Significance tests

Significance tests decide whether the abnormal returns around an event are statistically distinguishable from zero. Parametric tests (t-test, Patell Z, BMP) assume abnormal returns are approximately normal and gain reliability as the sample grows; non-parametric tests (sign, generalized sign, Corrado rank) use only the signs or ranks of abnormal returns and stay valid under non-normality, skewness, and heteroskedasticity. Because the two families fail under different conditions, applied researchers typically report at least one of each and flag any disagreement (Brown and Warner, 1985; Kothari and Warner, 2007). The spin-off study of Schipper and Smith (1983) is one applied example of this practice.

Every statistic on this page is the same object viewed from a different angle: a ratio whose numerator is a mean abnormal return and whose denominator is an estimate of that mean's standard error. The tests differ only in how the denominator is built. Read in sequence, they form an evolutionary chain in which each newer test replaces a denominator that is too small under some real-world violation with one that is correctly sized. Hold that idea and the formulas below stop being a list and become a single argument.

What is being tested

The null hypothesis is that the event has no effect on returns: \(H_0: E(AR) = 0\) at the firm-day level, or \(H_0: E(CAR) = 0\) and \(H_0: E(CAAR) = 0\) at the cumulative level. A test can fail in two directions. It can over-reject, declaring an effect that is not there (a specification problem, controlled by the test's size), or it can under-reject, missing a real effect (a power problem). Most of the refinements below exist to control over-rejection without sacrificing too much power. The reported \(p\)-value answers a conditional question: it is the probability of seeing an abnormal return at least this extreme if the event had truly had no effect, not the probability that the event had no effect, and statistical significance is not the same as economic materiality.

A subtler caveat is the joint-hypothesis, or bad-model, problem: every abnormal return is measured relative to a chosen return-generating model (constant-mean, market model, or a factor model), so a "significant" result jointly tests the event and the model used to define normal returns (Kothari and Warner, 2007; Fama, 1998). The tests on this page are designed for short event windows of a few days, where this concern is mild and the statistics are well behaved. Long-horizon inference (buy-and-hold abnormal returns, calendar-time portfolios) is a separate and substantially harder problem, treated in its own section below.

Three failure modes the tests exist to fix

The plain one-sample t-test on a column of abnormal returns is the natural starting point, and it is correct under three assumptions that daily stock data routinely break. Naming the violations once makes every refinement that follows legible as the patch for a specific defect (Brown and Warner, 1985).

1. Non-normality and fat tails. Daily returns are leptokurtic and skewed, so a single extreme day can dominate a mean and the normal reference distribution mis-sizes the test. This is the motivation for the non-parametric sign and rank tests, which discard the misbehaving magnitudes and keep only direction or order.
2. Event-induced variance. The event itself raises announcement-day return volatility, so the pre-event (estimation-window) variance understates the true denominator and the test over-rejects. This is the defect that the standardized cross-sectional (BMP) test was designed to remove (Boehmer, Musumeci, and Poulsen, 1991).
3. Cross-sectional correlation under event-date clustering. When many firms share the same event date their abnormal returns move together, the effective sample size is far below \(N\), and the naive standard error is too small. This is the failure mode the Kolari-Pynnonen adjustments correct (Kolari and Pynnonen, 2010), and it is the single most under-appreciated source of false positives in applied event studies.

The house rule that follows is the one serious referees expect: run one parametric and one non-parametric test that target the threats your sample actually faces, report both, and treat agreement as the real evidence.

Foundations

For a single instance, the random variable is the abnormal return on the event day (\(AR\)) or the cumulative abnormal return over the event window (\(CAR\)). Abnormal returns are the residuals of a benchmark model fitted on a pre-event estimation window that is disjoint from the event window (MacKinlay, 1997). The most common benchmark is the market model, \(R_{i,t} = \alpha_i + \beta_i R_{m,t} + \varepsilon_{i,t}\), estimated by OLS, so that \(AR_{i,t} = R_{i,t} - (\hat{\alpha}_i + \hat{\beta}_i R_{m,t})\). Across \(N\) firms or repeated events:

\[ AAR_{t} = \frac{1}{N} \sum_{i=1}^{N} AR_{i,t} \quad CAR_{i} = \sum_{t=T_{1}+1}^{T_{2}} AR_{i,t} \quad CAAR = \frac{1}{N} \sum_{i=1}^{N} CAR_{i} \]

