Boehmer–Musumeci–Poulsen test: Definition & Examples

The Boehmer–Musumeci–Poulsen test, usually called BMP, is a standardised cross-sectional test for a zero average standardised abnormal return in an event study. Its practical purpose is to allow the event itself to increase return variance: announcement-day noise should not be mistaken for evidence of a mean price response. Boehmer, Musumeci and Poulsen demonstrated that ignoring even modest event-induced variance can cause excessive rejection of a true null. Original paper.

BMP combines two operations. First, put each firm's abnormal return on its own forecast-error scale. Then estimate the uncertainty of their average from their dispersion across firms at the event, rather than assuming the historical scale remains correct.

First obtain the right standardised returns

For a market-model benchmark fitted by ordinary least squares,

AR_{i,t}=R_{i,t}-\hat\alpha_i-\hat\beta_iR_{m,t}.

Here $t$ is the firm's event date in calendar time. Each observation is subsequently aligned at event time zero, even if firms announce on different calendar dates. Estimate the benchmark in a separate, uncontaminated estimation window $\mathcal E_i$ with $M_i>2$ observations, positive residual variance and nonzero market-return variation. Under the classical market-model assumptions, the one-period forecast-error adjustment is

\begin{aligned} SAR_{i,t}&=\frac{AR_{i,t}}{\hat s_{i,t}},\\ \hat s_{i,t}^{\,2} &=s_i^2\left(1+\frac{1}{M_i} +\frac{(R_{m,t}-\bar R_{m,i})^2} {\sum_{u\in\mathcal E_i}(R_{m,u}-\bar R_{m,i})^2}\right). \end{aligned}

Here

s_i^2

is the estimation-window residual sum of squares divided by

M_i-2

, and

\bar R_{m,i}

is that window's mean market return. The extra terms reflect uncertainty in the estimated intercept and beta; the market-return term grows when the event-day benchmark lies far from its estimation-window mean. Dividing $AR_{i,t}$ by $s_i$ alone omits this forecast-error adjustment. EventStudyTools: Patell standardisation. The formula assumes a stable benchmark and serially uncorrelated, constant-variance estimation-period disturbances. The separate windows keep the event response out of the benchmark fit. MacKinlay, §5.

Then use the event cross section

Write the $N$ event-time-zero observations as $SAR_1,\ldots,SAR_N$ . Define

\overline{SAR}=\frac{1}{N}\sum_{i=1}^{N}SAR_i, \qquad s_{SAR}^2=\frac{1}{N-1}\sum_{i=1}^{N}(SAR_i-\overline{SAR})^2.

For $N>1$ and $s_{SAR}>0$ , the BMP statistic is

T_{BMP}=\frac{\overline{SAR}}{s_{SAR}/\sqrt N} =\frac{\sqrt N\,\overline{SAR}}{s_{SAR}}.

The denominator is an estimated standard error of the mean standardised return. Greater event-date dispersion raises this denominator. That is the adjustment which allows a variance increase without automatically treating it as a mean effect. The statistic is dimensionless; report raw abnormal returns separately to describe economic magnitude. Cowan, Appendix A.2, equation A.34.

Four observations: the arithmetic

Suppose four fictional, one-period abnormal returns have already received the forecast-error standardisation above:

(SAR_1,SAR_2,SAR_3,SAR_4)=(1,2,0,1).

Their mean is $1$ . The deviations from the mean are $0,1,-1,0$ , so their squared deviations sum to $2$ . Therefore

\begin{aligned} s_{SAR}^2&=\frac{2}{4-1}=\frac{2}{3},\\ T_{BMP}&=\frac{\sqrt4\times1}{\sqrt{2/3}} =\sqrt6\approx2.4495. \end{aligned}

These are standardised units, not returns of

1\%

2\%

. Four stipulated inputs demonstrate the calculation; they establish no empirical significance result.

Fictional one-period standardised abnormal returns for firms A to D are 1, 2, 0 and 1. Mean 1; squared deviations 0, 1, 1, 0; sample variance 2/3; BMP square root of 6, approximately 2.4495. Arithmetic only.

The BMP denominator uses dispersion across the event observations, after firm-specific forecast-error standardisation.

Changing the window changes the input

For a multiple-period question, standardise each firm's cumulative abnormal return by its window-level forecast-error standard deviation, then apply the cross-sectional construction to those standardised CARs. Shared intercept and beta estimates make forecast errors within the window covary. Simply adding daily SARs, or dividing a CAR by a one-day residual standard deviation, does not generally give this input. The window variance must reflect its length and any relevant covariance assumptions. MacKinlay, §5.B–C; Cowan, Appendix A.2, standardised CAR construction.

Increased variance and correlated firms are different problems

BMP's cross-sectional variance adjustment does not generally correct cross-sectional dependence. Firms with overlapping calendar windows can share shocks, so the average contains covariance terms that the ordinary cross-sectional standard error misses. Kolari and Pynnönen developed a modification of BMP specifically for this issue. Original correlation-adjustment paper.

Nor does the one-sample

t

algebra alone establish an exact reference distribution. An exact

t_{N-1}

result would require independent, identically distributed normal standardised observations. Estimated scales, heterogeneous event effects and small samples make calibration an additional question. BMP also cannot repair a misspecified expected-return model or separate the focal announcement from confounding news. Method and assumptions.

For the distinction between event-window dispersion and covariance between firms, see Cross-Sectional Dependence (public preview; full lesson requires membership).

Sources

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. Original paper's abstract: variance increases and test rejection rates.
Cowan, A. R. (2007). Eventus 8.0 User’s Guide, Appendix A.2, pp. 80–85: prediction-error standardisation, window correction and standardised cross-sectional test.
Kolari, J. W., & Pynnönen, S. (2010). Event Study Testing with Cross-sectional Correlation of Abnormal Returns. The Review of Financial Studies, 23(11), 3996–4025. Original paper's abstract: correlation under event-date clustering and the BMP modification.
EventStudyTools. Significance tests, sections “Patell Z” and “Standardized Cross-Sectional T”: one-period forecast-error correction, BMP construction and assumptions.
MacKinlay, A. C. (1997). Event Studies in Economics and Finance. Journal of Economic Literature, 35(1), 13–39. §5, pp. 19–21: event-time indexing, separate estimation window and parameter-estimation contributions to abnormal-return variance.

First obtain the right standardised returns

Then use the event cross section

Four observations: the arithmetic

Changing the window changes the input

Increased variance and correlated firms are different problems

Sources

Related terms

Ordinary least squares

Standard error