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ENT000177 Submitted: March 28, 2012 The Econometrics of Financial Markets John Y. Campbell AndrewW.Lo A. Craig MacKinlay ICi':;(

Princeton University Press Princeton, New Jersey

To Susanna, .Vaney, and Tina Copyright © 1997 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New jersey 08540 In the United Kingdom: Princeton University Press, Chichester,

\Vest Sussex All Rights Reserved Library of Congress Cataloging-in-Publication Data Campbell,john Y.

The econometrics of financial markets / john Y Campbell, Andrew

\V. Lo, A. Craig :vfacKinlay.

p. cm.

Includes bibliographical references and index.

ISBN 0-691-04301-9 (cloth alk. paper)

1. Capital market-Econometric models. I. La, Andrew W. (Andrew Wen-OlUan). II. MacKinlay, Archie Craig, 1955- IlL Title.

HG4523.Cn 1997 332'.09414--dc20 96-27868 This book \faS composed in ITC :--Jew Baskerville with l~TEX by Archetype Publishing Inc.,

15 Turtle Pointe Road, Monticello, IL 61856.

Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources.

Sc(ond printing. with corrections. 1997 http://pup.princctOn.eclu Printed in the Cnited States of America II) 9 8 7 fi 5 4

Event-Study Analysis 4

ECONOMISTS ARE FREQUENTLY ASKED to measure the effect of an economic event on the value of a firm. On the surface this seems like a difficult task, but a measure can be constructed easily using financial market data in an event study. The usefulness of such a study comes from the fact that, given rationality in the marketplace, the effect of an evcnt ,viII be reflected immediately in asset prices. Thus the event's economic impact can be measured using asset prices observed over a relatively short "time period. In contrast, direct measures may require many months or even years of observation.

The general applicability of the event-study methodology has led to its wide use. In the academic accounting and finance field, event-study methodology has been applied to a variety of firm-specific and economy-

,vide events. Some examples include mergers and acquisitions, earnings an-nouncements, issues of new debt or equity, and announcements of mac roe-conomic variables such as the trade deficit. 1 However, applications in other fields are also abundant. For example, event studies are used in the field of law and economics to measure the impact on the value ofa firm ofa change in the regulatory environment,2 and in legal-liability cases event studies are used to assess damages. s In most applications, the focus is the effect of an event on the price of a particular class of securities of the firm, most often common equity. In this chapter the methodology 'vfill be discussed in terms of common stock applications. However, the methodology can be applied to debt securities 'with little modification.

Event studies have a long history. Perhaps the first published study is Dolley (1933). Dolley examined the price effects of stock splits, studying nominal price changes at the time of the split. Using a sample of 95 splits 1'~Ne ~~ill further discuss the first three examples later in the chapter. McQueen and Roley (1993) provide an illustration using macroeconomic news announcements.

2See Schwert (1981) .

.'ISee Mitchell and Netter (1994).

14q

4. Event-Study Analysis 4.1. Outline of an Event Study 151 150.

as having seven steps:

from 1921 to 1931, he found that the price increased in 57 oflhc cases and the price declined in only 26 instances. There was no effect in the other 12 1. Event definition. The initial task of conducting an event study is to de-cases. Over the decades from the early 1930s until the late 1960s the level of fine the event ofinterest and identify the period over which the security sophistication of event studies increased. YIyers and Bakay (1948), Barker prices of the firms involved in this event '1,'1'111 be examined-the event (1956,1957,1958), and Ashley (1962) are examples of studies during this window. For example, if one is looking at the information content_of time period. The improvements include removing general stock market an earnings announcement 'With daily data, the event will be the earn-price movements and separating out confounding events. In the lalc 19605 ings announcement and the event window might be the one day of the seminal studies by Ball and Brown (1968) and Fama, Fisher, Jensen, and announcement. In practice, the event windm\' is often expanded to Roll (1969) introduced the methodology that is essentially still in use today. wo days, the day of the announcement and the day after the announce~

Ball and Brov.;n considered the information content of earnings, and Fama, ment. This is done to capture the price effects of announcements w,hich Fisher,Jensen, and Roll studied the effects of stock splits after removing the occur after the stock market closes on the announcement day. The pe-effects of simultaneous dividend increases. riod prior to or after the event may also be of interest and included In the years since these pioneering studies, several modifications of the separately in the analysis. For example, in the earnings-announcement basic methodology have been suggested. These modifications handle com~ case, the market may acquire information about the earnings prior to plications arising from violations of the statistical assumptions used in the the actual announcement and one can investigate this possibility by early work, and they can accommodate more specific hypotheses. Brown examining pre-event returns.

and Warner (1980, 1985) are useful papers that discuss the practical im- 2. Selection criteria. After identifying the event of interest, it is necessary portance of many of these modifications. The 1980 paper considers imple- to determine the selection criteria for the inclusion of a given firm in mentation issues for data sampled at a monthly interval and the 1985 paper the study. The criteria may involve restrictions imposed by data avail-deals \'o/ith issues for daily data. ability such as listing on the NYSE or AMEX or may involve restrictions This chapter explains the econometric methodology of event studies. such as membership in a specific industry. At this stage it is useful to Section 4.1 briefly outlines the procedure for conducting an event study. summarize some characteristics of the data sample (e.g., firm market Section 4.2 sets up an illustrative example of an event study. Central to capitalization, industry representation, distribution of events through any event study is the measurement of the abnormal return. Section 4.3 time) and note any potential biases which may have been introduced details the first step-measuring the normal performance-and Section 4.4 through the sample selection.

follows \\tith the necessary tools for calculating the abnormal return, mak- 3. Nonnal and abnonnal returns. To appraise the event's impact "ie require ing statistical inferences about these returns, and aggregating over many a measure of the abnormal return. The abnormal return is the actual event observations. In Sections 4.3 and 4.4 the discussion maintains the ex post return of the security over the event window minus the normal null hypothesis that the event has no impact on the distribution of returns. return of the firm over the event 'Window. The normal return is defined Section 4.5 discusses modifying the null hypothesis to focus only on the as the return that would be expected if the event did not take place. For mean of the return distribution. Section 4.6 analyzes of the power of an each firm i and event date r we have event study. Section 4.7 presents a non parametric approach to event stud-ies which eliminates the need for parametric structure. In some cases theory c7, = R;, - E[R;, I X,], (4.1.1) provides hypotheses concerning the relation between the magnitude of the event abnormal return and firm characteristics. In Section 4.8 we consider where <t' Rtt, and E(Rtt) are the abnormal, actual, and normal returns, cross-sectional regression models which are useful to investigate such hy- respectively, for time period t. Xt is the conditioning information for potheses. Section 4.9 considers some further issues in event-study design the normal performance model. There are two common choices for modeling the normal return-the constant-mean-retum model where Xl and Section 4.10 concludes.

is a constant, and the market model "There Xl is the market return. The constant-mean-return model, as the name implies, assumes that the 4.1 Outline of an Event Study mean return of a given security is constant through time. The market model assumes a stable linear relation bet1\'een the market return and At the outset it is useful to give a brief outline of the structure of an event the security return.

study. vVhile there -is no unique structure, the analysis can be viewed

• 152 4. Event-Study Analysis 4.3. Models for Measuring Normal Performance 153

4. Estimation procedure. Once a normal performance model has been se- each firm and quarter, three pieces of information are compiled: the date lected, the parameters of the model must be estimated using a subset of the announcement, the actual announced earnings, and a measure of of the data known as the estimation window. The most common choice, the expected earnings. The source of the date of the announcement is when feasible, is to use the period prior to the event \vindow for the esti- Datastream, and the source of the actual earnings is Compustat.

mation \vindow. For example, in an event study using daily data and the If earnings announcements convey information to investors, one would market model, the market-model parameters could be estimated over expect the announcement impact on the market's valuation of the firm's the 120 days prior to the event. Generally the event period itself is not equity to depend on the magnitude of the unexpected component of the included in the estimation period to prevent the event from influencing announcement. Thus a measure of the deviation of the actual announced the normal performance model parameter estimates. earnings from the market's prior expectation is required. We use the mean

5. Testing procedure. \'Vith the parameter estimates for the normal perfor- quarterly earnings forecast from the Institutional Brokers Estimate System mance model, the abnormal returns can be calculated. Next, we need (IIB/E/S) to proxy for the market's expectation of earnings. IIB/E/S com-to design the testing framework for the abnormal returns. Important piles forecasts from analysts for a large number of companies and reports considerations are defining the null hypothesis and determining the summary statistics each month. The mean forecast is taken from the last techniques for aggregating the abnormal returns of individual firms. month of the quarter. For example, the mean third-quarter forecast from

6. Empirical results. The presentation of the empirical results follows the September 1990 is used as the measure of expected earnings for the third formulation of the econometric design. In addition to presenting the quarter ofl990.

basic empirical results, the presentation of diagnostics can be fruitful. In order to examine the impact of the earnings announcement on the Occasionally, especially in studies ""ith a limited number of event obser- value of the firm's equity, we assign each announcement to one of three vations, the empirical results can be heavily influenced by one or two categories: good news, no news, or bad news. We categorize each an-firms. Knowledge of thi~ is important for gauging the importance of nouncement using the deviation of the actual earnings from the expected the results. earnings. If the actual exceeds expected by more than 2.5% the announce-

7. Interpretation and conclusions. Ideally the empirical results \-vill lead to ment is designated as good news, and if the actual is more than 2.5% less insights about the mechanisms by which the event affects security prices. than expected the announcement is designated as bad news. Those an-Additional analysis may be included to distinguish benveen competing nouncements where the actual earnings is in the 5% range centered about explanations. the expected earnings are designated as no news. Of the 600 announce-ments, 189 are good nev. 's, 173 are no news, and the remaining 238 are bad news.

