33.11 Aggregation

33.11.1 Over Time

To assess the impact of events on stock performance over time, we calculate the Cumulative Abnormal Return (CAR) for the event windows.

Hypotheses:

• $H_0$: The standardized cumulative abnormal return (SCAR) for stock $i$ is 0 (i.e., the event has no effect on stock performance).
• $H_1$: The SCAR is not 0 (i.e., the event does have an effect on stock performance).

33.11.2 Across Firms and Over Time

In addition to evaluating CAR for individual stocks, we may want to aggregate results across multiple firms to determine whether events systematically affect stock prices.

Additional Assumptions:

• Uncorrelated Abnormal Returns: The abnormal returns of different stocks are assumed to be uncorrelated. This is a strong assumption, but it holds reasonably well if event windows for different stocks do not overlap.
• Overlapping Event Windows: If event windows do overlap, follow the methodology proposed by Bernard (1987) and Schipper and Thompson (1983), Schipper and Smith (1983).

Hypotheses:

• $H_0$: The mean abnormal return across all firms is 0 (i.e., there is no systematic effect of the event).
• $H_1$: The mean abnormal return across all firms is different from 0 (i.e., the event has a systematic effect).

33.11.3 Statistical Tests

Two broad categories of statistical tests can be applied: parametric and non-parametric.

33.11.3.1 Parametric Tests

These tests assume that abnormal returns are normally distributed. Empirical evidence suggests that either of the following approaches can work well:

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

33.11.3.2 Non-Parametric Tests

Non-parametric tests provide robustness by avoiding assumptions about the distribution of abnormal returns.

1. Sign Test
   • Assumes abnormal returns and CAR are independent across stocks.
   • Under $H_0$, we expect 50% of stocks to have positive abnormal returns and 50% to have negative abnormal returns.
   • If there is a systematic relationship between the event and abnormal returns, we should observe a significant deviation from this 50-50 split.
   • If the alternative hypothesis suggests a negative relationship, the null hypothesis must be adjusted accordingly.
   • Important Note: In the presence of skewed distributions (common in daily stock return data), the size of the test may not be reliable. In such cases, a rank-based test is preferred.
2. Rank Test
   • More robust than the sign test when distributions are skewed.
   • Null Hypothesis ($H_0$): There is no abnormal return during the event window.
   • Alternative Hypothesis ($H_1$): There is an abnormal return associated with the event.

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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.

Schipper, Katherine, and Rex Thompson. 1983. “Evidence on the Capitalized Value of Merger Activity for Acquiring Firms.” Journal of Financial Economics 11 (1-4): 85–119.