The estimation window \(\{T_0, \ldots, T_1\}\) has length \(L_1 = T_1 - T_0 + 1\) (typically 120 to 250 trading days); the event window \(\{T_1+1, \ldots, T_2\}\) has length \(L_2 = T_2 - T_1\). The firm-level estimation-window variance:

\[ S^{2}_{AR_{i}} = \frac{1}{M_{i} - K} \sum_{t=T_{0}}^{T_{1}} AR_{i,t}^{2} \]

where \(M_i\) is non-missing returns in the estimation window and \(K\) is the model's degrees of freedom (1 for constant mean, 2 for market model, 4 for FF3). There is no \(\overline{AR}\) term inside the sum because estimation-window residuals from a model with an intercept are mean-zero by construction. The cumulative variance \(S^{2}_{CAR_{i}} = L_{2}\, S^{2}_{AR_{i}}\) used below assumes the event-window abnormal returns are serially uncorrelated and homoskedastic; in practice it understates the true \(CAR\) variance, which also absorbs estimation error and serial correlation (precisely what the generalized rank test addresses).

Cross-sectional versus time-series variance

A recurring choice separates two families of denominator. A time-series standard error estimates the spread of the portfolio's daily abnormal return over the estimation window (used by the crude dependence adjusted t and as a backbone for the rank test's null variance); it collapses cross-firm dependence into one number but cannot see firm-specific variance differences. A cross-sectional standard error estimates the spread of the abnormal returns across firms on the event day (used by the cross-sectional t and BMP); it lets the event-day data report any volatility the event itself induced, but assumes the firms are mutually independent on that day. The whole sequence below can be read as a search for a denominator that captures both the event-day cross-sectional spread and the cross-firm dependence at the same time, which is exactly what the adjusted BMP and generalized rank statistics deliver.

Why so many test statistics? The evolutionary chain

The tests are best understood as a chain in which each newer test changes one piece of the denominator to neutralize a specific failure of the previous one. The plain t-test over-rejects when firms have heterogeneous variances, so Patell (1976) standardizes each abnormal return by its own forecast error before aggregating, replacing one pooled denominator with \(N\) firm-specific ones. Patell still assumes the event does not change return variance, but events routinely spike announcement-day volatility, which makes Patell over-reject; the BMP test of Boehmer, Musumeci, and Poulsen (1991) fixes this by re-dividing the standardized returns by their own event-day cross-sectional spread, so the denominator widens automatically when the event makes everyone volatile. BMP in turn assumes the abnormal returns are cross-sectionally independent, which fails when events cluster on the same calendar dates; the adjusted tests of Kolari and Pynnonen (2010) deflate the statistic for that correlation. Finally, all parametric tests can break under non-normality, motivating the rank and sign tests, and rank tests themselves lose power on multi-day cumulative returns, motivating the generalized rank test that handles non-normality, induced variance, and clustering together.

Clustering matters more than intuition suggests, and this is the differentiator most teaching treatments omit. Even a small average pairwise correlation \(\bar{r}\) among abnormal returns inflates the true standard error of the mean by a factor of approximately \(\sqrt{1 + (N-1)\bar{r}}\) (Kothari and Warner, 2007, eq. 10). With \(\bar{r} = 0.02\) and \(N = 100\) the true standard error is about \(1.4\) times the naive estimate, so a test that ignores the correlation rejects the null far more often than its nominal size; as \(N\) grows the over-rejection gets worse, not better, because the inflation term scales with \(N\). This inflation factor is exactly what the \((1 + (N-1)\bar{r})\) denominators in the adjusted tests are correcting.

When to use which test

The matrix below summarizes the robustness profile of each test; "robust to event-induced variance" means the test holds its size when the event itself raises return volatility, and "robust to cross-sectional correlation" means it holds its size when events cluster in calendar time.

