What is the Patell Z-test?

The Patell Z-test standardizes each event's abnormal return by its own prediction-error standard deviation before aggregating, making it robust to cross-sectional variation in event-induced variance.

What is the BMP test?

The Boehmer-Musumeci-Poulsen (BMP) test extends the Patell test by using the cross-sectional standard deviation of standardized abnormal returns, accounting for event-induced variance increases.

When should I use non-parametric tests?

Use non-parametric tests (Sign test, Generalized Sign test, Rank test) when abnormal returns are not normally distributed or when your sample size is small.

What is the Kolari-Pynnonen test?

The Kolari-Pynnonen test adjusts the BMP test for cross-correlation of abnormal returns, which is important when events cluster in calendar time.

AAR & CAAR Test Statistics

AAR and CAAR test statistics determine whether the average abnormal return across a sample of firms is significantly different from zero. Eight tests are available, from the cross-sectional t-test to the Kolari-Pynnonen test that handles event-induced variance and cross-sectional correlation.

Part of the methodology guide

Part of the Event Study Methodology Guide.

Multi-event test statistics assess whether the average abnormal return across a sample of firms is significantly different from zero. These are the tests reported in most published event studies.

How Do the Multi-Event Tests Compare?

AAR and CAAR test statistics are the backbone of multi-event inference in financial event studies. They aggregate abnormal returns across a sample of firms to determine whether an event type produces a statistically significant market reaction on average. First formalized by Patell (1976) and extended by Boehmer, Musumeci, and Poulsen (1991), these tests range from simple cross-sectional t-tests to advanced procedures that correct for event-induced variance and cross-sectional correlation.

Test	R Class	Type	Assumes Normality	Handles Event-Induced Variance	Handles Clustering
Cross-Sectional t	CSectTTest	Parametric	Yes	No	No
Patell Z	PatellZTest	Parametric	Yes	No	No
BMP	BMPTest	Parametric	Yes	Yes	No
Sign test	SignTest	Non-parametric	No	No	No
Generalized Sign	GeneralizedSignTest	Non-parametric	No	No	No
Rank test	RankTest	Non-parametric	No	Yes	No
Kolari-Pynnonen	KolariPynnonenTest	Parametric	Yes	Yes	Yes
Calendar-Time Portfolio	CalendarTimePortfolioTest	Parametric	Yes	No	Yes

How Do I Run All Tests in R?

Run with all 8 multi-event tests

# Run with all 8 multi-event tests
ps <- ParameterSet$new(
  multi_event_statistics = MultiEventStatisticsSet$new(
    tests = list(
      CSectTTest$new(),
      PatellZTest$new(),
      BMPTest$new(),
      KolariPynnonenTest$new(),
      SignTest$new(),
      GeneralizedSignTest$new(),
      RankTest$new(),
      CalendarTimePortfolioTest$new()
    )
  )
)

task <- EventStudyTask$new(firm_data, index_data, request)
task <- run_event_study(task, ps)

Parametric Tests

Cross-Sectional t-test (CSect T)

The simplest multi-event test, used in over 60% of published event studies. Computes the AAR as the mean of abnormal returns across firms, then tests whether it differs from zero using the cross-sectional standard deviation.

AAR:

AAR_t = \frac{1}{N}\sum_{i=1}^{N} AR_{i,t}

t_{AAR_t} = \frac{\sqrt{N} \cdot AAR_t}{\hat{S}_{AAR,t}} \sim t_{N-1}

where $\hat{S}_{AAR,t}^2 = \frac{1}{N-1}\sum_{i=1}^{N}(AR_{i,t} - AAR_t)^2$ .

CAAR:

t_{CAAR} = \frac{\sqrt{N} \cdot CAAR}{\hat{S}_{CAAR}} \sim t_{N-1}

where $\hat{S}_{CAAR}^2 = \frac{1}{N-1}\sum_{i=1}^{N}(CAR_i - CAAR)^2$ .

Pros	Cons
Simple, intuitive	Assumes equal variance across firms
No model estimates needed for SE	Does not account for event-induced variance
Robust to different return models	Sensitive to outliers

Patell Z Test

Introduced by Patell in 1976, this test standardizes each firm's abnormal return by its own estimation-window standard deviation before aggregating. This gives less weight to volatile stocks and has been cited in over 2,500 studies according to Google Scholar.

