MacKinlay:Event Studies in Economics and Finance 21 dow market returns,the abnormal re- overall inferences for the event of inter- turns will be jointly normally distributed est.The aggregation is along two dimen- with a zero conditional mean and condi- sions-through time and across securi- ional variance o2(ARnt)where ties.We will first consider aggregation through time for an individual security o2R)=呢+ 1 (Bt-m) 6品 (8) and then will consider aggregation both across securities and through time.The From (8),the conditional variance has concept of a cumulative abnormal return two components.One component is the is necessary to accommodate a multiple disturbance variance o2 from(3)and a period event window.Define CAR(t,t2) second component is additional variance as the sample cumulative abnormal re- due to the sampling error in ai and Bi. turn (CAR)from t to t2 where This sampling error,which is common T1<t1≤t2≤T2.The CAR from t1tot2is for all the event window observations, the sum of the included abnormal re- also leads to serial correlation of the turns, abnormal returns despite the fact that the true disturbances are independent CAR,(1t)=∑AR (10) through time.As the length of the esti- T=T mation window LI becomes large,the second term approaches zero as the sam- pling error of the parameters vanishes. Asymptotically (as Li increases)the vari- ance of CAR,is The variance of the abnormal return will be o and the abnormal return observa- (t1,t2)=(亿2-t1+1)2 (11) tions will become independent through time.In practice,the estimation window This large sample estimator of the vari- ance can be used for reasonable values of can usually be chosen to be large enough to make it reasonable to assume that the L1.However,for small values of LI the contribution of the second component to variance of the cumulative abnormal re- the variance of the abnormal return is turn should be adjusted for the effects of the estimation error in the normal model zero. Under the null hypothesis,Ho,that parameters.This adjustment involves the second term of (8)and a further related the event has no impact on the be- adjustment for the serial covariance of havior of returns (mean or variance) the distributional properties of the the abnormal return. abnormal returns can be used to draw The distribution of the cumulative ab- inferences over any period within the normal return under Ho is event window.Under Ho the distribu- CAR(t1,t2)~N(0,t1,t2) (12) tion of the sample abnormal return of a given observation in the event window is Given the null distributions of the abnor- mal return and the cumulative abnormal ARit-N(0,2(AR)). (9) return,tests of the null hypothesis can be conducted. Next(9)is built upon to consider the ag- However,tests with one event obser- gregation of the abnormal returns. vation are not likely to be useful so it is C.Aggregation of Abnormal Returns necessary to aggregate.The abnormal re- turn observations must be aggregated for The abnormal return observations the event window and across observa- must be aggregated in order to draw tions of the event.For this aggregation