Two refinements the chooser table and the matrix do not capture. Under thin trading without a variance change, the Corrado rank test is best specified and most powerful, but when variance also rises it becomes misspecified and the generalized sign test is the fallback (Cowan and Sergeant, 1996). And as a default progression, the modern literature treats the plain Patell Z and plain cross-sectional t as over-rejecting under conditions that are the norm rather than the exception, so the preferred defaults are BMP, the Adjusted BMP, and, for robustness to non-normality, event-induced variance, and clustering at once, the generalized rank test of Kolari and Pynnonen (2011).

Parametric tests

T Test

Asks whether a single firm's abnormal return is large relative to its own estimation-window scatter. Tests \(H_0: E(AR_{i,0}) = 0\) (event day) or \(H_0: E(CAR_i) = 0\) (event window). Assumptions: the firm's abnormal returns are approximately normal, serially uncorrelated, and homoskedastic across the estimation and event windows. Simplest test; sensitive to event-induced volatility and to non-normality (Brown and Warner, 1985). Distributed Student-t.

\[ t = \frac{AR_{i,t}}{S_{AR_{i}}}, \qquad t = \frac{CAR_{i}}{S_{CAR_{i}}}, \quad S^{2}_{CAR_{i}} = L_{2}\, S^{2}_{AR_{i}} \]

Cross-Sectional T (CSect T)

Asks whether the average abnormal return is large relative to its spread across firms, which lets the event-day data report any event-induced volatility itself. Tests \(H_0: E(AAR_0) = 0\) or \(H_0: E(CAAR) = 0\). Assumptions: abnormal returns are cross-sectionally (i) independent, (ii) identically distributed with a common variance, and (iii) approximately normal, so the statistic is \(t_{N-1}\). When it fails: it equal-weights every firm, so one extremely volatile firm can dominate the mean; and because it never standardizes per firm, it loses power under heterogeneous variances and over-rejects under event-date clustering, where the \(N\) returns are not independent and the true standard error exceeds \(S_{AAR,0}/\sqrt{N}\). Distributed \(t_{N-1}\).

\[ t = \sqrt{N}\, \frac{AAR_{0}}{S_{AAR,0}}, \quad S^{2}_{AAR,0} = \frac{1}{N-1} \sum_{i=1}^{N}(AR_{i,0} - AAR_{0})^{2} \] \[ t = \sqrt{N}\, \frac{CAAR}{S_{CAAR}}, \quad S^{2}_{CAAR} = \frac{1}{N-1} \sum_{i=1}^{N}(CAR_{i} - CAAR)^{2} \]

Crude Dependence Adjusted T (CDA T)

Collapses cross-sectional dependence into a single number by using the time-series standard deviation of the portfolio mean over the estimation window, at the cost of ignoring firm-level variance differences. Assumptions: a constant (not necessarily zero) cross-correlation that the estimation-window portfolio variance already absorbs; firm variances treated as homogeneous. Same null. Introduced in Brown and Warner (1980) and carried into the daily-returns setting by Brown and Warner (1985).

\[ t = \frac{AAR_{0}}{S_{AAR}}, \quad S^{2}_{AAR} = \frac{1}{M-1} \sum_{t=T_{0}}^{T_{1}}\!\left( AAR_{t} - \overline{AAR} \right)^{2} \]

Patell Z (standardized residual test)

Divides each abnormal return by its own forecast-error-corrected standard deviation so that noisier firms count less, then aggregates. The key technical point most treatments skip: the event-day abnormal return is an out-of-sample forecast from the estimation regression, so its variance is larger than the in-sample residual variance and must be inflated. The three terms below are the base residual variance (the \(1\)), the estimation error in the fitted mean (the \(1/M_i\)), and a forecast-extrapolation penalty that grows when the event-day market return \(R_{m,0}\) sits far from its estimation-window average \(\overline{R}_m\) (Patell, 1976). Assumptions: the standardized abnormal returns are cross-sectionally independent, normal, and (critically) homoskedastic, that is, no event-induced variance; under the null each has unit variance. Standardization buys efficiency, not robustness, so the test is still sensitive to event-induced volatility and to cross-sectional correlation. Distributed \(N(0,1)\).

