Choosing the Right Test Statistic

The choice of test statistic determines whether your event study produces valid inference. Different tests make different assumptions about the distribution of abnormal returns, the behavior of variance around the event, and the independence of events across firms. Using the wrong test can lead to over-rejection (finding effects that are not there) or under-rejection (missing real effects). This page provides a systematic decision framework for selecting the right test.

Decision Framework

Selecting a test statistic involves answering four questions about your study design. Each question narrows the set of appropriate tests.

Parametric vs. Non-Parametric Tests

The first major distinction is between parametric and non-parametric tests. Each class has strengths and weaknesses.

Parametric Tests

Parametric tests assume that abnormal returns follow a known distribution (typically normal). They use the estimated variance of abnormal returns to construct test statistics. When the distributional assumptions hold, parametric tests are more powerful than non-parametric alternatives.

Non-Parametric Tests

Non-parametric tests make minimal distributional assumptions. They are particularly useful when returns are skewed, heavy-tailed, or contain outliers — conditions that are common in practice, especially for small-cap stocks and emerging markets.

Handling Event-Induced Variance

Many events cause return variance to increase on the event date. For example, earnings announcements typically double or triple daily return variance. M&A announcements can increase variance even more. This phenomenon is called event-induced variance.

Standard tests like the Patell Z assume that the variance of abnormal returns in the event window equals the estimation-window variance. When event-induced variance is present, this assumption is violated, and the Patell Z test over-rejects the null hypothesis — it reports significant effects even when there are none.

The BMP test (Boehmer, Musumeci, and Poulsen, 1991) is the most widely recommended solution. It first standardizes each firm's abnormal return by its estimation-window standard deviation, then computes the cross-sectional standard deviation of these standardized abnormal returns. This two-step approach captures event-induced variance in the cross-sectional step.

# Configure with BMP test
ps <- ParameterSet$new(
  multi_event_statistics = MultiEventStatisticsSet$new(
    tests = list(
      BMPTest$new(),           # Robust to event-induced variance
      CSectTTest$new(),        # Comparison (not robust)
      PatellZTest$new(),       # Comparison (not robust)
      GeneralizedSignTest$new()  # Non-parametric robustness check
    )
  )
)

task <- run_event_study(task, ps)

Handling Cross-Correlation (Event Clustering)

When multiple firms experience the event on the same date (or within overlapping event windows), their abnormal returns are cross-sectionally correlated. This violates the independence assumption of most tests and causes over-rejection.

Common scenarios with clustered events include:

• Regulatory changes: A new law affects all firms in an industry on the same date.
• Macroeconomic announcements: Interest rate decisions, GDP releases, or employment reports affect all stocks simultaneously.
• Industry-wide events: A product recall or safety incident that affects all competitors.

# Kolari-Pynnonen: robust to both variance change and clustering
ps <- ParameterSet$new(
  multi_event_statistics = MultiEventStatisticsSet$new(
    tests = list(
      KolariPynnonenTest$new(),   # Robust to variance + clustering
      CalendarTimePortfolioTest$new(), # Portfolio approach
      BMPTest$new(),              # Robust to variance only (comparison)
      RankTest$new()              # Non-parametric check
    )
  )
)

task <- run_event_study(task, ps)

Recommended Tests by Scenario

The table below provides concrete recommendations for common event study scenarios. Each recommendation includes a primary test and a robustness check.

Decision Tree

Follow this decision tree to select your test statistics. Start at the top and follow the path that matches your study.

How Do I Configure Multiple Tests in R?

The EventStudy package allows you to run multiple tests simultaneously. This makes it easy to compare results and assess robustness.

# Recommended setup: parametric + non-parametric + variance-robust
ps <- ParameterSet$new(
  # Single-event tests
  single_event_statistics = SingleEventStatisticsSet$new(
    tests = list(
      ARTTest$new(),
      CARTTest$new()
    )
  ),
  # Multi-event tests
  multi_event_statistics = MultiEventStatisticsSet$new(
    tests = list(
      # Parametric
      CSectTTest$new(),       # Baseline
      PatellZTest$new(),      # Standardized
      BMPTest$new(),          # Variance-robust
      KolariPynnonenTest$new(), # Variance + clustering robust

      # Non-parametric
      SignTest$new(),
      GeneralizedSignTest$new(),
      RankTest$new()
    )
  )
)

task <- run_event_study(task, ps)

# View all test results
task$get_aar()

# One-sided tests (e.g., testing for positive abnormal returns only)
ps <- ParameterSet$new(
  multi_event_statistics = MultiEventStatisticsSet$new(
    tests = list(
      CSectTTest$new(confidence_type = "one-sided"),
      BMPTest$new(confidence_type = "one-sided")
    )
  )
)

Common Mistakes

• Using only the Patell Z test: The Patell Z is popular because it is well-known, but it is not robust to event-induced variance. Always include the BMP test.
• Ignoring clustering: If your events share event dates (even partially overlapping windows), standard tests will produce spuriously significant results.
• Relying on a single test: No single test is best in all scenarios. Report at least two tests (one parametric, one non-parametric) to demonstrate robustness.
• Using parametric tests with small samples: With fewer than 20 events, asymptotic properties of Patell Z and BMP may not hold. Prefer the Cross-Sectional t-test and non-parametric tests.
• Not checking normality: Run a Shapiro-Wilk test on the estimation-window residuals or inspect a QQ-plot before relying on parametric tests.

Literature

• Boehmer, E., Masumeci, J. & Poulsen, A.B. (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.
• Corrado, C.J. (1989). A nonparametric test for abnormal security-price performance in event studies. Journal of Financial Economics, 23(2), 385-395.
• Patell, J.M. (1976). Corporate forecasts of earnings per share and stock price behavior. Journal of Accounting Research, 14(2), 246-276.

What Should I Do Next?

• AR & CAR Test Statistics — formulas and code for single-event tests
• AAR & CAAR Test Statistics — formulas and code for multi-event tests
• Variance-Based Tests — deep dive into event-induced variance handling
• Inference & Robustness — wild bootstrap and multiple testing corrections