'Every number is guilty unless proven innocent.' (Rao, 1997: 152)
Event studies are generally the first step in a twostep analysis process that aims at identifying the determinants of stock market responses to distinct event types. They produce as an outcome abnormal returns (ARs), which are cumulated over time to cumulative abnormal returns (CARs) and then 'averaged'  in the case of so called sample studies  over several observations of identical events to AARs and CAARs  where the second 'A' stands for 'average'. These event study results are then oftentimes used as dependent variables in regression analyses.
\begin{equation}H_0: μ = 0 \end{equation}
\begin{equation}H_1: μ \neq 0 \end{equation}
Event studies may imply a hierarchy of calculations, with ARs being compounded to CARs, which can again be 'averaged' to CAAR in crosssectional studies (sometimes also called 'sample studies'). In longrun Event Studies instead of CAR the BuyandHold abnormal return (BHAR) is widely use. BHAR can alos be 'averaged' to ABHAR in crosssectional studies. There is a need for significance testing at each of these levels. μ in the above mentioned equations may thus represent ARs, CARs, BHARs, AARs, CAARs, and ABHARs. Let's shortly revisit these six different forms of abnormal return calculations, as presented in the introduction and longrun eventstudies:
\begin{equation}AR_{i,t}=R_{i,t}E[R_{i,t}\Omega_{i,t}] \end{equation}
\begin{equation}AAR_{t}= \frac{1}{N} \sum\limits_{i=1}^{N}AR_{i,t} \end{equation}
\begin{equation}CAR_{i}=\sum\limits_{t=T_1 + 1}^{T_2} AR_{i,t} \end{equation}
\begin{equation}BHAR_{i}=\prod\limits_{t=T_1 + 1}^{T_2} (1 + R_{i,t})  \prod\limits_{t=T_1 + 1}^{T_2} (1 + E[R_{i,t}\Omega_{i,t}])\end{equation}
\begin{equation}CAAR=\frac{1}{N}\sum\limits_{i=1}^{N}CAR_i\end{equation}
\begin{equation}ABHAR=\frac{1}{N}\sum\limits_{i=1}^{N}BHAR_{i}\end{equation}
To preserve the portfolio interpretation that CAAR offers, but the average of standardized cumulative abnormal returns not (e.g. in the Patell Test), we report a precisionweighted CAAR value
\begin{equation}PWCAAR=\sum\limits_{i=1}^{N}\sum\limits_{t=T_1 + 1}^{T_2}\omega_i AR_{i, t}\end{equation}
where $$\omega_i = \frac{\left(\sum\limits_{t=T_1 + 1}^{T_2} S^2_{AR_{i,t}}\right)^{0.5}}{ \sum\limits_{1=1}^{N}\left(\sum\limits_{t=T_1 + 1}^{T2}S^2_{AR_{i,t}}\right)^{0.5}}$$
and $S^2_{AR_{i,t}}$ is the forecasterror corrected standard deviation defined in EQ $\ref{EQ:FESD}$.
The literature on event study test statistics is very rich, as is the range of significance tests. Generally, significance tests can be grouped in parametric and nonparametric tests (NPTs). Parametric tests assume that individual firm's abnormal returns are normally distributed, whereas nonparametric tests do not rely on any such assumptions. In research, scholars commonly complement a parametric test with a nonparametric tests to verify that the research findings are not due to eg. an outlier (see Schipper and Smith (1983) for an example). Table 1 provides an overview and links to the formulas of the different test statistics.
Table 1: 'Recommended' Significance Tests per Test Level (note: work in progress/ draft)
Null hypothesis tested  Parametric tests  Nonparametric tests  Test level 

$H_0: AR = 0$  AR ttest  Individual Event  
$H_0: AAR = 0$  AAR ttest, Patelltest, adjusted Patelltest, BMPtest, adjusted BMPtest, and skewness adjusted ttest  Cowan GSIGNtest, GRANK tTest, and GRANK zTest  Sample of Events 
$H_0: CAR = 0$  Individual Event  
$H_0: CAAR = 0$ 
CAAR ttest, Patelltest, adjusted Patelltest, BMPtest, adjusted BMPtest, and skewness adjusted ttest 
Sample of Events