With the announcements categorized, the next step is to specify the 4.2 An Example of an Event Study sampling interval, event window, and estimation \\'indow that will be used to analyze the behavior of firms' equity returns. For this example we set the The Financial Accounting Standards Board (FASB) and the Securities Ex-sampling interval to one day; thus daily stock returns are used. We choose a change Commission strive to set reporting regulations so that financial state-41*day event window, comprised of 20 pre-event days, the event day, and 20 ments and related information releases are informative about the value of post-event days. For each announcement we use the 250-trading-day period the firm. In setting standards, the information content of the financial dis-prior to the event window as the estimation window. After we present the closures is of interest. Event studies provide an ideal tool for examining Lhe methodology of an event study, \',re use this example as an illustration.

information content of the disclosures.

In this section we describe an example selected to illustrate the event-study methodology. One particular type of disclosure-quarterly earnings announcements-is considered. \Ve investigate the information content of 4.3 Models for Measuring Normal Performance quarterly earnings announcements for the thiny firms in the Dow Jones Industrial Index over the five-year period from January 1989 to December A number of approaches are available to calculate the normal return of a 1993. These announcements correspond to the quarterly earnings for the given security. The approaches can be loosely grouped into hv-o categories-last quarter of 1988 through the third quarter of 1993. The five years of statistical and economic. Models in the first category follow from statistical data for thirty firms provide a total sample of 600 announcements. For assumptions concerning the behavior of asset returns and do not depend on

154 4. Event-Study Analysis 4.3. Modelsfor Measuring Normal Performance 155 any economic arguments. In contrast, models in the second category rely 4.3.21I1arketAlodel on assumptions concerning investors' behavior and are not based solely on The market model is a statistical model which relates the return of any statistical assumptions. It should, however, be noted that to use economic giyen security to the return of the market portfolio. The model's linear models in practice it is necessary to add statistical assumptions. Thus the specification follows from the assumed joint normality of asset returns. 4 potential advantage of economic models is not the absence of statistical For any security i we have assumptions, but the opportunity to calculate more precise measures of the normal return using economic restrictions.

For the statistical models, it is conventional to assume that asset re-Rit = ai + f3iRmt + fit (4.3.2) 9 turns are jointly multivariate normal and independently and identically dis- E[E,,] = 0 Var[E,,] = O'E-;'

tributed through time. Formally, we have:

where Rit and Rml are the period-t returns on security i and the market portfolio, respectively, and fit is the zero mean disturbance term. ai. fJi' (AI) Let R t be an (Nx 1) vector of asset returns for cawndar time period t. R t is and a~~ are the parameters of the market model. In applications a broad-independently multivariate normally distributed with mean J.L and covariance matrix based stock index is used for the market portfolio, with the S&P500 index, o for all t. the CRSP value-weighted index, and the CRSP equal-weighted index being popular choices.

This distributional assumption is sufficient for the constant-mean-return The market model represents a potential improvement over the con-model and the market model to be correctly specified and permits the de- stan t-mean-return model. By removing the portion of the return that is velopment of exact finite-sample distributional resulL<; for the estimators related to variation in the market's return, the variance of the abnormal and statistics. Inferences using the normal return models are robust to return is reduced. This can lead to increased ability to detect event effects.

deviations from the assumption. Further, we can explicitly accommodate The benefit from using the market model will depend upon the R' of the deviations using a generalized method of moments framework. market-model regression. The higher the R2, the greater is the variance re-duction of the abnormal return, and the larger is the gain. See Section 4.4.4 for more discussion of this point.

4.3.1 Constant-Mean-Return Model Let J1.i. the ith element of {.t, be the mean return for asset i. Then the 4.3.3 Other Statistical Models constant-mean-return model is A number of other statistical models have been proposed for modeling the normal return. A general type of statistical model is the factor modeL R.;t = + Sit (4.3.1) Factor models potentially provide the benefit of reducing the variance of E[SiI] = 0 J.'i Var[Si,] =

(ft.;,

the abnormal return by explaining more of the variation in the normal return. Typically the factors are portfolios of traded securities. The market model is an example of a one-factor model, but in a multifactor model one where Ri!, the ith element ofRt> is the period-t return on security i, ~il is the might include industry indexes in addition to the market. Sharpe (1970) disturbance term, and a(~ is the (i, z) element of O. and Sharpe, Alexander, and Bailey (1995) discuss index models with factors Although the constant-mean-return model is perhaps the simplest based on industry classification. Another variant of a factor model is a model, Brown and Warner (1980, 1985) find it often yields results simi- procedure which calculates the abnormal return by taking the difference lar to those of more sophisticated models. This lack of sensitivity to the between the actual return and a portfolio of firms of similar size, where size model choice can be attributed to the fact that the variance of the abnormal is measured by market value of equity. In this approach typically ten size return is frequently not reduced much by choosing a more sophisticated groups are considered and the loading on the size portfolios is restricted model. Wnen using daily data the model is typically applied to nominal returns. \""ith monthly data the model can be applied to real returns or 4The specification actually requires the asset weights in the market portfolio to remain excess returns (the return in excess of the nominal riskfree return generally constant. However, changes over time in the market portfolio weights are small enough that measured using the US Treasury',bill) as well as nominal returns. they have little effect on empirical work.

~:'

156 4. Event-Stud), Analysis 4.4. MeasUring and Analyzing Abnormal Returns 157 Time Line:

to unity. This procedure implicitly assumes that expected return is directly related to the market value of equity.

esti.mation ] event ] po~t-event ]

In practice the gains from employing multifactor models for event stud- ( wmdow ( windO\\' ( \vmdow ies are limited. The reason for this is that the marginal explanatory power of additional factors beyond the market factor is small, and hence there is little To 1j o 12 1':,

reduction in the variance of the abnormal return. The variance reduction "Will typically be greatest in cases where the sample firms have a common r characteristic, for example they are all members of one industry or they are all firms concentrated in one market capitalization group. In these cases Figure 4.1. Time Line for an Event Study the use of a multifactor model warrants consideration.

Sometimes limited data availability may dictate the use of a restricted model such as the market-adjusted-return model. For some events it is not feasi-ble to have a pre-event estimation period for the normal model parameters, cross-section of mean returns, as sho"\\-TI by Fama and French (1996a) and and a market-adjusted abnormal return is used. The market-adjusted-return others, so a properly chosen APT model does not impose false restrictions model can be viewed as a restricted market model with (Xi constrained to be on mean returns. On the other hand the use of the APT complicates the o and {3i constrained to be 1. Since the model coefficients are prespecified, implementation of an event study and has little practical advantage relative an estimation period is nOt required to obtain parameter estimates. This to the unrestricted market model. See, for example, Brov.'Il and Weinstein model is often used to study the underpricing of initial public offerings.' (1985). There seems to be no good reason to use an economic model rather A general recommendation is to use such restricted models only as a last than a statistical model in an event study.

resort, and to keep in mind that biases may arise if the restrictions are false.

4.3. 4 Economic Models 4.4 Measuring and Analyzing Abuormal Returns Economic models restrict the parameters of statistical models to provide In this section we consider the problem of measuring and analyzing abnor-more constrained normal return models. Two common economic models mal returns. We use the market model as the normal performance return which provide restrictions are the Capital Asset Pricing Model (CAPM) and model, but the analysis is virtually identical for the constant-mean-return exact versions of the Arbitrage Pricing Theory (APT). The CAPM, due to modeL Sharpe (1964) and Lintner (1965b), is an equilibrium theory where the \Ve first define some notation. We. index returns in event time using expected return of a given asset is a linear function of its covariance ,vith r. Defining r = 0 as the event date, r = T} + I to r = T2 represents the return of the market portfolio. The APT, due to Ross (1976), is an asset the event window, and r = To + I to T = TJ constitutes the estimation pricing theory where in the absence of asymptotic arbitrage the expected "indow. Let LJ = T J - To and L, = T2 - TJ be the length of the estimation return of a given asset is determined by its covariances \vith multiple factors. \\;ndow and the event windm\lT, respectively. If the event being considered Chapters 5 and 6 provide extensive treatments of these two theories. is an announcement on a given date then T, = TJ + I and L, = J. If The Capital Asset Pricing Model was commonly used in event studies applicable, the post-event window will be from "[ = T2 + 1 to "[ = T3 and its during the 1970s. During the last ten years, hmvever, deviations from the length is L3 = T3 - T 2. The timing sequence is illustrated on the time line CAPM have been discovered, and this casts doubt on the validity of the in Figure 4.J.

restrictions imposed by the CAPM on the market model. Since these re- \Ve interpret the abnormal return over the event window as a measure strictions can be relaxed at little cost by using the market model, the use of of the impact of the event on the value of the firm (or its equity). Thus, the the CAPM in event studies has almost ceased. methodology implicitly assumes that the event is exogenous with respect to Some studies have used multifactor normal performance models mo- the change in market value of the security. In other words, the revision in tivated by the Arbitrage Pricing Theory. The APT can be made to fit the value of the firm is caused by the event. In most cases this methodology is appropriate, but there are exceptions. There are examples where an event

See Ritter (1990) for an example. is triggered by the change in the market value of a security, in which case

158 4. Event-Study Analysil 4.4. MPflswing and Analyzing Abnormal Returns 159 the event is endogenous. For these cases, the usual interpretation v.'ill be properties of abnormal returns. First we consider the abnormal return incorrect. properties of a given security and then we aggregate across securities.

