Abnormal Return Tests and the BMP Statistic

You've run an event study: earnings surprises, say, and you've computed each stock's abnormal return — the return left over after subtracting what the market would have predicted anyway — on the announcement day. Averaged across hundreds of events, the abnormal return is positive. Is that a real effect, or could it show up by chance? The obvious move is a standard $t$-test, treating each stock's abnormal return as a draw from a fixed, known variance estimated from the calm period before the event. The problem is that events often aren't calm: earnings announcements, mergers, and other news frequently make a stock's returns more volatile on the event day itself, and a $t$-test that ignores this event-induced variance spike will call the surprise "highly significant" far too often, even when the true average abnormal return is zero.

An analogy: judging a shot from an unusually shaky camera

Imagine you're testing whether a photographer's hand shakes more on scary shoots. You measure the blur (a stand-in for return volatility) on quiet, uneventful shoots and get a baseline. Then you look at blur on scary shoots and it's higher on average — but so is the spread of blur values, because on scary shoots some sessions barely shake and others shake wildly. If you judge "is the average blur different?" using only the baseline's (calm) spread, you'll be shocked by how far scary-shoot blur seems to deviate — you're comparing against a ruler that's too fine for the noisier situation. The right fix is to measure the spread during the scary shoots themselves, not borrow it from the calm ones.

The two test statistics, one symbol at a time

Let $AR_{i}$ be stock $i$'s abnormal return on the event day and $\widehat{\sigma}_i$ be its estimated abnormal-return standard deviation from a pre-event estimation window (a period of normal trading before the event). The standard event-study $t$-test standardizes each stock's abnormal return by this pre-event volatility and averages:

$$t_{\text{standard}} = \frac{1}{\sqrt{N}} \sum_{i=1}^{N} \frac{AR_i}{\widehat{\sigma}_i}, \qquad \text{compared against a standard normal distribution.}$$

In plain English: divide each stock's abnormal return by its "normal" volatility, then average those standardized values across all $N$ events — if the true effect is zero, this should behave like an average of standard normal draws. The trouble is that $\widehat{\sigma}_i$ is estimated before the event, so if volatility jumps on the event day itself, the denominator understates the true event-day noise, making every $AR_i/\widehat{\sigma}_i$ look artificially large.

The BMP statistic fixes this by using the cross-sectional standard deviation of the standardized abnormal returns, computed from the event day itself, instead of assuming it equals one:

$$t_{\text{BMP}} = \frac{\overline{SAR}}{s_{SAR}/\sqrt{N}}, \qquad \text{where } SAR_i = \frac{AR_i}{\widehat{\sigma}_i},\ \overline{SAR}=\frac{1}{N}\sum_i SAR_i,\ s_{SAR} = \text{std. dev. of the } SAR_i \text{ across stocks}.$$

In plain English: instead of trusting that the standardized returns naturally have a spread of one, BMP measures how spread out they actually are on the event day, across the whole cross-section of events, and uses that measured spread in the denominator of the test. If the event inflated volatility, $s_{SAR}$ will be bigger than one, correctly widening the test's tolerance and preventing a false alarm.

Worked example 1: a calm case where both tests agree

Suppose 100 earnings events each have standardized abnormal returns $SAR_i$ with mean $\overline{SAR}=0.30$ and, because volatility didn't spike, a measured cross-sectional standard deviation $s_{SAR} = 1.05$ (close to the theoretical 1). The standard test gives $t_{\text{standard}} = \sqrt{100} \times 0.30 = 3.0$. BMP gives $t_{\text{BMP}} = 0.30 / (1.05/10) = 0.30/0.105 \approx 2.86$. Both are comfortably significant (above roughly 1.96), and they roughly agree — because event-induced variance was mild here.

Worked example 2: a volatile case where they diverge

Now suppose the same mean $\overline{SAR}=0.30$, but the earnings events are noisy enough that the cross-sectional spread is $s_{SAR} = 2.10$ — double the calm-period assumption, because earnings day genuinely shook things up. The naive standard test still reports $t_{\text{standard}} = \sqrt{100}\times 0.30 = 3.0$, unchanged, since it never looked at event-day dispersion at all. But BMP gives $t_{\text{BMP}} = 0.30/(2.10/10) = 0.30/0.21 \approx 1.43$ — no longer significant at conventional levels. The naive test was reporting false confidence: it measured how far 0.30 was from zero using a ruler (spread of 1) that was half the true event-day noise (spread of 2.10).

What this means in practice

Any event study on announcements known to move volatility — earnings, M&A news, regulatory decisions, guidance changes — should default to BMP or an equivalent variance-corrected test rather than the naive standardized-return $t$-test, especially with modest sample sizes where the cross-sectional standard deviation is the main defense against event-induced variance inflating false positives. Reporting only the naive $t$-statistic in these settings is a well-known way to overstate the reliability of an "abnormal return" finding, and reviewers and practitioners specifically look for a variance-adjusted test before trusting an event-study result.