## 28.6 Aggregation

### 28.6.1 Over Time

We calculate the cumulative abnormal (CAR) for the event windows

$$H_0$$: Standardized cumulative abnormal return for stock $$i$$ is 0 (no effect of events on stock performance)

$$H_1$$: SCAR is not 0 (there is an effect of events on stock performance)

### 28.6.2 Across Firms + Over Time

Additional assumptions: Abnormal returns of different socks are uncorrelated (rather strong), but it’s very valid if event windows for different stocks do not overlap. If the windows for different overlap, follow and Schipper and Smith (1983)

$$H_0$$: The mean of the abnormal returns across all firms is 0 (no effect)

$$H_1$$: The mean of the abnormal returns across all firms is different form 0 (there is an effect)

Parametric (empirically either one works fine) (assume abnormal returns is normally distributed) :

1. Aggregate the CAR of all stocks (Use this if the true abnormal variance is greater for stocks with higher variance)
2. Aggregate the SCAR of all stocks (Use this if the true abnormal return is constant across all stocks)

Non-parametric (no parametric assumptions):

1. Sign test:
• Assume both the abnormal returns and CAR to be independent across stocks

• Assume 50% with positive abnormal returns and 50% with negative abnormal return

• The null will be that there is a positive abnormal return correlated with the event (if you want the alternative to be there is a negative relationship)

• With skewed distribution (likely in daily stock data), the size test is not trustworthy. Hence, rank test might be better

2. Rank test
• Null: there is no abnormal return during the event window

### References

Bernard, Victor L. 1987. “Cross-Sectional Dependence and Problems in Inference in Market-Based Accounting Research.” Journal of Accounting Research, 1–48.
Schipper, Katherine, and Abbie Smith. 1983. “Effects of Recontracting on Shareholder Wealth: The Case of Voluntary Spin-Offs.” Journal of Financial Economics 12 (4): 437–67.