# How to calculate t-statistic for one day abnormal return (event study)?

How can I calculate if one day (event day) abnormal return is statistifically significant. I am trying to calculate statistical significance (using t-statistics but other statistics could be fine also).

I have stock market index daily price data and need to is an event day abnormal return statistically significant.

I cannot calculate t-statistic in the same way I calculated cumulative abnormal return (CARs) for 6 and 12 days because I cannot calculate standard deviation.

I am calulating t-statistic for CARs with following formula: S.E. refers to the standard deviations of ARs during the event window which is different from my estimation window for normal returns. The problem is that for one day, S.E. cannot be calulated.

How can I calculate significance of one day abnorlmal return?

• The more fundamental problem concerns the definition of a "one day abnormal return." Could you please explain what that is and what data it might be based on? – whuber Jul 30 '18 at 16:14
• The data is closing prices of FTSE 100 index. The normal return for the index is the average daily return of the FTSE 100 calculated from estimation period 100 days before the event. What I mean by "one day abnormal return" is the abnormal return on the event day. Calculated with following formula: AR(t)=R(t)-E(R), where AR(t) is the abnormal return for the day, R(t) is the real return of the index in that day and E(R) is the expected/normal return of the index. – Finance student Jul 30 '18 at 16:29
• You describe a residual return, but you haven't provided the essential information: exactly how do you determine that a day is "abnormal"? This issue can potentially affect any p-value by orders of magnitude: it's crucial to be clear about it. – whuber Jul 30 '18 at 18:20

Or else, go to the basic: introduce the variable $X=0$ everyday except on the abnormal day for which $x=1$, and call $Y$ the stock market index. The function $aX+b$ is equal to $b$ on normal days (because $X0=$), and $a+b$ on the abnormal day (because $X=1$). So when you do the regression $Y \approx aX+b$, you'll get the $b=$ the average stock market index and $a$ = the difference between normal and abnormal day.
Additionally, you'll a t-value and a p-sign for $b$ which is exactly the test you are looking for. If you are patient and good at math, you can prove that the above test is actually the Student t-test with equal variance on the two sub-population.
This test works even when you have more than one abnormal value, provided that the abnormal value is the same for all abnormal day. As it may not be the case, you can introduce one dummy variables $X_1$, $X_2$,... per abnormal day, and do the same. You will then make a Fisher F-test to test if globally, the abnormality is statistically significant.