showed the three moment CAPM explained asset prices more adequately
than the two moment CAPM did.
In empirical applications such as the study of the abnormal performances o f
securities, the market model is used quite frequently as the return generating model
(Peterson, 1989). The implicit assumption is that the assets are priced by the two moment
CAPM. However, if the empirical findings show that the assets are priced by the three
moment CAPM then the underlying return generating process should be consistent with
this asset pricing model. Since, the underlying return generating process for the three
moment CAPM is the quadratic characteristic lines model1, then the abnormal
performance should also be measured using this model. If this is the case, it will be
interesting to empirically examine whether the abnormal returns obtained using the
quadratic characteristic lines model will result in an outcome different than the one
obtained using the market model in event studies in finance. Note that throughout in this
dissertation we use the term "event study" to delineate only those studies where assets are
assumed to be priced by theoretically developed equilibrium pricing model and the
abnormal returns pertaining to an event is measured by a return generating process
commensurate with the assumed equilibrium asset pricing model2. To achieve this

objective we replicate the original study by Fama, Fisher, Jensen and Roll (FFJR, 1969)
on stock splits with the market model (that they used) as well as with the quadratic
1The QCL is given by
t ~Rft = CQi + Cu(Rmt - Rj } ) + C2i(.Rmt - R m)~ + e it
where Rit, R^ and Rml are the rates o f return on the security i, the risk-free rate, and the market rate o f
return. Coi,Cu and C2Il-sate the regression parameters.
1 Even the simplest o f all the market adjusted returns model where the ex post abnormal return on any
security i is given by the difference between its return and that on the market portfolio is based on the
assumption that securities have a systematic risk o f unity (Brown and Warner, 1980)

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3
characteristic lines model. The results we obtained for the market model using our data
set are strikingly similar to the original FFJR3 study. But the results using the quadratic
characteristic lines model are statistically significantly different than the one obtained
using the market model.
In what follows we provide the theoretical justification and background o f our
approach and the organization o f this dissertation.
(ii).

Theoretical Justification and Background
The Sharpe (1964) and Lintner (1965) capital asset pricing model (CAPM) based

on two parameters (mean and variance) has numerous applications in the finance
literature: performance measurement, tests of security market efficiency etc. The twomoment CAPM assumes that the underlying return generating process is the single index
model. This single index represents the entire economy (the market), hence the more
popular name for the single index model is the market model.
Tests of the CAPM are based on the assumption that the market model is the

appropriate underlying return-generating process (Gibbons, 1982; Stambaugh, 1982;
Shanken, 1985). Ex ante both the CAPM and the market model are single period models.
To test the validity o f the CAPM in ex post form, however, the CAPM is treated as a
cross sectional model (assuming in equilibrium all the firms in the economy are priced by
the CAPM for a given period). The market model in its ex post form is treated as a time
series model for every individual firm in the economy. Thus, the CAPM is a crosssectional model that seeks to explain the price o f an asset in terms o f the risk-free return
available, the return on the market portfolio, and the beta (P ) factor or the relative

3 We intentionally use a different data set to check whether the findings of FFJR (1969) still hold. In other
words we wanted to make sure that the findings are not just unique to the data set FFJR used.

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4
volatility of the asset compared to the market. The market model is thus supposed to
provide a proxy for the estimate of this parameter.4
Several studies such as Blume and Friend (1970, 1973), Black, Jensen and
Scholes (1972), Blume and Husick (1973), Fama and Macbeth (1973 ), Basu (1977),
Reinganum (1981), Litzenberger and Ramaswamy (1979 ) and Banz (1981 ) etc. have
appeared in the literature regarding the validity o f the CAPM, but no single empirical
work has provided evidence that the two-moment CAPM correctly represents the pricing
o f assets in the economy. In general, all the above cited studies found for aggressive
stocks ( p > 1) the CAPM tends to underestimate the required rate o f return. For
defensive stocks (/? < 1), the CAPM overestimates the required rate o f return (Copeland
and Weston, 1992, Pg:215). To correct for this overestimation/underestimation problem,
Kraus and Litzenberger (1976) argue that the investor’s risk preference should be
measured not only by the variance of the underlying stock, but by variance and a
preference for positive skewness.5 Their study extends the Sharpe-Lintner two-moment
CAPM by incorporating the effect o f skewness on valuation. Kraus and Litzenberger

