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%From: bparks@wuecona.wustl.edu (Bob Parks)
%Date: Wed, 28 Jul 1993 11:52:08 -0500 (CDT)

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\title{A Simulation Investigation of Seemingly Unrelated Regression\\
as Used in Accounting Information Event Studies}


\author{Walter Teets\thanks{Assistant Professor, University of
Illinois at Urbana--Champaign.} \and
Robert Parks\thanks{Associate Professor, Washington
University in St. Louis.}
}

\date{\protect\vfill August 21, 1992\\ \protect\vfill
\protect\begin{flushleft} Comments welcome.  \\ The authors thank
Cray Research, Inc., for the generous grant of time on a
CRAY-2S/4-128 at the National Center for Supercomputing Applications,
University of Illinois at Urbana-Champaign.  We thank Dick Dietrich,
Nick Dopuch, John Fellingham, Grace Pownall, Barb Wheeling, and Dave
Ziebart for helpful discussions on this topic.
\protect\end{flushleft}}

\maketitle
\thispagestyle{empty}

Researchers studying stock price reactions to accounting information
releases can choose among several statistical
methods/models.  Firm-specific equation methods appear to be
particularly appropriate when the research hypotheses involve
possible differences across firms.  However, the firm-specific
equation methods make more demands on the data, requiring estimation
of
firm-specific coefficients and (possibly) covariance parameters.
Therefore,
it is not clear whether the researcher realizes net gains by using
firm-specific equation methods.  In this paper, we examine the
empirical behavior of test statistics arising from one firm-specific
equation method, seemingly unrelated regression (SUR), and
alternative test statistics based on the same set of equations, but
not incorporating estimates of cross-sectional
correlation.\footnote{In the interests of readability, we will use
``SUR'' to refer to both of the firm-specific equation models we
discuss.  It should be recognized that ``SUR'' properly refers only
to the model that utilizes estimates of cross-sectional correlation.
In the remainder of the text, when it is necessary to distinguish
between the two methods, we will use the terms ``consecutive equations''
when the equations are estimated using OLS firm by firm, and
``true SUR'' when discussing simultaneous estimation incorporating
estimates of cross-sectional correlation.} Evidence on the empirical
distributions of these statistics may guide researchers in designing
and interpreting research.

Understanding the empirical behavior of SUR is essential before
accounting researchers can correctly interpret results of existent
research or take full advantage of its conceptually
desirable features when testing hypotheses involving possible
differences across firms.  If characteristics of the data used by
accountants lead to high Type I error rates or poor power, inference
based on normal theory statistics derived from SUR may be faulty.
However, use of critical values based on empirical distributions may
overcome the difficulties and lead to appropriate inferences.

We provide evidence on the empirical distributions of several SUR
statistics through a simulation of accounting information releases
and associated stock price responses.  Following Brown and Warner
[1980,~1985]\nocite{brown_w1980,brown_w1985}, we start with actual
stock returns, randomly select event dates and introduce abnormal
performance on those dates.  Using this data, we estimate a SUR model
and calculate statistics.  Multiple repetitions of this process
generate empirical distributions that provide evidence on Type I
error rates and power.  We generate empirical distributions of the
statistics for a number of different scenarios corresponding to
different types of accounting information releases.  We vary the
number of firms used in the estimation procedure, the level of
abnormal performance introduced on an event date, the number of event
dates per firm in the estimation period, and the number of days
included in the event window.  This allows us to assess whether SUR
should be expected to work better in some accounting research
contexts than in others.  We use NYSE, ASE, and NASDAQ firms.
Briefly, we find (1) the null hypothesis that all coefficients
relating stock returns to information events are simultaneously equal
to zero is rejected far too often when no abnormal performance has
been introduced into the returns; (2) after correcting for high Type
I error rates, powers of statistics testing all coefficients
simultaneously equal to zero are low, especially if there are few
events per firm; (3) the hypothesis that the average of the
coefficients relating stock returns to information events equals zero
is sometimes rejected too often when no abnormal performance is
introduced, but the over-rejection is not severe; and (4) event date
uncertainty, as it is typically addressed in SUR, severely reduces
the powers of the tests in most situations.

The paper is related to research that has used simulation techniques
to examine the behavior of statistical methods used in event studies.
Using monthly and/or daily returns data for American and New York
stock exchange firms, Brown and Warner
[1980,~1985]\nocite{brown_w1980,brown_w1985} and Dyckman, Philbrick,
and Stephan [1984]\nocite{dyckman_ps1984} investigate how well
various abnormal return metrics, pooled cross-sectionally, are able
to identify significant average abnormal returns.  Campbell and
Wasley [1991]\nocite{campbell_w1991} examine the same issue using
daily returns for NASDAQ securities.  None of these studies examine
methods that utilize firm-specific equations relating information
variables to firm returns.  Hence, they do not examine specifically the
situation where effects may differ across firms, nor do they address
the incorporation of estimates of cross-sectional correlation into
the estimation procedure.

Collins and Dent [1984]\nocite{collins_d1984} propose and examine via
simulations a technique that incorporates cross-sectional correlation
in the case where all events affect all firms on the same day(s),
using NYSE and ASE firms.  Malatesta [1986]\nocite{malatesta1986} and
McDonald [1987]\nocite{mcdonald1987} both examine SUR in event study
frameworks, but both use only one event per firm, and use only
samples of 30 firms.  In Malatesta, the independent variables are the
same across all firms---the so-called multivariate analysis.  We
extend previous research on SUR on several dimensions.  First, we
simulate 2, 5, and 20 events per firm, instead of only 1.  We also
examine how the number of firms in the model affects the statistics
by using samples of 25, 50, and 75 firms, rather than only 1 sample
size of 30 firms.  Finally, we investigate the effect of event date
uncertainty on test statistics by examining 1, 2, and 5 day event
windows.

The remainder of the paper is organized as follows.
Section~\ref{sec:statmeth} discusses the use of SUR in accounting and
finance event studies, and reasons for concern about the test
statistics generated.  Section~\ref{sec:sim} presents the approach
used in the simulations, including the scenarios simulated and how
those scenarios relate to accounting research situations.
Section~\ref{sec:stats} covers the hypotheses tested in event
studies, the statistics collected from each simulation, and the
theoretical distributions of these statistics.
Sections~\ref{sec:results} and \ref{sec:interp} present the empirical
results of the simulations and implications for design and
interpretation of accounting research.  Finally,
section~\ref{sec:conclude} gives directions for future work.

\section{SUR as used in accounting event studies}
\label{sec:statmeth}

Accounting information event studies assess stock price reactions to
releases of accounting information.  The researcher attempts to
estimate the relation between new information in an accounting number
and the change in stock price (the stock return).  Early event
studies examined (cumulative) abnormal returns from the event period
to see whether there were significant abnormal returns on average
across firms during the event period.  To assess differential returns
across different groups of firms, firms were assigned to portfolios
based on some firm characteristic, and the abnormal returns for
different portfolios were compared.  A problem with this approach to
assessing differential returns in
some research contexts is that either (1) the levels of
information in the accounting numbers must be assumed to be the same
for all firms, so a firm's abnormal return captures the
(cross-sectionally varying) stock price effect of the ``unit'' of
information, or (2) the abnormal return for each firm must be assumed
to be a composite, due partly to the level of new information in the
firm's accounting number, which could vary cross-sectionally, and
partly to the coefficient mapping a unit of information into the
stock return, which could also vary cross-sectionally.  In other
words, in cases where the level of information in an accounting
number may vary across firms, and the stock price response to a unit
of information may vary across firms, abnormal returns give composite
measures that are not easily separated into their constituent parts.
Researchers must ignore part of the information available in the
accounting release if portfolio abnormal return analysis is used.
Alternatively, abnormal returns can be regressed on measures of
information in cross-sectional pooled regressions.  In this case, the
researcher implicitly assumes the coefficient relating information
to abnormal returns is cross-sectionally constant.

To incorporate both different levels of information, and different
mappings across firms of stock price response to a ``unit'' of
information, an extended market model can be used.  The extended
market model is
\begin{equation} r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it} \label{eq:retgen}
\end{equation}
where $r_{it}$ and $r_{mt}$ are daily returns to firm
{\em i'}s stock and the market portfolio for day~{\em t},
$\varepsilon_{it}$ is a random error term, and $\alpha_i$ and
$\beta_i$ are market model parameters.  The terms $\alpha_i$ and
$\beta_i\ r_{mt}$ are included to control for market wide movements
in stock prices unrelated to the accounting information releases of
interest.  The term $\varepsilon_{it}$ is included to allow for other
events affecting stock price that are not related to the accounting
information release of interest.  $A_{i}$ is a vector of
accounting information releases whose elements take on nonzero values
only for those days on which there is a
release of interest.  $\gamma_i$ is a firm-specific coefficient
relating accounting information  to stock returns.

If there is only one nonzero element per firm, equal to unity, in
each $A_i$ vector, the $\gamma_i$ coefficients are equivalent to
market model abnormal returns.  If there are multiple nonzero
elements in the $A_i$ vector, each equal to unity, the $\gamma_i$
coefficient is equal to the average of all event period abnormal
returns for firm $i$.  Finally, if the nonzero elements of the $A_i$
vectors  vary cross-sectionally
 and through time, the $\gamma_i$
coefficients are firm-specific response coefficients that map
different levels of information into stock returns.  It is this last
situation that we simulate in this paper.

Equation (\ref{eq:retgen}) can be estimated using ordinary least
squares for each firm.  However, if there is contemporaneous
correlation across firms among the errors $\varepsilon_i$, stated
significance levels of statistical tests may be incorrect,
potentially leading to incorrect inference.  SUR may be used to
incorporate estimates of cross-sectional correlation in the
estimation process and in statistical tests.

The general SUR model is
\begin{equation}
\left[\begin{array}{c}{\bf Y}_1\\{\bf Y}_2\\ \vdots\\{\bf Y}_n
\end{array} \right] = \left[\begin{array}{cccc}{\bf X}_1 & {\bf 0}&
\ldots &{\bf 0}\\{\bf 0}&{\bf X}_2&\ldots&{\bf 0}\\
\vdots&\vdots&\ddots&\vdots\\
{\bf 0}&{\bf 0}&\ldots&{\bf X}_n\end{array}\right]
\left[\begin{array}{c}{\bf \Gamma}_1\\{\bf \Gamma}_2\\\vdots\\{\bf \Gamma}_n
\end{array}\right] + \left[\begin{array}{c}{\bf \varepsilon}_1 \\{\bf
\varepsilon}_2 \\ \vdots \\{\bf \varepsilon}_n\end{array}\right]
\label{eq:genmod}
\end{equation}
\indent where

\begin{center}
\begin{tabular}{rcp{3.5in}}
$n$ & = & number of firms in the sample \\
${\bf Y}_i$ & = & $t\times 1$ vector of $t$ time series observations
on the dependent variable for the $i$th firm;  this corresponds to
the $r_i$ vectors in equation (\ref{eq:retgen})\\
${\bf X}_i$ & = & $t \times k$ matrix of explanatory variables for
firm $i$;  from (\ref{eq:retgen}), this matrix consists of the
constant, $r_{mt}$, and $A_{i}$ vectors\\
${\bf \Gamma}_i$ & = & $k \times 1$ vector of firm-specific
coefficients ($\alpha_i$, $\beta_i$, and $\gamma_i$ from
(\ref{eq:retgen})) relating the dependent variable to the explanatory
variables, and \\
${\bf \varepsilon}_i$ & = & $t \times 1$ vector of errors.
\end{tabular}
\end{center}

\noindent This set of equations may be written
\[{\bf Y} = {\bf X \Gamma} +
{\bf \varepsilon}. \]  The most efficient estimator of ${\bf \Gamma}$ is
\begin{equation}
\hat{\bf \Gamma} = ({\bf X}^{\prime} ({\bf \hat{\Sigma}^{-1} \otimes I}) {\bf
X})^{-1}({\bf X}^{\prime} ({\bf \hat{\Sigma}^{-1} \otimes I}) {\bf Y}),
\label{eq:estim}
\end{equation}
where ${\bf \hat{\Sigma}}$
is an $n \times n$ matrix of pairwise covariances among the $n$
firms.  The elements of ${\bf \hat{\Sigma}}$ are obtained by estimating the
firm
specific equations in (\ref{eq:genmod}) equation by equation,
obtaining the $n$ error vectors $\varepsilon_i$, and calculating
covariances between all pairs of error vectors.

