Event Study: T-Stat for Cumulative Abnormal Returns

#1
I am performing an event study using the market model theory.

Rit = αi + βiRmt + εit

With Abnormal Returns(AR)

ARiτ = Riτ − αi − βiRmτ.

So to test the significance of the Cumulative Abnormal Return's(CAR's), one must calculate the variance of the aggregated AR's across firms and then sum this number for each observation in the event window to achieve the variance of the CAR's, and then use the sqrt of this as the denominator in the t-stat.


However, my problem is that I want to test the CAR's across different points in the event window. i.e from the event date, t=0 to one day after the event t=1. Then also from t=0 up to t=5.

My question is, does the denominator of the t-stat for the CAR's change when using a different period?


For example if testing the CAR's from t=0 up to t=5, one sums the AR's between these two points for the numerator of the tstat, howevever for the denominator does one also only sum the variance of the AR's between t=0 and t=5, or does the denominator remain constant regardless of which points along the event window I am testing and remain the square root of the sum of all the variances across the evt window?

Many thanks! Hope my question was clear enough!

(All formula for relevant tests and variances are attached)
 
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