$H_0: BHAR = 0$  Individual Event  
$H_0: ABHAR = 0$  ABHAR ttest and skewness adjusted ttest  Sample of Events 
N.B.: test results are included in the output of EventStudyTools' abnormal return calculator
Parametric test statistics ground on the classic ttest. Yet, scholars have further developed the test to correct for the ttest's prediction error. The most widely used of these 'scaled' tests are those developed by Patell (1976) and Boehmer, Musumeci and Poulsen (1991). Among the nonparametric tests, the ranktest of Corrado (1989), and the signbased of Cowan (1992) are very popular. EST provides these test statistics (soon) in its analysis results reports.
Why different test statistics are needed
The choice of test statistic should be informed by the research setting and the statistical issues the analyzed data holds. Specifically, eventdate clustering poses a problem leading to (1) crosssectional correlation of abnormal returns, and (2) distortions from eventinduced volatility changes. Crosssectional correlation arises when sample studies focus on (an) event(s) which happened for multiple firms at the same day(s). Eventinduced volatility changes, instead, is a phenomenon common to many event types (e.g., M&A transactions) that becomes problematic when events are clustered. As consequence, both issues introduce a downward bias in the standard deviation and thus overstate the tstatistic, leading to an overrejection of the null hypothesis.
Comparison of test statistics
There have been several attempts to address these statistical issues. Patell (1976, 1979), for example, tried to overcome the ttest's proneness to eventinduced volatility by standardizing the event window's ARs. He used the dispersion of the estimation interval's ARs to limit the impact of stocks with high return standard deviations. Yet, the test too often rejects the true null hypothesis, particularly when samples are characterized by nonnormal returns, low prices or little liquidity. Also, the test has been found to be still affected by eventinduced volatility changes (Campbell and Wasley, 1993; Cowan and Sergeant, 1996; Maynes and Rumsey, 1993, Kolari and Pynnonen, 2010). Boehmer, Musumeci and Poulsen (1991) resolved this latter issue and developed a test statistic robust against volatilitychanging events. Furthermore, the simulation study of Kolari and Pynnonen (2010) indicates an overrejection of the null hypothesis for both the Patell and the BMP test, if crosssectional correlation is ignored. Kolari and Pynnonen (2010) developed an adjusted version for both test statistics that accounts for crosssectional correlation.
The nonparametric rank test of Corrado and Zivney (1992) (RANK) applies restandardized event window returns and has proven robust against induced volatility and crosscorrelation. Sign tests are another category of tests. One advantage the tests’ authors stress over the common ttest is that they are apt to also identify small levels of abnormal returns. Moreover, scholars have recommend the used of nonparametric sign and rank tests for applications that require robustness against nonnormally distributed data. Past research (e.g. Fama, 1976) has argued that daily return distributions are more fattailed (exhibit very large skewness or kurtosis) than normal distributions, what suggests the use of nonparametric tests.
Several authors have further advanced the sign and ranked tests pioneered by Cowan (1992) and Corrado and Zivney (1992). Campbell and Wasley (1993), for example, improved the RANK test by introducing an incremental bias into the standard error for longer CARs, creating the CampbellWasley test statistic (CUMRANK). Another NPT is the generalized rank test (GRANK) test with a Student tdistribution with T2 degrees of freedom (T is the number of observations). It seems that GRANK is one of the most powerful instruments for both shorter and longer CARwindows.
The Cowan (1992) sign test (SIGN) is also used for testing CARs by comparing the share of positive ARs close to an event to the proportion from a normal period. SIGN's null hypothesis includes the possibility of asymmetric return distribution. Because this test considers only the sign of the difference between abnormal returns, associated volatility does not influence in any way its rejection rates. Thus, in the presence of induced volatility scholars recommend the use of BMP, GRANK, SIGN.
Most studies have shown that if the focus is only on single day ARs, the means of all tests stick close to zero. In the case of longer event windows, however, the mean values deviate from zero. Compared to their nonparametric counterparts, the Patell and the BMPtests produce means that deviate quite fast from zero, whereas the standard deviations of all tests gravitate towards zero. For longer event windows, academics recommend nonparametric over parametric tests.
Therefore, the main idea is that in case of longer eventwindows, the conclusions on the tests power should be very carefully drawn because of the many over or underrejections of the null hypothesis. Overall, comparing the different test statistics yields the following insights:
 Parametric tests based on scaled abnormal returns perform better than those based on nonstandardized returns
 Generally, nonparametric tests tend to be more powerful than parametric tests
 The generalized rank test (GRANK) is one of the most powerful test for both shorter CARwindows and longer periods
Table 2 provides a short summary of the individual test statistics discussed above.
Table 2: Summary Overview of Main Test Statistics
#  Name [synonym] 
Key Reference  Antecedent  Strengths  Weaknesses 



