4.3.1 Constant-Mean-Return Model

Let nh the ith element of fx, be the mean return for asset i. Then the conslant-mean-return model is

Ra = 14 + 4 (4.3.1)

Щи) = 0 VarU ] = er*.

where the ith element orR is the period-/ return on security i, £ is the disturbance term, and of is the (i, г) element of П.

Although the constanl-mcan-rcturn model is perhaps the simplest model, Brown and Warner (1980, 1985) find il often yields results similar to those of more sophisticated models. This lack of sensitivity to the model choice can be attributed to the faci dial the variance of the abnormal return is frequently not reduced much by choosing a more sophisticated model. When using daily data the model is typically applied to nominal returns. With monthly data the model can be applied to real returns or excess returns (the return in excess of the nominal riskfree return generally measured using the US Treasury bill) as well as nominal returns.

4.3.2 Market Model

The market model is a statistical model which relates the return of any given security to the return of the market portfolio. The models linear specification follows from the assumed joint normality of asset returns.1 For any security i wc have

rt = <*, +fl,!< + < (4.3.2)

E[e ] = 0 Varc = ol

where R and R arc the period-/ returns on security i and the market portfolio, respectively, and <5 is the zero mean disturbance term. a f) and <r(- are the parameters of the market model, hi applications a broad-based stock index is used for the market portfolio, with the S&P500 index, the CRSP value-weighted index, and the CRSP equal-weighted index being popular choices.

The market model represents a potential improvement over the con-stanl-mean-return model. By removing the portion of the return that is related to variation in the markets return, the variance of the abnormal return is reduced. This can lead lo increased ability to delect event effects. The benefit from using the market model will depend upon the R1 of the market-model regression. The higher the if, the greater is the variance reduction of the abnormal return, and the larger is the gain. See Section 4.4.4 for more discussion of this point.

4.3.3 Other Statistical Models

A number of other statistical models have been proposed for modeling the normal return. A general type of statisiical model is the factor model Factor models potentially provide the benefit of reducing ihe variance of the abnormal return by explaining more of die 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 mullifactor model one might include industry indexes in addition lo the market. Sharpe (1970) and Sharpe, Alexander, and Bailey (1995) discuss index models with factors based on industry classification. Another variant of a faclor model is a procedure which calculates the abnormal return by taking die difference between the actual return and a portfolio of firms of similar size, where size-is measured by market value of equity. In this approach typically ten size groups arc considered and die loading on the size portfolios is restricted

The specification actually requites lite asset weights in the market portfolio lo remain constant. However, changes over time in the market portfolio weights are small enough that they have little effect on empirical work.

any ;conomic arguments. In contrast, models in the second category rely on assumptions concerning investors behavior and are not based solely on statistical assumptions. It should, however, be noted that to use economic models in practice it is necessary to add statistical assumptions. Thus the potential advantage of economic models is not the absence of statistical assumptions, but the opportunity to calculate more precise measures of the normal return using economic restrictions.

ror the statistical models, it is conventional to assume that asset return i are jointly multivariate normal and independently and identically distributed through time. Formally, we have:

{Al, Let R, be an (N x 1) vector of asset returns for calendar time period t. R, is independently multivariate normally distributed with mean ц and covariance matrix fl for all t.

This distributional assumption is sufficient for the constant-mean-rcturn modpl and the market model to be correctly specified and permits the development of exact finite-sample distributional results for the estimators and statistics. Inferences using the normal return models are robust lo deviations from the assumption. Further, wc can explicitly accommodate deviations using a generalized method of moments framework.

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

In practice the gains from employing multifacior models for event studies 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 reduction in the van.икс of the abnormal return. The variance reduction will typically be greatest in cases where the sample (inns have a common characteristic, for example they are all members of one industry or they are all firms concentrated in one market capitalization group. In these cases the use of a multifactor model warrants consideration.

Sometimes limited data availability may dictate the use of a restricted model such as the marketmljusted-return model. For some events it is not fcasi-j hie to have a prc-cvenl estimation period for the normal model parameters, and a market-adjusted abnormal return is used. The market-adjusled-return model can be viewed as a restricted market model with or, constrained to be j 0 and (i, constrained to be 1. Since the model coefficients are prcspcctficd, j an estimation period is not required to obtain parameter estimates. This I model is often used to study the imdei pricing of initial public offerings/ j Л general recommendation is to use such restricted models only as a last resort, and to keep in mind that biases may arise if the restrictions are false.

4. 1,4 Economic Models

Economic models restrict the parameters of statistical models to provide more constrained normal return models. Two common economic models which provide restrictions are the Capital Asset Pricing Model (CAPM) and exact versions of the Arbitrage Pricing Theory (APT). The CAPM, due to Sharpe (1904) and I.intner (19051)), is an equilibrium theory where the expected return of a given asset is a linear function of its covariance with the return of the market portfolio. The APT, tine to Ross (1976), is an asset pricing theory where in the absence of asymptotic arbitrage the expected return of a given asset is determined by its rovariances with multiple factors. Chapters f> and (i provide extensive treatments of these two theories.

The Capital Asset Pricing Model was commonly used in event studies during the 1970s. During the last ten years, however, deviations from the CAPM have been discovered, and this casts doubt on the validity of the restrictions imposed by the CAPM on the market model. Since these restrictions can he relaxed at little cost by using the market model, the use of the CAPM in event studies has almost ceased.