Hypotheses about the relations between returns and accounting
information events can be tested using linear combinations of the
$\gamma_i$ coefficients, either the OLS estimates from the individual
firm-specific equations in (\ref{eq:retgen}), or the EGLS estimates
from equations (\ref{eq:genmod}) and (\ref{eq:estim}).  If the
researcher believes {\em ex ante} that cross-sectional correlation is
negligible, s/he may choose to use the OLS estimates, hence avoiding
the possible introduction of estimation error in
$\hat{\Sigma}$.  If the probability of significant
cross-sectional correlation is high, however, the researcher may
choose to use the EGLS estimates, even though estimation error may be
present in $\hat{\Sigma}$.  We present evidence on several OLS and EGLS
statistics that may be used in hypothesis testing.

Although the capability to estimate firm-specific coefficients and
incorporate estimates of cross-sectional correlation is desirable in
many accounting contexts, these capabilities do not come costlessly.
There are four potential problems.  First, SUR assumes normally
distributed error terms, but it is well known that market model daily
abnormal returns (essentially the $\varepsilon_{i}$ in equation
(\ref{eq:retgen})) are leptokurtic.  It is not known how sensitive
the SUR statistics are to departures from normality.  Second, the
test statistics frequently used in SUR are only asymptotically
correct.  While a typical accounting event study using SUR uses a
time series of approximately five years of daily data per firm, there
is no theory specifying how many observations are needed before the
asymptotic statistics are relevant.  Third, although the researcher may
use five years of daily returns, the number of non-zero observations
per firm in the event vector itself is generally much
smaller---perhaps as few as~1 or~2 in the case of management
forecasts of earnings.  While the total number of observations per
firm is large, the number of meaningful observations used to estimate
the main parameters of interest is very small.  But the degrees of
freedom used to establish rejection regions are based on the total
number of observations, not the meaningful observations for the
parameters of interest.  Fourth, the cross-sectional correlation
matrix is not known, but must be estimated from the data.  Another
source of estimation error is therefore introduced into the model.

In this paper, we provide evidence on the joint effect on the test
statistics of these four potential problems.  Determining the
contribution of each effect individually is beyond the scope of this
paper.  Parks and Teets [1993]\nocite{parks_t1992} provide additional
evidence on the contribution of the individual elements.

\section{Simulation overview, rationale, and implementation details}
\label{sec:sim}

The objective of this study is to furnish evidence on the empirical
behavior of SUR by simulating SUR in contexts representative of those
accounting researchers might encounter.  This section gives a broad
overview of the simulation process, followed by a more detailed
examination of the rationale for and the implementation details of
each step of the process.

\subsection{Overview of simulation procedures}
\label{ssec:simproc}

All of the simulations reported in this paper started with actual
firm and market returns.  There were four different factors that we
manipulated across the simulations: (1) the number of events per
firm, (2) the magnitude of the abnormal performance added to the
actual return on an event date, (3) the number of firms in the model,
and (4) the length of the event window around each event date.  All
combinations of these four factors (detailed in the following
sections) gave us 108 different scenarios:


\begin{center}
\begin{tabular}{rlrr}
&Ranges of abnormal performance introduced && 4\\
&\quad 0, [.00125,.00375], [.00375,.00625], [.0025,.0075] \\
$\times$&Numbers of events per firm && 3\\
&\quad 2, 5, or 20\\
$\times$&Numbers of firms in model && 3\\
&\quad 25, 50, or 75\\
$\times$&Length of event window &\hspace{2em}& 3\\
&\quad 1, 2, or 5 trading days\\ \cline{4-4}
&\hspace{3em} Total number of scenarios&&108
\end{tabular}
\end{center}


\subsection{Returns}
\label{ssec:rets}
The underlying returns used in these simulations are actual firm
returns.  Starting with actual returns is important, as it is
well-known that daily returns are leptokurtic.  The departure from
normality may affect the empirical distributions of the
SUR statistics used in hypothesis tests.  We gathered returns data
for~30 different sets of~75 firms, and
corresponding market returns.  For each firm, 1,280 returns were
used.  This represents approximately~five years of daily
returns.\footnote{The number~1,280 was chosen due
to hardware considerations on a Cray supercomputer used in the simulations.}

One potential benefit of SUR is that it incorporates estimates of
cross-sectional correlation of residuals in coefficient estimates and
statistical tests.  A reason for this correlation may be that firms
in an industry are affected similarly by events for which the market
model provides insufficient control.  Therefore, we selected firms
from related industries for each set of 75 firms.  First, listings of
NYSE/ASE and NASDAQ firms in each two digit SIC code were generated
from the CRSP daily stock return files.  Separate lists were
generated for 1975--79, 1980--84, and 1985--89.  To be included in a
list, a firm could have no missing data during the respective time
period.  Next, we combined lists of firms from consecutive two digit
SIC codes until we had lists containing 75 firms.  For example, one
set contained returns from 1975--79 for NYSE/ASE firms in SIC codes
34--35.  The 30 sets of firms included 10 sets from each of the three
time periods.  Twenty sets were composed of NYSE/ASE firms and ten
were from NASDAQ.  Each set of returns was used as the basis for 55
sets of simulations, resulting in 1,649 simulations for each of the
108 different scenarios.\footnote{We generated 1650 simulations for
each scenario, but one set of results was inadvertently erased.}

\subsection{Event dates}
\label{ssec:numev}

Two broad classes of events are of interest to accounting
researchers.  The first comprises those accounting-related
announcements that occur irregularly.  The second
comprises events that occur regularly.  SUR may not be equally effective in
these different situations, because they give rise to different
numbers of non-zero elements in the information event vectors.
Management forecasts may give rise to only one or two announcements
in a five year period.  There will be five annual reports issued in a
five year period, and 20 quarterly earnings announcements.  To
investigate whether the number of events per firm affects the
distributions of test statistics, we ran
simulations where each firm experienced 2, 5, or 20 events.

Two sets of event dates were generated for each set of simulations.
The first set was used for the 5 and 20 event simulations.  Twenty
dates, corresponding to quarterly announcements, were generated for
each firm.  The first date for each firm
fell within the first 77 observations.  The next 19 dates were
obtained by adding 63 days to the previous date (there are
approximately 63 trading days per quarter).  For the five events per
firm simulations (yearly events), the first, fifth, etc.,
dates were used.  Having the five events be a subset of the 20 events
allows us to assess the effect of adding events to an existing set of
events.  This corresponds to the research situation where the
researcher must decide whether to use quarterly announcements as well
as annual announcements.

For the two events per
firm simulations, different dates were generated.  The only
requirement for these dates was that they not be within thirty days of
each other for the same firm.

The maximum number of events per firm in a given simulation was 20.
Over the 55 sets of simulations based on a single set of firms, this
implies that 1,100 dates per firm were used as event dates.  No
attempt was made to insure that the dates for a given firm were
different across simulations.

\subsection{Levels of abnormal performance}
\label{ssec:level}

In order to determine the Type~I error rates of the various
statistics, simulations were run where no abnormal performance was
introduced into the returns vectors.  Simulated values used in the
$A_i$ vectors on event dates as measures of (false) information were
uniformly distributed [.00125,.00375] (mean of .0025).

To assess the powers of different statistics under the alternative
hypothesis, three ranges of abnormal performance and associated
information events were used: [.00125,.00375] (mean of .0025),
[.00375,.00625] (mean of .005), and [.0025,.0075] (mean of .005).
Three different ranges were used to assess how different levels of
abnormal performance affect the ability of SUR to detect stock price
responses to information events.  In these simulations, the simulated
values were used in the $A_i$ vectors on event dates and
were added to the respective firms' returns vectors on corresponding
days.  This implies that the true coefficient relating the
information variables to the abnormal returns was unity for all
firms.

\subsection{Numbers of firms}
\label{ssec:numfirm}

Accounting researchers are faced with questions regarding adequacy of
sample size.  In SUR, there are two sample size issues.  The first
one, how many events per firm are needed, has been discussed in a
previous section.
The second one involves the number of firms that are needed for
effective use of SUR.  We did simulations with 3 different sample
sizes, in terms of numbers of firms.  Each of the 30 sets of 75 firms
was broken down into subsets of 25, 50, and 75 firms.  The models
using~50 firms added~25 firms to the models
using~25 firms, and the models using all~75 firms
added~25 firms to the~50 firm models.  When~25 additional firms were
added to a model, the information event dates and values for the
firms already in the model remained the same.  This allows us to
assess the effect of adding additional firms to an existing data set,
rather than confounding the addition of firms with the effects of new
event dates and abnormal returns for existing firms.

\subsection{Event windows}
\label{ssec:window}

Assessing the stock price change associated with an accounting
information release is made more difficult because the researcher may
not know exactly when a piece of accounting information became known.
This difficulty was overcome in studies based on market model
abnormal returns by using a cumulative abnormal return that covered a
multiday window assumed to include the date of actual release of the
information.  Generally, longer event windows in SUR are implemented
by assigning the information value for a given event to several
consecutive elements in the information event vector.  However, this
does not accomplish the same thing that was accomplished with a
cumulative abnormal return.  Underlying the use of cumulative
abnormal returns is the idea that the effect of (possibly
unidentified) confounding events occurring during the cumulation
period will average out to zero, so the expected value of the
cumulative abnormal return will be the abnormal return associated
with the event of interest.  Assigning the information value to
several consecutive event dates in SUR may accomplish quite another
thing.  In effect, the researcher assumes that the entire stock price
change due to an accounting disclosure takes place rapidly, but is
unable to identify exactly when the disclosure was
made.\footnote{There may be cases when the researcher assumes that
the reaction to the announcement takes place over several days.  In
these cases, having non-zero information values for consecutive days
may be appropriate.} Assume that all event windows are two days in
length (the information event vector has two consecutive non-zero
values for each event), and that the return of interest occurs on one
of the two days.  In terms of equation (\ref{eq:retgen}), the portion
of $r_{it}$ not explained by $\alpha_i + \beta_i r_{mt}$ has an
expected value of zero on non-event days, and is the desired abnormal
return on the true event day.  Hence, the $\gamma_i$ coefficient will
be estimated using observations half of which have the desired
relation, and half of which in effect associate the
information variable with a value of zero (or noise).  Because
$\gamma_i$ is estimated by minimizing the sum of squared residuals,
it will be biased toward zero, and its standard error will be biased
upwards, relative to the situation where the information variable is
non-zero only on the ``correct'' date.

In order to assess the behavior of SUR in situations where the event
date cannot be identified exactly, two day and five day event windows are
used, in addition to the one day window.  In the two (five) day simulations,
the information variable is assigned the non-zero value two (five)
consecutive days, while the non-zero value is added as an abnormal
return to the actual return on only one of the two (five)
days.\footnote{We always added the value to the first return in the
multiday window.  As the event dates are randomly generated, this
should not affect the generalizability of our results to situations
where the event window is constructed to precede or surround the
uncertain event date.}

\section{Hypotheses tested and statistics used in the tests}
\label{sec:stats}

There are two basic hypotheses tested in accounting event studies,
both having to do with the association between stock price changes
and information events.  The first null hypothesis is that each of
the $\gamma_i$ coefficients in (\ref{eq:retgen}) is equal to zero.
The second null hypothesis is that the average stock price response
to the information events, across all firms and events, is equal to
zero; i.e., there is no information disclosed in the information
releases that is associated with stock price changes, on average.
Likelihood ratios, $\chi^2$ statistics, several F statistics, and
statistics based on sums of coefficients and associated standard
errors have been proposed to test these hypotheses.  Parks and Teets
[1993]\nocite{parks_t1992} presents evidence on several of
these statistics.  Here, we concentrate on three sets of F-statistics
and statistics based on sums of coefficients and associated standard
errors.

\subsection{$H_{01}\!: \gamma_i = 0\; \forall i$}

The first null hypothesis is that each of the
$\gamma_i$ coefficients in (\ref{eq:retgen}) is equal to
zero.\footnote{The alternative hypothesis is that one or more of the
$\gamma_i$ coefficients is not equal to zero.  It is not that none
are equal to zero.}  This null hypothesis is useful in two
situations.  The first is when the researcher expects some firms to
experience positive returns and others to have negative returns to
similar announcements.  The average of the $\gamma_i$'s may be
insignificantly different from zero, because positive coefficients
for some firms are offset by negative coefficients for others.  For
example, if the event of interest to the accounting researcher is a
change in tax law which will benefit some firms, but will harm
others, a test for an effect of the law {\em on average} is likely
to lead the researcher to conclude that the change in tax law does
not significantly affect firms, while a simultaneous test of the
individual coefficients may lead to the opposite conclusion.  The
second situation is when an announcement has major implications for
only one or two companies, and minor implications for others.  Again,
the {\em average} effect may be insignificant, but the effect is not
insignificant for all firms.