1  ttest [ORDIN] 



2  Standardized residual test [Patelltest] 
Patell (1976)  ORDIN 


3  Adjusted Patelltest  Kolari and Pynnönen (2010)  Patelltest 


4 
Standardized crosssectional test 
Boehmer, Musumeci and Poulsen (1991)  Patelltest 


5  Adjusted BMPtest 
Kolari and Pynnönen (2010)  BMPtest 


6  Skewness adjusted test  Hall (1992) 




7 
Corrado Rank test 
Corrado and Zivney (1992) 


8  Generalized rank t test [GRANKT] 
Kolari and Pynnönen (2011)  RANK 
Accounts for


9  Generalized rank z test [GRANKZ]  Kolari and Pynnönen (2011)  RANK  see GRANKT  
10  Sign test [SIGN] 
Cowan (1992) 



11  Cowan Generalized sign test [GSIGN] 
Cowan (1992)  SIGN  
12  Jackknife test  Giaccotto and Sfiridis (1996)  ORDIN  
13  Wilcoxon signedrank test  Wilcoxon (1945) 

Insights about strenghts and weaknesses were compiled from Kolari and Pynnonen (2011)
Formulas, acronyms, and the decision rule applicable to all test statistics
Let $L_1 = T_1  T_0 + 1$ the estimation window length with $T_0$ as the 'earliest' day of the estimation window, and $T_1$ the 'latest' day of the estimation window relative to the event day and $L_2 = T_2  T_1$ the event window length with $T_2$ as the 'latest day' of the event window relative to the event day. Define $N$ as the sample size (i.e. number of events / observations); $S_{AR_i}$ represent the standard deviation as produced by the regression analysis over the estimation window according to the following formula
Note: * For the ABHAR ttest replace in the equations $CAAR$ by $ABHAR$ and $CAR_i$ by $BHAR_i$.
[2] Patelltest (standardized residual test)
[2] Kolari and Pynnönen adjusted Patelltest (adjusted standardized residual test)
Kolari and Pynnönen (2010) propose a modification to the Patelltest to account for crosscorrelation of the abnormal returns. Using the standardized abnormal returns ($SAR_{i,t}$) defined as in (EQ: $\ref{eq:sar}$), and defining $\bar r$as the average of the sample crosscorrelation of the estimation period abnormal returns, the test statistic for $H_0: AAR = 0$ of the adjusted Patelltest is
$$z_{patell, t}=z_{patell, t} \sqrt{\frac{1}{1 + (N  1) \overline r}},$$
where $z_{patell, t}$ is the Patell test statistic. It is easily seen that if the correlation $\overline r$ is zero, the adjusted test statistic reduces to the original Patell test statistic. Assuming the squareroot rule holds for the standard deviation of different return periods, this test can be used when considering Cumulated Abnormal Returns ($H_0: CAAR = 0$):
$$z_{patell}=z_{patell} \sqrt{\frac{1}{1 + (N  1) \overline r}}.$$
[4] BMPtest (standardized crosssectional test)
Similarly, Boehmer, Musumeci and Poulsen (1991) proposed a standardized crosssectional method which is robust to the variance induced by the event. Test statistics on day $t$ ($H_0: AAR = 0$) in the event window is given by
$$z_{bmp, t}= \frac{ASAR_t}{\sqrt{N}S_{ASAR_t}},$$
with $ASAR_t$ defined as for Patelltest [2] and with standard deviation
$$S^2_{ASAR_t} = \frac{1}{N1}\sum\limits_{i=1}^{N}\left(SAR_{i, t}  \frac{1}{N} \sum\limits_{l=1}^N SAR_{l, t} \right)^2.$$
Furthermore, EST API provides the test statistic for testing $H_0: CAAR = 0$ given by
$$z_{bmp}=\sqrt{N}\frac{\overline{CSAR}}{S_{\overline{CSAR}}},$$
where $\overline{CSAR}$ is the averaged cumulated standardized abnormal returns across the $N$ firms,
$$\overline{CSAR} = \frac{1}{N}\sum\limits_{i=1}^{N}CSAR_i,$$