Some studies have used inullilactor normal performance models motivated by the Arbitrage Pricing Theory. Ihe APT can be made lo lit the

\Scr Kiitrr ( I Will Iim .in ex.....>l<-.

Time Line:

/ estimation I window

-1--1-1-1-

Г V, 0 7i

Figure 4.1. Time Line for an Event Study

cross-section of mean returns, as shown by Fama and French (1996a) and others, so a properly chosen APT model does not impose false restrictions on mean returns. On the other hand the use of the APT complicates the implementation of an event study and has little practical advantage relative to the unrestricted market model. See, for example, Brown and Weinstein (1985). There seems to be no good reason to use an economic model rather than a statistical model in an event study.

4.4 Measuring and Analyzing Abnormal Returns

In this section we consider the problem of measuring and analyzing abnormal returns. We use the market model as the normal performance return model, but the analysis is virtually identical for the constant-mean-return model.

We first define some notation. We index returns in event time using r. Defining r = 0 as the event date, r = T\ + 1 to г = Ti represents the event window, and r = % + 1 to г = T\ constitutes the estimation window. Let L\ = Т\ - Тй and Li - 1\ - T\ be the length of the estimation window and the event window, respectively. If the event being considered is an announcement on a given date then T<i - T\ + 1 and hi - 1. If applicable, the post-event window will be from r = 7i + 1 to r = T$ and its length is L-s = Tii - 1\. The timing sequence is illustrated on the time line in Figure 4.1.

We interpret the abnormal return over the event window as a measure of the impact of the event on the value of the firm (or its equity). Thus, the methodology implicitly assumes that the event is exogenous with respect to the change in market value of the security. In other words, the revision in 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 is triggered by the change in the market value of a security, in which case

event window

(post-event window J

4. Event-Study Analysis

the event is endogenous. For these cases, the usual interpretation will he incorrect.

It is typical lor the estimation window and the event window not to overlap. Thisdcsign provides estimators for the parameters of the normal return nodel which arc not influenced by the event-related returns. Including the :vcnt window in the estimation of the normal model parameters could lead о the event returns having a large influence on the normal return measure. In this situation both the normal returns and the abnormal returns would reflect the impact of the event. This would be problematic since the jnctliodology is built around the assumption that the event impact is captured by the abnormal returns, ln Section 4.5 wc consider expanding the null hypothesis to accommodate changes in the risk of a firm around the event. In this case an estimation framework which uses the event window returns will be required.

4.4.1 Estimation of the Market Model

Recall that the market model Tor security i and observation r in event time is

R = , + P,Rm, +< H-4.1)

The estimation-window observations can be expressed as a regression system,

r, = хд+е (4.4.2)

where r, = [R,r +\ Rn\ Y is an (L\ x 1) vector of estimation-window returns, X, = [(. rm] is an x2) matrix with a vector of ones in the first column and the vector of market return observations r, = [It r +i Rmr, Y in the second column, and 6, = [cr./ijis the (2x 1) parameter vector. Xhas a subscript because the estimation window may have timing that is specific to firm i. Under general conditions ordinary least squares (OI.S) is a consistent estimation procedure for the market-model parameters. Further, given the assumptions of Section 4.3, OI.S is efficient. The OlS estimators of the market-model parameters using an estimation window of L\ observations arc

w\ next show how to use these OI.S estimators to measure the statistical

-/. /. Measuring and Analyzing Abnormal Returns

15!)

properties of abnormal returns. First we consider the abnormal return properties of a given security and then we aggregate across securities.

4.4.2 Statistical Rro/ierties of Abnormal Returns

Given the market-model parameter estimates, we can measure anil analyze the abnormal returns. Let £* be the (/ях1 sample vector of abnormal returns for linn i from the event window, 4) + I to 7j>. Then using the market model lo measure the normal return and the OI.S estimators from (4.4.15), we have for the abnormal return vector:

e* = r; -a,L-PX ,

= R-XOj. (4.4.7)

where R* = I /(V/, + 1 R,i,Y is an ( > x 1) vector of event-window returns, x* = [t Rm\ is an ( jx2) matrix with a vector of ones in the first column and the vector of market return observations R*, = [R rl + \ R r*Y in the second column, and 9, - [a,Pi\ is the (2x 1) parameter vector estimate. Conditional on the market return over the evenl window, ihe abnormal returns will be jointly normally distributed with a zero conditional mean and conditional covariance matrix v, as shown in (4.4.8) and (4.4.9), respectively.

f.[r; -х;б>, i x;i f.kr; -x;o,) -х; /, - о,) \ x;i

0. (4.4.8)

Kle*i; i x;i

E[[ei -XiO, - 0\le: - Хв, - в,)\ x;] f.[e* e* - e{0, - 0,)X - x;(0, - 0,)t,

+ x(d, - o,h<), - o,yx I КI

lal+Xt.X\X,)-,Xa;. (4.4.9)

I is the (Ьх/л) identity matrix.

From (4.4.8) we see that the abnormal return vector, with an expectation of zero, is unbiased. The covariance matrix ol the abnormal return vector from (4.4.9) has two parts. The first term in the sum is the variance due lo the Inline disturbances and the second term is the additional variance due lo the sampling error in 0,. This sampling error, which is common

f.i ; i x;i =