for all ilie elements oldie abnormal return vector, will lead to serial correlation of the abnormal returns despite the fact that the true disturbances are independent through lime. As the length of the estimation window I.\ becomes large, the second term will approach zero as the sampling error of the parameters vanishes, and the abnormal returns across lime periods will become independent asymptotically.

Under the null hypothesis, II , that the given event has no impact on the mean or variance of returns, we can use (4.4.8) and (4.4.9) and the joint normality of the abnormal returns lo draw inferences. Under Hu, for the vector of event-window sample abnormal returns we have

к* ~ JV((),V(). (4.4.10)

Equation (4.4.10) gives us the distribution for any single abnormal return observation. We next build on this result and consider the aggregation of abnormal returns.

4.1. .7 Aggregation of Abnormal Returns

The abnormal return observations must be aggregated in order to draw overall inferences for the event ol interest. The aggregation is along two dimensions-through time and across securities. We will first consider aggregation through time for an individual security and then will consider aggregation both across securities and through time.

We introduce the cumulative abnormal return to accommodate multiple sampling intervals within the event window. Define CAR,(r, r2) as the cumulative abnormal return for security i from T to r2 where 7 ) < T < r2 < / >. Let 7 be an (h>x 1) vector with ones in positions rj - 7*i to т2 - 7) and zeroes elsewhere. Then we have

(!AR,(r r2) = ie] (4.4.11).

Var(XR,(T. r,.) = o;(TbT2) = -fVri- (4.4.12)

Il follows from (1.4.10) thai under 11ц,

(iAR,(r,.r2) ~ Л/ (0.гт;(Г1, r2)). (4.4.13)

We can construe! a lest of 11 for securily / from (4.4.13) using the standardized cumulative abnormal return,

- (AR,(t, , т..)

SCAR,(r,. r.,) = . --, (4.4.14)

0,(T, r2)

where (7((Г, г ) is < ah ulaled with <7(- from (4.4.4) substituted forcr,-. Under the null hv])oihesis ihe distribution of S(.AR;(T, т2) is Student I with /.) - 2

degrees of freedom. From the properties of the Student t distribution,

the expectation of SCAR,(ti, t2) is 0 and the variance is (£rf). For a large estimation window (for example, /. > 30), the distribution of SCARj(ri, r2) will be well approximated by the standard normal.

The above result applies to a sample of one event and must be extended for the usual case where a sample of many event observations is aggregated. To aggregate across securities and through time, we assume that there is not any correlation across the abnormal returns of different securities. This will generally be the case if there is not any clustering, that is, there is not any overlap in the event windows of the included securities. The absence of any overlap and the maintained distributional assumptions imply that (the abnormal returns and the cumulative abnormal returns will be independent across securities. Inferences with clustering will be discussed later. I

The individual securities abnormal returns can be averaged using £* from (4.4.7). Given a sample of A7 events, defining £* as the sample average of the N abnormal return vectors, we have

Varfr] = V = -L£v (4.4.16)

We can aggregate the elements of this average abnormal returns vectjor through time using the same approach as we did for an individual securitys vector. Define CAR(ti , r2) as the cumulative average abnormal return from Ti to т2 where T\ < x\ < т2 < T% and 7 again represents an (Lxl) vector with ones in positions ri - Tj to r2 - T\ and zeroes elsewhere. For the cumulative average abnormal return we have

CAR((1,r2) = 7ё* (4.4.17)

Var[CAR(r r2)] = 04x1,4) = VV7. (4.4.18)

F.quivalently, to obtain CAR(T(, t2), we can aggregate using the sample cumulative abnormal return for each security i. For N events we have

CAR(r r2) = iCAR,(r r2) (4.4.19) 1=1

Var[CAR(T T2)] = 5*(г т2) = a,2(r т2). (4.4.20)

In (4.4.16), (4.4.18), and (4.4.20) wc use die assumption that the event windows of the N securities do not overlap to set the covariance terms to zero. Inferences about the cumulative abnormal returns can be drawn using

CAR(t!,T2) ~ (O.aTbtii)), (4.4.21)

since under the null hypothesis the expectation of the abnormal returns is zero. In practice, since al(x\, r2) is unknown, we can use д*(т\, r2) = fit 5Zi=t fVt. Tz) as a consistent estimator and proceed to test 11 using

CAR(r r2) н 1Л

/> = --~ МО. I). (4.4.22)

[а (гы,)]3

This distributional result is for large samples of events and is not exact because an estimator of the variance appears in the denominator.

