When doing an event study one will have to choose between using J1 or J2 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 J2 does. On the other hand if the true abnormal return is larger for securities with higher variance, then the better choice will give equal weight to the realized cumulative abnormal return of each security, which is what J does. In most studies, the results are not likely to be sensitive to the choice of J1 versus J2 because the variance of the CAR is of a similar magnitude across securities.