For each simulation, we calculated three F-statistics that have been
suggested to test $H_{01}$.  Two of the statistics are from the true
SUR model, while the third is from the consecutive equations model.
The F-statistics from the true SUR model are both based on the
restricted and unrestricted system mean square errors (SMSE) from the
SUR system in equation (\ref{eq:genmod}).  The first F-statistic is
that calculated by the SAS statistical package.  It is defined
\[\frac{\mbox{SMSE}_R - \mbox{SMSE}_U}{q} \left/
\frac{\mbox{SMSE}_U}{(T - k) * N},\right. \] where $_R$ and $_U$
denote restricted and unrestricted, $q$ is the number of
restrictions, $N$ is the number of firms (equations) in the model,
$T$ is the number of time-series observations used for each firm, and
$k$ is the number of coefficients estimated for each firm.  For
testing $H_{01}$, $q$ is equal to the number of firms in the
model, corresponding to the restriction under the null that all
$\gamma_i$ in equation (\ref{eq:retgen}) are equal to zero.  It is
distributed asymptotically F($N$,$(T-k)*N$).

The second F-statistic, which we denote the Schipper and Thompson
F-statistic (see Schipper and Thompson
[1985]\nocite{schipper_t1985}) is also based on the difference in
SMSE's, and is exact in situations where the independent variables
are the same across all firms---the multivariate case.  It is
commonly thought to be more conservative than the SAS F.  The
Schipper-Thompson (hereafter ST) F to test $H_{01}$ is
defined to be \[(\mbox{SMSE}_R - \mbox{SMSE}_U) * \left(\frac{T - k -
N + 1}{(T - k) * N}\right),\] and is distributed F($N$,$T-k-N+1$) in
the multivariate case.

In cases where the researcher decides not to incorporate estimates of
cross-sectional correlation into coefficient estimates or statistical
tests, an F-statistic based on the OLS estimates of equation
(\ref{eq:retgen}) can be used to test the hypotheses.
This F-statistic is equivalent to the SAS F statistic when the
covariance matrix is diagonal.  However, it can also be expressed as
\[ \frac{1}{N} \sum_{i=1}^N \frac{\gamma_i^2}{\sigma^2_{\gamma_i}},
\] where $N$ is the number of firms, the $\gamma_i$ are from equation
(\ref{eq:retgen}), and $\sigma^2_{\gamma_i}$ are the OLS estimates of
the variances of the $\gamma_i$.\footnote{See Theil, pp. 314-317,
particularly problem 3.1.} This statistic, which we denote the Theil
F, is distributed asymptotically F($N$,$(T-k)*N$).

\subsection{$H_{02}\!: \sum_{i=1}^N \gamma_i = 0$}
The second null hypothesis is that the average\footnote{Tests of
$\sum_{i=1}^N \gamma_i = 0$ and $\frac{1}{N}\sum_{i=1}^N \gamma_i =
0$ result in the same F statistic.  It is convenient to drop the
$\frac{1}{N}$ term.}
stock price response to the information events, across all firms and
events, is equal to zero; i.e., there is no information disclosed in
the information releases that is associated with stock price changes,
on average.  In cases where the effects of the information events are
expected to be in the same direction for all firms, this may provide
a more powerful test than the previous test.  This is because the
accumulation of many small effects, none significant in themselves,
may be significant in total.\footnote{Earlier abnormal returns
studies generally tested this hypothesis.  While offsetting effects
could be handled by creating separate portfolios
for firms expected to be affected positively or negatively,  this
would still give statistics based on groups of firms, rather than a
statistic testing all firms individually, but simultaneously.}

To test $H_{02}$, we again calculated two F-statistics from the
true SUR model.  The SAS F used to test $H_{02}$ is again defined as
\[\frac{\mbox{SMSE}_R - \mbox{SMSE}_U}{q} \left/
\frac{\mbox{SMSE}_U}{(T - k) * N}.\right. \]  For testing
$H_{02}$, $q$ is one, the only constraint under the null being
that the sum of the $\gamma_i$ coefficients be zero.  This statistic
is distributed asymptotically F(1,$(T-k)*N$).

The Schipper and Thompson F-statistic used to test $H_{02}$ is
simply \[(\mbox{SMSE}_R - \mbox{SMSE}_U),\] and is distributed
F(1,$T-k$) in the multivariate case.

There are three statistics based on the consecutive equations model
that can be used to test $H_{02}$.  The first, which we again
call the Theil F, is asymptotically
distributed F(1,$(T-k)*N$).  It can be calculated as
\[
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\frac{\left(\sum_{i=1}^{N}\gamma_i\right)^2}{\sum_{i=1}^{N}\sigma^2_{\gamma_i}}. \]

The final two statistics, which we call SUMT and SUMC, were used by
Malatesta\footnote{SUMT and SUMC are the W** and Z** statistics in
Malatesta.  McDonald reports the W** statistic, but not the Z**
statistic. } [1986]\nocite{malatesta1986}.
SUMT is a sum of t-statistics, divided by the square root of the
number of t-statistics in the sum: \[ \mbox{SUMT} =
\left(\sum_{i=1}^{N} \frac{\gamma_i}{\sigma_{\gamma_i}}\right) \left/
\sqrt{N}, \right. \] \noindent where $N$ is the number of firms.
SUMT is distributed N(0,1) under the null hypothesis of no stock
price effect, on average, associated with the information events, and
ignoring the covariance structure of the firm returns.

SUMC is
the sum of the $\gamma_i$ coefficients, divided by the
square root of the sum of their OLS variances: \[ \mbox{SUMC} =
\left(\sum_{i=1}^{N} \gamma_i \right) \left/ \left(\sum_{i=1}^{N}
\sigma_{\gamma_i}^2\right)^{(1/2)}\right. .\] \noindent SUMC is also
distributed N(0,1), again assuming that firm returns are independent.

\section{Results of simulations}
\label{sec:results}

In this section, we report on two aspects of the results of the
simulations.  First, we discuss the Type I error rates of the various
statistics for each of the two hypotheses.  Next, we discuss the
power of the statistics.

\subsection{Type I errors: Rejections when no abnormal performance is
introduced}

\subsubsection{$H_{01}\!: \gamma_i = 0\; \forall i$}
In table \ref{tab:all0null} we present the Type~I error rates for the
three F-statistics used to test $H_{01}$.  A Type I error occurs when
the null hypothesis is rejected when no abnormal performance was
introduced into the returns series on the event dates.\footnote{It
may be that the apparent over-rejections are due to our having
non-zero elements in the $A_i$ vectors that happen to coincide with
dates on which there actually were significant information events for
the firms used in the simulations.  However, that is a strength of
basing the simulations on real returns data, not a weakness.  The
researcher can never get a truly clean sample, where nothing else has
affected the firms on the identified event dates.  Only by using
returns drawn from such an information rich environment can we get a
realistic idea about how the statistical methods will behave in
practice.}  All of the
rejection frequencies
are significantly above the nominal levels.  Using a normal
approximation to the binomial distribution, and n=1649, the 95\%
confidence intervals should be [3.9\%,~6.1\%] for the nominal 5\%
rejection level, and [.5\%,~1.5\%] for the 1\% rejection level.
The null hypothesis is rejected far too
often.  At the nominal 5\% level, rejection rates range from 9.2\% to
33.9\% for the SAS F, 8.2\% to 23.5\% for the ST F, and 6.9\% to
20.3\% for the Theil F.  The best case at the 5\% level occurs for the
Theil F statistic for 20 events and 25 firms, using a 1 day window.
The rejection rate of 6.9\% is about 3.5 standard
deviations away from the 5\% level.  For the nominal 1\% level,
rejection rates are also high.  For the SAS F, rejection rates range
from a low of 3.5\% to a high of 19.5\%.  Similar ranges for the ST F
and the Theil F are 2.9\% to 11.9\% and 2.7\% to 9.6\%. The best case
at the 1\% level again occurs for the Theil F, for 20 events and 25
or 75 firms, using a 1 day window.  The rejection rate of 2.7\% is
almost 7 standard deviations from the nominal 1\% level.

Note that the 2 event scenarios are independent of the 5 event
scenarios, while the 5 event scenarios are selected by choosing every
4th event from the 20 event scenarios.  Hence, the 5 and 20 event
scenarios are not independent.  Also, the statistics themselves are
not independent.  They are calculated from the same models, using the
same returns and simulated events.  The statistics differ in whether
they incorporate estimates of cross-sectional covariance.  They also
differ as to degrees of freedom used in finding critical values used
to determine significance levels of the calculated statistics,
although the statistics may be asymptotically nearly the same.

There are a few regularities to be seen across the three panels.
First, the more firms there are in the model, the worse
over-rejection is. This holds true across all windows for almost all
numbers of events.  Second, the Theil F statistic almost always has
the lowest over-rejection rate, followed by the ST F, followed by the
SAS F, regardless of number of events, firms, or window length.
Third, for a given window length and number of firms, the highest
over-rejection rate is generally for the 5 event scenario.
Generally, the next highest is with 2 events, with 20 events being
the lowest, but still very high compared to the nominal Type I error
rates.  The ``best'' window length is 5 for scenarios with 2 or 5
events, but is 1 for scenarios with 20 events.

\subsubsection{$H_{02}\!: \sum_{i=1}^N \gamma_i = 0$}

Table \ref{tab:sum0null} contains the empirical Type I error rates
for the 5 statistics used to test $H_{02}$, that the average of
the $\gamma_i$ coefficients is zero.  In general, the empirical Type~I error
rates for all of these statistics are much closer to the nominal
levels than the error
rates for the statistics used to test $H_{01}$.  For scenarios
with 2 or 20 events, only 14 out of the 180 different statistics
are outside the 95\% confidence intervals.  If the statistics were
independent, we would expect 9 to be outside the limits of a
95\% confidence interval, strictly due to chance, and these
statistics are not independent.  Across all scenarios and all 5
statistics, rejection rates at the 5\% level, 2 or 20 events, range
from 3.5\% to 6.9\%.  At the 1\% level, respective values are .4\% to
1.7\%.

For the scenarios with 5 events, the situation is very different.  Only
14 of the 90 statistics presented are within the 95\% confidence
interval limits.  For 5 events, all of the statistics exhibit
high Type~I error rates, ranging from 5.2\% to 17.2\% at the nominal
5\% level and 1.4\% to 6.7\% at the 1\% level.  Most of them lie
outside the 95\% confidence intervals of [3.9\%,6.1\%] and
[.5\%,1.5\%].  We have no explanation for why
the statistics do relatively well with 2 and 20 events, but very
poorly with 5 events.  (Recall that the 20 event scenarios
essentially take the 5 event scenarios and add an additional 15
events.  Therefore, the 5 and 20 event scenarios are not independent.
Yet the 20 event statistics appear to be reasonably well specified,
while the 5 event statistics reject too often.)

Comparing the true SUR statistics, the rejection rates for the SAS F
and the ST F are identical for all scenarios presented.  (See
Appendix A for an explanation of this apparent equality.)  Comparing
the statistics calculated using the consecutive equation model, the
Theil F and SUMC statistics have essentially identical rejection
rates, differing in only 2 cells presented.

At the 5\% level, the 95\% confidence interval for n=1649 is about 4\%
to 6\%.  As shown in table \ref{tab:sum0null}, the SUMC statistics
reject the null less often than the SUMT statistics but for most of
the scenarios with 2 or 20 events per firm, the rejection frequencies
for both statistics are within the 95\% band.  Again, for the 5
events per firm scenarios, both statistics reject significantly more
often than would be expected based on the nominal size of the tests.
The SUMT rejection rates are particularly high for some scenarios.
There is no uniform behavior with respect to the number of firms, or
the window length.

Probably these statistics fail (lie outside the 95\% confidence band)
due to ignoring the covariance structure.  The statistics only fail
by being above the top of the 95\% confidence band, probably
indicating that for those cases ignoring the covariance structure
produces statistics a little too large.

\subsection{Empirical distributions when no abnormal performance is
introduced}

In this section, we present the first and fifth percentiles of the
empirical distributions of the statistics when no abnormal
performance is introduced.
This is done in order to present meaningful power tables in the next
section.  Powers of statistical tests must be evaluated for specific
Type~I error levels.  Researchers are generally interested in powers
of tests given Type~I error rates of 5\% or 1\%.  Since statistics
for some scenarios had Type~I error rates much higher than indicated
by the nominal size, power tables for nominal Type~I error rates of
5\% and 1\% would not be very informative.  To provide more
meaningful power tables in the next section, we used as critical
values the first and
fifth percentiles of the empirical distributions generated when no
abnormal performance was introduced.  We present those empirical
values in this section.

Rather than using the calculated F- or z-statistics, we work
with the p-values of those statistics.  The p-values are those
associated with the calculated statistics and their theoretical
distributions and degrees of freedom.  This allows uniform
interpretation of the empirical critical values given.  That is, all
empirical critical values are p-values, ranging between 0 and 1,
rather than being points from different distributions with different
degrees of freedom.