empirically show that there is no underestimation or overestimation problem in asset
valuation, as is the case with the two-moment CAPM. The extension o f the two-moment
CAPM to a three-moment CAPM incorporates the behavioral assumptions of preference
for positive skewness.
Empirical studies by Kane (1982), Sears, and Wei (1988) and Harvey and
Siddique (2000) also support a preference for positive skewness.6,7 Thus, if the three4 Note that
estimated from the market model is an input in the security market line (SML) form o f the
CAPM
5 Note that one o f the sufficient conditions for the two-moment CAPM to be valid is that investors base
their choices on expected return and the variance o f the underlying rate o f return probability distribution. In
the case o f the three-moment CAPM, however, it is assumed that investors base their preferences on
expected rate of return, variance, and preference for positive skewness o f the underlying rate of return
probability distribution.
6 The two-moment CAPM can be obtained by assuming that an investor has a quadratic utility function and
that the underlying rates o f return distribution is Gaussian (normal). The assumption o f normality can be

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5
moment CAPM is the appropriate pricing mechanism, the return-generating process must
be the quadratic characteristic lines model (QCL) and not the linear characteristic lines
model (LCL). What would be the effect, if any, if one uses the QCL rather than the LCL
to compute abnormal returns in event studies? This is the research question we address in
this dissertation .In general, event studies assume that the two-moment CAPM is the
appropriate pricing mechanism and that the correct model to ascertain the effect of an
event on the rates o f return is the LCL.
In this study we examine whether the use o f the QCL will result in a different
outcome than that obtained using the LCL. Our study is performed in a comparative
framework. We select stock split as our event because Fama, Fisher, Jensen, and Roll’s

(FFJR hereafter) (1969) seminal event study was done on stock splits. To facilitate
comparison, we use the logarithmic form o f the market model (as used by FFJR) on our
data set. Also, we repeat the FFJR event study methodology for selecting an event and

discarded if one assumes that the investor's utility function belongs to the HARA class o f utility functions.
Note that in the development o f the three-moment CAPM the assumption o f quadratic utility will not be
valid by construction. Therefore, here also it is assumed that the investor’s utility function belongs to the
HARA class. The only difference between the two moment and the three moment CAPM is that in the case
of the earlier the expansion of the utility function is done to the quadratic term, whereas in the case o f the
former the expansion is carried out to the cubic term (Rubinstein, 1973).
7 As mentioned earlier, the expansion of any of the utility functions from the HARA class by Taylor’s
theorem can be carried out to any number of terms, as all are infinite series. In the CAPM literature to date,
however, the expansion is done either up to the quadratic (for the two-moment CAPM) or to the cubic
levels (for the three-moment CAPM). As demonstrated by Rubinstein (1973) and Stephens and Proffitt
(1991), it is easy to derive an n-moment (n>3) CAPM. As Kraus and Litzenberger observe, there is no
behavioral justification beyond three moments. One cannot explain how an investor will view the fourth
moment (kurtosis) or any o f the higher central moments.

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6
estimation window.8 We will also empirically obtain and compare the results using
widely practiced standard event study procedures.9
FFJR hypothesize that investors might interpret a stock split as a message about
future changes in the firm’s expected cash flows (and therefore as a message about a
change in dividends). A dividend increase shows the manager’s confidence about
maintaining the firm’s cash flow at a higher level in the long run. To test this hypothesis,
their sample is divided into firms that increased their dividend beyond the average for the
market after the split and those who paid lower dividends. The FFJR results show that the

stocks in the dividend increase class have positive abnormal returns, whereas the
cumulative average returns (CARS) for the split stocks with poor dividend performance
decline until about a year after the split. This FFJR hypothesis is known as the signaling
hypothesis (Lakonishok and Lev, 1987). In this dissertation we examine whether this
hypothesis still holds when positive skewness preference is recognized and whether there
is a significant difference in the market reactions for the dividend increase and dividend
decrease groups. Will this hypothesis hold if QCL is used as the appropriate process to
measure the abnormal returns? Broadly speaking we address this question in this
dissertation
Kraus and Litzenberger (1976) also provide conditions under which the QCL10
will be the appropriate return-generating process compared to the LCL:11

8 For a detailed description o f the event studies procedures, see Chapter 3. FFJR eliminate the 30-month
period surrounding the effective split date for firms subsequently announcing dividend decreases. Only IS
months preceding the split month for splits followed by dividend increases are eliminated. Their event
window is 60 months surrounding the split date both for the dividend increase and decrease class, thereby
leading to an overlapping estimation and event window.
9 The more standard practice is to separate the event and the estimation window to eliminate any
announcement effect on the parameter estimation o f the normal performance measure (Campbell, Lo, and
MacKinley, 1997).
10 A detailed analytical examination o f these conditions is provided in Chapter 4 o f this dissertation.
11 The results are in terms o f CARs.

