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.
\(N\) is the number of events examined in a test. All following test statistics are given at event date, but are valid for all \(t \in [t_2, t_3]\).
Parametric Tests
Cross-Sectional Test for Averaged Abnormal Returns (AAR)
The Cross-Sectional Test (CSect T) is a statistical tool employed in event studies to evaluate the null hypothesis that the average abnormal return at the event date is zero. The test statistic for CSect T is given by:
Formula
The cross-sectional t-statistic for the averaged abnormal return (AAR) is calculated as follows:
\[ t_{AAR_{0,t}} = \frac{AAR_{0,t}}{\hat{S}_{AAR,0}}, \]
with
\[ \hat{S}_{AAR,0}^2 = \frac{1}{N-1} \sum\limits_{i=1}^N (AR_{i,0}-AAR_0)^2 \]
where \(N\) is the number of events, \(AAR_0\) is the average abnormal return at the event date, \(\hat{S}_{AAR,0}\) and is the standard deviation of the abnormal returns at the event date. The test statistic follows a t-distribution with \(N-1\) degrees of freedom under the null hypothesis.
Advantages
Event Impact Assessment: The CSect T test provides a mechanism to evaluate the significance of an event’s impact on average abnormal returns.
Broad Applicability: This test is applicable in a wide range of scenarios where an event’s impact on financial returns is of interest.
Robustness: The CSect T test is robust to a variety of different underlying return generating processes.
Disadvantages
Sensitivity to Event Assumptions: The CSect T test is sensitive to the assumption that the events are independent and identically distributed. If this assumption is violated, the test may yield biased results.
Single-Point Assessment: This test focuses on the event date and does not consider the cumulative effect of an event over a longer window.
Distributional Assumptions: The validity of the CSect T test relies on the assumption that the test statistic follows a t-distribution, which may not always hold true.
In conclusion, the CSect T test is a crucial statistical tool in event studies for assessing the significance of average abnormal returns at the event date. However, it is essential to consider its limitations and use it in conjunction with other tests to obtain a comprehensive understanding of the event’s impact on financial returns.
Cross-Sectional Test for Cumulative Averaged Abnormal Returns (CAAR)
The Cumulative Average Abnormal Return Cross-Sectional Test is a statistical test used in event studies. The null hypothesis of interest for this test is \(E(CAAR) = 0\) which asserts that the expected value of the Cumulative Average Abnormal Return (CAAR) is zero.
Formula
The t-statistic for the cumulative abnormal return (CAR) is calculated as follows:
\[ t_{CAAR_{0,t}} = \frac{CAAR_{0,t}}{\hat{S}_{CAAR,0}}, \]with
\[ \hat{S}_{CAAR}^2 = \frac{1}{N-1} \sum\limits_{i=1}^N (CAR_{i}-CAAR)^2 \]
The approximate null distribution for the test statistic is \(t \stackrel{\cdot}{\sim} t_{N-1}\), implying that under the null hypothesis, the test statistic follows a t-distribution with \(N-1\) degrees of freedom.
Advantages
Cumulative Impact Assessment: The CAAR CSect T test provides a mechanism to evaluate the significance of an event’s impact on cumulative average abnormal returns, allowing for the analysis of event effects over a period of time rather than just at a single point.
Broad Applicability: This test is applicable in a wide range of scenarios where an event’s impact on cumulative financial returns is of interest.
Robustness: The CAAR CSect T test is robust to a variety of different underlying return generating processes.
Disadvantages
Sensitivity to Event Assumptions: The CAAR CSect T test is sensitive to the assumption that the events are independent and identically distributed. If this assumption is violated, the test may yield biased results.
Distributional Assumptions: The validity of the CAAR CSect T test relies on the assumption that the test statistic follows a t-distribution, which may not always hold true.
Dependence on Estimation Window: The calculation of cumulative abnormal returns and thus the CAAR CSect T test statistic depends on the selection of an appropriate estimation window. Inaccurate selection can lead to biased results.
In conclusion, the CAAR CSect T test is a vital statistical tool in event studies for assessing the significance of cumulative average abnormal returns. However, it’s important to consider its limitations and use it in conjunction with other tests to get a comprehensive understanding of the event’s impact on financial returns.