# Test statistic of event study

Following the event study paper USING DAILY STOCK RETURNS The Case of Event Studies

let us suppose that I have daily stock returns for 50 companies from the date 2012-01-01 until 2014-01-01. and I want to exmine the effect of a single day event (i.e. some announcement on 2013-02-01) and I use 250 day range as training set of the regression model which will be used to estimate the abnormal returns.

Therefore we get;

$$Estimation\ Window\ [T_1,T_2] \ = \ 250 \ days$$

$$Evemt \ Window \ [t_1 , t_2] = 20 \ days$$

where:

$$t_1 = (T_2+1) \ and \ Event \ Date = t_0$$

According to the attachet paper;

OLS market model

$$A_{i,t} \ = R_{i,t} \ - \ \hat\alpha_i \ - \ \hat\beta_i R_{m,t'}$$

where

$$A_{i,t}$$ as the excess return for security $$i$$ at day $$t$$. (abnormal returns)

$$R_{i,t}$$ designate the observed arithmetic return for security $$i$$ at day $$t$$.

Now, I want to test the statistical significance of the event date $$t_0$$. The null hypothesis is that the event day $$t_0$$ has no abnormal returns or $$A_{i,t_0} \ = \ 0$$. According to the paper, the test statistic for any event day $$t$$ is $$\bar A_t/\hat S(\bar A_t)$$.

Question

How to Calculate the test statistic in my case?

If I understand it rightly, I have to take the mean of the abnormal returns for the 50 companes at $$t_0$$ as the numerator. for example: $$\bar A_{t} \ = \ \frac{1}{N_t}\sum_{i=1}^{n_t}A_{i,t'}$$ where $$N_t \ = \ 50 \ and \ t \ = \ t_0$$

but how to calculate the $$\hat S(\bar A_t)$$ part?

Should I use the OLS regression model that I trained on the dataset $$[T_1,T_2]$$ to perform in-sample prediction on the same set $$[T_1,T_2]$$ then calculate the mean of predicted returns of 50 companies $$(\bar A_t)$$ for every $$t$$, then calculate the overall mean $$(\bar{\bar A})$$ and as a result;

$$\hat S(\bar A_t) \ = \ \sqrt{\bigg(\sum_{t=1}^{t=250}(\bar A_t \ - \ \bar{\bar A})^2\bigg)/249}$$

or I just have to take the standard deviation of the normal returns for the $$[T_1,T_2]$$ dataset?

If I understand it rightly, I have to take the mean of the abnormal returns for the 50 companes at $$t_0$$ as the numerator. for example: $$\bar A_{t} \ = \ \frac{1}{N_t}\sum_{i=1}^{n_t}A_{i,t'}$$ where $$N_t \ = \ 50 \ and \ t \ = \ t_0$$

That's absolutely right. $$\bar{A_{t}}$$ is the cross-sectional average of abnormal returns on day $$t$$.

But how to calculate the $$\hat S(\bar A_t)$$ part?

I may not understand your stated explanation, but the equation on p. 7 is quite straightforward. I assume you are using 250 days for the estimation period and 20 additional days for the event period; the latter equally distributed around the event day $$t_0$$:

1. Calculate the above $$\bar{A_{t}}$$ for each of your 270 points in time, i.e. for the 250 days in the estimation period $$[t_{-250};t_{-11}]$$ and the 21 days of the event period $$[t_{-10};t_{+10}]$$. If using the OLS market model then yes, the expected return calculation would be an in-sample estimation.

2. Calculate the mean of $$\bar{A_{t}}$$, i.e. $$\bar{\bar{A}}$$ for the interval $$[t_{-250};t_{-11}]$$, i.e. the average of the daily abnormal return during the estimation period.

3. $$\hat S(\bar A_t)$$ is calculated as the standard deviation of $$\bar{A_{t}}$$ during the estimation period $$[t_{-250};t_{-11}]$$. For an unbiased estimator, you are right to use the value of 249 in the denominator.

4. Step (2) and (3) both refer to the estimation period $$[T_1,T_2]$$. The final test statistic is the ratio of of $$\bar{A_t}$$ for each $$t$$ of the event window (i.e.$$[t_1,t_2]$$) and the constant value of $$\hat S(\bar A_t)$$ from step (3).