1. Econometric Details -- the market model Assume that asset returns are jointly multivariate normal and independently and identically distributed through time - under this assumption, the market model is correctly specified - a linear relationship between R it and R mt follows from joint normality where This model improves on the constant mean return model by controlling for R mt - the variance of abnormal returns is reduced as a result - if other variables explain some of the variation in R it, then they could be included on the RHS too -the other variables could be industry indexes or macro variables - including additional variables is most useful when firms in the sample are similar in some respect- like if they are all in the same industries - in most cases additional variables have little explanatory power so they don’t help describe returns or affect the results of the event study

2. Measuring and Analyzing Abnormal Returns -introduce the following notation L 1 is the length of the estimation windowL 1 = T 1 - T 0 L 2 is the length of the event window L 2 = T 2 - T 1  = 0 is the event date - use the data in the estimation window to run OLS on the market model - under general conditions the OLS estimator is consistent - under normality, it is efficient, too denote the OLS estimates of  i and  i by and  =0 T2T2 T1T1 T 0 +1 L1L1 L2L2

3. First consider just one security AR it is the abnormal for security i at time t - it is the error term of the market model calculated on an out of sample basis - it is basically the forecast error - the difference between the actual R it and the forecast UnderH 0 : the event has no effect, conditional on the event window market returns, the abnormal returns are jointly normally distributed with a 0 conditional mean and conditional variance given by:

4. where In practice, is used in place ofwhere

5. This is the standard formula for computing the variance of a forecast error, applied to the market model - as long as R mt in the event window is similar to R mt in the estimation window is small -further, as L 1 increases, In practice, the estimation window can usually be chosen to be large enough so that - if this is not the case then the abnormal returns in the even window will be serially correlated - because the sampling error is estimating alpha hat and beta hat affects every abnormal return -assume that(so L 1 is large enough)

6. Under H 0 : the event has no impact These abnormal returns must be aggregated in order to draw inferences - sum across periods in the event window - sum across firms -first consider summing across time - this is necessary if the event window has multiple periods -the cumulative abnormal return is defined as follows: sum the abnormal returns on security i from period t1 to period t2, where asymptotically, as

7. -Note that we are ignoring the covariance terms between AR it and AR it+1 and the variance due to sampling error in computing and - if L1 is small, then additional terms would be included to allow for these effects Under H 0, -next, aggregate across firms -assume that there is no clustering - that is, assume that the event window of the different securities do no overlap ex. firm X is acquired in Nov, 1984 firm Y is acquired in May, 1985 - different time periods

8. -the role that this assumption plays is that it ensures that the CAR’s are independent across securities - we could just assume independence -in some settings independence is not a reasonable assumption - then the analysis becomes more complicated to aggregate across firms, simply sum the individual CAR i terms where N is the # of firms Given the assumption that the CARs are independent across firms, - the covariance terms are 0

9. Under H 0 : the event has no effect A test statistic to test H 0 is asymptotic with N and L 1,, so use the stat normal tables

10. 1) The Sign Test - assume that the CAR i ’s are independent across securities - under H 0, is is equally likely that CAR i is positive or negative If the alternative is that the event has a positive effect, then where Let N + denote the # of firms for which the test stat is asymptotically Additional tests that can be used along with the above

11. 2) The Rank Test - suppose you have a sample of L 2 abnormal returns for each of N securities - for each security, rank the abnormal returns from 1 to L 2, lowest to highest Let K it denote the rank of the abnormal return of security i for the event time period   ranges from T 1 + 1 to T 2 and  =0 is the event day Under H 0 : the expected rank of the event day is - so the abnormal return on the event day is expected to be in the middle of the group the test stat is on the following slide: Additional Tests Con’t

12. 2) The Rank Test “test stat” is where under H 0, asymptotically Additional Tests Con’t

13. If multiple hypotheses exist for the source of abnormal returns, a cross- sectional regression might be able to distinguish between them Suppose you have N abnormal returns observations and M possible explanatory variables x 1, x 2, x 3, …x n you could run the regression where and perform hypothesis test on the  i ’s Cross Sectional Models