Tables \ref{tab:emprejall} and \ref{tab:emprejsum} present the first
and fifth percentiles of
the empirical distributions of the p-values of the statistics
generated in the simulations where no abnormal performance was introduced.
Table \ref{tab:emprejall} presents the empirical critical p-values for the
statistics used to test $H_{01}\!: \gamma_i=0\;\forall i$; table
\ref{tab:emprejsum} presents similar values for testing $H_{02}:
\sum_{i=1}^N \gamma_i =0$.

Each cell entry is based on the empirical distribution of the 1,649
statistics generated for a specific scenario.  Consider the cell in
table \ref{tab:emprejall}, panel A, for the SAS F statistic testing whether all
of the
$\gamma_i$ are equal to 0, for 2 events, 25 firms, window length 1
day, 5\% level test.  The p-values based on theoretical distributions
were obtained for each of
the SAS F statistics from the 1,649 simulations using 2 events, 25
firms, and a window length of 1 day, where
no abnormal performance was added to the firm return vectors on the
(false) event dates.  The value of 0.219 indicates that 5\% of the
p-values for these F~statistics were smaller than 0.219\%.  An
F~statistic with a p-value larger than 0.219\% would not represent a
rejection of the null hypothesis at the 5\% level, based on the
empirical distribution.

Another way of looking at tables \ref{tab:emprejall} and
\ref{tab:emprejsum} is that if the
p-values listed in each cell were used as the critical values for
rejecting the null hypothesis, the Type~I error rates would be
exactly 5\% (1\%) for all scenarios where no abnormal performance was
introduced in the simulation process.

\subsubsection{$H_{01}\!: \gamma_i=0\;\forall i$}

The high Type I error rates presented in table \ref{tab:all0null}
imply that the first and fifth percentiles of the empirical
distributions of nominal p-values must be smaller than 1\% and 5\%.
Table \ref{tab:emprejall} shows that, except for scenarios with 20
events per firm, the first and fifth percentiles are at least an
order of magnitude smaller than the nominal 1\% and 5\% levels.  Of
the true SUR statistics, the ST F statistic is in general closer to
the nominal values than is the SAS F.  Empirical 5\% rejection rates
were achieved for the ST F by using p-values ranging from .047\% to
2.421\%.  To achieve a 1\% rejection rate, p-values ranging from
9E-6\% to .404\% were needed.  For the SAS F, empirical p-values used
to achieve 5\% (1\%) empirical rejection rates ranged from .005\%
(1E-7\%) to 1.868\% (.276\%).  The empirical p-values for the Theil F
from the consecutive equations model are closer to the nominal
p-values than are the ST F at the 5\% nominal level, but neither
statistic dominates at the 1\% nominal level.

For all three statistics, for a given
number of firms and window length, increasing the number of events
per firm brings the empirical 5\% and 1\% points closer to the
respective nominal points.  However, for a given number of events,
adding firms moves the empirical critical values farther from the
nominal values.  That is, as more firms are added, smaller critical
p-values must be used to achieve the same nominal size test.
Finally, for scenarios with 2 or 5 events per firm, increasing the
window length brings the empirical critical p-values up towards the
nominal values, but for the 20 event scenarios, increasing window
lengths push the empirical critical p-values down, farther from the
nominal values.

\subsubsection{$H_{02}\!: \sum_{i=1}^N \gamma_i = 0$}

Table \ref{tab:emprejsum} presents the first and
fifth percentiles of the distributions of p-values for the statistics
used to test whether the average of the coefficients is significantly
different from zero.  These points from the empirical distributions
are closer to the nominal values than were
the respective points for the statistics testing $H_{01}$.  The empirical
values are generally below the nominal values, but not always.
For scenarios with 20 events per firm, the empirical values are
sometimes larger than the nominal values.

For a given number of firms and
window length, scenarios with 2 or 20 events have empirical values
close to the nominal values, while the empirical values for 5 event
scenarios are generally smaller.  There is no consistency in the
effect of adding firms for a given number of events and window
length.  For 2 event scenarios, for a given number of firms,
increasing the window length brings the empirical critical values
closer to the nominal values.  There is no such consistency for the 5
and 20 event scenarios.

In summary, from both the tables of Type~I error rates and the tables
of the 5\% and 1\% points of the empirical distributions, it is clear
that all of the statistics testing $H_{01}$ reject too often
when no abnormal performance was introduced.  The rejection rates for the
statistics
used to test $H_{02}$ are much closer to the nominal
levels, although there is a slight tendency to over-reject here as well.
The next section looks at the other side of the coin---how often the
statistics reject the null of no abnormal performance when abnormal
performance was added to the returns.

\subsection{Power: Rejections of the null when abnormal performance
is introduced}

Given that the Type~I error rates are generally excessive for the
statistics examined, simply calculating power as the percentage of
rejections at the nominal 5\% and 1\% levels, when abnormal
performance is introduced,
would be misleading.  Therefore, we present in tables
\ref{tab:powstall} through \ref{tab:powsumc} corrected power tables.
The critical p-values used to determine rejection or non-rejection of
the null hypothesis are those presented in tables \ref{tab:emprejall}
and \ref{tab:emprejsum},
discussed in the previous section.  All corrected power tables
contain three panels, as three different ranges of abnormal performance
were used in the simulations used to assess power.  Abnormal
performance metrics used in the simulations reported in Panels A were
drawn from a uniform distribution over the range
[.00125,.00375], with a mean of .0025.  The second range, reported in
panels B, was [.00375,.00625], with a mean of .005.  The third range
also had a mean of .005, but had a larger range, [.0025,.0075].

For
all statistics for both hypotheses, for a given number of firms,
number of events per firm, and window length, the power of the tests
increased as the mean level of abnormal return added on the event
dates increased.  That is, rejection rates for all scenarios and all
statistics, panels A,
where the abnormal returns have a mean of .0025, are lower than
corresponding cells from panels B and C, where the mean of the
abnormal returns added is .005, with differing ranges.  The
statistics are differentially sensitive to the variance of abnormal
returns added, as rejection rates are sometimes higher in panels B
than C, and sometimes lower.

One final regularity can be seen across all corrected power tables.
Up to this point, we have discussed number of events per firm and
number of firms in the model as separate variables.  We have done
this primarily because these are separate concepts in the realm of
accounting research.  However, from a purely statistical point of
view, changes in either (or both) change the total number of events
in the model.  An additional column has been inserted in table
\ref{tab:powstall}, showing the total number of events in any given
scenario.  For all of the statistics for both tests, the rejection
rates generally increase with the number of total events.  However,
the rates of increase differ across hypotheses tested, and will be
discussed under the appropriate hypothesis.

\subsubsection{$H_{01}\!: \gamma_i=0\;\forall i$}
The first noteworthy item is that, after correcting for the different
empirical rejection rates under the null hypothesis by using the
scenario specific rejection rates given in table \ref{tab:emprejall},
the corrected power rates for the SAS F and the ST F are identical to
three decimal places.  (See Appendix A for a reconciliation of these
statistics.)  Therefore, we present a combined table for the SAS and
ST F statistics.

Tables \ref{tab:powstall} and \ref{tab:powtall} both show that, for a
given number of events per firm and window
length, adding additional firms increases the power of the tests.
However, the increase in power achieved by adding events for a given
number of firms is much greater.  For example, for the ST F, using 75
firms instead of 25 firms, each with 2 events, increases the
rejection rate from 6.2\% to 6.6\%.  However, having 5 events rather
than 2 events for 25 firms increases the rejection rate from 6.2\% to
11.0\%.  In terms of total number of events in the model, 2 events, 25
firms has 50 total events.  Moving to 2 events, 75 firms gives 150
total events, while moving to 5 events, 25 firms only gives 125 total
events.  Yet the increase in power is greater by increasing the
number of events per firm.

Finally, window length has a dramatic effect on the powers of the
test statistics in the instances where the power for the (correctly
specified) one day window is reasonably good.  For example, the Theil
F in panel B of table \ref{tab:powtall} rejects 45.5\% of the time at
the 5\% level for the 5 events, 75 firms scenario when a one day
window is used.  That drops to 26\% when a two day window is used,
and to 11.2\% when a five day window is used.

\subsubsection{$H_{02}\!: \sum_{i=1}^N \gamma_i = 0$}
The corrected power rates for the SAS F and ST F for testing $H_{02}$
are again identical to three decimal places, after correcting for the
difference between nominal and empirical Type~I error rates.
Therefore, we again present a combined table for the SAS and ST F
statistics.

As was true for the statistics testing $H_{01}$, the total number of
events in the model is important.  However, for $H_{01}$, the number
of events per firm was very important.  Here, that seems to be less
true.  For example, examine table \ref{tab:powsumc}, panel A, the 1\%
column under the one day window.  The power with 2 events per firm,
75 firms, is 16.4\%.  For 5 events, 25 firms, it is 13.1\%.  The
total number of events goes down for the 5 events, 25 firms scenario,
and the power declines.  However, numbers of events per firm is not
unimportant, either.  In the 5\% column for the same numbers of
events, firms, and window length, the rejection rate increases
modestly, from 31.4\% to 32.9\%, even though the total number of
events has decreased.  Overall, the total number of events seems more
important in determining power for tests of $H_{02}$ than
$H_{01}$.  In any case, increasing either the number of events for a
given number of firms or increasing the number of firms for a given
number of events per firm both result in increases in power.

The effect of increases in window length is not as consistent for
statistics testing $H_{02}$ as for those testing $H_{01}$.
Increasing the window length results in
substantial declines for all scenarios for the lowest level of added
abnormal performance.  For the higher level of abnormal performance,
both ranges, there is substantial decline for scenarios with 2 or 5
events, or 20 events and 25 firms.  However, for 20 events and 50 or
75 firms, the decreases in power are small when moving from a 1 day
window to a 2 day window.  Moving to a 5 day window results in
substantial declines for all scenarios.

Finally, after correcting for the difference in Type~I error rates,
SUMT appears to be the dominant statistic for testing $H_{02}$.
It always has higher power than the other statistics examined.

\section{Research design and interpretation in light of simulation
results}
\label{sec:interp}

In light of our simulations, what can one say about designing and
interpreting results of accounting research studies using true SUR or
the consecutive equations models?  Basically, if one wishes to test
hypotheses similar to our $H_{01}$, that all firm-specific response
coefficients are equal to zero, one must use something like the
models studied here.  However, our evidence indicates that one must
use {\em very} conservative rejection regions, or the probability of
Type~I errors will be large.  Furthermore, if one corrects for high
Type~I error rates by using rejection regions based on the empirical
distributions outlined here, power may be poor, especially if the
research examines infrequent events.  For example, in table
\ref{tab:powstall}, panel A, for scenarios with 2 events per firm,
5\% level test, rejection rates range from 6.2 to 6.6\%, if the exact
event date was known.  With event date uncertainty, that range
declines to 5.1 to 6.4\%.  Since rejections of the null hypothesis in
this table are based on the empirical critical p-values from table
\ref{tab:emprejall}, approximately 5\% of the simulations would have
resulted in rejections {\em even if no abnormal performance had been
introduced}.  For scenarios with few events per firm, the power is
barely above the number of rejections expected in the absence of
introduction of any
abnormal performance.  However, if the study focusses on recurring
events such as quarterly announcements, and there is only slight
event date uncertainty, the power is much better.  For example, in
the same column, if there are 20 events per firm, rejection rates
range from 42.6\% to 61.2\%.

If the researcher is interested only in average effects, the true SUR
and the consecutive equations statistics are much better behaved.
Type~I error rates reported in table \ref{tab:sum0null}, while still
generally too high, are nowhere near as high as reported in table
\ref{tab:all0null} for tests of $H_{01}$.  The powers of the tests
are also higher.

If one is only interested in the average effects, the tendency may be
to use a more traditional event study method, such as portfolio
abnormal return analysis or cross-sectional pooled regressions of
abnormal returns on information variables.  However, these methods
also have drawbacks.  In portfolio abnormal return analysis, the
levels of (surprise in) accounting information cannot be included as
explanatory variables.  In cross-sectional pooled regressions,
constraining the coefficients to be the same for all firms or for
groups of firms can create problems.\footnote{Teets
[1992]\nocite{teets1992b} presents an example where inference from
SUR and inference from a pooled cross-sectional regression are
opposite.  Several differences between SUR and the pooled
cross-sectional approach are examined to see which one(s) drive the
difference in inference.  The constraint of equality of coefficients
under the pooled cross-sectional method appears to drive the
difference in inference.} Our simulations indicate that true SUR or
the consecutive equations model, both of which permit inclusion of
levels of information and firm-specific coefficients, may be
acceptable alternatives, in terms of Type I error rates and power,
for tests of average effects.