It is typical for the estimation \\lindow and the event \\i1ndmv not to over-lap. This design provides estimators for the parameters of the normal return 4.4.2 Statistical Properties of Abnormal Returns model which are not influenced by the event-related returns. Including the event window in the estimation of the normal model parameters could lead Given the market-model parameter estimates, live can measure and analyze to the event returns having a large influence on the normal return mea- the abnormal returns. Let E; be the (L2x1) sample vector of abnormal sure. In this situation both the normal returns and the abnormal returns returns for firm i from the event windm..', TI + 1 to T2* Then using the would reflect the impact of the event. This would be problematic since the market model to measure the normal return and the OLS estimators from methodology is built around the assumption that the event impact is cap- (4.4.3), we have for the abnormal return vector:

tured by the abnormal returns. In Section 4.5 \ve consider expanding the null hypothesis to accommodate changes in the risk of a firm around the E; Ri -

~

Cti L -

~,.

f3iRm event. In this case an estimation framework which uses the event \vindO\v returns will be required. R; -X;Oi, (4.4.7)

,\*here R; = [Rrrl+I'-'~Y~J' is an (Lzxl) vector of event-window returns, 4.4.1 Estimation of the Marhet Model X7 = [~R:;J is an (L2x2) matrix \vi.th a vector of ones in the first column Recall that the market model for security i and observation r in event time and the vector of market return observations R:! = [RrnT1+l ... Rml~}' in the IS second column, and Bi = [ai ~i]' is the (2xl) parameter vector estimate.

Rr Cti + f3iRnr + Eir- ( 4.4.1) Conditional on the market return over the event '"indow, the abnormal re-turns \vill bejointly normally distributed with a zero conditional mean and The estimation-window observations can be expressed as a regression sys- conditional covariance matrix Vi as shown in (4.4.8) and (4.4.9), respec-tem, tiyely.

Ri X/}i+j, (4.4.2) where Ri = [R;Yo+l' - - R.;Y1]' is an (L I x 1) vector of estimation-windmv re- W7 I X;J = E[R;-X;e i I X;J turns, Xi = [~ RmJ is an (LI x 2) matrix with a vector of ones in the first col-

= E[(R; - X;Oi) - X;U'}i - Oi) I X;J umn and the vector of market return observations Rill = [Rm7()+l . - . RrnTI J' in the second column, and 8 i = [Cti f3d' is the (2x 1) parameter vector. X has o. ( 4.4.8) a subscript because the estimation vvindo'iN may have timing that is specific to firm i. Under general conditions ordinary least squares (OLS) is a consis- Vi E[~ 7' I X~]

tent estimation procedure for the market-model parameters. Further, given E((< - X;(Oi - Oi)][t7 - X;(Oi - Oil]' I X;]

=

the assumptions of Section 4.3, OLS is efficient. The OLS estimators of the market-model parameters using an estimation window of LI observations = E[E'I E" - E'(O - O)'X" I I ! I I

- X'(O I I

- 0)1£1M are

+ X;(Oi - o;)(e i - O;)'X;' I X;J Oi = (X~Xi) -I X:Ri (4.4.3)

= I a,2 + X'I (X'X)-'X I I M

2 1 a Ei *

( 4.4.9)

,2 1 ,',

a£; --.. ( 4.4.4)

L I -2 ! 1 I is the (L2 XL2) identity matrix.

From (4.4.8) we see that the abnormal return vector, with an expecta-Ei Ri -X/}i (4.4.5 ) tion of zero, is unbiased. The covariance matrix of the abnormal return Var[O;] (X'X )-' ai*

2 (4.4.6) Yector from (4.4.9) has two parts. The first term in the sum is the variance

= i i due to the future disturbances and the second term is the additional vari-We next show how to use these OLS estimators to measure the statistical ance due to the sampling error in 8i. This sampling error, which is common

, 160 4, Event-Study AnalJsis 4,4, Measuring and Analyzing Abnormal Returns 161 for all the elements of the abnormal return vector, will lead to serial corre~ degrees of freedom. From the properties of the Student t distribution, lation of the abnormal returns despite the fact that the true disturbances the expectation of SCARi(Tt, T2) is 0 and the variance is (z: =~). For a large are independent through time. As the length of the estimation window Ll estimation window (for example, LI > 30), the distribution ofSCAR;('I, ,,)

becomes large, the second tCfm ,.,.ill approach zero as the sampling error of will be well approximated by the standard normal.

the parameters vanishes, and the abnormal returns across time periods will The above result applies to a sample of one event and must be extended become independent asymptotically for the usual case where a sample of many event observations is aggregated.

Under the null hypothesis, Ho, that the given event has no impact on To aggregate across securities and through time, we assume that there is the mean or variance of returns, we can use (4.4,8) and (4.1,9) and the joint not any correlation across the abnormal returns of different securities. This normality of the abnormal returns to draw inferences. Under Ho, for the \\ill generally be the case if there is not any clustering, that is, there is not vector of event-window sample abnormal returns \ve have any overlap in the event v..i.ndows of the included securities. The absence of any overlap and the maintained distributional assumptions imply that the E; ~ N(O, V;), (4.1.10) abnormal returns and the cumulative abnormal returns will be independent Equation (4.4.10) gives us the distribution for any single abnormal return across securities. Inferences 'with clustering 'will be discussed later.

observation. Vve next build on this result and consider the aggregation of The individual securities' abnormal returns can be averaged using E7 abnormal returns. from (4 7), Given a sample of N events, defining.' as the sample average of the N abnormal return vectors, we have 4,4,3 Aggregation of Abnormal Returns N 1

The abnormal return observations must be aggregated in order to draw

• N L" i=I i

(4.1,15) overall inferences for the event of interest. The aggregation is along two dimensions-through time and across securities. "\Ie \",ill first consider ag- 1 N gregation through time for an individual security and then will consider Var[E'] = V = _'\'V, (4.1,16)

N2~ 1 aggregation both across securities and through time. i=l

\Ve introduce the cumulative abnormal return to accommodate multi- '1Ne can aggregate the elements of this average abnormal returns vector ple sampling intervals \\Tithin the event window. Define CARi(il, i2) as the through time using the same approach as we did for an individual security'S cumulative abnormal return for security i from i l to i2 v[here Tl < rl :::: vector. Define CAR( Tl, T2) as the cumulative average abnormal return from T2 ::: T2 - Let I be an (~x 1) vector with ones in positions rI - TI to T2 - T] T] to T2 where Tl < TI :::: T2 :::: T2 and I again represents an (L-z x I) vector and zeroes else\vhere. Then we have with ones in positions Tl - Tl to T2 - TJ and zeroes elsev. . here. For the OO;('j, ,,) ~  !""- cumulative average abnormal return we have lEi (4.1.11)

Var[CAR;('I, ,,)] = "; ('I, ,,) ,'Vi" (4.1.12) CAR (rl, ,,) '" I'.' (4.4,17)

It follows from (4.1,10) that under Ho, Var[CAR(rl, ,,)] ,,'('I, ,,) = I'V1 , (4.1,18) 00;('1, ,,) ~ N(O, ",'('1, ,,)), (4.1.13) Equivalently, to obtain CAR(rl, ,,), we can aggregate using the sample cumulative abnormal return for each security i. For N events we have We can construct a test ofHo for security i from (4.1,13) using the standard, ized cumulative abnormal return, N OO;(rl, ,,) CAR('I, r,) hLOO;('I, ,,) (4.1,19)

SCAR, ('I , ,,) = (4.1,14) i=l

&;('1, r,)

N where ai2 (TJ, T2) is calculated with af.~ from (4.4.4) substituted for (5f.~' Under Var[CAR(rl, ,,)] ,,'('I, ,,) I '\' 2 N~ ~ (Ji (T1' T2)* (4.4,20) the null hypothesis the distribution of SCAR;('I, ,,) is Student t with LI - 2 i=l

162 4. Event-Study Analysis 4.4. Measuring and Analyzing Abnormal Returns 163 In (4.4.16), (4.4.18), and (4.4.20) we use the assumption that the event constant-mean-return model will lead to a reduction in the abnormal re-windows of the N securities do not overlap to set the covariance terms to turn variance. This point can be shown by comparing the abnormal return zero. Inferences about the cumulative abnormal returns can be drawn using variances. For this illustration we take the normal return model parameters

-CAR(T" as given.