with standard deviation
$$S^2_{\overline{CSAR}} = \frac{1}{N1} \sum\limits_{i=1}^{N} \left(CSAR_i  \overline{CSAR}\right)^2.$$
[5] Kolari and Pynnönen adjusted BMPtest (adjusted standardized crosssectional test)
Kolari and Pynnönen (2010) propose a modification to the BMPtest to account for crosscorrelation of the abnormal returns. Using the standardized abnormal returns ($SAR_{i,t}$) defined as in the previous section, and defining $\bar r$as the average of the sample crosscorrelation of the estimation period abnormal returns, the test statistic for $H_0: AAR = 0$ of the adjusted BMPtest is
$$z_{J, t}=z_{bmp, t} \sqrt{\frac{1 \overline r}{1 + (N  1) \overline r}},$$
where $z_{bmp, t}$ is the BMP test statistic. It is easily seen that if the correlation $\overline r$ is zero, the adjusted test statistic reduces to the original BMP test statistic. Assuming the squareroot rule holds for the standard deviation of different return periods, this test can be used when considering Cumulated Abnormal Returns ($H_0: CAAR = 0$):
$$z_{J}=z_{bmp} \sqrt{\frac{1 \overline r}{1 + (N  1) \overline r}}.$$
The skewnessadjusted ttest, introduced by Hall 1992, corrects the crosssectional ttest for skewd abnormal return distribution. This test is applyable for averaged abnormal return ($H_0: AAR = 0$), the cumulative averaged abnormal return ($H_0: CAAR = 0$), and the averaged buyandhold abnormal return ($H_0: ABHAR = 0$). In the following, we limit us on the situation of cumulative averaged abnormal returns. First, reviside the crosssectional standard deviation
$$S^2_{CAAR} = \frac{1}{N1} \sum\limits_{i=1}^{N}(CAR_i  CAAR)^2.$$
The skewness parameter is then defined as
$$\gamma = \frac{N}{(N2)(N1)} \sum\limits_{i=1}^{N}(CAR_i  CAAR)^3S^{3}_{CAAR} .$$
Furthermore, define
$$S = \sqrt{N}\frac{CAAR}{S_{CAAR}}.$$
Then the skewness adjusted test statistic for CAAR is
$$t_{skew} = S + \frac{1}{3}\gamma S^2 + \frac{1}{27}\gamma^2S^3 + \frac{1}{6N}\gamma.$$
For a further discussion see Hall (1992).
In a first step, the Corrado's (1989) rank test transforms abnormal returns into ranks. Ranking is done for all abnormal returns of both the event and the estimation period. If ranks are tied, the midrank is used. For adjusting on missing values Corrado and Zyvney (1992) suggested a standardization of the ranks by the number of nonmissing values $M_i$ plus 1
$$K_{i, t}=\frac{rank(AR_{i, t})}{1 + M_i} $$.
The rank statistic for testing on a single day ($H_0: AAR = 0$) is then given by
$$t_{rank} = \frac{\overline{K}_t  0.5}{S_{\overline{K}}},$$
where $\overline{K}_t = \frac{1}{N_t}\sum\limits_{i=1}^{N_t}K_{i, t}$, $N_t$ is the number of nonmissing returns across firms, and
$$S^2_{\overline{K}} = \frac{1}{L_1 + L_2} \sum\limits_{t=T_0}^{T_2} \frac{N_t}{N}\left(\overline{K}_t  0.5 \right)^2$$.
When analyzing a multiday event period, Campell and Wasley (1993) define the RANKtest considering the sum of the mean excess rank for the event window as follows ($H_0: CAAR = 0$):
$$t_{cumrank} =\sqrt{L_2} \left(\frac{\overline{K}_{T_1, T_2}  0.5}{S_{\overline{K}}}\right),$$
where $\overline{K}_{T_1, T_2} = \frac{1}{L_2} \sum\limits_{t=T_1 + 1}^{T_2}\overline{K}_t$ is the mean rank across firms and time in event window. This test is also known as cumrank test. By adjusting the last day in the event window $T_2$, one can get a series of test statistics as definded by Campell and Wasley (1993).
