Variance Tests

Variance-based test statistics for event studies account for event-induced changes in return variance. Standard t-tests may be biased when events increase volatility; adjusted tests like the Boehmer-Musumeci-Poulsen (BMP) test and standardized cross-sectional test correct for this heteroskedasticity.

What Are the Key Definitions?

Variance tests in event studies are statistical procedures that account for changes in return volatility caused by the event itself. When a corporate event such as an earnings announcement or merger increases stock price variance by 2–5 times its normal level, standard t-tests that assume constant variance produce inflated test statistics and false rejection rates exceeding 15–20%. Variance-adjusted tests like the BMP test correct this bias by standardizing abnormal returns using event-window rather than estimation-window variability.

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

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. With a typical estimation window of 120 trading days and degrees of freedom around 118, the test follows a Student-t distribution that closely approximates the standard normal. 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:

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

1. Simplicity: The AR t-test is a simple and straightforward method for testing the significance of individual abnormal returns.
2. 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.
3. 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

1. 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.
2. 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. Use diagnostics and export tools to validate your model assumptions.
3. 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. For a complete overview of single-event significance tests, see AR & CAR Test Statistics.

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:

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 $t_3$.
• $t_{CAR_i(t_2, t_3)}$ 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.

Advantages

1. 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.
2. 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.
3. 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

1. 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.
2. 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.
3. 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. When working with multiple events, the AAR & CAAR test statistics extend these ideas to cross-sectional aggregation.

How Does Event-Induced Variance Affect Tests?

Corporate events — such as earnings announcements, M&A deals, or regulatory actions — can increase stock price volatility around the event date. As demonstrated by Boehmer, Musumeci, and Poulsen (1991), event-day return variance can be 3–4 times higher than the estimation-window average. When volatility increases during the event window, the actual standard deviation exceeds the estimated one, producing inflated t-statistics and spurious rejections that can push Type I error rates above 20% at the nominal 5% level.

Event-Induced Variance

The increase in return volatility caused by new information entering the market at the event date, violating the constant-variance assumption of standard t-tests.

Standardized Abnormal Return (SAR)

An abnormal return divided by the firm's estimation-window standard deviation, ensuring that each firm contributes equally to the aggregate test statistic regardless of its baseline volatility.

Cross-Sectional Standard Deviation

The standard deviation computed across firms at a single event time, used by the BMP test to capture event-window variability rather than relying on estimation-window estimates.

Robust Tests for Variance Changes

The BMP test (Boehmer, Musumeci & Poulsen, 1991) adjusts for event-induced variance by standardizing each firm's abnormal return by its own estimation-window standard deviation, then using the cross-sectional standard deviation of these standardized abnormal returns in the test statistic.

Let $\hat{S}_i$ be the estimation-window standard deviation of firm $i$. The standardized abnormal return is:

The BMP test statistic uses the cross-sectional variance of these $SAR$ values, ensuring that firms with higher event-window variance do not disproportionately influence the result.

The Kolari-Pynnonen test extends the BMP test by additionally adjusting for cross-sectional correlation among abnormal returns. This is particularly important when firms in the sample share a common event date (e.g., a regulatory change affecting an entire industry).

library(EventStudy)

# Configure parameter set with variance-robust tests
ps <- ParameterSet$new(
  multi_event_statistics = MultiEventStatisticsSet$new(
    tests = list(
      BMPTest$new(),            # Robust to event-induced variance
      KolariPynnonenTest$new()  # Also adjusts for cross-sectional correlation
    )
  )
)

# Run event study with robust tests
task <- EventStudyTask$new(firm_data, index_data, request)
task <- run_event_study(task, ps)

Nonparametric Tests

Non-parametric tests do not require normally distributed returns. See the AAR & CAAR Test Statistics page for the full set of non-parametric tests including the Sign test, Generalized Sign test, and Rank test.

Literature

• Boehmer, E., Musumeci, J. & Poulsen, A. (1991). Event-study methodology under conditions of event-induced variance. Journal of Financial Economics, 30(2), 253–272.
• Kolari, J.W. & Pynnonen, S. (2010). Event study testing with cross-sectional correlation of abnormal returns. Review of Financial Studies, 23(11), 3996–4025.
• Campbell, J.Y., Lo, A.W. & MacKinlay, A.C. (1997). The Econometrics of Financial Markets. Chapter 4.

What Should I Read Next?

• AAR & CAAR Test Statistics — full comparison of all eight multi-event tests
• AR & CAR Test Statistics — single-event significance testing
• Diagnostics & Export — validate your results and export
• Inference & Robustness — wild bootstrap and multiple testing corrections