# Significance Tests

by Dr. Simon Müller | 29 Nov 2013

### Dr. Simon Müller

PhD in Maths (Stuttgart)

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Event studies are generally the first step in a two-step analysis process that aims at identifying the determinants of stock market repsonses 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.

Explaining abnormal returns by means of regression analysis, however, is only meaningful if the abnormal returns are significantly different from zero, and thus not the result of pure chance. This assessment wil be made by hypothesis testing. Following general principles of inferential statistics, the null hypothesis ($H_0$) thus maintains that there are no abnormal returns within the event window, whereas the alternative hypothesis ($H_1$) suggests the presence of ARs within the event window. Formally, the testing framework reads as follows:

$$H_0: μ = 0$$

$$H_1: μ \neq 0$$

Event studies may imply a hierarchy of calculations, with ARs being compounded to CARs, which can again be 'averaged' to CAARs in cross-sectional studies (sometimes also called 'sample studies'). There is a need for significance testing at each of these levels. μ in the abovementioned equations may thus represent ARs, CARs, and CAARs. Let's shortly revisit these three different forms of abnormal return calcuations, as presented in the introduction:

$$AR_{i,t}=R_{i,t}-E[R_{i,t}|\Omega_{i,t}]$$

$$AAR_{t}= \frac{1}{N} \sum\limits_{i=1}^{N}AR_{i,t}$$

$$CAR_{i}=\sum\limits_{i=T_1 + 1}^{T_2} AR_{i,t}$$

$$CAAR=\frac{1}{N}\sum\limits_{i=1}^{N}CAR_i$$

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 t-test*   Individual Event
$H_0: AAR = 0$ AAR t-test, Patell-test, BMP-test, J-test GRANK Sample of Events
$H_0: CAR = 0$ GRANK, SIGN Individual Event
$H_0: CAAR = 0$ Sample of Events

N.B.: *-labeled test results are included in the output of EventStudyTools' abnormal return calculator

Parametric test statistics ground on the classic t-test. Yet, scholars have further developed the test to correct for the t-test'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 rank-test of Corrado (1989), and the sign-based 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, event-date clustering poses a problem leading to (1) cross-sectional correlation of abnormal returns, and (2) distortions from event-induced volatility changes. Cross-sectional correlation arises when sample studies focus on (an) event(s) which happened for multiple firms at the same day(s). Event-induced 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 t-statistic, leading to an over-rejection 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 t-test's proneness to event-induced 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 non-normal returns, low prices or little liquidity. Also, the test has been found to be still affected by event-induced 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 volatility-changing events. Furthermore, the simulation study of Kolari and Pynnonen (2010) indicates an over-rejection of the null hypothesis for both the Patell and the BMP test, if cross-sectional correlation is ignored. Kolari and Pynnonen (2010) developed an adjusted version for both test statistics that accounts for cross-sectional correlation.

The nonparametric rank test of Corrado and Zivney (1992) (RANK) applies re-standardized event window returns and has proven robust against induced volatility and cross-correlation. Sign tests are another category of tests. One advantage the tests’ authors stress over the common t-test 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 non-normally distributed data. Past research (e.g. Fama, 1976) has argued that daily return distributions are more fat-tailed (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 Campbell-Wasley test statistic (CUM-RANK). Another NPT is the generalized rank test (GRANK) test with a Student t-distribution with T-2 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 CAR-windows.

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 BMP-tests 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 event-windows, the conclusions on the tests power should be very carefully drawn because of the many over- or under-rejections of the null hypothesis. Overall, comparing the different test statistics yields the following insights:

1. Parametric tests based on scaled abnormal returns perform better than those based on non-standardized returns
2. Generally, nonparametric tests tend to be more powerful than parametric tests
3. The generalized rank test (GRANK) is one of the most powerful test for both shorter CAR-windows 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 Type
(P/NP)
Antecedent Strengths Weaknesses
1 t-test
[ORDIN]
P
• Simplicity
• Prone to cross-sectional correlation and volatility changes.
2 Standardized residual test
[Patell-test]
Patell (1976) P ORDIN
• Immune to the way in which ARs are distributed across the (cumulated) event window.
• Prone to cross-sectional correlation and event-induced volatility.
3 Adjusted Patell-test Kolari and Pynnönen (2010) P Patell-test
• Same as Patell
• Immune to cross-sectional correlation

4

Standardized cross-sectional test
[BMP-test]

Boehmer, Musumeci and Poulsen (1991) P Patell-test
• Immune to the way in which ARs are distributed across the (cumulated) event window.
• Accounts for event-induced volatility.
• Prone to cross-sectional correlation.
[J-test]
Kolari and Pynnönen (2010) P BMP-test
• Accounts for cross-correlation and event-induced volatility.