T,) - Ar(0, ij 2 (T" T2) ) , (4.4.21) The variance of the abnormal return for the market model is since under the null hypothesis the expectation of the abnormal returns is zero. In practice, since 0-2(r1' '[2) is unknovvn, we can use a (Ll' (2)

., =

a; Var[Rit - ai - ,BiRm,l j0~ L~~l 8}Cr l, '[2) as a consistent estimator and proceed to test Ho using = Var[R;,l - ,BfVar[Rm,l (4.4.25)

~(T" T2~ ~

= (l - Rf) Var[R;,],

J, = N(O, I). (4.4.22)

[ij (T" T,)j' where Ri is the R2 of the market-model regression for security i.

For the constant-mean-return model, the variance of the abnormal re-,

This distributional result is for large samples of events and is not exact turn ~it is the variance of the unconditional return, Var[Ri/], that is, because an estimator of the variance appears in the denominator.

A second method of aggregation is to give equal weighting to the indi-vidual SCARi's. Defining SCAR(!"l, 1:"2) as the average over N securities from a£ = Var[R;, - f.'il = Var[R;,l. (4.4.26) event time TI to T2, we have Combining (4.4.25) and (4.4.26) we have SCAR(T" T2) =

,~-

.\1 N L..,SCARi(Tj, T2)' (4.4.23) a;' = (l - Rf) ar (4.4.27) i=l Since R; lies betvveen zero and one, the variance of the abnormal return Assuming that the event "Windows of the N securities do not overlap in using the market model \\lill be less than or equal to the abnormal return calendar time, under H Q , SCAR(TI, T2)"Will be normally distributed in large variance using the constant-mean-return model. This lower variance for samples with a mean of zero and variance CVT{~!4>>)' vVe can test the null the market model """ill carry over into all the aggregate abnormal return hypothesis using measures. As a result, using the market model can lead to more precise N(L, -

J2 = ( L, _ 2 4))', --

SCAR(T" T2) -

, N(O, I). (4.4.24) inferences. The gains will be greatest for a sample of securities \..ith high market-model R2 statistics.

In principle further increases in R2 could be achieved by using a multi-Vlhen doing an eventstudy one will have to choose between using II or 12 factor model. In practice, however, the gains in R2 from adding additional for the test statistic. One "vould like to choose the statistic ,,\-"ith higher power, factors are usually small.

and this "ill depend on the alternative hypothesis. If the true abnormal return is constant across securities then the better choice v.;ill give more 4.4.5 CARs for the Earnings-Announcement Example weight to the securities with the lower abnormal return variance, which is what 12 does. On the other hand if the true abnormal return is larger for The earnings-announcement example illustrates the use of sample abnor-securities \-'.'ith higher variance, then the better choice \vill give equal weight mal returns and sample cumulative abnormal returns. Table 4.1 presents to the realized cumulative abnormal return of each security, which is what JI the abnormal returns averaged across the 30 firms as \vell as the averaged does. In most studies, the results are not likely to be sensitive to the choice cumulative abnormal return for each of the three earnings nev,rs categories.

of II versus 12 because the variance of the CAR is of a similar magnitude Two normal return models are considered: the market model and, for across securities. comparison, the constant-mean-return model. Plots of the cumulative ab-normal returns are also included, with the CARs from the market model in Figure 4.2a and the CARs from the constant-mean-return model in Fig-4.4.4 Sensitivity to Normal Return Model ure 4.2b.

Vve have developed results using the market model as the normal return The results of this example are largely consistent with the existing lit-model. As previously* noted, using the market model as opposed to the erature on the information content of earnings. The evidence strongly

4.4. Measuring and Analyzing Abnormal Returns 165 0.03 I'~~~~~~~~~~~~~~~~~~--,

Table 4.1. Abnormal returns for an event stud), of the information content of earnings an-nouncements. 0.02 Good-:\:cws Firms Event Day Good )jews

'" CAR Market :'\1odcl No Xews

'" CAR Bad News

'" CAR Constant-Mean-Rcturn Model Good News CAR J\O News

'" CAR Bad News

'" CAR 0.01 0

\---/)

~'------,

No-l\cws Firm.'

-20

-19

.093 .093

-.177 -.084

.080

.018

.080

.098

-.107

-.180

.107

-.286

.105 .105 .019 .019 -.077 -.077 U ' .. -

-.235 -.129 -.048 -.029 -.142 -.219

-18

-17

.088 .004 .012

.024 .029 -.151 -.041

.llO .029 -.258 .069 -.060 -.086 -.115 -.043 -.262 -0.01 " r-/''' ........ _........

-.079 -.337 -.026 -.086 -.140 -.255 -.057 -,319 /.,/

-16 -.018 .011 -.019 -.060 -.010 -.346 -.086 -.172

-15 -.040 -.029 .013 -.047 -.054 -.101 -.183 -.355

.039

.099

-.216

-.117

-.075

-.037

-.394

-.431 Bad-News Firm~

-14 .038 .008 .040 -.007 -.021 -.421 -.020 -.375 -.150 -.266 -.101

-0.02

-.532

-13 .056 .064 -.057 -.065 .007 -.414 -.025 -.399 -.191 -.458 -.069 -.601

-12 .065 .129 .146 .081 -.090 -.504 .101 -.298 .133 -.325 -.106 -.707

-II .059 .199 -.020 .051 -.088 -.592 .125 -.172 .005 -.319 -.169 -.876 -0.03

-10 .028 .227 .025 .087 -.092 -.683 .134 -.038 .103 -.216 -.009 -.885 -20 -10 o 10 20

-9 .155 .382 .l1S .202 -.040 -.724 .210 .172 .022 -.194 .011 -.874 Event Time

-8 .057 .438 .070 .272 .072 -.652 .106 .2i8 .163 -.031 .135 -.738

-7 -.010 .428 -.106 .166 -.026 -.677 -.002 .277 .009 -.022 -.027 -.765

-6 .104 .532 .026 .192 -.013 -.690 .011 .288 -.029 -.051 .030 -.735 Figure 4.2a. Plot of Cumulative Market-Alodel Abnormal Return for Earning Announce-

-5 .085 .616 -.085 .107 .164 -.527 .061 .349 -.068 -.120 .320 -.415

-4 .099 .715 .040 .147 -.139 -.666 .031 .379 .089 -.031 -.205 -.620 ments

-3 .117 .832 .036 .183 .098 -.568 .067 .447 .013 -.018 .085 -.536

-2 .006 .838 .226 .409 -.1l2 -.680 .010 .456 .311 .294 -.256 -.791

-I .164 1.001 -.168 .241 -.180 -.860 .198 .654 -.170 .124 -.227 -1.018 0 .965 1.966 -.091 .150' -.679 -1.539 1.034 1.688 -.164 -.040 -.643 -1.661 I .251 2.217 -.008 .142 -.204 -1.743 0.03

.357 2.045 -.170 -.210 -.212 -1.873 2 -.014 2.203 .007 .148 .072 -1.672 -.013 2.033 .051 -.156 .078 -1.795 3 -.164 2.039 .042 .190 .083 -1.589 -.088 1.944 -.121 -.277 .146 -1.648 4 -.014 2.024 .000 .190 .106 -1.483 .041 1.985 .023 -.253 0.02

.149 -1.499 5 .135 2.160 -.038 .152 .194 -1.289 .248 2.233 -.003 -.256 .286 -1.214 Good-Kews Firms 6 -.052 2.107 -.302 -.150 .076 -1.213 -.035 2.198 -.319 -.575 .070 -1.l43 7

8 9

.060

.155

-.008 2.167 2.323 2.315

-.199

-.108

-.146

-.349

-.457 -.041

-.603 -.069

.120 -1.093

-1.l34

-1.203

.017

.1l2

-.052 2.215 2.326 2.274

-.112

-.187

-.057

-.687

-.874

-.931

.102

.056

-.071

-1.041

-.986

-1.056 O.oJ

\ , ~ - --~

---

N()-N<.:w~ Firms 10 .164 2.479 .082 -.521 .130 -1.073 .147 2.421 .203 -.728 .267 -.789 0 r---..