We chose the numbers of events per firm to be representative of
situations encountered in accounting research.  First, the two events
per firm scenario corresponds to infrequent events, such
as management forecasts of earnings.  Second, regular accounting
announcements occur annually or quarterly.  Over a five year period,
that will give rise to 5 or 20 events per firm.  To avoid problems
with structural change in the sample firms, accountants have
frequently restricted their analysis to not more than 5 year periods.
Our results suggest that the powers of the test statistics are low
for scenarios with only 2 events per firm.  This suggests that SUR
may not have sufficient power to correctly identify significant
information events, {\em after correcting for high Type~I error
rates}, in studies of infrequently occurring events.
However, for studies using quarterly announcements, SUR may work
acceptably.\footnote{This discussion assumes that the average
abnormal returns associated with the different frequency events are
similar.  If the infrequent events are associated with large abnormal
returns, and the quarterly announcements are associated with small
abnormal returns, these conclusions may not hold.}

If the researcher has to choose between adding firms to the sample,
where each firm will have only a few events, or identifying
additional events for firms already in the sample, our results
suggest that either strategy will provide gains in power.  If the
research question has to do with average effects, either strategy may
work equally well.  However, if the question has to do with simultaneous
tests for all firms (our $H_{01}$), finding additional events for
the existent sample firms will give greater increases in power.

We used different window lengths to determine the extent of the
problem caused by event date uncertainty when using SUR.  Our results
suggest the way that event date uncertainty is typically addressed in
SUR can severely affect the test statistics.  There is a substantial
reduction in power upon moving to a 2 day window from a 1 day window,
and a further reduction upon moving to a 5 day window.  This
reduction in power suggests that SUR as typically implemented may not
be appropriate when there is major event date
uncertainty.\footnote{There are several ways that multiple day event
windows could be implemented using firm-specific models.  First, if
the researcher doesn't need to incorporate cross-sectional
correlation, firm-specific models based on equation (\ref{eq:retgen})
could be estimated with single day returns outside the event periods
and multiple day event period returns, using WLS instead of OLS.  If
the researcher wants to incorporate cross-sectional correlation, and
events are on the same day for all firms, SUR can be used after
appropriately scaling the multiple day firm and market returns.
Finally, if events happen at different times for different firms,
leading to nonsynchronous multiple day event period returns, a method
suggested by Marais [1986]\nocite{marais1986} may be used.}

Finally, what do our results imply about the choice between the true
SUR and consecutive equations models?  On the whole, there doesn't
seem to be a lot to recommend one over the other, empirically.  For
testing $H_{01}$, the ST F statistic generally has higher
corrected power than the Theil F from the consecutive equations
model, but in many of the scenarios, they are very similar.  In tests
of $H_{02}$, the SUMT statistic from the consecutive
equations model had the highest power in all scenarios.  The near
equality of the methods may be due to the noise in the
cross-sectional covariance estimation process offsetting the gains
from incorporating covariance estimates in the coefficient estimates
and statistical tests.

\section{Directions for future work}
\label{sec:conclude}

This simulation study of SUR examined situations where events
occurred for different firms at (possibly) different times.  It did
not examine the situation where a sequence of
events (possibly) affects a number of firms on the same event
dates.  It is in this situation that the Schipper Thompson F
statistics are theoretically exact.  It would be interesting to simulate this
situation, to see if the ST statistics have better Type~I error rates
and power.  This would provide evidence on whether the over-rejection
when no abnormal performance was introduced is due to the theoretical
distributions holding only asymptotically, or due to leptokurtosis of
the returns.

Alternatively, one could generate normally distributed data to use in
place of the actual returns data used in this study.  Simulations
based on this data could provide evidence on the effects of
non-normality.  One could also use much longer time series of
generated data, without having to worry about nonstationarity.  This
could provide evidence on the effects of the distributions holding
only asymptotically.

In these simulations, all coefficients were
positive, and equal to unity.  In this case tests of the second
hypothesis, on the average of the coefficients, should be more
powerful than tests of the first hypothesis.
Simulations where there is variation across the coefficients relating
information to stock prices would give additional evidence on the use
of SUR in information event studies.

Finally, comparing pooled cross-sectional methods, portfolio abnormal
return analysis, and SUR under a variety of simulated conditions
might provide evidence about when researchers could rely on a method,
and when assumptions of that method are violated seriously enough
that different methods must be contemplated.

\clearpage

\appendix

\section{Reconcilation of SAS F and ST F}

The SAS F used to test $H_{01}\!: \gamma_i = 0\; \forall i$ is defined
as $\frac{\mbox{SMSE}_R -
\mbox{SMSE}_U}{q} \left/ \frac{\mbox{SMSE}_U}{(T - k) * N},\right.$
and is distributed asymptotically F($N$,$(T-k)*N$), while the ST F is
defined $\mbox{ST} = (\mbox{SMSE}_R - \mbox{SMSE}_U) * \left(\frac{T
- k - N + 1}{(T - k) * N}\right)$ and is distributed asymptotically
F($N$,$T-k-N+1$) in the multivariate case.

Define $\mbox{SMSEDF} =
\frac{\mbox{SMSE}_U}{(T - k) * N}$.
Consider the ratio of the SAS F to the ST F.  It is
$\frac{1}{\mbox{SMSEDF}} \frac{1277}{1278-N}$, since $T$ is 1280 for
all simulations, and $k$ is 3.  Based on our simulations, the means
(standard deviations) of the ratio $\frac{\mbox{SAS F}}{\mbox{ST F}}$
for $N=25$, 50, and 75
are 1.0192 (.000022), 1.040 (.000044) and 1.0616 (.000037).
The slight variation in
the ratio is due to the $\frac{1}{\mbox{SMSEDF}}$ factor.
Asymptotically, $\mbox{SMSEDF} \rightarrow 1$.  The mean (standard
deviation) of SMSEDF for $N=25$, based on 59,364 simulations
using 25 firms, is .999967 (.000021).  Respective numbers for 50 and
75 firms are .999935 (.000042) and .999916 (.000035).
The SAS F is essentially a constant multiple of the ST F; the
multiple depends only on the $N$ parameter.

The SAS F is compared to the F($N,(T-k)*N$) distribution, and the ST
F is compared to the F($N,T-k-N+1$) distribution.  However, if the
denominator degrees of freedom are large (greater than 1000), the F
distribution is essentially only dependent on its numerator, a
$\chi^2$ divided by its degrees of freedom.  Hence, both the SAS F
and ST F are compared to a $\chi^2_N$ divided by $N$.  For $N=25$,
50, and 75, the critical values for a
nominal 5\% test are 1.506, 1.350, and 1.293.

Since $\mbox{SAS F} \approx 1.0192\ \mbox{ST F}$ for $N=25$, and both
statistics are compared to the same critical value, the empirical
rejection rates are different, and always higher for $\mbox{SAS F}$.
Assume that a simulation using 25 firms resulted in an ST F of 1.505.
Based on the nominal 5\% critical value of 1.506, this ST F would not
lead to rejection of the null.  However, the corresponding SAS F
would be approximately 1.535 (since SMSEDF is not exactly 1), and
would lead to rejection of the null.

The reason the rejection rates based on the empirical distributions
are the same is that the empirical critical values compensate for the
factor relating the SAS F to the ST F.  Consider the 5\% critical
values from table \ref{tab:emprejall} for the SAS F and the ST F, for
the scenario with 2 events per firm, 25 firms, using a 1 day window.
The SAS F critical p-value is .219\%, which translates to an F of
1.9973, while the ST F critical p-value is .325\%, which has a
corresponding F value of 1.9596.  The ratio of these critical values
is approximately 1.0192, which is the same as the mean multiplication
factor (for 25 firms) used to translate the ST F into the SAS F.  The
empirical critical values offset the multiplication factor by which
the two statistics differ.

The SAS F and ST F statistics used to test $H_{02}\!: \sum_{i=1}^N
\gamma_i = 0$ are even more closely related.  The SAS F is as defined
for the test of $H_{01}$, but $q$ is now equal to unity.  The ST F is
simply $\mbox{SMSE}_R - \mbox{SMSE}_U$.  As indicated previously,
$\mbox{SMSEDF} \rightarrow 1$ asymptotically.  Therefore, the SAS F
and ST F have almost identical values.  Although the degrees of
freedom used to determine critical values differ across the two
statistics (1 and $(T-k)*N$ for the SAS F and 1 and $T-k$ for the ST
F), the denominator degrees of freedom are large enough both
statistics are essentially compared to a $\chi^2_1$.  The minor effect of
SMSEDF is offset by the small differences in the p-values of the
empirical critical values presented in table \ref{tab:emprejsum}.

\clearpage

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\end{thebibliography}

\renewcommand{\thefootnote}{\alph{footnote}}

\begin{table}\centering
\caption{
Type I error rates: Percentage of 1,649 simulations where}
$H_{01}: \gamma_i = 0\; \forall i$ was rejected when no abnormal
performance was introduced.
\label{tab:all0null}

\[\mbox{Model\footnotemark[1]: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]
%\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: SAS F Statistic\footnotemark[4]}\\[1ex]
\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window\footnotemark[2]} & \multicolumn{2}{c||}{2 Day
Window\footnotemark[2]} &
\multicolumn{2}{c}{5 Day Window\footnotemark[2]}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\footnotemark[3]\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 13.2 & 7.7 & 12.7 & 7.3 & 10.7 & 5.5\\
 2&50& 19.6 & 11.5 & 18.1 & 9.8 & 13.9 & 7.8\\
 2&75& 25.5 & 15.0 & 23.4 & 13.9 & 19.3 & 11.6\\[1ex]
 5&25& 14.9 & 7.6 & 15.1 & 7.0 & 12.5 & 4.8\\
 5&50& 23.4 & 12.0 & 24.9 & 12.3 & 18.5 & 8.4\\
 5&75& 31.2 & 18.3 & 33.9 & 19.5 & 25.5 & 13.2\\[1ex]
20&25& 9.2 & 3.5 & 11.3 & 3.8 & 12.6 & 4.7\\
20&50& 13.6 & 5.0 & 18.0 & 6.9 & 16.7 & 7.3\\
20&75& 21.2 & 9.7 & 23.7 & 9.9 & 22.0 & 10.6
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Schipper-Thompson F
Statistic\footnotemark[5]}\\[1ex]
\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 12.0 & 7.0 & 11.6 & 6.9 & 9.7 & 5.0\\
 2&50& 15.7 & 9.0 & 14.4 & 7.7 & 11.5 & 5.9\\
 2&75& 17.9 & 10.4 & 16.2 & 9.7 & 13.7 & 7.2\\[1ex]
 5&25& 13.7 & 6.5 & 13.7 & 6.0 & 10.9 & 4.3\\
 5&50& 18.1 & 8.7 & 19.3 & 9.0 & 13.6 & 6.1\\
 5&75& 21.9 & 10.7 & 23.5 & 11.9 & 16.6 & 8.4\\[1ex]
20&25& 8.2 & 3.0 & 9.6 & 2.9 & 10.9 & 3.5\\
20&50& 9.7 & 3.5 & 12.7 & 4.2 & 13.4 & 5.0\\
20&75& 12.4 & 4.1 & 14.1 & 5.2 & 12.9 & 5.5
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Theil F-statistic\footnotemark[6]}\\[1ex]
\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 12.2 & 7.0 & 11.0 & 6.6 & 9.3 & 4.8\\
 2&50& 14.3 & 8.5 & 13.6 & 7.5 & 10.4 & 5.4\\
 2&75& 14.5 & 8.7 & 13.2 & 7.9 & 11.9 & 6.2\\[1ex]
 5&25& 12.9 & 6.4 & 13.3 & 5.8 & 11.5 & 5.0\\
 5&50& 14.9 & 7.9 & 17.5 & 8.4 & 14.7 & 7.2\\
 5&75& 17.2 & 8.6 & 20.3 & 9.6 & 16.1 & 7.9\\[1ex]
20&25& 6.9 & 2.7 & 9.4 & 3.0 & 11.7 & 4.1\\
20&50& 8.5 & 3.0 & 11.2 & 3.5 & 12.9 & 4.7\\
20&75& 8.9 & 2.7 & 10.9 & 3.8 & 11.0 & 3.8
\end{tabular}
\end{table}

\clearpage


\indent \footnotemark[1] Models were estimated using 1,280 daily firm
($r_{it}$) and market ($r_{mt}$) returns from the periods
12/06/74--12/31/79, 12/10/79--12/31/84, or 12/06/84--12/29/89.
Within a simulation, all returns were from the same time period.  To
simulate accounting information events, randomly selected elements of
the $A_{it}$ vector were assigned randomly generated non-zero values
from one of the ranges [.00125,.00375], [.00375,.00625], or
[.0025,.0075].  Values varied across firms and events, but all were
generated from the same range within a simulation.  In simulations used
to determine powers of the tests, abnormal performance was introduced
into firms' returns by adding the non-zero values to the firms' returns
corresponding to the first date in each event window.