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7
(a) The parameter associated with the squared market term (in the deviation
from mean form) is significantly different from zero, and
(b) The return on the market portfolio is asymmetrically distributed.

In a preliminary check, we find that these conditions hold for our data set. The
parameter described above (in (a)), for each security in our sample is significantly
different from zero and the return on the market is asymmetric as well.
Having ascertained that the conditions for QCL hold, we find that the C ARs
obtained using QCL dominate the CARs obtained using LCL in event time and CAR
space for the dividend increase sub-sample. Furthermore, the standardized abnormal
returns for the QCL model are significantly different than those obtained using the LCL
model. For the dividend decrease group, the CARs for the FFJR model dominate the
CARs for the QCL model. The standardized abnormal returns for the QCL model also are
significantly different than those of the LCL model. Neither the FFJR model nor the QCL
paradigm reveals any statistically significant abnormal return for the dividend decrease
group. Using QCL we do find support for the dividend hypothesis12 o f the FFJR study.
Also the extent of investor reaction obtained using QCL is statistically significantly
different than that obtained using the FFJR methodology.
Our study examines the validity o f the Kraus and Litzenebrger conditions for
QCL to be the appropriate return generating process in an event study framework.
According to their argument if these conditions hold then skewness preference exists and
thus QCL and LCL results should be significantly different. If these conditions do not
hold, then the results o f LCL and QCL should be identical. Thus our hypothesis of
interest is if the Kraus and Litzenberger conditions are valid for our data set then the

12 FFJR argue that a large price increase at the time o f stock split is due to altered expectations concerning
future dividends rather than to any intrinsic effects o f the splits themselves.

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8
abnormal returns adjusted for skewness preference via QCL should be significantly
different than the market model abnormal returns.

The rest of this study is organized as follows. Chapter 2 reviews the literature on
the market model and its applications in event studies and the literature on the threemoment CAPM and the QCL model. Chapter 3 describes the FFJR and standard event
studies methodologies. Chapter 4 develops a conceptual framework on the relationship
between QCL and LCL. The chapter explores the possibilities of surmising a priori
results that the QCL will obtain, given that we know the result that has been obtained
using LCL. Chapter 5 presents the empirical results. Chapter 6 summarizes our findings
and explores the possibilities for further research.

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9

Chapter 2
Literature Review
(i).

Introduction
Many recent empirical studies, such as those by Kraus and Litzenberger (1976),

Sears and Wei (1988), and Harvey and Siddique (2000), show that pricing o f assets can
be explained better by the three-moment CAPM than by the two-moment CAPM. The
three-moment CAPM is an equilibrium model based on mean, variance, and the skewness
o f the rate o f return probability distribution, while the two-moment CAPM is based on
only mean and variance. All event studies to date have used the LCL model, or a variant
o f it, as the underlying return-generating process in the computation of event-specific
abnormal returns as suggested by FFJR (1969). The implicit assumption in these studies
is that the two-moment CAPM is the underlying pricing mechanism. If assets are
correctly priced according to the three-moment CAPM, however, then the underlying
return-generating process should be the QCL model. QCL should be used to incorporate

the effect o f skewness in the estimation o f abnormal returns.
In this dissertation we use the QCL model to repeat FFJR’s (1969) pioneering
event study on stock splits. The purpose o f this empirical study is twofold. First, we test
if QCL is a better fitting return-generating model than the market model. Then we
compare the pattern o f abnormal returns using QCL with that o f the FFJR methodology.
Our intent is to see if the results using QCL are significantly different than the market
model. If we find the abnormal returns with QCL to be statistically significantly different,
then further refinements in event study methodology may be needed. For example, it may
be prudent to check data for the appropriate return-generating model before conducting
an event study.
This chapter reviews the literature on the market model and its applications in
event study techniques. The chapter also examines the literature on the three-moment
CAPM and the QCL model. We focus our review on the empirical findings and

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10
theoretical developments o f the three-moment CAPM done by Kraus and Litzenberger
and further refinements and additions by others. The theoretical underpinnings for the
current research are found in the literature in the following areas:
(a) The Event Study,
(b) Model Specification,
(c) Empirical Application of the Market Model and Specification Error
Biases,
(d) The Single Index Market Model Versus Multifactor Models: The Issue
of The Associated Pricing Mechanism,
(e) Beyond the Two-Moment CAPM,
(f) Recent Support for the Three-Moment CAPM.

