Excess Returns Methodology (the basics)

We often ask whether some event, like a merger announcement, dividend omission, or stock split, has an impact on stock prices.  Since we have CRSP data available, it is pretty easy to construct a test to answer this question.  In general the idea is simple; first determine what a "normal" return ought to be for the stock in question, and then see whether the actual return during the "event window" is more or less than this "normal" return.  Of course the devil is in the details:  What is a "normal" return? Over what time period do you determine this "normal" return? How long is the event window? Are there characteristics of reported stock returns that make these measurements unreliable?  I'm not going to answer all of these questions for this problem set.  You'll get some answers in your seminar.  Instead, I'm going to focus on how to construct some of the more basic tests so you can get experience programming them.

All of this can be done in SAS--albeit with some difficulty.  I think it is easier to do it in Fortran, if you know enough Fortran.  Over the semester we'll look at both ways.

You should read "Using Daily Stock Returns; The Case of Event Studies" by Stephen J. Brown and Jerold B. Warner in Journal of Financial Economics Volume 14, Issue 1 (March, 1985) pages 3-31, and also "Valuation Effects of Security Offerings and the Issuance Process" by Wayne H. Mikkelson and Meghan Partch in Journal of Financial Economics Volume 15, Issue 1 (March 1986) pp. 31-60.  I have a master that you can use to make copies.

Terminology

Event date--the date the event occured
Event window--several days before and after the event.  This is the time period over which you think you will observe some market reaction to the event.   We use some days before the event because lots of time news leaks out before the public announcement.  We use some days after the event because sometimes it takes a day or two for the market to fully impound the effects of the event into the stock price.
Event time--trading days relative to the event date.  Usually the event date is considered day 0 and days before the event date are negative.  So day -5 is 5 days before the event, day +5 is 5 days after the event.
Normal return--the return you would expect on the stock during the event window if there had been no event.
Abnormal return--the actual stock return less the predicted normal return.

What types of returns should be used?

We have daily and monthly returns available.  If you really want, you can construct other interval returns--weekly, two-weekly, annual, etc.  Since most of the events we look at are of short duration you will usually use daily, or at most weekly, returns.  For these exercises we'll use daily returns.

What is a "normal" return?

There are at least three ways to identify a "normal" return. 

1. The average return during some estimation period that doesn't include the event window.   This is called the comparison period.
2. The market return on a given day.
3. The return that would be predicted by the CAPM on a given day..
4. There are others that use more sophisticated models of the return-generating process, for example, the Fama-French 3 factor model or Scholes-Williams betas, or Dimson betas.

We won't talk about the pros and cons of these methods here.  You will get some of that in your seminar.  Once you have determined what is a normal return, the abnormal return is just the actual return minus the predicted normal return. We'll estimate the first three measures. 

Over what time period should the normal return be estimated?

You can estimate the normal return before, during, or after the event.  For the purposes of our exercises we will use days -104 to -5 for the estimation period.   We'll use days -4 to +1 for the event window.

How do you test whether the abnormal return is significant?

There are a number of ways to test for significance, depending on how you characterized normal above.  I'll specify a test for each of your exercises.

Portfolios or individual stocks?

Identifying abnormal returns is usually easier in a portfolio than in individual stocks.  For that reason the tests will be on portfolios of stocks rather than on individual stocks.

Test statistics--Brown and Warner

Brown and Warner used the following test statistics for the null hypothesis that excess returns are zero.  These tests assume you are testing a collection of firms' abnormal returns.

One day event window, N days in estimation period:

Let A_i,t be firm i's abnormal return (either mean adjusted, market adjusted, or market model adjusted) on day t.
Let Abar_t be the mean (over all the companies in the sample) abnormal return on day t.
Let s be the standard deviation of the Abar_t over all days in the estimation period (using N-1 weighting, so it is an estimate of the population standard deviation).
The test statistic for a given mean (over all the companies) abnormal return on a given day, Abar_t, is

T = Abar_t/s

If enough days are in the estimation period--over 200, then the statistic is assumed to be normal.  Otherwise it is Student-t.

Multiple day event window (n days in the event window):

Let SAbar be the sum of the average abnormal returns during the event window.
Since s is the estimate of the standard deviation of one day's abnormal return, sn^1/2 is an estimate of the standard deviation of the sum of n independant abnormal returns.

The test statistic is

z = SAbar/(sn^1/2 ) 

and is assumed to be unit normal.  n is the number of days in the event window.

Test statistics--Mikkelson and Partch

Mikkelson and Partch only use market model abnormal returns so A_i,t = R_i,t - (alpha + beta R_M,t )--they call these prediction errors, PE  (We'll still use the Abar terminology).  On day t, Abar_t is calculated just as above--the mean abnormal return over all firms on day t. 

The difference between Mikkleson and Partch's test statistic, and that of Brown and Warner, is in how the expected standard deviation of abnormal returns is calculated and the fact that they allow for different numbers of days in the event window for each firm. Mikkleson and Partch allow for each calendar day's abnormal return to  have a different standard deviation.  So corresponding to each abnormal return A_i,t is a standard deviation s_i,t calculated as follows:

Let v_i² be the variance of the residual from firm i's market model regression (the one you used to find alpha and beta).  Assume you used N observations in this regression (i.e. N days in the estimation period) then:

is the standard deviation of each day t's abnormal return for company i, where the R_M with the bar over it is the mean market return over the N-day estimation period.  Note that this will differ for each firm because the average of the market return is over different days, unless of course some of the firms share the same event date.

Then the standardized prediction errors for each day and each company,

SPE_i,t =  A_i,t /s_i,t

should all be Student-t with variance N/(N-2). 

One day event window:

For a given day, the average of these prediction errors over all the firms in the sample (there are M firms in the sample), ASPE_t is approximately normal with variance 1/(# firms in sample) = 1/M. This gives us the test statistic for an abnormal return on a given day, t, as