Calculating the Z-statistic to measure significance of cumulative abnormal returns (CAR)

[Lotte Brands]

Hello everybody,

I am searching for a while on the internet and on this forum but cannot find what I am looking for.
For my thesis, I calculated the cumulative abnormal returns (CARs) with an event study around IPOs. Now I want to test the null hypothesis that the mean of CAR is equal to zero. I know it is possible with a ttest, but in many similar papers the z-statistic is used to show significance. My question is how can test the null hypothesis based on the z-statistic?
Hope someone can help.

Best regards,

Lotte Brands

[Attaullah Shah]

Are there any theoretical justification for using z-statistics. Also, tell us how the standard errors need to be ajusted, becuase someone in the forum might be able to give you formula or suggestions on the adjustments to be made.

[Roberto Liebscher]

Adding to Attaulah's comment, I guess the question boils down to what sample size you have at hand. In the papers you read -- which you could have linked for us to see your sources -- I reckon the authors had a sample size greater than 30. In this case the test statistics $$\frac{\bar{x}-\mu}{\frac{s}{\sqrt{N}}}$$ follows a normal distribution under the null hypothesis. That's why the authors looked up the z-score in a table of the cumulative distribution function of a standard normal. However, a t-distribution is more reasonable for smaller samples. But as the difference between both distributions diminishes as the sample size grows I can see no reason not to use a t-test for your analysis.

There is one nice example out there that shows how average abnormal returns are tested for significance in Stata. Maybe this might be of interest for you: http://dss.princeton.edu/online_help...ventstudy.html

[Lotte Brands]

Hello,

The authors had a sample size greater than 30 and so have I. My final sample contains about 300 IPOs. The example of Princeton is very clear, I also used their method to conduct the event study. However that regression gives a t-statistic in the output that tests the CAR. I attached two articles that also test the significance of CARs, but they show a z-statistic. Therefore I am not sure if it will be correct if I report the t-statistic and matching p-value. Yesterday I found how to calculate the z-score, but I am not sure if that is needed for testing CAR based on the z-statistic.

Regards,

Lotte Brands

[Roberto Liebscher]

Lotte, I guess you are not allowed to attach Journal of Finance or Financial Review articles since they are protected by copyright law.

What is evident from the paper is that while we were talking about the z-statistic, they actually mean Patell's z-statistic as proposed by Patell (1976) http://www.jstor.org/discover/10.2307/2490543?uid=3737864&uid=2129&uid=2&uid=70&uid=4&si d=21104852560837 . This is something different. I do not think that Stata has a built-in solution for this but you could compute the test statistics on your own. For instance, you could follow the instructions here: http://www.eventstudytools.com/signi...e-tests#Patell Once you have your test statistics calculated, Stata's normal command makes it easy to derive the associated p-value.

For instance, you obtain the p-value for a Patell z of -1.90 (the lower right value in Table III of the JoF paper) by simply typing

display normal(-1.9)*2

Hope this helps,
Roberto

[Lotte Brands]

Oh that's true, did not think of that, do you know how I can edit a post so I can change it? I cannot find how to do it.

I am not sure if I want to use the Patell Z yet, because in some other papers they use a regular Z-statistic.

I have a last question,when I calculate the z-statistic for all my CARs I know if each individual CAR is significantly different from zero, however I want to know it for the mean CAR. Can I just average the z-statistics of all CARs to achieve this?

Regards,

Lotte

[Roberto Liebscher]

No, I would not do that. Think again of the z-statistic: Plug in the mean CAR for $$\bar{x}$$ and an estimate of the standard error of x (the denominator).

When you remind yourself of the Princeton example you already read the simple Stata command to test for the significance of the mean CAR is:

reg cumulative_abnomal_return if dif==0, robust

But note that this command basically computes the t-statistic (and not the z-statistic) for the mean CAR (the constant in the regression).

[Mar Gaux]

Hello,

I am also conducting an event study using the Princeton code but only for one firm. In the code, we test the significance of average abnormal return (AAR). Then, the following code

reg cumulative_abnormal_return if dif==0, robust

gives us "the statistical significance of the cumulative abnormal return accross all companies" as precised by the authors. So, if i am correct and as i am studying only one firm, by running this code I can know if the CAR of my firm is statistically different from zero (I am not sure).
However, i don't understand why i) we have "cumulative_abnormal_return if dif==0" as we have only one CAR per firm and ii) when running this line i do not obtain results (there is no pvalue but only the coef in the output) ... if you have any observation or answer about it I would be very glad !

Best regards,

Margaux