Note 1: The web based request form separates between rank and cumrank test statistic. In the API the cumrank test is classed into rank test (CAAR).
Note 2: The adjustment for event induced variance as done by Campell and Wasley (1993) is omitted here and may be implemented in a future version. In a such case we recommend the GRANKT or GRANKZ test.
In order to account for possible eventinduced volatility, the GRANK test squeezes the whole event window into one observation, the socalled 'cumulative event day'. First, define the standardized cumulative abnormal returns of firm $i$ in the event window
$$SCAR_{i}=\frac{CAR_{i}}{S_{CAR_{i}}},$$
where $S_{CAR_{i}}$ is the standard deviation of the prediction errors in the cumulative abnormal returns of firm $i$. The standardized CAR value $SCAR_{i}$ has an expectation of zero and approximately unit variance. To account for eventinduced volatility $S_{CAR_{i}}$ is restandardized by the crosssectional standard deviation
$$SCAR^*_{i}=\frac{SCAR_{i}}{S_{SCAR}},$$
where
$$S^2_{SCAR}=\frac{1}{N1} \sum\limits_{i=1}^N \left(SCAR_{i}  \overline{SCAR} \right) \quad \text{ and } \quad \overline{SCAR} = \frac{1}{N} \sum\limits_{i=1}^N SCAR_{i}.$$
By construction $SCAR^*_{i}$ has again an expectation of zero with unit variance. Now, let's define the generalized standardized abnormal returns ($GSAR$):
$$GSAR_{i, t} = \left\{ \eqalign{ SCAR^*_i &\text{ for t in event window} \\ SAR_{i ,t} &\text{ for t in estimation window}} \right.$$
The CAR window is also considered as one time point, the other time points are considered GSAR is equal to the standardized abnormal returns. Define on this $L_1 + 1$ points the standardized ranks:
$$K_{i, t}=\frac{rank(GSAR_{i, t})}{L_1 + 2}0.5$$
The generalized rank tstatistic is then defined as:
$$t_{grank}=Z\left(\frac{L_1  1}{L_1  Z^2}\right)^{1/2}$$
with
$$Z=\frac{\overline{K_{0}}}{S_{\overline{K}}},$$
$t=0$ indicates the cumulative event day, and
$$S^2_{\overline{K}}=\frac{1}{L_1 + 1}\sum\limits_{t \in CW}\frac{N_t}{N}\overline{K}_t^2$$
with CW representing the combined window consisting of estimation window and the cumulative event day, and
$$\overline{K}_t=\frac{1}{N_t}\sum\limits_{i=1}^{N_t}K_{i, t}.$$
$t_{grank}$ is tdistributed with $L_1  1$ degrees of freedom.
Using some facts about statistics on ranks, we get the standard deviation of $\overline{K_{0}}$
$$S^2_{\overline{K_{0}}} =\frac{L_1}{12N(L_1 + 2)}.$$
By this calculation following test statistic can be defined
$$z_{grank} = \frac{ \overline{K_{0}} }{ S_{\overline{ K_{0} } } } = \sqrt{ \frac{12N(L_1+ 2)}{L_1}} \overline{K_{0}},$$
which converges under null hypothesis fast to the standard normal distribution as the firms $N$ increase.
This sign test has been proposed by Cowan (1991) and builds on the ratio of positive cumulative abnormal returns $\hat{p}$ present in the event window. Under the null hypothesis, this ratio should not significantly differ from 0.5.
$$t_{sign}= \sqrt{N}\left(\frac{\hat{p}0.5}{\sqrt{0.5(10.5)}}\right)$$
Under the Null Hypothesis of no abnormal returns, the number of stocks with positive abnormal cumulative returns ($CAR$) is expected to be in line with the fraction $\hat{p}$ of positive $CAR$ from the estimation period. When the number of positive $CAR$ is significantly higher than the number expected from the estimated fraction, it is suggested to reject the Null Hypothesis.