(ii).

A Short Description o f the Event Study
Event study methodology is a frequently used analytical tool in financial research.

The goal o f an event study is to determine whether security holders earn abnormal returns
because o f specific corporate events such as merger announcements, earnings
announcements, stock splits, etc. The excess return, or abnormal return (Peterson, 1989),
is the difference between the observed return and the predicted return assuming that
returns stem from a given return-generating process. Assuming that the participants
exhibit rational behavior in the market place, the effect o f an event will be reflected
immediately in asset prices. The event’s economic impact can be measured using asset
prices observed over a relatively short time period.
Most event studies rely exclusively on Sharpe’s (1963) return-generating model.
The original market model and its various different specifications have been used to
obtain abnormal returns in event studies. We describe the original form and frequently
used specifications o f the market model below.

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11
(iii).

Model Specification
The original return-generating process proposed by Markowitz (1959) and Sharpe

(1963) is a statistical hypothesis that states that the expected rate o f return on a stock i in
time period /is a linear function o f the expected rate of return on a global market
portfolio. The ex ante form o f the market model hypothesizes a stochastic linear process

that generates security returns and is given by:
EiRu) = a i + fitE(Rml)i t = 1,

,T

(1)

where ctj and pi are the intercept and the slope, respectively, of the straight line defined on
the [ E(Rmt), £(/?„) ] plane, and Rit and Rmt are the rates o f return on the iIh security and
the market portfolio during the /th time period.
In ex ante!ex post form, the market model can be written as:
£ (£ J

(2)

where the left side is the expected return on security i in period t, given the market rate o f
return. The testable ex post form o f the model may be written as:
Rit = a ( + P,Rmt

t = 1, . . . , T

(3)

where S.t is the random error term or the residual portion that is unexplained by the
regression of the z',h stock on the market rate o f return during the /th time period.
The market model assumes that investors are single period, risk-averse, and that
they maximize the expected utility of terminal period wealth. In its original ex ante form
the market model is a single period model. But in the ex post form it is assumed that the
slope and intercept terms are constant over the time period so that estimates o f a and P
can be obtained. The market model is a statistical model, but it can be theoretically

related to the CAPM (Fama, 1976; Gibbons, 1982).
Equation (3) is a Type IV regression o f Rit on Rml that is used to empirically
obtain estimates o f a . and Pi (Press, 1972). In this simple linear regression model, the
independent variable is known and assumed to be nonstochastic. But in the market

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12
model, the independent variable is random, not nonstochastic. Therefore, equation (3) is a
regression equation o f Type IV;13 that is, equation (3) is an error-in-variables regression,
not a Type I (the functional regression usually seen in statistical textbooks) regression.
Estimation o f alpha and beta, however, is treated as a simple linear regression equation o f
Type I. The stochastic random term is presumed to follow wide-sense stationarity
assumptions:
(a) E(eit) = 0(zero mean)
(b) \a r(e it) = c r c (homoscedastic)
(c) cov(£it,ei ti.k) = 0 for all Ar^Oand
(d) cow(eu,Rmt) = 0
The sits are not assumed to be independent; rather, they are assumed to be
uncorrelated—there is no linear relationship between any two error terms. No assumption
regarding the distribution o f error terms is made. (Eventually the error terms are assumed
to be normally distributed to facilitate hypothesis testing.) The covariance between
Rm and s it is zero, market returns are observed without error. The above assumptions
imply that the conditional expectation o f Rit given Rmt is E(Rit | Rmr) = a { + PtRmt, the ex
ante/ex post specification o f the model.
There are several re-specifications o f the market model that are frequently used in
empirical research:
(a) The logarithmic form o f the price relatives is specified as:
loge rjt = a, + /?, loge lt +eit,

(4)

where rJt and /, are the price relatives for security j and the equally
weighted market portfolio. For small values o f rJt and I, , the natural
logarithm o f the security price relative is approximately the rate o f return
13 Unlike the simple linear regression where the independent variable is assumed to be a known variable, in
the case o f a Type IV regression the independent variable is a stochastic random variable. Hence,

i

of

2

If it were a Type I regression,
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13
in period t. This specification of the market model is used in FFJR’s
seminal event study.
(b) The standard specification of the market model is:
Ru = a i +PiRml+eit

(5)

This form is used by Scholes and Williams (1977), Brown and Warner
(1980), Brown and Warner (1985), and others.