II 12

-.081

-.058 2.398 2.341

.040

.246

-.481 -.009

-.235 -.038

-1.082

-1.l19

-.013

-.054 2.407 2.354

.045

.299

-.683

-.384

.006 -.783 v

'T

.017 -.766 13 -.165 2.176 .014 -.222 .071 -1.048 -.246 2.107 -.067 -.451 .114 -.652 -0.01 ,,'

14 -.081 2.095 -.091 -.312 .019 -1.029 -.011 2.096 -.024 -.475 .089 -.561 15 -.007 2.088 -.001 -.314 -.043 -1.072 -.027 2.068 -.059 -.534

\ /

-.022 -.585 Bad-N<.:ws Firms """,/

16 .065 2.153 -.020 -.334 -.086 -1.l59 .103 2.171 -.046 -.580 -.084 -.670 -0.02 17 .081 2.234 .017 -.317 -.050 -1.208 .066 2.237 -.098 -.677 -.054 -.724 18 .172 2.406 .054 -.263 .066 -1.l42 .lIO 2.347 .021 -.656 -.071 -.795 19 -.043 2.363 .119 -.114 -.088 -1.230 -.055 2.292 .088 -.568 .026 -.769 -0.03 L.'~~~~~~~~~~~~~~=~

20 .013 2.377 .094 -.050 -.028 -1.258 .019 2.311 .013 -.554 -.115 -.884 -20 -10 0 10 20 Event Time Tbe sample consists of a total of 600 quarterly announcements for tbe thirty companies in the DowJones Industrial Index for the five-year period January 1989 to December 1993. Two mod-els are considered for tbe normal returns, tbe market model using the CRSP value-weighted Figure 4.2b. Plot of Cumulative Constant-Mean-Retum-1Wodel Abnormal Return for Earn-index and tbe constant-mean-return model. Tbe announcements are categorized into three ing Announcements groups, good news, no news, and bad news. " is tbe sample average abnormal return for the speCified day in event time and CAR is the sample average cumulative abnormal return for day

-20 to the speCified day. Event time is measured in days relative to the announcement date. supports the hypothesis that earnings announcements do indeed convey in-formation useful for the valuation of firms. Focusing on the announcement day (day zero) the sample average abnormal return for the good-news firm

i 167 166 4. Event-Study Analysis 4.5. Modifying tlu Null Hypothesis using the market model is 0.965%. Since the standard error of the one-day overlap, the covariances between the abnormal returns may differ from good-news average abnormal return is 0.104%, the value Of}l is 9.28 and zero, and the distributional results presented for the aggregated abnormal the null hypothesis that the event has no impact is strongly rejected. The returns are not applicable. Bernard (1987) discusses some of the problems story is the same for the bad-news firms. The event day sample abnormal related to clustering.

return is -0.679%, with a standard error of 0.098%, leading to.h equal to vVhen there is one event date in calendar time, clustering can be ac-

-6.93 and again strong evidence against the null hypothesis. As would be commodated in !:\vo different ways. First, the abnormal returns can be expected, the abnormal return of the no-news firms is small at -0.091 % aggregated into a portfolio dated using event time, and the security level and, 'with a standard error 0[0.098%, is less than one standard error from analysis of Section 4.4 can be applied to the portfolio. This approach allows zero. There is also some evidence of the announcement effect on day one. for cross correlation of the abnormal returns.

The average abnormal returns are 0.251 % and -0.204% for the good-news A second way to handle clustering is to analyze the abnormal returns and the bad-news firms respectively. Both these values are more than two without aggregation. One can test the null hypothesis that the event has no standard errors from zero. The source of these day-one effects is likely to be impact using unaggregated security-by-security data. The basic approach is that some of the earnings announcements are made on event day zero after an application of a multivariate regression model \"'1th dummy variables for the close of the stock market. In these cases the effects "ill be captured in the event date; it is closely related to the multivariate F-test of the CAPM pre-the return on day one. sented in Chapter 5. The approach is developed in the papers of Schipper The conclusions using the abnormal returns from the constant-mean- and Thompson (1983, 1985), Malatesta and Thompson (1985), and Collins return model are consistent 'With those from the market modeL Hmfever, and Dent (1984). It has some advantages relative to the portfolio approach.

there is some loss of precision using the constant-mean-return model, as the First, it can accommodate an alternative hypothesis where some of the firms variance of the average abnormal return increases for all three categories. have positive abnormal returns and some of the firms have negative abnor-

'When measuring abnormal returns 'With the constant-mean-return model mal returns. Second, it can handle cases where there is partial clustering, the standard errors increase from 0.104% to 0.130% for good-news firms, that is, where the event date is not the same across firms but there is overlap from 0.098% to 0.124% for no-news firms, and from 0.098% to 0.131 % in the event windows. This approach also has some drawbacks, however. In for bad-news firms. These increases are to be expected when considering many cases the test statistic has poor finite-sample properties, and often it a sample of large firms such as those in the Dow Index since these stocks has little power against economically reasonable alternatives.

tend to have an important market component whose variability is eliminated using the market model.

The CAR plots show that to some extent the market gradually learns 4.5 Modifying the Null Hypothesis about the forthcoming announcement. The average CAR of the good-news firms gradually drifts up in days -20 to -I, and Lhe average CAR of the Thus far we have focused on a single null hypothesis-that the given event bad-news firms gradually drifts down over this period. In the days after the has no impact on the behavior of security returns. vVith this null hypothesis announcement the CAR is relatively stable, as would be expected, although either a mean effect or a variance effect represents a violation. However, there does tend to be a slight (but statistically insignificant) increase for the in some applications we may be interested in testing only for a mean effect.

bad-news firms in days two through eight. In these cases, we need to expand the null hypothesis to allow for changing (usually increasing) variances.

To accomplish this, we need to eliminate any reliance on past returns 4.4.6 Inferences with Clustering in estimating the variance of the aggregated cumulative abnormal returns.

In analyzing aggregated abnormal returns, we have thus far assumed that Instead, we use the cross section of cumulative abnormal returns to form the abnormal returns on individual securities are uncorrelated in the cross an estimator of the variance. Boehmer, Musumeci, and Poulsen (1991) section. This will generally be a reasonable assumption if the event windO\\'s discuss this methodology, \vhich is best applied using the constant-mean-of the included securities do not overlap in calendar time. The assumption return model to measure the abnormal return.

allows us to calculate the variance of the aggregated sample cumulative The cross-sectional approach to estimating the variance can be applied abnormal returns without concern about covariances between individual to both the average cumulative abnormal return (CAR(r\, r2>> and the av-sample CARs, since ~hey are zero. However, when the event windows do erage standardized cumulative abnormal return (SCAR(TJ, T2)) . Using the

'~'"

~

. 168 4. Event-Stu.dy Anal),si5 4.6. Analysis of Power 169 cross section to form estimators of the variances we have Given an alternative hypothesis HA and the CDF of II for this hypothesis, we can tabulate the power of a test of size 0; using Vai'[CAR(rl, r2)] =

1 N V 2 :L(CAR;(rl, r,) - CAR(rl. r,))2 (4.5.1 )

Pia, H A) Pr(jl < ",-I mI H A) 1 i=l

+ Pr (jl > ",-I (1 -~) I HA)' (4.6.1) 1 N With this framework in place, we need to posit specific alternative hy-Vai'[SCAR(rl, r,)] - , :L(SCAR;(rl, r,) - SCAR(rl, r2)t (4.5.2)

N i=l potheses. Alternatives are constructed to be consistent 'with event studies using data sampled at a daily intervaL We build eight alternative hypotheses using four levels of abnormal returns, 0.5%, 1.0%, 1.5%. and 2.0%, and two For these estimators of the variances to be consistent we require the levels for the average variance of the cumulative abnormal return of a given abnormal returns to be uncorrelated in the cross section. An absence of security over the sampling interval, 0.0004 and 0.0016. These variances cor*

clustering is sufficient for this requirement. Note that cross-sectional ho- respond to standard deviations of2% and 4%, respectively. The sample size, moskedasticity is not required for consistency. Given these variance estima- that is the number of securities for which the event occurs, is varied from tors, the null hypothesis that the cumulative abnormal returns are zero can 1 to 200. We document the power for a test with a size of 5% (et = 0.05) then be tested using large sample theory given the consistent estimators of giving values of -1.96 and 1.96 for ",-I (a/2) and ",-I (I-a/2), respectively.

the variances in (4.5.2) and (4.5.1). In applications, of course, the pmver of the test should be considered when One may also be interested in the impact of an event on the risk of a selecting the size.

firm. The relevant measure of risk must be defined before this issue can I The power results are presented in Table 4.2 and are plotted in Figures 4.3a and 4.3b. The results in the left panel of Table 4.2 and in Figure 4.3a II be addressed. One choice as a risk measure is the market-model beta as implied by the Capital Asset Pricing Model. Given this choice, the market are for the case where the average variance is 0.0004, corresponding to a model can be formulated to allow the beta to change over the event windO\\* standard deviation of 2%. This is an appropriate value for an event which and the stability of the beta can be examined. See Kane and Unal (1988) does not lead to increased variance and can be examined using a one-day for an application of this idea. event ,'lindow. Such a case is likely to give the event-study methodology its i highest power. The results illustrate that when the abnormal return is only 0.5% the power can be low. For example, 'oI1ith a sample size of20 the power of a 5% test is only 0.20. One needs a sample of over 60 firms before the 4.6 Analysis of Power power reaches 0.50. However, for a given sample size, increases in power are substantial when the abnormal return is larger. For example, when the To interpret an event study, we need to know what is our ability to detect abnormal return is 2.0% the power of a 5% test with 20 firms is almost LOO the presence of a nonzero abnormal return. In this section we ask what is "ith a value of 0.99. The general results for a variance of 0.0004 is that the likelihood that an event-study test rejects the null hypothesis for a given when the abnormal return is larger than I % the power is quite high even level of abnormal return associated ,'lith an event, that is, we evaluate the for small sample sizes. 'When the abnormal return is small a larger sample pO\'I'er of the test. size is necessary to achieve high power.

