How to Interpret CAAR, AAR and the Patell Z-Statistic
A plain-language guide to what each event-study number means, which test to trust, and how to avoid over-claiming.
In short
AAR is the average abnormal return across firms on a single event-time day; CAAR is the AAR cumulated across the event window. A significant CAAR means the sample, on average, earned returns above (positive) or below (negative) the model benchmark around the event. The Patell Z tests that significance after standardising each abnormal return by its estimation error.
AR vs AAR vs CAR vs CAAR: definitions and how they nest
Abnormal return (AR)
The abnormal return $AR_{i,t}$ is firm $i$'s realised return on day $t$ minus the return predicted by the normal-return model: $AR_{i,t}=R_{i,t}-E(R_{i,t})$. It is the event-attributable part of the return for one firm on one day.
Average abnormal return (AAR)
The average abnormal return $AAR_t=\frac{1}{N}\sum_{i=1}^{N}AR_{i,t}$ is the cross-sectional mean of abnormal returns across the $N$ firms on a single event-time day $t$. It isolates the typical reaction on each day around the event.
Cumulative abnormal return (CAR)
The cumulative abnormal return $CAR_i=\sum_{t=T_1+1}^{T_2}AR_{i,t}$ sums one firm's abnormal returns over the event window, capturing the total event-attributable return for that firm rather than its day-by-day path.
Cumulative average abnormal return (CAAR)
The cumulative average abnormal return $CAAR=\frac{1}{N}\sum_{i=1}^{N}CAR_i$ averages CAR across firms (equivalently, cumulates AAR across the window). It is the headline number of a multi-firm event study: the average total reaction to the event.
The four quantities nest cleanly: average AR across firms to get AAR; cumulate AR across days to get CAR; do both to get CAAR. See the formal definitions and every test statistic on Significance tests.
What a significant CAAR actually tells you
A statistically significant CAAR says the average return over the event window departed from the model's prediction by more than sampling variation would explain. It does not by itself establish causation: it establishes an association between the event window and abnormal performance, conditional on the chosen normal-return model. The economic claim depends on clean event-date definition, a credible benchmark, and the absence of confounding news.
Read three things together: the sign (did the market judge the event good or bad), the magnitude in percent (a 0.3% CAAR and a 5% CAAR are different stories), and the test statistic (is the effect distinguishable from zero). A large but insignificant CAAR usually means a noisy or small sample; a tiny but significant CAAR means a precisely estimated but economically minor reaction.
The Patell Z explained
The plain t-test on CAAR treats every firm's abnormal return as equally precise. It is not: a firm with a short or volatile estimation window has a noisier abnormal return. The Patell Z fixes this by standardising each abnormal return by its own forecast-error standard deviation before aggregating (Patell, 1976):
$$SAR_{i,t}=\frac{AR_{i,t}}{S_{AR_{i,t}}},\qquad S_{AR_{i,t}}^2=S_{AR_i}^2\left(1+\frac{1}{M_i}+\frac{(R_{m,t}-\bar R_m)^2}{\sum_{s=T_0}^{T_1}(R_{m,s}-\bar R_m)^2}\right).$$
The bracketed term is a prediction-interval inflation: it grows the variance for event days whose market return is far from the estimation-window average, because the model extrapolates less reliably there. Aggregating the standardised abnormal returns gives, for the event window, $z=\frac{1}{\sqrt N}\sum_i CSAR_i/S_{CSAR_i}$ with $S_{CSAR_i}^2=L_2\frac{M_i-2}{M_i-4}$, which is approximately standard normal under the null.
Patell Z in one sentence
The Patell Z is a standardized-residual test that divides each firm's abnormal return by its forecast-error standard deviation before averaging, so firms estimated with more precision count for more and the statistic corrects for estimation error.
Patell Z vs BMP vs simple t: when each is appropriate
| Test | Corrects for | Breaks when | Use when |
|---|---|---|---|
| Cross-sectional t | Cross-sectional dispersion of CAR | Cross-sectional correlation (event clustering) | Quick read, independent events, equal precision |
| Patell Z | Estimation error, unequal firm variances | Event-induced volatility; cross-sectional correlation | Firms differ in estimation precision, no volatility shift |
| BMP (standardized cross-sectional) | Estimation error and event-induced volatility | Strong cross-sectional correlation | The event may have changed return volatility (the common case) |
| Kolari-Pynnonen adjusted | Cross-sectional correlation on top of the above | Very small samples | Clustered events sharing calendar dates |
A practical default: report the BMP test as the primary statistic and the Patell Z and a non-parametric rank test as robustness, because BMP is robust to the event-induced volatility that the Patell Z assumes away. Full formulas for all of these are on Significance tests.
Worked example: an earnings-surprise CAAR
Suppose 120 firms with positive earnings surprises have, over the $[-1,+1]$ window, $CAAR = +1.4\%$. The cross-sectional t is 3.1, the Patell Z is 3.8, and the BMP t is 2.6. Interpretation: the market reaction is positive and economically meaningful (1.4% in three days), and it survives every test. The BMP being lower than the Patell Z is expected and informative: it tells you event-induced volatility inflated the Patell Z, and the BMP is the number to quote. You would conclude that positive earnings surprises produced a significant positive announcement-window reaction, and report BMP as the headline test.
Get these statistics on your own sample. ARC computes AAR, CAAR, the Patell Z and BMP from a CSV upload and lays them out in a results table.
Run it free in ARC →Common misreadings and how to avoid over-claiming
- "Negative CAAR means the event was bad for the firm." It means the market priced the news negatively relative to the benchmark. Mind the benchmark: a negative CAAR against a roaring market can still be a positive raw return.
- "p < 0.05, so the effect is real." Significance is conditional on the model and on no confounding news. Always screen for same-window events.
- "The Patell Z is the strictest test." No. The Patell Z assumes constant volatility; when the event raises volatility it over-rejects, so it can be the most permissive. BMP is the safer headline test.
- Non-normality. Daily abnormal returns are fat-tailed. For small samples pair a parametric test with a rank or generalized sign test.
Compute these on your own data, free
The ARC calculator returns AAR, CAAR and the full test battery from a CSV, so you can reproduce every number in this guide on your own sample. Choose the benchmark on Expected-return models.
FAQ
What does a negative CAAR mean?
A negative CAAR means the sample, on average, earned returns below the normal-return benchmark over the event window: the market priced the event unfavourably relative to expected performance. It is measured against the model, not against zero raw return.
What p-value threshold should I use?
Two-sided 5% is conventional, with 1% and 10% reported alongside. Because daily abnormal returns are fat-tailed, support a borderline parametric p-value with a non-parametric rank or generalized sign test before concluding.
Patell Z or BMP, which should I report?
Report the BMP test as the headline statistic. It keeps the Patell standardisation but adds robustness to event-induced volatility, which the plain Patell Z assumes away and which is present in most real events.