t-test in event studies

[Kleopatra Koulikidou]

Dear Statalist members,

Using the http://dss.princeton.edu/online_help...ventstudy.html example while performing an event study, I have reached the point where I should perform the test for significance (t-statistic). In this section is mentioned that the formula is: TEST= ((ΣAR)/N) / (AR_SD/sqrt(N)), although when imported in Stata the example uses the follow command:
gen test =(1/sqrt(number of days in event window)) * ( cumulative_abnormal_return /ar_sd).

So, my question here is how from the first formula we conclude the Stata command, cause I can not figure out the rationale (e.g. is ΣAR actually the CAR and what N stands for?-the number of observations or something else?-)
I have also read MacKinley(1997) popular paper for event studies but it seems to use when calculating the t-test- the number of observations (CAR) and not the number of days in the event window.
Is Princeton using here a different t-test and how this test is actually called (since it seems to exist several t-tests in event studies e.g. standardized t-test, cross-sectional t-test, portfolio t test)

Thank you in advance for your valuable help.

[Nikos Tsileponis]

Hi Kleopatra,

ΣΑR is the sum of abnormal returns, i.e. CAR. Also, N is the number of days in the event window.

TEST= ((ΣAR)/N) / (AR_SD/sqrt(N)) = (ΣAR*sqrt(N)) / (N*ΑR_SD) = ΣAR / (sqrt(N)*AR_SD) = (1/sqrt(number of days in event window)) * ( cumulative_abnormal_return /ar_sd).

[Given that sqrt(x)/x = 1/sqrt(x) ]

[Kleopatra Koulikidou]

Thank you so much Nikos for your prompt response. It makes sense now.
Regarding now the test name, do you have any idea how it is called cause it is not probably the regular t-test from MacKinley(1997)?

[Nikos Tsileponis]

I am not certain about the name of this test. I thought it was a t-test, but haven't really read MacKinley(1997). Let's see if someone else can help you out with this one.

In any case, you can trust this website from Princeton for sure. There is no reason to think that something might be wrong there. Everyone is following their steps in doing event studies in Stata.

[Rich Goldstein]

1. Kleopatra really needs to give a full citation - it is not reasonable to expect to either know what "McKinley 1997" is or to look it up when you already know; please read the FAQ for more on this

2. Markus - please start a new thread after you read the FAQ - as asked, your question is not answerable since we don't know what you actually typed into Stata

[Mar Gaux]

Hello Everyone,
As Kleopatra, i am asking the same question about the N in the TEST of Princeton ("Testing of significance" in https://dss.princeton.edu/online_hel...ventstudy.html). The code from Princeton uses N as the number of days in the event window. However, Jacobs et al. (2010) use the same very test but with N=number of announcements in the sample and they are far from being the only ones to use N as the number of announcements. As you can imagine, this changes the value of the test.
I was wondering if, in the spirit of the TEST of Princeton, it is possible to calculate the significance of average abnormal return for each day in the event window (not the aggregated test which corresponds to the TEST of Princeton) but using the CAR for J-1, J-0 and J+1 (for a 3-day event window for instance) instead of the CAR for all aggregated days in the event window ? Is this sound ok ?
Thank you so much for your answer.
Regards,
Margaux

[Deniz Civril]

Hi Margaux,

I think the code from Princeton might be wrong and it seems to me that a lot of people have been using this code without paying enough attention.

sort id date

gen abnormal_return = ret-predicted_return if event_window == 1

by id: egen cumulative_abnormal_return = total(abnormal_return)
 
sort id date

by id: egen ar_sd = sd(abnormal_return) 

gen test = (1/sqrt(number of days in event window)) * ( cumulative_abnormal_return /ar_sd) 

list company_id cumulative_abnormal_return test if dif == 0

In this code, abnormal return is calculated only for the event_window. Later when they calculate abnormal return sd, it only uses the values from the event window. As far as I understand from the literature, it should be using the values from the estimation window.

Because:

T-stat for an abnormal return for firm i at point t:
tstat_ARt = AR_i,t/S_ARi
where S_ARi is the standard deviation of the abnormal return in the estimation window.

T-stat for the cumulative abnormal returns for firm i:
tstat_CAR = CAR_i/S_CAR

where S_CAR = sqrt(L) S_ARi and L is number of days in the event window (because variance of abnormal returns for L days = L* Variance of abnormal returns in the estimation window.) I am using L here is because the website is using N for two different things: first the number of company-event combinations and later in the formula of the test.

The test in the website is for "the average abnormal return for each stock" TEST= ((ΣAR)/N) / (AR_SD/sqrt(N))
average of CAR = CAR/L
sd of average of CA = sqrt(L) S_ARi/ L = S_ARi/sqrt(L)

In short, I think you should change the code such that S_ARi is the standard deviation of the abnormal return in the estimation window.
Maybe there is another test that they are conducting and I am misunderstanding it but here is my take!

Good luck!