The fraction $\hat{p}$ is estimated as
$$\hat{p}=\frac{1}{N}\sum\limits_{i=1}^{N}\frac{1}{L_1}\sum\limits_{t=T_0}^{T_1}\varphi_{i, t},$$
where $\varphi_{i,t}$ is $1$ if the sign is positive and $0$ otherwise. The Generalized sign test statistic ($H_0: CAAR = 0$) is
$$z_{gsign}=\frac{(wN\hat{p})}{\sqrt{N\hat{p}(1\hat{p})}},$$
where $w$ is the number of stocks with positive cumulative abnormal returns during the event period. For the test statistic, a normal approximation of the binomial distribution with the parameters $\hat{p}$ and $N$, is used.
Note 1: This test is based on the paper of Cowan, A. R. (1992).
Note 2: EST provides GSIGN test statistics also for single days ($H_0: AAR = 0$) in the event time period.
Note 3: The GSIGN test is based on the traditional SIGN test where the null hypothesis assumes a binomial distribution with parameter $p=0.5$ for the sign of the $N$ cumulative abnormal returns.
Note 4: If $N$ is small, the normal approximation is inaccurate for calculating the pvalue, in such case we recommend to use the binomial distribution for calculating the pvalue.
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References and further readings
Boehmer, E., Musumeci, J. and Poulsen, A. B. 1991. 'Eventstudy methodology under conditions of eventinduced variance'. Journal of Financial Economics, 30(2): 253272.
Campbell, C. J. and Wasley, C. E. 1993. 'Measuring security performance using daily NASDAQ returns'. Journal of Financial Economics, 33(1): 7392.
Campbell, J., Lo, A., MacKinlay, A.C. 1997. 'The econometrics of financial markets'. Princeton: Princeton University Press.
Corrado, C. J. and Zivney, T. L. 1992. 'The specification and power of the sign test in event study hypothesis test using daily stock returns'. Journal of Financial and Quantitative Analysis, 27(3): 465478.
Cowan, A. R. (1992). 'Nonparametric event study tests'. Review of Quantitative Finance and Accounting, 2: 343358.
Cowan, A. R. and Sergeant, A. M. A. 1996. 'Trading frequency and event study test specification'. Journal of Banking and Finance, 20(10): 17311757.
Fama, E. F. 1976. Foundations of Finance. New York: Basic Books.
Giaccotto C. and Sfiridis J. M. 'Hypothesis testing in event studies: The case of variance changes'. Journal of Econometrics and Business, 48(4): 349370.
Hall, P., 1992, 'On the Removal of Skewness by Transformation'. Journal of the Royal Statistical Society. Series B (Methodological) Vol. 54, No. 1, pp. 221228
Kolari, J. W. and Pynnonen, S. 2010. 'Event study testing with crosssectional correlation of abnormal returns'. Review of Financial Studies, 23(11): 39964025.
Kolari, J. W. and Pynnonen, S. 2011. 'Nonparametric rank tests for event studies'. Journal of Empirical Finance, 18(5): 953971.
Maynes, E. and Rumsey, J. 1993. 'Conducting event studies with thinly traded stocks'. Journal of Banking and Finance, 17(1): 145157.
Patell, J. A. 1976. 'Corporate forecasts of earnings per share and stock price behavior: Empirical test'. Journal of Accounting Research, 14(2): 246276.
Rao, R. 1997. 'Statistic and truth. Putting chance to work'. World Scientific Publishing Co Ltd: Singapore.
Schipper, K. and Smith, A. 1983. 'Effects of recontracting on shareholder wealth: The case of voluntary spinoffs.' Journal of Financial Economics, 12(4): 437467.
Wilcoxon, F. (1945). 'Individual comparison by ranking methods'. Biometrics Bulletin, 1(6): 8083.