"We consider a two~sided test of the null hypothesis using the cumulative- In the right panel of Table 4.2 and in Figure 4.3b the power results I

abnormal*return*based statistic II from (4.4.22). We assume that the abnor' are presented for the case "where the average variance of the cumulative mal returns are uncorrelated across securities; thus the variance of CAR is abnormal return is 0.0016, corresponding to a standard deviation of 4%.

a'(rl, r,), where a'(rl. r2) = 1/ N 2 I:: 1 o-;(rl, r,) and N is the sample size. This case corresponds roughly to either a multi-day event window or to a Under the null hypothesis the distribution of II is standard normal. For a one-day event window with the event leading to increased variance 'which two*sided test of size a we reject the null hypothesis if II < ",-I(a/2) or if ~ is accommodated as part of the null hypothesis. Here we see a dramatic II > ",-I (I-a/2) where ",(.) is the standard normal cumulative distribution i decline in the power of a 5% test. When the CAR is 0.5% the power is only function (CDF). I 0.09 with 20 firms and only 0.42 with a sample of200 firms. This magnitude I

I

170 4. Event-Study Analysis 4.6. Analysis oj Power 171 a

Table 4.2. Power of event-stud)' test statistic JI to reject the null hypothesis that the abnormal I(X:X ' ' ' "

I/ /../ ---,--,"'-'-'-'

return is zero. 00 / , /" Abno! ,n,' Rwnn' 0%

o Sample Size 03%

Abnonnal Return 1~% 15% 2.0% 0.5%

Abnormal Return 1.0% 1.5% 2.0%

oal

... 10 5

f I ll ,'- ' ,,' Abnor maIR(\urnl-')f 00 c.. If,, " A.bnormal R

(! = 2% (f = 4% ":t' " e\urn20%

1 0.06 0.08 0.12 0.]7 0.05 0.06 0.Q7 0.08 o I, '

2 0.06 0.11 0.19 0.29 0.05 0.06 0.08 0.11 I, 3 0.07 0.14 0.25 0.41 0.06 0.07 0.10 0.14 *1, Abnormal Return 0.5%

4 0.08 0.17 0.32 0.52 0.06 0.08 0.12 '" 11, :

5 0.09 0.20 0.39 0.61 0.06 0.09 0.13 0.17 o [I,:'

0.20 6 0.09 0.23 0.45 0.69 0.06 0.09 0.15 0.23 7 0.10 0.26 0.51 0.75 0.06 0.10 0.17 0.26 8 0.11 0.29 0.56 0.81 0.06 0.11 0.19 0.29 o 10 20 30 40 50 60 70 80 90 100 9 0.12 0.32 0.61 0.85 0.07 0.12 0.20 0.32 Sample Size 10 0.12 0.35 0.66 0.89 0.Q7 0.12 0.22 0.35 11 0.13 0.38 0.70 0.91 0.07 0.13 0.24 0.38 (a) 12 0.14 0.41 0.74 0.93 0.07 0.14 0.25 0.41 13 0.15 0.44 0.77 0.95 0.07 0.15 0.27 0.44 ~ "~~ __ 1'=* ~ __

14 15 16 0.15 0.]6 0.17 0.46 0.49 0.52 0.80 0.83 0.85 0.96 0.97 0.98 0.08 0.08 0.08 0.15 0.16 0.17 0.29 0.31 0.32 0.46 0.49 0.52 00 6 ---

~

/

/ / .,., .,., Abnorm,ll R<;twn 2 0%

17 0.18 0.54 0.87 0.98 0.08 0.18 0.34 0.54 / ,-

18 0.]9 0.56 0.89 0.99 0.08 0.19 0.36 0.56 <0 / / /

.,., "Abnormal Return 1.5%, _----

19 0.19 0.59 0.90 0.99 0.08 0.19 0.37 0.59 ~ c:i

/

/

20 0.20 0.61 0.92 0.99 0.09 0.20 0.39 0.61 £ I /

25 0.24 0.71 0.96 1.00 0.10 0.24 0.47 0.71 I /

o I 30 0.28 0.78 0.98 1.00 0.11 0.28 0.54 0.78 I /~ Abnormal Rctllrn 1.0%

35 0.32 0.84 0.99 1.00 0.11 0.32 0.60 0.84 I / _- /

40 0.35 0.89 1.00 1.00 0.12 0.35 0.66 0.89 / /

45 0.39 o / /

0.92 1.00 1.00 0.13 0.39 0.71 0.92 U 50 0.42 0.94 1.00 1.00 0.14 0.42 0.76 0.94 k/ Abnormal Return 0.5%

60 0.49 0.97 1.00 1.00 0.16 0.49 0.83 0.97 70 0.55 0.99 1.00 1.00 0.18 0.55 0.88 0.99 80 0.61 0.99 1.00 1.00 0.20 o 10 W W @ ~ ~ N W 00 100 0.61 0.92 0.99 90 0.66 1.00 1.00 1.00 0.22 0.66 0.94 Sample Size 100 120 0.71 0.78 1.00 1.00 1.00 1.00 1.00 1.00 0.24 0.28 0.71 0.78 0.96 0.98 1.00 1.00 1.00 I (b) 140 160 180 200 0.84 0.89 0.92 0.94 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.32 0.35 0.39 0.42 0.84 0.89 0.92 0.94 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 I

I Figure 4.3. Power of Event-Study Test Statistic]l to Reject the Null Hypothesis that the Abnormal Return Is Zero, lVhen the Square Root of the Average Variance of the Abnormal RetumAcrossFirms is (a) 2% and (b) 4%

The power is reponed for a test ,~itb a size of 5%. The sample size is the number of event observations included in the study, and (f is the square root of the average variance of the abnormal return across finns. there is a sample size of 30, the power is 0.54. Generally if the abnormal return is large one will have little difficulty rejecting the null hypothesis of no abnormal return.

of abnormal return is difficult to detect with the larger variance of 0.0016. We have calculated power analytically using distributional assumptions.

In contrast, when the CAR is as large as 1.5% or 2.0% the 5% test still has If these distributional assumptions are inappropriate then our power calcu-reasonable power.. For example, when the abnormal return is 1.5% and lations may be inaccurate. However, Brown and Warner (1985) explore this

i 172 4. Event-Stud) Analysis *' 4.8. Cross-Sectional Models 173 issue and find that the analytical computations and the empirical power are event day zero. The framework can be easily altered for events occurring very close. over multiple days.

It is difficult to reach general conclusions concerning the the ability Drawing on notation previously introduced, consider a sample of iJ.J.

of event-study methOdology to detect nonzero abnormal returns. 'Vhen abnormal returns for each of N securities. To implement the rank test it conducting an event study it is necessary to evaluate the pmver given the is necessary for each security to rank the abnormal returns from 1 to l&..

parameters and objectives of the study. If the power seems sufficient then Define Kir as the rank of the abnormal return of security i for event time one can proceed, otherwise one should search for "\vays of increasing the period T. Recall that r ranges from T} + 1 to T2 and T = 0 is the event day.

power. This can be done by increasing the sample size, shortening the event The rank test uses the fact that the expected rank under the null hypothesis windo\v, or by developing more specific predictions of the null hypothesis. is L-lt . The test statistic for the null hypothesIs of no abnormal return on event day zero is:

4.7 Nonparametric Tests 1< =

1,,(

N N

L,,+l)

(::;j K,o - -2 - / s(L,,) (4.7.1)

-l&.1 LT, (1NLN(

The methods discussed to this point are parametric in nature, in that specific assumptions have been made about the distribution of abnormal returns, s(L,,) = K"-- L9+1))' (4.7.2)

Alternative non parametric approaches are available which are free of spe- r=Tj+l i=l cific assumptions concerning the distribution of returns. In this section we discuss t\vo common non parametric tests for event studies, the sign test and Tests of the null hypothesis can be implemented using the result that the the rank test. asymptotic null distribution of J4 is standard normal. Corrado (1989) gives The sign test, which is based on the sign of the abnormal return, re- further details.

quires that the abnormal returns (or more generally cumulative abnormal Typically, these non parametric tests are not used in isolation but in returns) are independent across securities and that the expected propor- conjunction vn.th their parametric counterparts. The nonparametric tests tion ofpositive abnormal returns under the null hypothesis is 0.5. The basis enable one to check the robustness of conclusions based on parametric of the test is that under the null hypothesis it is equally probable that the tests. Such a check can be worthwhile as illustrated by the work of Campbell CAR will be positive or negative. If, for example, the alternative hypothe- and Wasley (1993). They find that for daily returns on NASDAQ stocks sis is that there is a positive abnormal return associated with a given event, the non parametric rank test provides more reliable inferences than do the the null hypothesis is Ho: P ::; 0.5 and the alternative is H A : P > 0.5 where standard parametric tests.

p = Pr(CAR, 2:: 0.0). To calculate the test statistic we need the number of cases \vhere the abnormal return is positive, N+, and the total number of 4.8 Cross-Sectional Models cases, N. Letting]?, be the test statistic, then asymptotically as N increases

\ve have Theoretical models often suggest that there should be an association be-N+ ] N'/2 tween the magnitude of abnormal returns and characteristics specific to J, = [- - 0.5 - ~ N(O, 1) .

