What assumptions do the various test statistics make?

Written by Prof. Wolf
Updated 4 months ago

Overview of the test statistics EST calculates

Test statistic Applicability Type
T-Test AR, CAR Parametric
CSect T AAR, CAAR Parametric
Skewness Corrected T AAR, CAAR Parametric
CDA T AAR, CAAR Parametric
Patell Z AAR, CAAR Parametric
Adjusted Patell Z AAR, CAAR Parametric
StdCSect T AAR, CAAR Parametric
Adjusted StdCSect T AAR, CAAR Parametric
Rank Z AAR, CAAR Non-parametric
Generalized Rank Z AAR, CAAR Non-parametric
Generalized Rank T AAR, CAAR Non-parametric
Sign Z AAR, CAAR Non-parametric
Generalized Sign Z AAR, CAAR Non-parametric
Wilcoxon AAR Non-parametric

Every test statistic is based on a list of assumptions, which ensure that the corresponding p-value can be trusted in practice.

For the technically inclined user: The p-value is based on the (approximate) distribution of the test statistic under the null hypothesis, and in order to derive this distribution certain assumptions are needed in each case.

Parametric test statistics

Test statistic Assumptions
T-Test

The abnormal returns AR, over both the estimation window and the event window, are independent and identically distributed (i.i.d.) according to a normal distribution with mean zero and unknown (but common) variance. 

CSect T

Across the N stocks, the abnormal returns AR are independent and identically distributed (i.i.d.) according to a normal distribution with mean zero and unknown (but common) variance. Note that this variance may differ from the variance(s) of the abnormal returns during the estimation window so that the test is robust to event-induced increases in variance(s). 

Skewness Corrected T

Same as in CSect T except that the common distribution does not have to be normal, and thus may exhibit skewness. 

CDA T

The average abnormal returns AAR, over both the estimation window and the event window, are independent and identically distributed (i.i.d.) according to a normal distribution with mean zero and unknown (but common) variance. This allows for the abnormal returns AR to have (i) cross-sectional dependence on any given day and (ii) different variances across stocks. However, the test is not robust to event-induced increases in the variance of the average abnormal returns AAR. 

Patell Z

Across stocks, the standardized abnormal returns SAR (for testing AAR), respectively the cumulative standardized abnormal returns CSAR (for testing CAAR), are independent and identically distributed according to a normal distribution with mean zero and unknown variance, which is the same as the variance during the estimation period. Hence, this test is not robust to event-induced increase in variance(s). 

Adjusted
Patell Z

Same as for Patell Z except that the abnormal returns AR are allowed to be correlated across stocks on any given day. The pairwise correlations are assumed to be constant through time and to be identical to their counterparts during the estimation window (which are also constant through time). 

StdCSect T

Same as for Patell Z except that for a given stock the variance of the (standardized) abnormal return for t = 0 can be different compared to the estimation window. Hence, this test is robust to event-induced increase in variance(s). 

Adjusted StdCSect T Same as for StdCSect T except that the abnormal returns AR are allowed to be correlated across stocks at any given day. The pairwise correlations are assumed to be constant through time and to be identical to their counterparts during the estimation window (which are also constant through time). 

Apart from Skewness Corrected T, all parametric test statistics assume that the stock returns follow a normal distribution. This assumption is hard to check in practice and does not hold for most stocks. The good news is that a violation of the normality assumptions does not matter (much), in the sense that the resulting p-values can still be trusted, as long as the relevant sample size is sufficiently large; with the exception of the T-test, "relevant sample size" means the number stocks always. There is no hard-and-fast rule as to what constitutes "sufficiently large" in practice but, as a rule of thumb, a sample size greater than 50 is typically enough, and even a sample size greater than 30 can suffice. Consequently, if the number of stocks is less than 30, we recommend not using parametric test statistics and, instead, switching to non-parametric test statistics. For the T-Test, the "relevant" sample size means the number of days in the event window; unless this number exceeds 30, we recommend not to use this test. (In particular, we recommend not to use this test for testing AR, since in this case, the number of days in the event window is only one.)

Non-parametric test statistics

Test statistic Assumptions
Rank Z For any given stock, the full sequence of abnormal returns AR, covering both the estimation and the event window, are i.i.d. according to an arbitrary distribution which need not be normal. Across stocks, the distributions may differ, allowing for different
variances, say. However, the assumptions do not allow for an event-induced increase in variance(s).
Generalized Rank Z Strictly speaking, same as Rank Z. However, this test works with standardized abnormal returns SAR instead of `simple' abnormal returns AR and is in practice more robust to event-induced increase in variance(s). Furthermore, Monte Carlo studies have shown that this test is also robust to mild serial correlations in returns, which can arise for some stocks. 
Generalized Rank T Same as Generalized Rank Z. But, in addition, Monte Carlo studies have shown that this test is (more) robust to cross-sectional dependence of stock returns. Therefore, this is the preferred Rank test of the three, for testing both AAR and CAAR.
Sign Z

For testing AAR, across stocks, the abnormal returns AR on the event day are independent and have the same probability p to be greater than zero; under the null p = 0.5. For testing CAAR, it's analogous to "CAR during the event window" in place of "AR on the event day".

Generalized Sign Z Same as Sign Z but the probability p under the null need not be equal to 0.5 and is estimated from the estimation window. (This is a useful generalization since if the distribution is skewed it can have a mean of zero but the probability of getting a number greater than zero must not be equal to 0.5 at the same time.)
Wilcoxon The sample of abnormal returns AR, across stocks, is i.i.d. and the probability of observing a positive AR under the null is 0.5. Therefore, this test is not robust to skewed distributions that have a mean of zero but a probability different from 0.5 resulting in a positive AR (under the null).

Unlike parametric tests, nonparametric tests do not rely on normality assumptions on the stock returns and, therefore, can also safely be used for small(er) sample sizes. Although, for the sake of completeness, we have several nonparametric tests in our menu, at the end of the day the recommendation from our side is quite simple: Use the Generalized Rank T test for testing both AAR and CAAR; it's "the latest and the best" of the tests in the menu. (Compared to other tests it is more complicated to code and implement, but this is of no concern to the user, since we have done the job for you.)

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