Standardized abnormal return:

SAR_{i,t} = \frac{AR_{i,t}}{\hat{S}_i}

Test statistic:

Z_{Patell} = \frac{1}{\sqrt{N}} \sum_{i=1}^{N} \frac{SAR_{i,t}}{\sqrt{\frac{M_i - 2}{M_i - 4}}} \stackrel{a}{\sim} N(0,1)

where $M_i$ is the estimation window length for firm $i$ .

Pros	Cons
Accounts for different firm volatilities	Assumes no event-induced variance change
More powerful than CSect T for heterogeneous samples	Requires estimation-window variance estimates
Well-established in the literature	Over-rejects when variance increases at event

Patell Z vs. CSect T

If firms in your sample have very different volatilities (e.g., mixing large-cap and small-cap), Patell Z is more appropriate because it standardizes by each firm's own variance.

BMP Test (Boehmer, Musumeci & Poulsen)

Addresses a key weakness of the Patell Z test: events often change the variance of returns (event-induced variance). The BMP test cross-sectionally standardizes the Patell-standardized AR, making it robust to variance changes at the event date.

t_{BMP} = \frac{\sqrt{N} \cdot \overline{SAR}_t}{S_{SAR,t}} \sim t_{N-1}

where $\overline{SAR}_t = \frac{1}{N}\sum_{i=1}^{N} SAR_{i,t}$ and $S_{SAR,t}$ is its cross-sectional standard deviation.

Pros	Cons
Robust to event-induced variance	Requires estimation-window variance estimates
Correctly sized even during volatile events	Slightly less powerful than Patell Z when variance is stable
Recommended by Kolari & Pynnonen (2010)	More complex to interpret

When variance changes matter

Earnings announcements, M&A, and crisis events typically increase return variance at the event date. The BMP test prevents the inflated variance from producing false positives. Use it alongside Patell Z as a robustness check.

Kolari-Pynnonen Test

An adjusted version of the BMP test that corrects for both event-induced variance and cross-sectional correlation (event clustering). When multiple events occur on the same calendar date, standardized abnormal returns are correlated across firms, causing the BMP test to over-reject. The Kolari-Pynnonen adjustment accounts for this.

t_{KP} = t_{BMP} \cdot \sqrt{\frac{1}{1 + (N-1)\bar{r}}}

where $\bar{r}$ is the average cross-sectional correlation of the standardized abnormal returns across firms.

Pros	Cons
Robust to event-induced variance and clustering	Requires estimation of cross-sectional correlation
Correctly sized for same-date events	Slightly less powerful than BMP when no clustering
Recommended for industry-wide event studies	Requires standardized residuals from estimation window

When to use Kolari-Pynnonen

Use when events cluster on the same or nearby calendar dates (e.g., regulatory announcements, industry-wide shocks). It combines the strengths of the BMP test with a correction for cross-sectional dependence.

Calendar-Time Portfolio Test

When multiple events cluster on the same calendar dates, the cross-sectional independence assumption is violated. The Calendar-Time Portfolio test aggregates returns into a single time series of portfolio returns and tests using time-series standard errors.

AAR_t = \frac{1}{N_t}\sum_{i=1}^{N_t} AR_{i,t}

t_{CalTime} = \frac{AAR_t}{\hat{S}_{TS}}

where $\hat{S}_{TS}$ is the time-series standard deviation of the portfolio AAR.

Pros	Cons
Handles temporal clustering of events	Less powerful with few calendar dates
Does not require cross-sectional independence	Sensitive to portfolio weighting
Better for industry-wide events	Uncommon in standard event studies

Non-Parametric Tests

Non-parametric tests do not require normally distributed returns. According to Corrado and Zivney (1992), non-parametric tests can improve rejection accuracy by 15–30% relative to parametric alternatives when daily returns exhibit skewness or excess kurtosis. Use them when diagnostics show non-normal residuals (Shapiro-Wilk p < 0.05) or when your sample contains extreme outliers.

Sign Test

Tests whether the proportion of positive abnormal returns exceeds 50%. Under the null hypothesis of zero abnormal returns, positive and negative AR are equally likely.

