Test statistic of event study

Question

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?

Would appreciate your help.

Answer 1

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 $t$.

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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).