6

[RANK]

• Loses power for longer CARs (e.g., [-10,10]).
7 Generalized rank t test
[GRANKT]
Kolari and Pynnönen (2011) NP RANK

Accounts for

• cross-correlation of returns,
• returns serial correlation
• and event-induced volatility.

8 Generalized rank z test [GRANKZ] Kolari and Pynnönen (2011) NP RANK see GRANKT
9 Sign test
[SIGN]
Cowan (1992) NP
• Accounts for skewness in security returns
• Poor performance for longer event windows
10 Generalized sign test
[GSIGN]
Cowan (1992) NP SIGN
11 Jackknife test Giaccotto and Sfiridis (1996) NP ORDIN
12 Wilcoxon signed-rank test Wilcoxon (1945) NP
• Considers that both the sign and the magnitude of ARs are important.

Notes: P = parametric, NP = nonparametric; 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

$$S^2_{AR_i} = \frac{1}{M_i - 2} \sum\limits_{t=T_0}^{T_1}(AR_{i,t})^2$$

$M_{i}$ refers to the number of non-missing (i.e., matched) returns.

[1] t-test

Type AR AAR CAR CAAR
t statistic $t_{AR_{i,t}}=\frac{AR_{i,t}}{S_{AR_i}}$ $t_{AAR_t}=\sqrt{N}\frac{AAR_t}{S_{AAR_t}}$ $t_{CAR}=\frac{CAR_i}{S_{CAR}}$ $t_{CAAR}=\sqrt{N}\frac{CAAR}{S_{CAAR}}$
Standard deviation $S^2_{AR_i} = \frac{1}{M_i-2} \sum\limits_{t=T_0}^{T_1}(AR_{i,t})^2$ $S^2_{AAR_t} =\frac{1}{N-1} \sum\limits_{i=1}^{N}(AAR_{i, t} - \overline{AAR}_t)^2$ $S^2_{CAR} = L_2 S^2_{AR_i}$ $S^2_{CAAR} = \frac{1}{N-2} \sum\limits_{i=1}^{N}(CAR_i - CAAR)^2$

Please note: There are alternative aproaches to calculate the standard deviations for CARs and CAARs (see, for example, Campbell, Lo and MacKinleay (1997)).

[2] Patell-test

The cross-sectional t-test does not account for event-induced variance and thus overstates significance levels. Patell (1976, 1979) suggested to correct for this overstatement by first standardizing each $AR_i$ before calculating the test statistic using the standardized $AR_i$.
$$SAR_{i,t} = \frac{AR_{i,t}}{S_{AR_{i,t}}} \label{eq:sar}$$
As the event-window abnormal returns are out-of-sample predictions, Patell adjusts the standard error by the forecast-error:
$$S^2_{AR_{i,t}} = S^2_{AR_i} \left(1+\frac{1}{M_i}+\frac{(R_{m,t}-\bar{R}_{m})^2} {\sum\limits_{t=T_0}^{T_1}(R_{m,t}-\bar{R}_{m})^2}\right)$$
with $\bar{R}_{m}$ as the mean of the market returns in the estimation window. $SAR_{i,t}$ is distributed as a t-distribution with ${M_i-2}$ degrees of freedom under the Null. Test statistic for testing $H_0: AAR = 0$ is then given by
$$z_{Patell, t} = \frac{ASAR_t}{\sqrt{N}S_{ASAR_t}},$$
where $ASAR_t$ is the sum over the sample of the standardized abnormal returns
$$ASAR_t = \sum\limits_{i=1}^N SAR_{i,t},$$
with expectation zero and variance
$$S^2_{ASAR_t} = \sum\limits_{i=1}^N \frac{M_i-2}{M_i-4}.$$

Test statistic for testing $H_0: CAAR = 0$ is given by
$$z_{Patell}=\frac{1}{\sqrt{N}}\sum\limits_{i=1}^{N}\frac{CSAR_i}{S_{CSAR_i}},$$
with $CSAR$ as the cumulative standardized abnormal returns
$$CSAR_{i} = \sum\limits_{t=T_1+1}^{T_2} SAR_{i,t}$$
with expectation zero and variance
$$S^2_{CSAR_i} = L_2\frac{M_i-2}{M_i-4}.$$
Under the assumption of cross-sectional independence and some other conditions (Patell, 1976), $z_{Patell}$ is standard normal distributed.