NO.5 the event observation. To investigate this association, an appropriate tool is a cross-sectional regression of abnormal returns on the characteristics of interest. To set up the model, define y as an (N x 1) vector of cumulative For a test of size (1 - a), H" is rejected if J, > <1>-' (a). abnormal return observations and X as an (N x K) matrix of characteris-A weakness of the sign test is that it may not be well specified if the tics. The first column of X is a vector of ones and each of the remaining distribution of abnormal returns is skewed, as can be the case \vith daily (K - 1) columns is a vector consisting of the characteristic for each event data. \-Vith skewed abnormal returns, the expected proportion of positive observation. Then, for the model, we have the regression equation abnormal returns can differ from one half even under the null hypothesis.

y = X(J+TJ, (4.8.1)

In response to this possible shortcoming, Corrado (1989) proposes a nOI1-parametric rank test for abnormal performance in event studies. VVe briefly where (J is the (Kxl) coefficient vector and TJ is the (Nxl) disturbance describe his test of the null hypothesis that there is no abnormal return on e vector. Assuming E[X'llJ = 0, we can consistently estimate using OLS.

1'/4 4. Event-Study Analysis ~

~

4.9. Further Issues 175 For the-'VLS estimator we have investors rationally use firm characteristics to forecast the likelihood of the e= (XX)-I Ky. (4.8.2) event occurring. In these cases, a linear relation between the firm charac-teristics and the valuation effect of the event can be hidden. Malatesta and Assuming the elements of 1] are cross-sectionally uncorrelated and homo- Thompson (1985) and Lanen and Thompson (1988) provide examples of skedastic, inferences can be derived using the usual OLS standard errors. this situation.

Defining ah as the variance of the elements of 1] we have Technically, the relation between the firm characteristics and the degree of anticipation of the event introduces a selection bias. The assumption Var[e]" I 1

= (XX)- ai,.

<) that the regression residual is uncorrelated with the regressors, E[X'7JJ = 0, (4.8.3) breaks down and the OLS estimators are inconsistent. Consistent estimators Using the unbiased estimator for ary, can be derived by explicitly allowing for the selection bias. Acharya (1988, 1993) and Eckbo, Maksimovic, and WiIliams (1990) provide examples of

-2

__ 1 AI" this. Prabhala (1995) provides a good discussion of this problem and the a1'/ (N _ K) 1'/1'/, (4.8.4) possible solutions. He argues that, despite misspecification, under weak conditions, the OLS approach can be used for inferences and the l-statistics r,

where = y - xe, we can construct t-statistics to assess the statistical signifi- can be interpreted as Imver bounds on the true Significance level of the cance of the elements of8. Alternatively, vvithout assuming homoskedastic- estimates.

ity, we can construct heteroskedasticity-consistent z-statistics using 4.9 Further Issues Var[e] = ~ (XX)-l N

[tx;X;i};]

i=l (X'X)-l, (4.8.5)

A number of further issues often arise when conducting an event study. We discuss some of these in this section.

\vhere x; is the ith row of X and Tti is the ith element off]. This expression for the standard errors can be derived using the Generalized Method ofMo-ments framev,;ork in Section A.2 of the Appendix and also follows from the 4.9.1 Role oj the Sampling Interval results of White (1980). The use of heteroskedasticity-consistent standard If the timing of an event is kno'WIl precisely, then the ability to statistically errors is advised since there is no reason to expect the residuals of (4.8.1) identify the effect of the event v.;-ill be higher for a shorter sampling interval.

to be homoskedastic. The increase results from reducing the variance of the abnormal return Asquith and Mullins (1986) provide an example of this approach. The without changing the mean. We evaluate the empirical importance of this wo-day cumulative abnormal return for the announcement of an equity issue by comparing the analytical formula for the power of the test statistic offering is regressed on the size of the offering as a percentage of the value 11 with a daily sampling interval to the power vvith a weekly and a monthly of the total equity of the firm and on the cumulative abnormal return in interval. We assume that a week consists of five days and a month is 22 days.

the eleven months prior to the announcement month. They find that the The variance of the abnormal return for an individual event observation is magnitude of the (negative) abnormal return associated with the announce- assumed to be (4%f~ on a daily basis and linear in time.

ment of equity offerings is related to both these variables. Larger pre-event In Figure 4.4, we plot the power of the test of no event-effect against cumulative abnormal returns are associated ,vith less negative abnormal the alternative of an abnormal return of I % for 1 to 200 securities. As returns, and larger offerings are associated with more negative abnormal one would expect given the analysis of Section 4.6, the decrease in power returns. These findings are consistent with theoretical predictions which going from a daily interval to a monthly interval is severe. For example, they discuss. ,vith 50 securities the power for a 5% test using daily data is 0.94, v*!hereas One must be careful in interpreting the results of the cross-sectional re- the power using weekly and monthly data is only 0.35 and 0.12, respectively.

gression approach. In many situations, the event-v*,rindmv* abnormal return The clear message is that there is a substantial payoff in terms of increased

",ill be related to firm characteristics not only through the valuation cffcCL~ power from reducing the length of the event window. Morse (1984) presents of the event but also through a relation between the firm characteristics detailed analysiS of the choice of daily versus monthly data and draws the and the extent to which the event is anticipated. This can happen \vhtn same conclusion.

