Definitions and Annotations
Let \(t_0\) be the starting point and \(t_1\) the end point of the estimation window. The event window is defined by \(t_2\) and \(t_3\) with \(t_2\leq t_3\) and \(t_2 \geq t_1 + 1\).
We denote by \(\hat{S}_i\) the sample standard deviation of the abnormal returns calculated on the estimation period, namely
\[ \hat{S}_i = \frac{1}{m - k}\sum_{i=t_0}^{t_1} \text{AR}_i^2, \]
where \(m=t_1 - t_0\) is the length of the estimation window and \(K\) the degree of freedom of the applied model for estimating the abnormal return, e.g. for the market model \(K=2\) and for the Comparison Period Mean Adjusted Model \(K=1\).
Parametric Tests
Abnormal Return (AR) t-test
The AR t-test is a statistical method used to determine whether the abnormal return of a security on a specific day is significantly different from zero. This test helps researchers identify whether the event of interest has a significant impact on the security’s return at a particular point in time.
Formula
The t-statistic for the abnormal return (AR) is calculated as follows:
\[ t_{AR_{i,t}} = \frac{AR_{i,t}}{\hat{S}_i}, \]
where:
\(t_{AR_{i,t}}\) is the t-statistic of firm i at time t.
\(t_{AR_{i,t}}\) is student-t distributed with \(M-K\) degrees of freedom, where \(M\) is the length of the estimation window and \(K\) is the degree of freedom of the applied model.
\(\hat{S}_i\) is the sample standard deviation in the estimation window.
The t-statistic is then compared to the critical value from the t-distribution with the appropriate degrees of freedom to determine whether the abnormal return is significantly different from zero.
Advantages
Simplicity: The AR t-test is a simple and straightforward method for testing the significance of individual abnormal returns.
Immediate Impact Analysis: This test enables researchers to assess the immediate impact of an event on a security’s return at a specific point in time.
Identification of Significant Dates: The AR t-test can help identify days on which the event had a significant impact on the security’s return.
Disadvantages
Ignores Cumulative Effects: The AR t-test only analyzes the significance of individual abnormal returns and does not consider their cumulative effects over a longer event window.
Sensitive to Model Choice: The accuracy of the AR t-test depends on the chosen model for estimating expected returns. An inappropriate model can lead to biased results.
Multiple Testing Problem: Conducting multiple AR t-tests on different dates may increase the probability of Type I errors (false positives) due to the multiple testing problem.
In conclusion, the AR t-test is a useful statistical tool for assessing the significance of individual abnormal returns at specific points in time. However, it is essential to consider its limitations and complement it with other tests, such as the CAR t-test, to obtain a comprehensive understanding of the event’s impact on security returns.
Cumulative Abnormal Return (CAR) t-test
The CAR t-test is a statistical method used to determine whether the cumulative abnormal return of a security over an event window is significantly different from zero. This test helps researchers identify whether the event of interest has a significant impact on the security’s return over the entire event window, considering the cumulative effects of the event.
Formula
The t-statistic for the cumulative abnormal return (CAR) is calculated as follows:
\[ t_{CAR_i(t_2, t_3)} = \frac{CAR_i(t_2, t_3)}{\sqrt{t_3 - t_2} \hat{S}_i} \]
where:
\(t_{CAR_i(t_2, t_3)}\) is the t-statistic for the cumulative abnormal return of firm i over the event window from \(t_2\) to \(t_3\).
\(t_{CAR_i(t_1, t_2)}\) is student-t distributed with \(M-K\) degrees of freedom, where \(M\) is the length of the estimation window and \(K\) is the degree of freedom of the applied model.
\(CAR_i(t_2, t_3)\) is the cumulative abnormal return from \(t_2\) to \(t_3\).
\(\hat{S}_i\) is the sample standard deviation in the estimation window.
The t-statistic is then compared to the critical value from the t-distribution with the appropriate degrees of freedom to determine whether the cumulative abnormal return is significantly different from zero.
Advantages
Cumulative Impact Analysis: The CAR t-test accounts for the cumulative effects of the event on the security’s return over the entire event window, providing a more comprehensive view of the event’s impact.
