1.7.4 Statistical Testing

Basic Prerequisites: Statistics & Inference (t-test, CLT, Type I/II error) and 03 · Abnormal Returns & CAR.

---

1. Intuition & Practical Objective

An event study is not finished when it computes a CAR - it must decide whether that CAR is statistically distinguishable from zero or could be a draw from noise. This page covers the test statistic, its assumptions, its power (ability to detect real abnormal performance), and the two inferential traps (cross-correlation and event-period variance) that break it.

The test statistic is the CAR divided by an estimate of its standard deviation (Kothari–Warner eq. 5):

$$J=\frac{CAR(t_1,t_2)}{\sqrt{{\sigma }^2(t_1,t_2)}},{\sigma }^2(t_1,t_2)=L{\sigma }^2(A{R}_t),$$

where $L=t_2-t_1+1$ and ${\sigma }^2(A{R}_t)$ is the variance of the one-period mean abnormal return (estimated from the estimation-window time series). Under the null of zero abnormal performance, $J$ is approximately unit normal. Two design facts matter more than the formula (Kothari–Warner §3.6):

1. Short horizons are well-specified and powerful; long horizons are poorly specified with low power.
2. Power depends on sample size, firm volatility, and whether the abnormal performance is concentrated in a known window. A 10% abnormal return on a known single day is detected with 100% power by just 6 stocks; spread over 6 months, 200 stocks detect it only 65% of the time (Kothari–Warner §3.6.4).

---

2. Mathematical Ground Truth & Derivations

The test statistic (Kothari–Warner eq. 5–6). For the CAR over window $[t_1,t_2]$ with $L$ periods:

$$J=\frac{CAR(t_1,t_2)}{\sqrt{L{\sigma }^2(A{R}_t)}}\sim \mathcal{N}(0,1)\text{under }{H}_0.$$

${\sigma }^2(A{R}_t)$ is estimated from the time series of portfolio (mean) abnormal returns in the estimation window, so it automatically captures cross-sectional dependence across firms (Brown–Warner 1985 eq. 5 uses the same “portfolio excess return” approach).

Cross-correlation inflation (Kothari–Warner eq. 10). If the sample firms’ abnormal returns are positively correlated (event-date clustering, common industry), the true SE is larger than the independence-based one by:

$$\frac{{\sigma }_{AR}(\text{dep})}{{\sigma }_{AR}(\text{ind})}=\sqrt{1+(N-1){\rho }_{ij}},$$

where ${\rho }_{ij}$ is the average pairwise correlation. At $\rho =0.02$ and $N=100$, this is $\sqrt{1+99\cdot 0.02}\approx 1.73$ - the t-stat is overstated by ~73%. This is why long-horizon tests that ignore cross-dependence “reject the null far more often than the size of the test” (Kothari–Warner §4.4.2).

Power. Power = $P(\text{reject }{H}_0\mid H_a\text{ true})$. For a cross-sectional mean abnormal return $\mu$ with SE $\sigma /\sqrt{N}$, the power of a two-sided 5% test is approximately

$$P(|Z|>1.96-\frac{\mu }{\sigma /\sqrt{N}}).$$

Power rises with $N$ (sample size) and falls with $\sigma$ (firm volatility). Crucially, power is high only if abnormal performance is concentrated in a precisely-known window (Table 2, Kothari–Warner §3.6.1).

---

3. Computational Implementation - the test, its traps, and power

Stdlib only. Three experiments: (a) the cross-correlation inflation factor; (b) the event-period-variance over-rejection trap; (c) power vs sample size.

Reading the output. (a) The $\sqrt{1+(N-1)\rho }$ factor reproduces Kothari–Warner’s headline number exactly: $\rho =0.02,N=100\Rightarrow 1.73$ - a ~73% overstatement of significance from even mild cross-dependence. (b) When the event-day variance doubles (Beaver 1968: volatility rises around earnings announcements), a naive 5% test rejects the null 16.5% of the time - a badly misspecified test. (c) Detecting a 1% abnormal return with average-volatility firms needs a large sample: ~200 firms for 76% power. For a concentrated signal, power is dramatically better - Kothari–Warner report 6 firms detect a 10% one-day abnormal return with 100% power.

---

4. Failure Modes & First-Principles Breakdowns

1. The joint-test problem (the deepest one). Every event-study test is a joint test of “abnormal returns are zero” and “the expected-return model is correct.” A rejection can mean either. Short windows make the model component negligible (~0.05% daily expected return); long windows are dominated by it (Kothari–Warner §3.5.1).
2. Event-period variance increase. Volatility rises around events, so estimation-window SEs understate event-period variability and the test over-rejects (Experiment b: 5% → 16.5%). Remedies: use event-period cross-sectional variance, nonparametric rank tests (Corrado 1989).
3. Cross-correlation / clustering. Ignoring event-date clustering inflates the t-stat by up to ~1.7× (Experiment a). The portfolio-time-series SE handles cross-sectional dependence; calendar-time portfolios handle it for long horizons.
4. Low power at long horizons. Even huge cumulative abnormal performance (25% over 5 years, 200 firms) is detected under 50% of the time (Jegadeesh–Karceski 2004). Long-window “no result” is often just low power.
5. Non-normality / skewness of BHARs. Long-horizon BHARs are right-skewed; the t-distribution is then asymmetric with mean below zero under the null (Brav 2000). Use skewness-adjusted or bootstrap t-stats.

---

5. References

• Kothari & Warner (2007), Econometrics of Event Studies, Handbook of Corporate Finance
• Brown & Warner (1985), Using Daily Stock Returns, JFE 14(1)
• Brown & Warner (1980), Measuring Security Price Performance, JFE 8(3)
• Corrado (1989), A Nonparametric Test for Abnormal Security-Price Performance in Event Studies, JFE 23(2)
• Jegadeesh & Karceski (2004), Long-Term Performance Evaluation, Working Paper

---

6. Connected Graph Bridges

• Back: 03 · Abnormal Returns & CAR · Index Hub
• Forward: 05 · Failure Modes · 06 · Advanced Extensions
• Base: Statistics & Inference (t-tests, power, CLT) · Bayesian Statistics
• Practice: Backtesting Hygiene (multiple-testing caveat on significance thresholds)