Z_{Sign} = \frac{N^+ - 0.5N}{0.5\sqrt{N}} \stackrel{a}{\sim} N(0,1)

where $N^+$ is the number of firms with positive AR at the event date.

Pros	Cons
No distributional assumptions	Low power (ignores magnitude)
Robust to outliers	Assumes symmetric return distribution
Simple to interpret	Cannot detect effects driven by magnitude

Generalized Sign Test (Cowan 1992)

Improves on the Sign test by estimating the expected proportion of positive returns from the estimation window, rather than assuming 50%.

\hat{p} = \frac{1}{N}\sum_{i=1}^{N}\left(\frac{1}{M_i}\sum_{t=t_0}^{t_1}\mathbf{1}[AR_{i,t} > 0]\right)

Z_{GSign} = \frac{N^+ - N\hat{p}}{\sqrt{N\hat{p}(1-\hat{p})}} \stackrel{a}{\sim} N(0,1)

Pros	Cons
Accounts for return asymmetry	Still ignores magnitude
Better sized than simple Sign test	Requires estimation-window data
Robust to skewed distributions	Slightly more complex

Rank Test (Corrado 1989)

Ranks all abnormal returns (estimation window + event window) and tests whether event-window ranks are systematically higher or lower than expected.

K_{i,t} = \frac{\text{rank}(AR_{i,t})}{T_i + 1} - 0.5

Z_{Rank} = \frac{\overline{K}_t}{\hat{S}_K} \stackrel{a}{\sim} N(0,1)

where $\overline{K}_t$ is the mean centered rank at event time $t$ and $\hat{S}_K$ is its standard deviation.

Pros	Cons
Uses magnitude information (via ranks)	Less intuitive than t-tests
Robust to non-normality and outliers	Requires combining estimation + event data
Well-powered across distributions	Sensitive to ties in rankings
Handles event-induced variance

Best non-parametric choice

The Rank test is generally the most powerful non-parametric test because it uses magnitude information (through ranks) while remaining robust to non-normality. Use it as your primary non-parametric robustness check.

Which Tests Should I Report?

AAR (Average Abnormal Return): The cross-sectional mean of abnormal returns at a given event time, measuring the average market reaction across all firms in the sample.
CAAR (Cumulative Average Abnormal Return): The sum of AARs over an event window, capturing the total average price impact of the event across the sample period.
Standardized Abnormal Return (SAR): An abnormal return divided by its estimation-window standard deviation, used in the Patell Z and BMP tests to weight each firm inversely by its volatility.

Most published event studies report 2–3 tests. A robust reporting strategy:

Scenario	Recommended Tests
Standard event study	CSect T + Patell Z + one non-parametric (Rank or Sign)
Event-induced variance likely	BMP + Rank test
Non-normal residuals	Sign + Generalized Sign + Rank
Clustered event dates	Kolari-Pynnonen + Calendar-Time Portfolio
Event-induced variance + clustering	Kolari-Pynnonen + Rank test
Maximum robustness	CSect T + Patell Z + BMP + Kolari-Pynnonen + Rank

Literature

Patell, J.M. (1976). Corporate forecasts of earnings per share and stock price behavior. Journal of Accounting Research, 14(2), 246–276.
Boehmer, E., Musumeci, J. & Poulsen, A.B. (1991). Event-study methodology under conditions of event-induced variance. Journal of Financial Economics, 30(2), 253–272.
Cowan, A.R. (1992). Nonparametric event study tests. Review of Quantitative Finance and Accounting, 2(4), 343–358.
Corrado, C.J. (1989). A nonparametric test for abnormal security-price performance in event studies. Journal of Financial Economics, 23(2), 385–395.
Campbell, J.Y., Lo, A.W. & MacKinlay, A.C. (1997). The Econometrics of Financial Markets.
Kolari, J.W. & Pynnonen, S. (2010). Event Studies for Financial Research.

Run this in R

The EventStudy R package lets you run these calculations programmatically with full control over parameters.

Get the R Package Or try the App

What Should I Read Next?

AR & CAR Test Statistics — single-event significance tests
Test Statistics Overview — choosing the right test
Assumptions — when tests break down
Diagnostics & Export — validate model fit