(c) Van Horne’s specification of the market model is:
ru = al + Pirm,+£i'-

(6)

Van Home suggests that in the estimation of parameters a, and /?,, the
excess rate o f return form of the market model rather than the standard
form o f the market model should be employed, where excess returns are
defined by:

n, = R» ~ Rf< and rmt =Rm - R fn
where Rfi is the risk free rate during the period t.
(d) Black’s specification of the market model argues that Rfi should be
replaced by Rq, , where R0t is the return on a zero beta portfolio.
These re-specifications of the market model assume that in equilibrium the correct
underlying pricing mechanism is the SML o f the two-moment CAPM, i.e.:
E(Rit) = Rf + 0 iE[(Rnt) - R f ]

(?)

The underlying return-generating process o f the CAPM is specified by one o f the
above four return-generating processes. The market model and CAPM will be equal if the
following conditions hold:
(a) a ^ R f d - f r )
(b) The variance o f the market index is the same as the variance of the
market return.14

14 Let it be the market index. Then condition (b) above implies

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14
(c) The investment horizons in both the models are the same, and all
sufficient conditions o f the CAPM hold.
Fama (1973) and Subrahmanyan and Stapleton (1983) report these conditions.
Fama (1973) and Subrahmanyam and Stapleton (1983) also show that a linear market
model is sufficient for deriving the CAPM. This makes the market model the true
underlying return-generating process for the CAPM.
Several other ad hoc return-generating processes have appeared in the literature
(Chang, 1991; Ndubizu, Arize and Chandy, 1989). These are based on the assertion that
the rate o f return on assets (i.e., the left side in the market model) can be better described
not only by market factors, but also by industry and other factors. For example Benjamin
(1966) argues for the inclusion of size, etc. No theoretically viable equilibrium pricing
mechanism can be ascribed to these ad hoc models.

(iv).

Empirical Application o f the Market Model and Specification Error Biases
The market model is a common specification o f the return-generating process for

assets (Fama, Fisher, Jensen and Roll, 1969; Smith, 1977; Dodd, 1980; etc.). The model
also is referred to as the one factor market model, the single index model, and the LCL.
Usually the ordinary least squares (OLS) technique is used to estimate the
parameters o f the single index market model to conduct event studies in finance (Fama,
Fisher, Jensen and Roll, 1969; Smith, 1977; Dodd, 1980 etc.). Under general conditions,
the ordinary least squares method is a BLUE (best linear unbiased) estimate o f the market
model parameters.
FFJR (1969) use the market model to study stock splits and their implications for
market efficiency. The FFJR model assumes that alpha and beta are constant in the pre-

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15
and post-event dates. The study uses the logarithmic specification o f the simple single
index market model [equation (4)] to test the effect of stock splits on the rate of return of
a security. The price relative for th e /h security for month t is rJt, /, is the link relative to
Fisher’s (1966) combination investment performance index, and uJt is the random
disturbance term satisfying the assumptions of the linear regression model. The natural
logarithm o f the security price relative is the rate o f return (with continuous
compounding) for the month in question. The log of the market index relative is
approximately the monthly rate o f return on a portfolio that includes equal dollar amounts
o f all securities in the market. Using the available time series on Rjt and I, , the
parameters a ] and

are calculated.

Based on these estimates o f the parameters, FFJR calculate abnormal returns and
cumulative abnormal returns to determine if stock splits affect stock price returns
disproportionately.13 Their evidence indicates that stocks in the dividend increase class
(the sample o f firms that increase dividends beyond the average for the market following
a split) have positive abnormal returns following the split. According to the FFJR study,
the market’s reaction to the split is the reaction to its dividend implications. Thus,
abnormal return around stock splits, when followed by an increase in dividends beyond
the market average, is due to improved performance prospects.
Brown and Warner (1980) employ stock returns data to examine various
methodologies employed in event studies to measure security price performance. They
compare various abnormal performance measures such as mean-adjusted returns, marketadjusted returns, and market- and risk-adjusted returns. The probabilities o f Type I and
Type II errors are assessed for each abnormal performance measure using both

parametric and nonparametric tests. In addition, Brown and Warner examine the
distributional properties o f test statistics generated by each methodology. Brown and

15 The exact computation methods are given in Chapter 3.

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