\indent \footnotemark[2] For an $x$ day window, the $A_{i}$
vector has $x$ consecutive nonzero elements for each event.  Under
the null hypothesis, no abnormal returns are added to the returns vectors.

\indent \footnotemark[3] Nominal size of the test---the expected
percentage of rejections of $H_{01}$ due to chance when no abnormal
performance is introduced, if the test statistic is well specified.

\indent \footnotemark[4] The SAS F-statistic is defined as
\[\frac{\mbox{SMSE}_R - \mbox{SMSE}_U}{q} \left/
\frac{\mbox{SMSE}_U}{(T - k) * N},\right. \] where SMSE denotes
system mean squared error, $_R$ and $_U$ denote restricted and
unrestricted, $q$ is the number of restrictions, $N$ is the number of
firms (equations) in the model, $T$ is the number of time-series
observations used for each firm, and $k$ is the number of
coefficients estimated for each firm.  The statistic to test $H_{01}$
is asymptotically distributed F($N,1277*N$).

\indent \footnotemark[5] The Schipper-Thompson F-statistic is
\[(\mbox{SMSE}_R - \mbox{SMSE}_U) * \left(\frac{T - k - N + 1}{(T -
k) * N}\right),\] and is asymptotically distributed F($N$,$1277-N+1$).

\indent \footnotemark[6] The Theil F-statistic is \[ \frac{1}{N}
\sum_{i=1}^N \frac{\gamma_i^2}{\sigma^2_{\gamma_i}}, \] where
$\sigma^2_{\gamma_i}$ are the OLS estimates of the variances of the
$\gamma_i$.  The statistic to test $H_{01}$ is asymptotically
distributed F($N$,$1277*N$).


\clearpage
\begin{table}\centering
\caption{
Type I error rates: Percentage of 1,649 simulations where}
$H_{02}: \sum_{i=1}^N \gamma_i = 0$ was rejected when no abnormal
performance was introduced

\label{tab:sum0null}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: SAS F Statistic\footnotemark[1]}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 5.6 & 1.7 & 5.5 & 1.5 & 4.5 & 1.1\\
 2&50& 5.6 & 1.6 & 5.1 & 1.3 & 4.7 & 1.0\\
 2&75& 5.9 & 1.4 & 5.2 & 1.4 & 4.5 & 1.0\\[1ex]
 5&25& 5.3 & 2.0 & 6.1 & 1.4 & 5.8 & 1.6\\
 5&50& 6.8 & 1.6 & 6.5 & 1.8 & 6.8 & 1.8\\
 5&75& 6.4 & 2.2 & 6.5 & 1.5 & 7.9 & 2.2\\[1ex]
20&25& 5.4 & 0.9 & 6.2 & 0.7 & 4.8 & 1.0\\
20&50& 5.3 & 1.2 & 5.5 & 1.0 & 5.4 & 1.3\\
20&75& 5.9 & 1.3 & 5.9 & 1.3 & 5.3 & 1.0
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Schipper Thompson F
Statistic\footnotemark[2]}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 5.6 & 1.7 & 5.5 & 1.5 & 4.5 & 1.1\\
 2&50& 5.6 & 1.6 & 5.1 & 1.3 & 4.7 & 1.0\\
 2&75& 5.9 & 1.4 & 5.2 & 1.4 & 4.5 & 1.0\\[1ex]
 5&25& 5.3 & 2.0 & 6.1 & 1.4 & 5.8 & 1.6\\
 5&50& 6.8 & 1.6 & 6.5 & 1.8 & 6.8 & 1.8\\
 5&75& 6.7 & 2.2 & 6.5 & 1.5 & 7.9 & 2.2\\[1ex]
20&25& 5.4 & 0.9 & 6.2 & 0.7 & 4.8 & 1.0\\
20&50& 5.3 & 1.2 & 5.5 & 1.0 & 5.4 & 1.3\\
20&75& 5.9 & 1.3 & 5.9 & 1.3 & 5.3 & 1.0
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Theil F
Statistic\footnotemark[3]}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 5.4 & 1.2 & 5.0 & 1.3 & 4.1 & 1.2\\
 2&50& 4.4 & 1.1 & 4.4 & 0.9 & 3.7 & 0.8\\
 2&75& 4.5 & 0.7 & 4.4 & 0.8 & 3.5 & 0.6\\[1ex]
 5&25& 5.2 & 1.8 & 5.8 & 1.7 & 6.7 & 2.1\\
 5&50& 6.3 & 2.2 & 7.3 & 2.4 & 10.0 & 3.2\\
 5&75& 7.0 & 2.0 & 7.7 & 2.4 & 12.4 & 4.4\\[1ex]
20&25& 4.7 & 0.6 & 5.2 & 0.4 & 4.5 & 1.0\\
20&50& 4.5 & 1.0 & 4.7 & 0.8 & 4.9 & 1.0\\
20&75& 4.9 & 0.8 & 4.9 & 0.8 & 4.6 & 1.0
\end{tabular}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel D: SUMT\footnotemark[4]}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 6.9 & 1.3 & 6.2 & 1.2 & 5.0 & 1.5\\
 2&50& 5.1 & 1.1 & 5.2 & 1.0 & 4.5 & 1.2\\
 2&75& 5.6 & 1.0 & 5.4 & 0.8 & 4.1 & 0.7\\[1ex]
 5&25& 6.5 & 1.7 & 7.2 & 2.1 & 8.6 & 3.0\\
 5&50& 7.0 & 2.5 & 9.3 & 3.0 & 13.2 & 4.9\\
 5&75& 7.0 & 2.4 & 10.5 & 2.7 & 17.2 & 6.7\\[1ex]
20&25& 5.0 & 1.0 & 4.9 & 1.0 & 5.8 & 1.3\\
20&50& 4.9 & 0.8 & 4.6 & 1.1 & 5.6 & 1.5\\
20&75& 4.1 & 1.0 & 5.4 & 1.0 & 6.6 & 1.4
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel E: SUMC\footnotemark[5]}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 5.4 & 1.4 & 5.0 & 1.3 & 4.1 & 1.2\\
 2&50& 4.4 & 1.1 & 4.4 & 0.9 & 3.7 & 0.8\\
 2&75& 4.5 & 0.7 & 4.4 & 0.8 & 3.5 & 0.6\\[1ex]
 5&25& 5.2 & 1.8 & 5.8 & 1.7 & 6.7 & 2.1\\
 5&50& 6.3 & 2.2 & 7.3 & 2.4 & 10.0 & 3.2\\
 5&75& 7.0 & 2.0 & 7.7 & 2.4 & 12.4 & 4.4\\[1ex]
20&25& 4.7 & 0.6 & 5.2 & 0.5 & 4.5 & 1.0\\
20&50& 4.5 & 1.0 & 4.7 & 0.8 & 4.9 & 1.0\\
20&75& 4.9 & 0.8 & 4.9 & 0.8 & 4.6 & 1.0
\end{tabular}
\end{center}

%\vspace{1ex}

\hspace*{2em} See notes to table \ref{tab:all0null} for definitions of
symbols and description of model.

\hspace*{2em} \footnotemark[1] See notes to table 1 for formula.  The
statistic to test $H_{02}$ is asymptotically distributed
F(1,$(T-k)*I$).

\hspace*{2em} \footnotemark[2] Statistic is calculated as SMSE$_R$ --
SMSE$_U$, and is asymptotically distributed F(1,$T-k)$.

\hspace*{2em} \footnotemark[3] Statistic is calculated
$\left(\sum_{i=1}^{N}\gamma_i\right)^2 \left/
\sum_{i=1}^{N}\sigma^2_{\gamma_i}\right.$, and is distributed
asymptotically F(1,$(T-k)*I$).

\hspace*{2em} \footnotemark[4] Statistic is calculated $\left(\sum_{i=1}^{N}
\frac{\gamma_i}{\sigma_{\gamma_i}}\right) \left/
\sqrt{N}, \right.$ and is distributed
asymptotically N(0,1).

\hspace*{2em} \footnotemark[5] Statistic is calculated $\left(\sum_{i=1}^{N}
\gamma_i \right) \left/ \left(\sum_{i=1}^{N}
\sigma_{\gamma_i}^2\right)^{(1/2)}\right.$ and is distributed
asymptotically N(0,1).
\end{table}

\clearpage

\begin{table}\centering
\caption{
First and fifth percentiles of empirical distributions of p-values of}
statistics testing $H_{01}: \gamma_i = 0\; \forall i$.  Each
distribution based on 1,649 \\
simulations where no abnormal performance was introduced.
\label{tab:emprejall}

\vspace{-.5ex}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\vspace{-.5ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: SAS F statistic}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&0.219\footnotemark[2]&$<$0.001&0.190&$<$0.001&0.735&$<$0.001\\
 2&50&0.017&$<$0.001&0.063&$<$0.001&0.272&$<$0.001\\
 2&75&0.005&$<$0.001&0.007&$<$0.001&0.062&$<$0.001\\[1ex]
 5&25&0.347& 0.002&0.485& 0.001&1.041& 0.001\\
 5&50&0.083&$<$0.001&0.089&$<$0.001&0.239& 0.002\\
 5&75&0.014&$<$0.001&0.017&$<$0.001&0.037&$<$0.001\\[1ex]
20&25&1.868& 0.128&1.430& 0.276&1.047& 0.093\\
20&50&0.966& 0.050&0.580& 0.029&0.438& 0.010\\
20&75&0.297& 0.023&0.211& 0.009&0.198& 0.003
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Schipper-Thompson F Statistic}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&0.325&$<$0.001&0.285&$<$0.001&1.010&   0.001\\
 2&50&0.059&$<$0.001&0.186&$<$0.001&0.658&   0.001\\
 2&75&0.047&$<$0.001&0.063&$<$0.001&0.357&$<$0.001\\[1ex]
 5&25&0.501&   0.004&0.685&   0.002&1.400&   0.002\\
 5&50&0.236&   0.001&0.251&   0.002&0.589&   0.010\\
 5&75&0.106&   0.001&0.129&   0.002&0.237&   0.004\\[1ex]
20&25&2.421&   0.197&1.885&   0.404&1.408&   0.146\\
20&50&1.974&   0.151&1.271&   0.096&0.996&   0.036\\
20&75&1.240&   0.162&0.947&   0.078&0.898&   0.033
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Theil F Statistic}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&0.386&$<$0.001&0.367&$<$0.001&1.140&$<$0.001\\
 2&50&0.126&$<$0.001&0.218&$<$0.001&0.758&0.001\\
 2&75&0.090&$<$0.001&0.092&$<$0.001&0.600&$<$0.001\\[1ex]
 5&25&0.464&0.002&0.707&0.002&0.997&0.002\\
 5&50&0.247&$<$0.001&0.262&0.001&0.481&0.003\\
 5&75&0.189&0.001&0.172&0.001&0.215&0.007\\[1ex]
20&25&2.964&0.156&1.968&0.340&1.517&0.116\\
20&50&2.582&0.128&1.703&0.110&1.219&0.079\\
20&75&2.195&0.227&1.638&0.096&1.455&0.029\\[1ex]
\multicolumn{8}{l}{See notes to table \ref{tab:all0null} for
definitions.}
\end{tabular}
\end{table}