\[ SAR_{i,0} = \frac{AR_{i,0}}{S_{AR_{i,0}}}, \quad S^{2}_{AR_{i,0}} = S^{2}_{AR_{i}}\!\left( 1 + \frac{1}{M_{i}} + \frac{(R_{m,0} - \overline{R}_{m})^{2}}{\sum_{t=T_{0}}^{T_{1}}(R_{m,t} - \overline{R}_{m})^{2}} \right) \]

The aggregated statistic divides the sum of standardized returns by the standard deviation of that sum, so \(S_{ASAR}\) is the scale of \(ASAR_0\), not a per-firm quantity. The variance term is written generically in the model degrees of freedom \(K\) (it reduces to \((M_i - 2)/(M_i - 4)\) for the market model, where \(K = 2\)):

\[ z = \frac{ASAR_{0}}{\sqrt{\sum_{i=1}^{N} \frac{M_{i}-K}{M_{i}-K-2}}}, \quad ASAR_{0} = \sum_{i=1}^{N} SAR_{i,0} \]

Adjusted Patell Z (Kolari-Pynnonen 2010)

Deflates the Patell statistic for cross-sectional correlation among estimation-period abnormal returns, the bias that arises when events cluster on the same dates. The deflation factor is the square root of \((1 - \bar{r})\) over the variance-inflation term \((1 + (N-1)\bar{r})\): when \(\bar{r} = 0\) the factor is \(1\) and nothing changes; as \(\bar{r}\) grows it shrinks the statistic toward zero, undoing exactly the standard-error understatement that clustering causes (Kolari and Pynnonen, 2010). Assumptions: a roughly constant average cross-correlation summarized by a single \(\bar{r}\); designed for same-date clustering (partially overlapping windows are a harder case). Distributed \(N(0,1)\).

\[ z_{\text{adj}} = z \cdot \sqrt{\frac{1 - \bar{r}}{1 + (N-1)\,\bar{r}}} \]

where \(\bar{r}\) is the average pairwise sample cross-correlation of estimation-period abnormal returns. (Note the \((1 - \bar{r})\) numerator: it is part of the correct Kolari-Pynnonen form, not an optional refinement.)

Standardized Cross-Sectional T (BMP, Boehmer-Musumeci-Poulsen 1991)

Takes the Patell-standardized abnormal returns and re-divides them by their cross-sectional standard deviation in the event window, so any common volatility spike caused by the event is differenced out: Patell's denominator is fixed at the theoretical value \(1\), while BMP estimates the yardstick live from the event-day cross-section, so it stretches automatically when the event makes everyone more volatile. This robustness to event-induced variance is the reason BMP has displaced the plain Patell test as the default short-window parametric test; if there is no induced variance, \(S(SAR_0) \approx 1\) and BMP collapses back toward Patell. Construction: step one computes the Patell SARs of the prediction-error-corrected form above; step two runs a cross-sectional t-test on those SARs (the "hybrid" or standardized cross-sectional test). Assumptions still required: cross-sectional independence of the SARs (it does not fix clustering, which needs the adjustment below) and approximate normality of the SAR cross-section (Boehmer, Musumeci, and Poulsen, 1991). Distributed \(t_{N-1}\). In its textbook form the statistic is the cross-sectional mean of the standardized returns over their standard deviation:

\[ t = \sqrt{N}\, \frac{\overline{SAR}_{0}}{S(SAR_{0})}, \quad S^{2}(SAR_{0}) = \frac{1}{N-1} \sum_{i=1}^{N}\!\left( SAR_{i,0} - \overline{SAR}_{0} \right)^{2} \]

(This is algebraically identical to \(t = ASAR_0 / (\sqrt{N}\, S(SAR_0))\) because \(ASAR_0 = N\,\overline{SAR}_0\).) For the event window the standardized cumulative return uses the forecast-error-corrected \(CAR\) standard deviation:

\[ t = \sqrt{N}\, \frac{\overline{SCAR}}{S(SCAR)}, \quad SCAR_{i} = \frac{CAR_{i}}{S(CAR_{i})}, \quad S^{2}(CAR_{i}) = L_{2}\, S^{2}_{AR_{i,0}} \]

Adjusted Standardized Cross-Sectional T (Kolari-Pynnonen)