Kolari and Pynnönen (2010) propose a modification to the Patell-test to account for cross-correlation 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 cross-correlation of the estimation period abnormal returns, the test statistic for $H_0: AAR = 0$ of the adjusted Patell-test 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 BMP test statistic. Assuming the square-root 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] BMP-test (standardized cross-sectional test)

Similarly, Boehmer, Musumeci and Poulsen (1991) proposed a standardized cross-sectional 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 Patell-test [2] and with standard deviation

$$S^2_{ASAR_t} = \frac{1}{N-1}\sum\limits_{i=1}^{N}\left(SAR_{i, t} - \frac{1}{N} \sum\limits_{l=1}^N SAR_{l, t} \right)^2.$$

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

$$S^2_{\overline{CSAR}} = \frac{1}{N-1} \sum\limits_{i=1}^{N} \left(CSAR_i - \overline{CSAR}\right)^2.$$

Kolari and Pynnönen (2010) propose a modification to the BMP-test to account for cross-correlation 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 cross-correlation of the estimation period abnormal returns, the test statistic for $H_0: AAR = 0$ of the adjusted BMP-test 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 square-root rule holds for the standard deviation of different return periods, this test can be used when considering Cumulated Abnormal Returns ($H_0: CAAR = 0$):

$$z_{J}=z_{bmp} \sqrt{\frac{1- \overline r}{1 + (N - 1) \overline r}}.$$

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 non-missing 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 non-missing 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 RANK-test 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 cum-rank 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 cum-rank test statistic. In the API the cum-rank 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 GRANK-T or GRANK-Z test.

[7] GRANK-T-test

In order to account for possible event-induced volatility, the GRANK test squeezes the whole event window into one observation, the so-called '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 the standard deviation of 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 event-induced volatility $S_{CAR_{i}}$ is re-standardized by the cross-sectional standard deviation

$$SCAR^*_{i}=\frac{SCAR_{i}}{S_{SCAR}},$$

where

$$S^2_{SCAR}=\frac{1}{N-1} \sum\limits_{i=1}^N \left(SCAR_{i} - \overline{SCAR} \right) \quad \text{ and } \quad \overline{SCAR} = \frac{1}{N} \sum\limits_{i=1}^N SCAR_{i}.$$

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 t-statistic 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 t-distributed with $L_1 - 1$ degrees of freedom.

[8] GRANK-Z-test

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.

[9] SIGN-test

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.

$$t_{sign}= \sqrt{N}\left(\frac{\hat{p}-0.5}{\sqrt{0.5(1-0.5)}}\right)$$

[10] GSIGN-test

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

$$z_{gsign}=\frac{(w-N\hat{p})}{\sqrt{N\hat{p}(1-\hat{p})}},$$

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: EST provides GSIGN test statistics also for single days ($H_0: AAR = 0$) in the event time period.

Note 2: 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 3: If $N$ is small, the normal approximation is inaccurate for calculating the p-value, in such case we recommend to use the binomial distribution.

[11] Jackknife-test

coming soon.

[12] Wilcoxon-test

coming soon.

coming soon.

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Campbell, J., Lo, A., MacKinlay, A.C. 1997. 'The econometrics of financial markets'. Princeton: Princeton University Press.

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Cowan, A. R. and Sergeant, A. M. A. 1996. 'Trading frequency and event study test specification'. Journal of Banking and Finance, 20(10): 1731-1757.

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Kolari, J. W. and Pynnonen, S. 2011. 'Nonparametric rank tests for event studies'. Journal of Empirical Finance, 18(5): 953-971.

Maynes, E. and Rumsey, J. 1993. 'Conducting event studies with thinly traded stocks'. Journal of Banking and Finance, 17(1): 145-157.

Patell, J. A. 1976. 'Corporate forecasts of earnings per share and stock price behavior: Empirical test'. Journal of Accounting Research, 14(2): 246-276.

Schipper, K. and Smith, A. 1983. 'Effects of recontracting on shareholder wealth: The case of voluntary spin-offs.' Journal of Financial Economics, 12(4): 437-467.

Wilcoxon, F. (1945). 'Individual comparison by ranking methods'. Biometrics Bulletin, 1(6): 80-83.

Dr. Simon Müller

Dr. Simon Müller studied mathematics and technical mechanics at the University of Stuttgart, Germany. He holds a Ph. D. in mathematics from the University of Stuttgart. After his Ph.D. thesis Simon worked as a Postdoc at the Dr. Margarete Fischer-Bosch Institute of Clinical Pharmacology in Stuttgart. Since 2012 he works as an independent statistical consultant. He is an expert R programmer and has working knowledge on statistic software SAS Base/Stat/Graph, and SPSS Statistics.