176 4. 9. Further Issues

4. Event-St.udJ Analysis 177

~~~~-r~~~=~~.-~~-.~.-~~-, Ball and Torous (1988) investigate this issue. They develop a maximum-00 likelihood estimation procedure which accommodates event-date uncer-

'" One-Day Intcn..-<ll tainty and examine results of their explicit procedure versus the informal procedure of expanding the event window. The results indicate that the t

co 6

t informal procedure works well and there is little to gain from the more

~ i elaborate estimation framework.

,I o

" One-Week !ntcrva!

~

4.9.3 Possible Biases

---, " - - Ont:-Month InterY;']

Event studies are subject to a number of possible biases. Nonsynchronous trading can introduce a bias. The nontrading or nonsynchronous trading I

effect arises when prices are taken to be recorded at time intervals of one o 20 40 60 80 100 120 140 160 180 200 length when in fact they are recorded at time intervals of other possibly Sample Size irregular lengths. For example, the daily prices of securities usually em-ployed in event studies are generally "closing" prices, prices at which the I

Figure 4.4. Power of Er.Jenl-Stud), Test Statistic JI to Reject the f..lull Hypothesis that the last transaction in each of those securities occurred during the trading day.

Abnormal Return is Zero, for Different Sampling Intervals, lVhen the Square Root of the These closing prices generally do not occur at the same time each day, but by Average Variance of the Abnormal Return Across Firms Is 4 % for the Daily Interval calling them "daily" prices, we have implicitly and incorrectly assumed that they are equally spaced at 24-hour intervals. As we showed in Section 3.1

\

of Chapter 3, this nontrading effect induces biases in the moments and A sampling interval of one day is not the shortest interval possible.

With the increased availability of transaction data, recent studies have used observation intervals of duration shorter than one day. The use of intra-daily data involves some complications, however, of the sort discussed in Chapter 3, and so the net benefit of very short intervals is unclear. Barclay Ii I

co-moments of returns.

The influence of the nontrading effect on the variances and covariances of individual stocks and portfolios naturally feeds into a bias for the market-model beta. Scholes and Williams (1977) present a consistent estimator of beta in the presence of nontrading based on the assumption that the true return process is uncorrelated through time. They also present some em-and Litzenberger (1988) discuss the use of intra-daily data in event studies. pirical evidence shOwing the nontrading-adjusted beta estimates of thinly traded securities to be approximately 10 to 20% larger than the unadjusted estimates. However, for actively traded securities, the adjustments are gen-4.9.2 Inferences with Event-Date Uncertainty erally small and unimportant.

Jain (1986) considers the influence of thin trading on the distribution Thus far we have assumed that the event date can be identified with certainty. of the abnormal returns from the market model ,vith the beta estimated Hmvcver, in some studies it may be difficult to identify the exact date. A using the Scholes-Williams approach. He compares the distribution of these common example is when collecting event dates from financial publications abnormal returns to the distribution of the abnormal returns using the usual such as the Wall Street Journal. When the event announcement appears in OLS betas and finds that the differences are minimal. This suggests that in the newspaper one can not be certain if the market l\raS informed before general the adjustment for thin trading is not important.

the close of the market the prior trading day. If this is the case then the The statistical analysis of Sections 4.3, 4.4, and 4.5 is based on the as-prior day is the event day; if not, then the current day is the event day. The sumption that returns are jointly normal and temporally lID. Departures usual method of handling this problem is to expand the event window to from this assumption can lead to biases. The normality assumption is im-two days-day 0 and day + I. While there is a cost to expanding the event portant for the exact finite-sample results. Without assuming normality, all

,vindow, the results in Section 4.6 indicate that the pmver properties of two- results would be asymptotic. However, this is generally not a problem for day event windows are still good, suggesting that it is worth bearing the cost event studies since the test statistics converge to their asymptotic distribu-i to avoid the risk of~issing the event. tions rather quickly. Brown and Warner (1985) discuss this issue.

~

J 4.10. Conclusion 179 178 4. Event~Study Analysis There can also be an upward bias in cumulative abnormal returns when return for target shareholders exceeds 20% for a sample of 663 successful these are calculated in the usual way. The bias arises from the observation- takeovers from 1960 to 1985. In contrast the abnormal return for acquirers by-observation rebalancing to equal weights implicit in the calculation of is close to zero at 1.14%, and even negative at -1.10% in the 1980's.

the aggregate cumulative abnormal return combined 'with the use of trans- Eckbo (1983) explicitly addresses the role of increased market power action prices which can represent both the bid and the ask side of the in explaining merger-related abnormal returns. He separates mergers of market. Blume and Stambaugh (1983) analyze this bias and show that it competing firms from other mergers and finds no evidence that the wealth can be important for studies using low-market-capitalization firms which effects for competing firms are different. Further, he finds no evidence that have, in percentage terms, "'ide bid-ask spreads. In these cases the bias can rivals of firms merging horizontally experience negative abnormal returns.

be eliminated by considering cumulative abnormal returns that represent From this he concludes that reduced competition in the product market buy~and~hold strategies.

is not an important explanation for merger gains. This leaves competition for corporate control a more likely explanation. Much additional empirical work in the area of mergers and acquisitions has been conducted. Jensen and Ruback (1983) andJarrell, Brickley, and Netter (1988) provide detailed 4.10 Conclusion surveys of this work.

A number of robust results have been developed from event studies In closing, we briefly discuss examples of event-study successes and limita- of financing decisions by corporations. 'VVhen a corporation announces tions. Perhaps the most successful applications have been in the area of that it \vill raise capital in external markets there is on average a negative corporate finance. Event studies dominate the empirical research in this abnormal return. The magnitude of the abnormal return depends on the area. Important examples include the wealth effects of mergers and acqui- source of external financing. Asquith and Mullins (1986) study a sample of sitions and the price effects of financing decisions by firms. Studies of these 266 firms announcing an equity issue in the period 1963 to 1981 and find events typically focus on the abnormal return around the date of the first that the two-day average abnormal return is -2.7%, while on a sample of announcement. 80 firms for the period 1972 to 1982 Mikkelson and Partch (1986) find that In the 1960s there was a paucity of empirical evidence on the wealth the two-day average abnormal return is -3.56%. In contrast, when firms effects of mergers and acquisitions. For example, Manne (1965) discusses decide to use straight debt financing, the average abnormal return is closer the various arguments for and against mergers. At that time the debate cen- to zero. Mikkelson and Partch (1986) find the average abnormal return tered on the extent to which mergers should be regulated in order to foster for debt issues to be -0.23% for a sample of 171 issues. Findings such as competition in the product markets. Manne argues that mergers represent these provide the fuel for the development of new theories. For example, a natural outcome in an efficiently operating market for corporate control these external financing results motivate the pecking order theory of capital and consequently provide protection for shareholders. He down plays the structure developed by Myers and Majluf (1984).

importance of the argument that mergers reduce competition. At the con- A major success related to those in the corporate finance area is the clusion of his article Manne suggests that the two competing hypotheses implicit acceptance of event~study methodology by the U.S. Supreme Court for mergers could be separated by studying the price effects of the involved for determining materiality in insider trading cases and for determining corporations. He hypothesizes that if mergers created market power one appropriate disgorgement amounts in cases of fraud. This implicit accep-would observe price increases for both the target and acquirer. In contrast tance in the 1988 Basic, Incorporated v. Levinson case and its importance if the merger represented the acquiring corporation paying for control of for securities law is discussed in Mitchell and Netter (1994).

the target, one '1Nould observe a price increase for the target only and not There have also been less successful applications of event-study method-for the acquirer. Hmvever, at that time Manne concludes in reference to ology. An important characteristic of a successful event study is the ability the price effects of mergers that " ... no data are presently available on this to identify precisely the date of the event. In cases where the date is difficult subject." to identify or the event is partially anticipated, event studies have been less Since that time an enormous body of empirical evidence on mergers and useful. For example, the wealth effects of regulatory changes for affected en-acquisitions has developed which is dominated by the use of event studies. tities can be difficult to detect using event~study methodology. The problem The general result is that, given a successful takeover, the abnormal returns is that regulatory changes are often debated in the political arena over time of the targets are large and positive and the abnormal returns of the acquirer and any accompanying wealth effects will be incorporated gradually into are close to zero. Jarrell and Poulsen (1989) find that the average abnormal

l, 180 4. l-vent-Study Analysis the value of a corporation as the probability of the change being adopted increases.

Dann and James (1982) discuss this issue in their study of the impact 5

of deposit interest rate ceilings on thrift institutions. They look at changes The Capital Asset Pricing Model in rate ceilings, but decide not to consider a change in 1973 because it was due to legislative action and hence was likely to have been an ticipated by the market. Schipper and Thompson (1983, 1985) also encounterthis problem in a study of merger-related regulations. They attempt to circumvent the problem of anticipated regulatory changes by identifYing dates when the probability of a regulatory change increases or decreases. How"ever, they find largely insignificant results, leaving open the possibility that the absence of distinct event dates accounts for the lack of\vealth effects.

Much has been learned from the body of research that uses event-study methodology. Most generally, event studies have shown that, as \'\'e \'>'Ould O~E OF THE IMPORTA~T PROBLEMS of modern financial economics is the expect in a rational marketplace, prices do respond to ne\\' information. "Ve quantification of the tradeoff bet\';een risk and expected return. Although expect that event studies V\iil1 continue to be a valuable and \-videly used tool common sense suggests that risky investments such as the stock market V\iill in economics and finance. generally yield higher returns than investments free of risk, it was only \..ith the development of the Capital Asset Pricing Model (CAPM) that economists

\\*ere able to quantify risk and the reward for bearing it. The CAPM implies Problems-Chapter 4 that the expected return of an asset must be linearly related to the covariance of its return with the return of the market portfolio. In this chapter we 4.1 Show that when using the market model to measure abnormal returns, discuss the econometric analysis of this model.

the sample abnormal returns from equation (4.4.7) are asymptotically inde- The chapter is organized as follows. In Section 5.1 we briefly review pendent as the length of the estimation windm*.,.- (L 1 ) increases to infinity. the CAPM. Section 5.2 presents some results from efficient-set mathemat-ics, including those that are important for understanding the intuition of 4.2 You are given the folloV\iing information for an event. Abnormal re-econometric tests of the CAPM. The methodology for estimation and testing turns are sampled at an interval of one day. The event-window length is is presented in Section 5.3. Some tests are based on large-sample statistical three days. The mean abnormal return over the event \\'indmv is 0.3% per theory making the size of the test an issue, as we discuss in Section 5.4. Sec-day. You have a sample of 50 event observations. The abnormal returns are tion 5.5 considers the power of the tests, and Section 5.6 considers testing independent across the event observations as well as across event days for a

\\;th weaker distributional assumptions. Implementation issues are covered given event observation. For 25 of the event observations the daily standard in Section 5.7, and Section 5.8 considers alternative approaches to testing deviation of the abnormal return is 3% and for the remaining 25 observa-based on cross-sectional regressions.

tions the daily standard deviation is 6%. Given this information, what would be the power of the test for an event study using the cumulative abnormal return test statistic in equation (4.4.22)? 'What would be the power using the 5.1 Review of the CAPM standardized cumulative abnormal return test statistic in equation (4.4.24)?

For the power calculations, assume the standard deviation of the abnormal

Vlarkowitz (1959) laid the groundwork for the CAPM. In this seminal re-returns is known.

search, he cast the investor's portfolio selection problem in terms of ex-4.3 vVhatwould be the answers to question 4.2 if the mean abnormal return pected return and variance of return. He argued that investors would opti-is 0.6% per day for the 25 firms with the larger standard deviation' mally hold a mean-variance efficient portfolio, that is, a portfolio with the highest expected return for a given level of variance. Sharpe (1964) and Lintner (l965b) built on Markowitz's work to develop economy-wide im-plications. They showed that if investors have homogeneous expectations 181