Augments the BMP test with the same cross-correlation adjustment as the Adjusted Patell, restoring correct size under event clustering. This is the headline contribution of Kolari and Pynnonen (2010): the single statistic now handles both event-induced variance (inherited from BMP) and cross-correlation (the new deflation term), and it extends cleanly to multi-day \(CAR\) testing, which is why it is the recommended default whenever events cluster (Kolari and Pynnonen, 2010). Distributed \(t_{N-1}\).

\[ t_{\text{adj}} = t \cdot \sqrt{\frac{1 - \bar{r}}{1 + (N-1)\,\bar{r}}} \]

Skewness Corrected T (Hall 1992)

Applies a polynomial transformation in the sample skewness so the t-statistic's distribution is closer to normal, improving size when abnormal returns are skewed (relevant for longer windows and smaller samples). The same skewness-adjusted polynomial originates with Johnson (1978). Assumptions: independent abnormal returns whose departure from normality is captured by third-moment skewness alone; it does not address kurtosis, induced variance, or clustering.

\[ t = \sqrt{N}\, \left( S + \tfrac{1}{3}\,\gamma\, S^{2} + \tfrac{1}{27}\,\gamma^{2}\, S^{3} + \tfrac{1}{6N}\,\gamma \right) \] \[ \gamma = \frac{N}{(N-2)(N-1)} \sum_{i=1}^{N} \frac{(AR_{i,0} - AAR_{0})^{3}}{S^{3}_{AAR,0}}, \quad S = \frac{AAR_{0}}{S_{AAR,0}} \]

Non-parametric tests

Sign Z

Asks whether the count of positive abnormal returns exceeds what chance alone (a 50/50 split) would produce; robust to skewness, weaker for longer windows. Assumptions: signs are independent across firms and the null positive probability is exactly \(0.5\), which fails when returns naturally drift, so the plain sign test can be misspecified for drifting samples (the defect the generalized version fixes). The basic sign test is classical and in event studies traces to Brown and Warner (1985). Distributed \(N(0,1)\).

\[ z = \frac{w - N \cdot 0.5}{\sqrt{N \cdot 0.5 \cdot 0.5}} \]

where \(w\) is the number of positive \(AR_{i,0}\) (or positive \(CAR_i\)).

Generalized Sign Z (Cowan 1992)

Replaces the 0.5 benchmark with \(\widehat{p}\), the empirical positive fraction estimated from the estimation window, which corrects for any natural asymmetry or drift in the return distribution: it judges the count of winners against the group's own historical win rate rather than a naive coin flip. Assumptions: independence of signs across firms; the estimation-window positive fraction is a valid null benchmark; a binomial-to-normal approximation for the count, which needs a reasonable \(N\). Documented properties: well specified for both exchange-listed and NASDAQ stocks, robust to right-skewness, and more stable than the rank test as the window lengthens, under higher variance, and under thin trading, at the cost of lower power than the rank test under ideal conditions; it ignores magnitude entirely, so it is weaker when effects are concentrated in size (Cowan, 1992). Following the standard implementation, \(\widehat{p}\) is the average across firms of each firm's estimation-window positive fraction. Distributed \(N(0,1)\).

\[ z = \frac{w - N \cdot \widehat{p}}{\sqrt{N \cdot \widehat{p}(1 - \widehat{p})}}, \quad \widehat{p} = \frac{1}{N}\sum_{i=1}^{N}\frac{1}{M_i}\sum_{t=T_0}^{T_1}\mathbf{1}\!\left[AR_{i,t} > 0\right] \]

Corrado Rank Test (Corrado 1989)

Pools each firm's estimation and event window, converts every abnormal return to a scaled rank, and asks whether the event-day mean rank sits unusually far from its midpoint of 0.5. The null standard deviation \(S_{\bar{K}}\) is computed from the time series of cross-sectional mean ranks over the whole pooled window, not from the event day alone, which is the source of its excellent specification under non-normality. Assumptions: abnormal returns are exchangeable under the null; crucially it does not require the cross-sectional distribution to be symmetric (an advantage over the sign test), and it makes no normality assumption. Documented properties and limits: better specified and more powerful than the t-test under non-normality and the best-specified non-parametric test for single-day windows, but power deteriorates for multi-day cumulative windows and it can be misspecified under event-induced variance and thin trading; the Corrado and Zivney (1992) re-standardization and the later generalized rank test are the fixes (Corrado, 1989; Corrado and Zivney, 1992). Distributed \(N(0,1)\).