\clearpage

\begin{table}\centering
\caption{
First and fifth percentiles of empirical distributions of p-values of}
statistics testing $H_{02}: \sum_{i=1}^N\gamma_i = 0$.  Each
distribution based on 1,649\\
simulations where no abnormal performance was introduced.
\label{tab:emprejsum}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: SAS F Statistic}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&4.044\footnotemark[1]&0.424&4.626&0.634&6.023&0.686\\
 2&50&4.154&0.810&4.947&0.815&5.237&1.022\\
 2&75&4.194&0.791&4.795&0.768&5.349&0.885\\[1ex]
 5&25&4.752&0.408&4.114&0.414&4.342&0.376\\
 5&50&3.530&0.497&3.872&0.458&3.592&0.518\\
 5&75&3.622&0.269&3.574&0.756&2.824&0.352\\[1ex]
20&25&4.435&1.156&4.331&1.296&5.313&0.904\\
20&50&4.661&0.852&4.649&1.063&4.574&0.699\\
20&75&4.285&0.707&4.260&0.650&4.423&1.019
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Schipper-Thompson F Statistic}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&4.064&0.430&4.646&0.642&6.046&0.695\\
 2&50&4.174&0.820&4.968&0.825&5.259&1.033\\
 2&75&4.215&0.801&4.817&0.778&5.372&0.895\\[1ex]
 5&25&4.772&0.415&4.134&0.421&4.363&0.382\\
 5&50&3.549&0.504&3.892&0.466&3.612&0.526\\
 5&75&3.643&0.275&3.594&0.766&2.842&0.358\\[1ex]
20&25&4.456&1.168&4.351&1.308&5.334&0.915\\
20&50&4.683&0.862&4.671&1.074&4.596&0.709\\
20&75&4.306&0.716&4.281&0.659&4.445&1.030
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Theil F Statistic}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&4.614&0.483&5.015&0.519&6.070&0.664\\
 2&50&5.434&0.948&5.685&1.032&6.937&1.386\\
 2&75&5.375&1.330&5.686&1.106&6.788&1.717\\[1ex]
 5&25&4.657&0.356&4.179&0.369&3.634&0.267\\
 5&50&3.340&0.385&2.932&0.234&1.674&0.165\\
 5&75&3.297&0.356&2.653&0.480&1.160&0.090\\[1ex]
20&25&5.477&1.375&4.934&1.578&5.509&1.046\\
20&50&5.680&0.922&5.153&1.354&5.158&1.000\\
20&75&5.329&1.228&5.091&1.044&5.174&1.002
\end{tabular}
\end{table}

\clearpage

\begin{table}\centering
\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel D: SUMT}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&3.903&0.880&3.841&0.804&5.004&0.576\\
 2&50&4.586&0.895&4.852&1.002&5.545&0.853\\
 2&75&4.102&0.924&4.900&1.344&5.703&1.435\\[1ex]
 5&25&3.582&0.399&3.506&0.407&2.567&0.157\\
 5&50&2.498&0.300&2.072&0.235&1.047&0.094\\
 5&75&3.376&0.302&2.047&0.462&0.579&0.061\\[1ex]
20&25&4.959&1.054&5.252&0.945&4.420&0.644\\
20&50&5.187&1.145&5.233&0.961&4.527&0.513\\
20&75&6.162&1.060&4.490&0.996&3.739&0.733
\end{tabular}

\vspace{2ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel E: SUMC}\\[1ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25&4.618&0.488&5.040&0.524&6.127&0.674\\
 2&50&5.436&0.989&5.688&1.040&6.979&1.418\\
 2&75&5.400&1.340&5.702&1.145&6.923&1.720\\[1ex]
 5&25&4.664&0.367&4.225&0.414&3.642&0.298\\
 5&50&3.360&0.391&2.940&0.264&1.690&0.172\\
 5&75&3.344&0.432&2.655&0.492&1.173&0.094\\[1ex]
20&25&5.506&1.376&4.936&1.607&5.527&1.103\\
20&50&5.706&1.014&5.193&1.378&5.188&1.067\\
20&75&5.361&1.238&5.100&1.059&5.245&1.007\\[2ex]
\multicolumn{8}{l}{\hspace*{2em}See notes to table \ref{tab:all0null} for model
description.}\\
\multicolumn{8}{l}{\hspace*{2em}See notes to table \ref{tab:sum0null} for
statistic definitions.}\\
\multicolumn{8}{l}{\hspace*{2em}\footnotemark[1]All p-values are expressed as
percentages}
\end{tabular}
\end{table}

\clearpage
\begin{table}\centering
\caption{Power: Percentage of 1,649 simulations where $H_{01}:
\gamma_i = 0\; \forall i$ was rejected by the}
SAS and Schipper Thompson F-statistics when abnormal performance
was introduced
\label{tab:powstall}

Rejections based on empirical distributions summarized in Table
\ref{tab:emprejall}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: Level 1, Abnormal returns from
[.00125,.00375]}\\[.5ex]\hline\hline
Number of&Number of&Events&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms (N)&$\times$ firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 50  & 6.2  & 1.3  & 5.1  & 1.0  & 5.2  & 1.1  \\
 2&50& 100 & 6.4  & 1.3  & 6.4  & 1.2  & 5.6  & 1.0  \\
 2&75& 150 & 6.6  & 1.2  & 5.9  & 1.2  & 6.0  & 1.3  \\[1ex]
 5&25& 125 & 11.0  & 2.5  & 8.9  & 1.3  & 6.7  & 1.2  \\
 5&50& 250 & 15.5  & 3.2  & 9.9  & 2.1  & 6.4  & 1.6  \\
 5&75& 375 & 17.6  & 4.7  & 10.9  & 2.7  & 7.0  & 1.6  \\[1ex]
20&25& 500 & 42.6  & 27.6  & 23.9  & 14.4  & 10.4  & 3.0  \\
20&50& 1000& 56.2  & 40.6  & 34.9  & 20.1  & 14.4  & 3.1  \\
20&75& 1500& 61.2  & 48.9  & 40.2  & 25.6  & 16.9  & 4.1
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Level 2, Abnormal returns from
[.00375,.00625]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 11.5  & 1.5  & 6.7  & 1.4  & 5.8  & 1.5  \\
 2&50& 13.8  & 2.9  & 9.2  & 1.5  & 6.9  & 1.3  \\
 2&75& 19.5  & 3.9  & 9.9  & 1.5  & 7.3  & 1.3  \\[1ex]
 5&25& 35.7  & 15.1  & 20.7  & 4.4  & 11.8  & 1.5  \\
 5&50& 48.2  & 27.9  & 30.1  & 11.9  & 13.8  & 3.2  \\
 5&75& 52.5  & 35.7  & 34.8  & 18.3  & 14.0  & 3.9  \\[1ex]
20&25& 87.0  & 73.9  & 62.3  & 51.1  & 31.5  & 18.2  \\
20&50& 94.5  & 89.5  & 76.0  & 60.2  & 41.3  & 25.7  \\
20&75& 97.2  & 94.2  & 81.5  & 70.5  & 47.6  & 32.0
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Level 3, Abnormal returns from
[.0025,.0075]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 12.7  & 1.8  & 6.9  & 1.6  & 5.9  & 1.3  \\
 2&50& 15.7  & 4.0  & 10.2  & 1.6  & 6.9  & 1.5  \\
 2&75& 20.4  & 4.9  & 10.6  & 1.8  & 7.2  & 1.5  \\[1ex]
 5&25& 37.6  & 17.2  & 21.5  & 4.9  & 12.3  & 1.5  \\
 5&50& 49.9  & 30.0  & 31.5  & 13.3  & 14.7  & 3.8  \\
 5&75& 54.4  & 37.7  & 36.6  & 20.3  & 15.7  & 4.6  \\[1ex]
20&25& 87.9  & 76.8  & 64.4  & 52.7  & 32.7  & 19.7  \\
20&50& 95.5  & 90.4  & 78.0  & 62.6  & 43.0  & 27.2  \\
20&75& 97.5  & 95.3  & 83.0  & 73.0  & 48.5  & 33.4  \\[1ex]
\multicolumn{8}{l}{See notes to table \ref{tab:all0null} for definitions.}
\end{tabular}
\end{table}
\clearpage

\begin{table}\centering
\caption{Power: Percentage of 1,649 simulations where $H_{01}:
\gamma_i = 0\; \forall i$ was}
rejected by Theil F-statistic when abnormal performance was introduced
\label{tab:powtall}

Rejections based on empirical distributions summarized in Table
\ref{tab:emprejall}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: Level 1, Abnormal returns from
[.00125,.00375]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 6.1  & 1.3  & 5.7  & 1.0  & 5.3  & 1.0  \\
 2&50& 7.0  & 1.4  & 5.8  & 1.2  & 5.6  & 1.0  \\
 2&75& 6.8  & 1.3  & 5.6  & 1.1  & 5.8  & 1.1  \\[1ex]
 5&25& 10.7  & 1.9  & 9.2  & 1.6  & 6.9  & 1.2  \\
 5&50& 15.0  & 2.2  & 9.9  & 1.6  & 7.5  & 1.3  \\
 5&75& 18.7  & 4.8  & 11.6  & 2.5  & 7.2  & 2.4  \\[1ex]
20&25& 42.6  & 25.6  & 23.7  & 12.9  & 11.1  & 2.9  \\
20&50& 54.0  & 37.7  & 33.5  & 18.2  & 13.6  & 4.0  \\
20&75& 59.9  & 46.0  & 40.1  & 22.4  & 16.2  & 3.3
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Level 2, Abnormal returns from
[.00375,.00625]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 9.0  & 1.3  & 6.4  & 1.5  & 5.5  & 1.0  \\
 2&50& 9.6  & 1.9  & 7.2  & 1.4  & 6.5  & 1.1  \\
 2&75& 11.3  & 2.2  & 6.9  & 1.2  & 6.9  & 1.2  \\[1ex]
 5&25& 24.4  & 5.8  & 16.2  & 2.5  & 9.6  & 1.5  \\
 5&50& 38.5  & 11.4  & 20.1  & 3.7  & 10.9  & 2.2  \\
 5&75& 45.5  & 21.8  & 26.0  & 7.6  & 11.2  & 3.8  \\[1ex]
20&25& 86.0  & 69.3  & 56.1  & 41.7  & 21.9  & 8.8  \\
20&50& 95.0  & 87.0  & 72.8  & 55.1  & 31.2  & 14.8  \\
20&75& 97.1  & 94.1  & 79.3  & 64.2  & 39.5  & 15.3
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Level 3, Abnormal returns from
[.0025,.0075]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 12.6  & 1.9  & 7.7  & 1.5  & 6.3  & 1.2  \\
 2&50& 16.1  & 3.1  & 10.1  & 1.5  & 7.0  & 1.5  \\
 2&75& 19.5  & 4.4  & 9.3  & 1.6  & 7.6  & 1.5  \\[1ex]
 5&25& 37.5  & 15.3  & 22.7  & 5.6  & 11.9  & 1.8  \\
 5&50& 48.8  & 26.9  & 32.3  & 10.0  & 15.7  & 3.2  \\
 5&75& 54.5  & 36.5  & 37.3  & 19.2  & 16.3  & 7.1  \\[1ex]
20&25& 88.1  & 74.3  & 63.0  & 51.2  & 32.4  & 17.9  \\
20&50& 95.9  & 89.1  & 77.5  & 61.1  & 42.6  & 28.0  \\
20&75& 97.6  & 94.6  & 82.7  & 69.8  & 48.6  & 30.7   \\[1ex]
\multicolumn{8}{l}{See notes to table \ref{tab:all0null} for definitions.}
\end{tabular}
\end{table}
\clearpage

\begin{table}\centering
\caption{Power: Percentage of 1,649 simulations where $H_{02}:
\sum_{i=1}^N\gamma_i = 0$ was rejected by}
the SAS and Schipper Thompson F-statistic when abnormal performance
was introduced
\label{tab:powstsum}

Rejections based on empirical distributions summarized in Table
\ref{tab:emprejsum}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: Level 1, Abnormal returns from
[.00125,.00375]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 13.6  & 4.3  & 10.0  & 3.3  & 7.2  & 1.5  \\
 2&50& 23.2  & 10.7  & 15.9  & 5.5  & 9.3  & 2.9  \\
 2&75& 31.3  & 15.8  & 20.2  & 7.6  & 10.9  & 2.7  \\[1ex]
 5&25& 32.9  & 13.8  & 20.9  & 6.2  & 13.6  & 3.3  \\
 5&50& 50.4  & 32.4  & 34.0  & 16.1  & 20.4  & 7.7  \\
 5&75& 60.1  & 37.2  & 43.6  & 26.8  & 24.3  & 9.6  \\[1ex]
20&25& 67.3  & 51.6  & 44.5  & 33.1  & 24.1  & 11.7  \\
20&50& 87.0  & 74.1  & 66.6  & 51.0  & 36.5  & 20.3  \\
20&75& 92.9  & 83.7  & 75.1  & 59.6  & 47.1  & 31.5
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Level 2, Abnormal returns from
[.00375,.00625]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 41.0  & 19.1  & 26.4  & 11.1  & 14.3  & 3.5  \\
 2&50& 60.5  & 44.5  & 43.5  & 24.8  & 21.6  & 10.0  \\
 2&75& 69.5  & 55.4  & 52.6  & 33.7  & 30.4  & 13.6  \\[1ex]
 5&25& 70.6  & 46.9  & 50.9  & 28.1  & 30.5  & 11.0  \\
 5&50& 86.4  & 73.1  & 70.2  & 50.5  & 48.0  & 28.0  \\
 5&75& 92.9  & 80.1  & 79.4  & 67.7  & 56.1  & 36.2  \\[1ex]
20&25& 95.5  & 91.4  & 82.7  & 74.0  & 57.1  & 39.4  \\
20&50& 99.5  & 98.5  & 95.6  & 90.5  & 75.4  & 58.5  \\
20&75& 99.8  & 99.8  & 98.2  & 95.1  & 85.3  & 73.5
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Level 3, Abnormal returns from
[.0025,.0075]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 38.9  & 17.8  & 25.5  & 10.4  & 13.5  & 3.3  \\
 2&50& 58.5  & 41.5  & 41.0  & 24.6  & 19.5  & 9.5  \\
 2&75& 67.7  & 52.8  & 49.8  & 32.1  & 28.9  & 12.5  \\[1ex]
 5&25& 71.4  & 48.2  & 51.8  & 28.1  & 30.7  & 11.3  \\
 5&50& 86.5  & 73.1  & 70.5  & 51.4  & 48.5  & 28.2  \\
 5&75& 93.1  & 80.5  & 80.5  & 67.9  & 56.1  & 36.9  \\[1ex]
20&25& 95.9  & 92.5  & 84.1  & 75.4  & 58.2  & 41.4  \\
20&50& 99.6  & 98.8  & 96.2  & 91.5  & 77.6  & 60.3  \\
20&75& 99.8  & 99.8  & 98.5  & 95.8  & 86.8  & 75.3   \\[1ex]
\multicolumn{8}{l}{See notes to table \ref{tab:sum0null} for definitions.}
\end{tabular}
\end{table}
\clearpage

\begin{table}\centering
\caption{Power: Percentage of 1,649 simulations where $H_{02}:
\sum_{i=1}^N\gamma_i = 0$ was}
rejected by Theil F-statistic when abnormal performance was introduced
\label{tab:powtsum}