Robustness: The CAR t-test is generally more robust than the AR t-test, as it considers the overall impact of the event rather than focusing on individual abnormal returns.
Reduced Sensitivity to Noise: By considering cumulative abnormal returns, the CAR t-test can be less sensitive to random fluctuations or noise in the data.
Disadvantages
Model Dependency: The accuracy of the CAR t-test depends on the chosen model for estimating expected returns. An inappropriate model can lead to biased results.
Ignores Individual Effects: The CAR t-test focuses on the cumulative impact of the event over the event window and may not capture significant individual abnormal returns on specific days.
Assumption of Independence: The CAR t-test assumes that abnormal returns are independent over time. If this assumption is violated, the test may produce inaccurate results.
In conclusion, the CAR t-test is a valuable statistical tool for assessing the significance of cumulative abnormal returns over an event window. It provides a more comprehensive view of the event’s impact on security returns by considering the cumulative effects. However, it is essential to consider its limitations and use it in conjunction with other tests, such as the AR t-test, to obtain a complete understanding of the event’s impact on security returns.
Nonparametric Tests
Past research indicates that distributions of abnormal returns exhibit fat tails and a rightward skew. When assessing positive abnormal performance, parametric tests tend to reject the hypothesis more frequently than is warranted, and less frequently for negative abnormal performance. These tests are poorly suited for situations where the assumption of normality in abnormal returns does not hold. In contrast, non-parametric tests are more accurately designed and demonstrate greater effectiveness in identifying incorrect null hypotheses that suggest the absence of abnormal returns.
Permutation Test
Therefore, one may consider using a nonparametric test, as e.g. applying a permutation test as published by Bugni et al. Permutation-based tests for discontinuities in event studies or Nguyen and Wolf A Note on Testing AR and CAR for Event Studies. The first paper is with more mathematical rigor and dives much deeper into the topic and also shows a wider application range for Event Studies. The second one focuses on applying the permutation test on above described test statistics for AR and CAR.
Permutation tests are highly flexible, making them applicable to a wide range of data types and research questions without requiring strict distributional assumptions. They provide exact p-values for small sample sizes, ensuring more accurate statistical inference compared to methods that rely on asymptotic approximations. Additionally, their nonparametric nature offers robustness against outliers and skewed data distributions, enhancing the reliability of research findings. In a practical application one just needs to perform drawing without replacement and the recalculation of the test statistics repeatedly (several thousand times).
How to Apply a Permutation Test
The p-value in a permutation test is calculated as follows (we use the same principle when comparing the variance of the estimation versus the event window). Let \(\textbf{X} = \{X_1, \cdots,X_m, X_{m+1}, \cdots, X_{m+n}\}\) with \(m\) as the length of the estimation period and \(n\) as the length of the event period and in the case of abnormal return calculation \(X_i = \text{AR}_i\), for \(i=1,\cdots, n+m\). Further, let \(T\) be the test statistic calculated on \(\textbf{X}\), for example the [Abnormal Return (AR) t-test] \(t_{AR}\) or the [Cumulative Abnormal Return (CAR) t-test] \(t_{car}\).
Let \(\textbf{X}^* = \{X_{i_1}, \cdots,X_{i_m}, X_{i_{m+1}}, \cdots, X_{i_{m+n}}\}\) be a permutation (drawn without replacement) of the set \(\textbf{X}\) and \(T^*_i\) be the test statistic on \(\textbf{X}^*\) for \(i=1,\cdots, \text{B}\) with e.g. \(B=10.000\). Then the estimation of the p value is calculated as follows:
\[ p_{permutation} = \frac{1}{B}\sum\limits_{i=1}^B 1_{[T^*_i > T]} \]
with \(1_{x}\) is the indicator function.
Note: This is just an estimation, as we do not calculate the p-value on all possible permutations due to large number of the permutations , namely \((m+n)!\).
Literature
Research Papers
- Bugni et al. Permutation-based tests for discontinuities in event studies
- Nguyen and Wolf A Note on Testing AR and CAR for Event Studies