\[ K_{i,t} = \frac{\operatorname{rank}(AR_{i,t})}{1 + M_{i} + L_{2,i}}, \quad \bar{K}_{t} = \frac{1}{N_{t}} \sum_{i=1}^{N} K_{i,t} \] \[ z = \frac{\bar{K}_{0} - 0.5}{S_{\bar{K}}}, \quad S_{\bar{K}}^{2} = \frac{1}{L_{1}+L_{2}} \sum_{t=T_{0}}^{T_{2}}\!\left( \bar{K}_{t} - 0.5 \right)^{2} \]

Generalized Rank T (Kolari-Pynnonen 2011)

Accounts for cross-sectional and serial correlation as well as event-induced volatility, the most robust of the lot (Kolari and Pynnonen, 2011). It works on generalized standardized abnormal returns (\(GSAR\)): the cumulative-return window is re-standardized by the cross-sectional standard deviation of the standardized abnormal returns, so for a single event day \(GSAR\) equals the Patell \(SAR\). The event day is indexed \(L_1 + 1\), one position after the \(L_1\) estimation-window observations. Ranks of the \(GSAR\) are demeaned and scaled, then transformed; the resulting statistic is Student-t with \(L_1 - 1\) (that is, \(T - 2\)) degrees of freedom. Assumptions: exchangeability of the generalized standardized returns under the null; it requires neither normality nor constant variance, which is what makes it the single most robust statistic in the battery and the natural non-parametric counterpart to the adjusted BMP.

\[ U_{i,t} = \frac{\operatorname{rank}(GSAR_{i,t})}{L_{1}+2} - 0.5, \quad \bar{U}_{t} = \frac{1}{N} \sum_{i=1}^{N} U_{i,t} \] \[ t = Z \cdot \sqrt{\frac{L_{1} - 1}{L_{1} - Z^{2}}}, \quad Z = \frac{\bar{U}_{L_{1}+1}}{S_{\bar{U}}} \]

Generalized Rank Z

Simpler normal-approximation variant of the Generalized Rank T, using the closed-form variance of the demeaned scaled rank. Distributed \(N(0,1)\).

\[ z = \frac{\bar{U}_{L_{1}+1}}{S_{\bar{U}_{L_{1}+1}}}, \quad S_{\bar{U}_{L_{1}+1}}^{2} = \frac{L_{1}}{12\, N\,(L_{1}+2)} \]

Wilcoxon Signed-Rank (Wilcoxon 1945)

Tests whether the distribution of \(AR_{i,0}\) is symmetric about zero (its central location is zero), using signed ranks so that both the sign and the magnitude of each abnormal return enter. It is preferred over the plain sign test when magnitudes are informative and over the Corrado rank test when only the event-day cross-section is of interest. Assumptions: the abnormal returns are independent and the distribution is symmetric under the null; a large-sample normal approximation is used. It is not available for the \(CAAR\) null.

\[ W = \sum_{i=1}^{N} \operatorname{sgn}(AR_{i,0}) \cdot \operatorname{rank}(|AR_{i,0}|), \quad E[W] = 0, \quad \operatorname{Var}[W] = \frac{N(N+1)(2N+1)}{6} \]

Permutation (randomization) Test

Resampling-based and distribution-free: the chosen statistic is recomputed many times under reshuffled event-day labels (or re-drawn pseudo-event days), and the \(p\)-value is the fraction of resampled statistics at least as extreme as the observed one. Concretely, recompute the statistic \(B\) times and set \(p = \#\{\,|\text{stat}^{*}| \ge |\text{stat}_{\text{obs}}|\,\}/B\). The approach is robust to non-normality (unlike the t-test) and exact at any sample size, but computationally more expensive; it follows the randomization argument of Corrado (1989) and the bootstrap-in-event-studies literature (Kramer, 2001).