A second method of aggregation is to give equal weighting to the individual SCAR.s. Defining SCAR(ri, r2) as the average over N securities from event time ri to r2, wc have

SCAR(r r2) = SCAR,(ti, r2). (4.4.23)

Assuming that the event windows of the N securities do not overlap in calendar time, under Н , SCAR(n, x2) will be normally distributed in large samples with a mean of zero and variance (77--). We can test the null hypothesis using

/Л/(Л:-4)У

h = ( j 2 j SCAR(r r2) ~ J\f(0, 1). (4.4.24)

When doing an event study one will have to choose between using Jx or /2 for the test statistic. One would like to choose the statistic with higher power, ~and this will depend on the alternative hypothesis. If the true abnormal return is constant across securities then the better choice will give more weight to the securities with the lower abnormal return variance, which is what }i does. On the other hand if the true abnormal return is larger for securities with higher variance, then the belter choice will give ecpial weight to the realized cumulative abnormal return of each security, which is what Jt does. In most studies, the results arc not likely to be sensitive to the choice of J\ versus Ji because the variance of the CAR is of a similar magnitude across securities.

4.4.4 Sensitivity to Normal lleturn Model

Wc have developed results using the market model as the normal return model. As previously noted, using the market model as opposed to the

constant-mean-rcturn model will lead to a reduction in the abnormal return variance. This point can be shown by comparing the abnormal return variances. For this illustration we take the normal return model parameters as given.

The variance of the abnormal return for the market model is al = Varlrt - a, - fhLA = Var[rt ] -tfVarl/< ]

= (1 -rt2)Var[rt ], (4.4.25)

where rt2 is the rt2 of the market-model regression for security t.

For the constant-mean-rcturn model, the variance of the abnormal return is the variance of the unconditional return, Vat (rt ], that is,

al = Var[rt -,] = Var[rt l. (4.4.26)

Combining (4.4.25) and (4.4.26) wc have

al = (1 -Rf)ol (4.4.27)

Since llf lies between zero and one, the variance of the abnormal return using the market model will be less than or equal lo the abnormal return variance using the constant-mean-rcturn model. This lower variance for the market model will carry over into all the aggregate abnormal return measures. As a result, using the market model can lead lo more precise inferences. The gains will be greatest lor л sample ot securities with high market-model rt2 statistics.

In principle further increases in rt2 could be achieved by using a multi-factor model. In practice, however, the gains in rt2 from adding additional factors arc usually small.

4.4.5 CARs for the Earnings-Announcement Example

The earnings-announcement example illustrates the use of sample abnormal returns and sample cumulative abnormal returns. Table 4.1 presents the abnormal returns averaged across the 30 firms as well as the averaged cumulative abnormal return for each of the three earnings news categories. Two normal return models are considered: the market model and, for comparison, the constant-mean-rcturn model. Plots of the cumulative abnormal returns are also included, with the CARs from the market model in Figure 4.2a and the CARs from the constant-mean-rcturn model in Figure 4.2b.

The results of this example are hugely consistent with the existing literature on the information content of earnings. The evidence strongly

Table 4.1. Abnormal returns for an iivnl study of the information content of earnings announcements.

Maikel Model CniiMaiil.Mc.iii-Rcluiii Model

l lu sani]iU-. i.nsisis ..I a l.Hal ul (UK) .juai (.i ly iumiHittcrilHills iiii llir lllli ly companies in lilt* Dow Jones Industrial Index lor ilie li\e-\eai period January 108.) to December I.H.CI. Two models ate considered, lor tUe notmat teuirns, die market model ttsinj. die (.RSI1 valuc-wcilucd index and ilie cnnsiant-itican-iciiirii moilel. Tlie annotnicements ;tre ralcori/cd inlo three loups, yiood news, no news, and had news. is the sample average abnormal return lor the specilied A,\\ in event time and (:AK i4 ihe sample average (innnlaiive abnot mat return lot dav -20 to the spei died dav. I.veul (iule is ineasuied in days relative to the announcement date.

4.4. Measuring anil Analyzing Abnormal Returns 0.03

0.02 0.01 0

-<!.() 1 -0.02

(;<KKl-Nrws Finns

NoNcwi Firm*

Bad-Nfwn Firms

-0.03

. t i i i i м 1

-10 0

Event Time

Figure 4.2a. Plot of Cumulative Market-Model Abnormal Return for Earning Announcements

0.03 0.02 0.01

3 о

-0.01 -0.02

-10 0 10

Event Time

Figure 4.2b. Plot of Cumulative Constant-Mean-Retum-Model Abnormal Return for Faming Announcements

supports (he hypothesis that earnings announcements do indeed convey information useful for the valuation of firms. Focusing on the announcement day (day zero) the sample average abnormal return for the good-news firni