Rejections based on empirical distributions summarized in Table
\ref{tab:emprejsum}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: Level 1, Abnormal returns from
[.00125,.00375]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 14.5  & 4.2  & 10.3  & 2.4  & 6.9  & 1.5  \\
 2&50& 24.8  & 10.2  & 16.5  & 5.5  & 10.2  & 3.2  \\
 2&75& 31.4  & 16.4  & 20.6  & 8.0  & 11.2  & 3.8  \\[1ex]
 5&25& 32.9  & 13.1  & 21.4  & 6.7  & 13.5  & 3.1  \\
 5&50& 50.9  & 29.3  & 33.7  & 13.3  & 17.0  & 6.4  \\
 5&75& 61.8  & 39.5  & 43.0  & 24.7  & 21.2  & 7.6  \\[1ex]
20&25& 68.0  & 52.6  & 44.9  & 32.3  & 23.2  & 10.7  \\
20&50& 87.1  & 71.7  & 64.0  & 50.2  & 34.2  & 19.0  \\
20&75& 92.6  & 84.9  & 73.6  & 58.5  & 41.6  & 25.1
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Level 2, Abnormal returns from
[.00375,.00625]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 33.4  & 11.7  & 20.6  & 5.7  & 11.1  & 2.1  \\
 2&50& 55.4  & 33.9  & 34.7  & 16.6  & 18.1  & 7.0  \\
 2&75& 66.0  & 49.5  & 43.8  & 23.9  & 23.7  & 10.2  \\[1ex]
 5&25& 67.3  & 36.7  & 44.1  & 18.8  & 25.3  & 7.3  \\
 5&50& 85.7  & 67.0  & 65.1  & 39.2  & 36.4  & 16.3  \\
 5&75& 92.8  & 81.7  & 77.0  & 61.0  & 44.3  & 21.7  \\[1ex]
20&25& 95.5  & 91.0  & 80.8  & 69.9  & 48.9  & 29.7  \\
20&50& 99.5  & 98.6  & 95.1  & 89.7  & 68.9  & 51.1  \\
20&75& 99.8  & 99.8  & 98.2  & 95.6  & 80.8  & 64.7
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Level 3, Abnormal returns from
[.0025,.0075]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 39.0  & 17.5  & 25.4  & 9.2  & 13.0  & 3.1  \\
 2&50& 60.2  & 41.0  & 41.2  & 22.7  & 21.8  & 9.8  \\
 2&75& 67.7  & 54.3  & 49.5  & 31.4  & 29.5  & 14.7  \\[1ex]
 5&25& 71.4  & 47.4  & 52.6  & 27.2  & 31.0  & 10.7  \\
 5&50& 87.0  & 70.3  & 70.3  & 47.5  & 44.3  & 23.2  \\
 5&75& 93.6  & 82.7  & 79.0  & 65.9  & 51.5  & 31.7  \\[1ex]
20&25& 96.1  & 92.2  & 84.1  & 75.3  & 57.4  & 40.8  \\
20&50& 99.6  & 98.7  & 96.2  & 91.0  & 73.6  & 59.6  \\
20&75& 99.9  & 99.8  & 98.5  & 96.2  & 83.1  & 69.6  \\[1ex]
\multicolumn{8}{l}{See notes to table \ref{tab:sum0null} for definitions.}
\end{tabular}
\end{table}
\clearpage

\begin{table}\centering
\caption{Power: Percentage of 1,649 simulations where $H_{02}:
\sum_{i=1}^N\gamma_i = 0$ was}
rejected by SUMT statistic when abnormal performance was introduced
\label{tab:powsumt}

Rejections based on empirical distributions summarized in Table
\ref{tab:emprejsum}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: Level 1, Abnormal returns from
[.00125,.00375]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 021.0  & 09.3  & 012.9  & 05.8  & 08.2  & 02.1  \\
 2&50& 36.5  & 21.2  & 23.1  & 10.1  & 12.9  & 3.5  \\
 2&75& 44.6  & 28.4  & 29.7  & 16.7  & 16.7  & 6.7  \\[1ex]
 5&25& 45.3  & 23.1  & 30.1  & 12.1  & 16.4  & 4.6  \\
 5&50& 63.3  & 43.5  & 44.8  & 25.2  & 23.2  & 10.2  \\
 5&75& 75.7  & 55.8  & 56.2  & 40.3  & 29.3  & 14.4  \\[1ex]
20&25& 81.4  & 67.1  & 60.3  & 43.7  & 31.0  & 14.3  \\
20&50& 94.5  & 89.5  & 78.7  & 63.3  & 46.9  & 27.0  \\
20&75& 98.6  & 94.7  & 86.2  & 75.3  & 56.1  & 40.4
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Level 2, Abnormal returns from
[.00375,.00625]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 54.9  & 38.1  & 35.1  & 20.4  & 19.6  & 5.3  \\
 2&50& 74.1  & 58.8  & 56.0  & 38.4  & 32.1  & 14.6  \\
 2&75& 82.2  & 69.6  & 65.9  & 51.8  & 41.9  & 26.2  \\[1ex]
 5&25& 80.7  & 64.2  & 64.6  & 44.0  & 39.5  & 15.2  \\
 5&50& 93.1  & 84.8  & 79.1  & 64.0  & 55.2  & 33.4  \\
 5&75& 98.5  & 92.5  & 87.8  & 79.4  & 61.0  & 46.0  \\[1ex]
20&25& 99.2  & 97.8  & 94.1  & 85.7  & 67.1  & 50.3  \\
20&50& 100.0  & 99.9  & 99.3  & 97.5  & 86.1  & 69.7  \\
20&75& 100.0  & 99.9  & 99.9  & 99.3  & 92.3  & 83.1
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Level 3, Abnormal returns from
[.0025,.0075]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 54.8  & 39.7  & 35.7  & 21.0  & 19.6  & 5.6  \\
 2&50& 74.8  & 60.5  & 56.8  & 39.7  & 33.3  & 14.5  \\
 2&75& 83.4  & 70.5  & 67.3  & 52.8  & 43.3  & 27.3  \\[1ex]
 5&25& 81.6  & 65.9  & 65.8  & 45.2  & 40.9  & 15.4  \\
 5&50& 93.5  & 85.7  & 80.3  & 65.5  & 56.4  & 34.9  \\
 5&75& 98.7  & 93.5  & 88.9  & 80.5  & 62.0  & 47.2  \\[1ex]
20&25& 99.4  & 98.2  & 95.0  & 86.8  & 69.3  & 51.5  \\
20&50& 100.0  & 99.9  & 99.5  & 97.8  & 87.9  & 71.9  \\
20&75& 100.0  & 99.9  & 99.9  & 99.5  & 93.3  & 85.0  \\[1ex]
\multicolumn{8}{l}{See notes to table \ref{tab:sum0null} for definitions.}
\end{tabular}
\end{table}
\clearpage

\begin{table}\centering
\caption{Power: Percentage of 1,649 simulations where $H_{02}:
\sum_{i=1}^N\gamma_i = 0$ was}
rejected by SUMC statistic when abnormal performance was introduced
\label{tab:powsumc}

Rejections based on empirical distributions summarized in Table
\ref{tab:emprejsum}

\[\mbox{Model: }r_{it} = \alpha_i + \beta_i\ r_{mt} + \gamma_i\
A_{it} + \varepsilon_{it}  \]

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel A: Level 1, Abnormal returns from
[.00125,.00375]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 14.5  & 4.2  & 10.4  & 2.4  & 7.0  & 1.5  \\
 2&50& 24.8  & 10.4  & 16.5  & 5.5  & 10.2  & 3.2  \\
 2&75& 31.4  & 16.4  & 20.6  & 8.2  & 11.3  & 3.8  \\[1ex]
 5&25& 32.9  & 13.1  & 21.5  & 7.1  & 13.6  & 3.3  \\
 5&50& 51.1  & 29.8  & 33.9  & 13.7  & 17.0  & 6.4  \\
 5&75& 62.0  & 40.9  & 43.0  & 24.9  & 21.3  & 7.8  \\[1ex]
20&25& 68.0  & 52.6  & 44.9  & 32.8  & 23.2  & 10.9  \\
20&50& 87.2  & 72.7  & 64.0  & 50.3  & 34.3  & 19.5  \\
20&75& 92.6  & 84.9  & 73.6  & 58.5  & 41.8  & 25.1
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel B: Level 2, Abnormal returns from
[.00375,.00625]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 41.3  & 18.8  & 27.2  & 9.2  & 13.8  & 3.0  \\
 2&50& 62.3  & 44.1  & 43.2  & 24.3  & 23.4  & 10.4  \\
 2&75& 69.3  & 57.0  & 52.3  & 33.7  & 31.2  & 16.6  \\[1ex]
 5&25& 71.3  & 46.6  & 52.7  & 28.1  & 31.4  & 11.0  \\
 5&50& 86.7  & 70.2  & 69.4  & 48.2  & 44.3  & 24.4  \\
 5&75& 93.1  & 83.5  & 78.8  & 66.1  & 52.0  & 32.0  \\[1ex]
20&25& 95.6  & 91.6  & 82.8  & 73.8  & 55.7  & 39.4  \\
20&50& 99.5  & 98.6  & 95.3  & 90.3  & 72.3  & 57.4  \\
20&75& 99.8  & 99.8  & 98.2  & 95.6  & 81.7  & 67.8
\end{tabular}

\vspace{1ex}

\begin{tabular}{c|c||r|r||r|r||r|r}
\multicolumn{8}{l}{Panel C: Level 3, Abnormal returns from
[.0025,.0075]}\\[.5ex]\hline\hline
Number of&Number of&\multicolumn{2}{c||}{1
Day Window} & \multicolumn{2}{c||}{2 Day Window} &
\multicolumn{2}{c}{5 Day Window}\\
events/firm&firms
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
(N)&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c||}{1\%}&\multicolumn{1}{c|}{\hspace*{.75em}5\%\hspace*{.75em}}&\multicolumn{1}{c}{1\%}\\\hline
 2&25& 39.0  & 17.5  & 25.4  & 9.2  & 13.0  & 3.1  \\
 2&50& 60.3  & 41.2  & 41.2  & 22.8  & 21.9  & 9.9  \\
 2&75& 67.7  & 54.4  & 49.5  & 31.8  & 29.8  & 14.7  \\[1ex]
 5&25& 71.5  & 47.6  & 52.9  & 27.9  & 31.0  & 11.3  \\
 5&50& 87.1  & 70.5  & 70.3  & 48.7  & 44.5  & 23.4  \\
 5&75& 93.7  & 83.7  & 79.0  & 66.0  & 51.7  & 31.8  \\[1ex]
20&25& 96.1  & 92.2  & 84.1  & 75.6  & 57.4  & 41.2  \\
20&50& 99.6  & 98.8  & 96.2  & 91.0  & 73.6  & 60.0  \\
20&75& 99.9  & 99.8  & 98.5  & 96.2  & 83.2  & 69.7  \\[1ex]
\multicolumn{8}{l}{See notes to table \ref{tab:sum0null} for definitions.}
\end{tabular}
\end{table}
\end{document}