Cumulative-window approaches

Each test above has a cumulative analog for the \(CAAR\) null: cumulative sign and generalized sign, cumulative BMP and Kolari-Pynnonen, and a cumulative rank statistic (CUMRANK-T/Z), matching the CAAR output of the analysis engine. The Wilcoxon test is the exception, as it is not defined for the CAAR null. Once abnormal returns are established as significant, a second stage typically explains their size: regressing \(CAR\) on firm characteristics with heteroskedasticity-robust standard errors, or comparing \(CAR\) across groups with a Welch t-test (two groups) or Welch ANOVA (three or more).

Long-horizon inference is a different problem

Everything above is built for short windows of a few days, where the bad-model problem is mild and the central-limit machinery behaves. Over horizons of one to five years the inference problem changes character, and the same tests are no longer reliable. Two tools dominate. Buy-and-hold abnormal returns (BHAR) compound each firm's return over the long window and subtract a matched benchmark, then test the cross-sectional mean; the difficulty is that long-run BHAR distributions are badly skewed and the t-test is misspecified, so skewness-adjusted or bootstrapped critical values are needed (Fama, 1998). The calendar-time portfolio approach (the Jaffe-Mandelker construction) instead forms a portfolio of all firms currently in event time each calendar month and tests its mean abnormal return against a factor model; because it aggregates in calendar time it automatically handles the cross-sectional dependence that overwhelms BHAR when events cluster, which is why it is the canonical remedy for heavy clustering. The deeper issue is the bad-model problem: at long horizons small errors in the expected-return model cumulate into large spurious abnormal returns, so long-run results are far more sensitive to the benchmark than short-run ones, and the literature treats them with corresponding caution. The expected-return benchmarks themselves (market, Fama-French, mean-adjusted, Scholes-Williams, GARCH) are detailed on the companion expected-return models page.

Worked example: from a formula to an ARC output

Consider a study of \(N = 50\) firms with an estimation-window length \(L_1 = 250\) and a one-day event window, where the average event-day abnormal return is \(AAR_0 = 1.2\%\). Suppose the cross-sectional standard deviation of the event-day abnormal returns is \(S_{AAR,0} = 3.3\%\), the cross-sectional standard deviation of the Patell-standardized returns is \(S(SAR_0) = 1.15\), the average standardized return is \(\overline{SAR}_0 = 0.37\), the summed standardized return gives a Patell aggregate, and 34 of the 50 abnormal returns are positive. These are exactly the quantities ARC reports in its test-statistics table, so each line below maps a formula on this page to one cell of that output.

• Cross-Sectional T: \(t = \sqrt{50}\,(0.012/0.033) \approx 2.57\). With \(t_{49}\) critical value \(\approx 2.01\), reject \(H_0\) at 5% two-sided.
• Patell Z: aggregating the standardized returns yields a Patell \(z \approx 2.6\). Against \(N(0,1)\), \(|z| > 1.96\) rejects \(H_0\) at 5% two-sided.
• BMP t: \(t = \sqrt{50}\,(0.37/1.15) \approx 2.27\). Against \(t_{49}\), this rejects \(H_0\) at 5%, but by a smaller margin, reflecting that BMP does not assume the event leaves variance unchanged.
• Sign Z: with \(w = 34\) positives, \(z = (34 - 25)/\sqrt{50 \cdot 0.25} = 9/3.54 \approx 2.55\), which also rejects at 5%.

Now suppose these 50 firms all reacted to one shared regulatory announcement, with an average pairwise correlation \(\bar{r} = 0.05\) among their estimation-window abnormal returns. The Kolari-Pynnonen deflation factor is \(\sqrt{(1 - 0.05)/(1 + 49 \cdot 0.05)} = \sqrt{0.95/3.45} \approx 0.525\), so the BMP statistic falls from \(2.27\) to about \(1.19\) and no longer rejects at 5%. The original "significance" was an artifact of treating 50 clustered firms as 50 independent observations. All four unadjusted tests agree here, which is the reassuring case; when tests disagree, the rule is to favor the more robust statistic that matches the suspected threat (BMP or Adjusted BMP under event-induced variance or clustering, a rank test under non-normality) and to report the disagreement rather than cherry-pick.