1.7.0.3 OLS estimation
In the …rst approach we follow the standard OLS estimation of the market model outlined in Campbell et al. (1997). For each …rm in our sample we use OLS to estimate the market model in 1.6. There we regress the daily returns of security i; on a constant, and on the daily market return Rm 44.
Rit = OLSi + OLSi Rmt+ it (1.6)
The model 1.6 is estimated in the estimation window represented in Figure 1.2 as the space between T0 and T1. There is no speci…c rule on how to choose the estimation window and for our study we take a window of 201 days that span from 230 to 30 days before the event realization. The length of the event window is usually chosen to be between 100 and 250 days45. It is also common not to extend the estimation window to the days just before the event as to avoid the model picking up movements in the returns due to information leakages.
Under general conditions OLS is a consistent estimator of the market model parameters and under the normality condition it provides the minimum variance among all unbiased estimators. From the OLS regression, for each …rm, we obtain the parameters [OLS and [OLS that we will use to estimate the expected normal return (Rnit in equation 1.1) for each day of the event window. For each day in the event window we can then compute the abnormal return AROLSit as in equation 1.7 (where Ri and Rm are respectively the actual return on the …rm i security and the actual return of the chosen market index).
AROLSit = Rit ( [OLSi + [OLSi Rmt) (1.7) 1.7.0.4 Theil’s method
OLS estimation of the market model is the traditional choice in the majority of event studies. However Dombrow et al. (2000) show that when the normality condition fails to hold other non-linear estimators may be preferred. The same authors argue for the adoption
44The market return is chosen to be the leading market index of the stock market in which the …rm security i is listed. The market indexes used are Milan Mibtel, DAX 200, FTSE 100, Dow Jones Industrials, Nikkei 225, Swiss Market, OMX Helsinki, SBF 250 and Brussels all share.
45Corrado (2010) suggests using an estimation window of 250 days that corresponds approximately to the number of trading days in a calendar year.
of robust statistics when the underlying distribution of the errors is uncertain. They then propose to use a nonparametric estimator, suggested by Theil (1950), for its high e¢ ciency and ease of computation and implementation46. In contrast to OLS the Theil’s estimator does not need any distributional assumptions and can be implemented as follows47:
for each case in the sample:
1. Sort the L1 (L1 = T1 T0+ 1) data pairs of (Rmt; Rit) in ascending order of Rmt: 2. Separate the data into two groups based on the median48 .
3. Calculate the slope parameters T heili;(j;j+L1
2 ); in 1:8, for all the L21 pair and choose the
4. Use the estimated \T heili to estimate the L1 parameters:
\T heilit = Rit \T heil
i Rmt
:
5. Choose \T heili as the median of the L1 \T heilit .
Then, similarly to equation 1.7, for each day and …rm we proceed to estimate the non-parametric abnormal returns as in equation 1.9.
ART heilit = Rit (\T heili + \T heili Rmt) (1.9) Notice that given the median based nature of this estimator the undue in‡uence of out-liers is removed. Both the OLS and Theil’s estimators are easy and fast to compute and implement. However the latter one does not need any distributional assumptions on the error term. Moreover, Dombrow, Rodriguez and Sirmans …nd that Theil’s nonparametric estima-tion has relatively greater power, than OLS, to detect abnormal performance in presence of non normally distributed errors and o¤ers comparable results to OLS under normality.
46For an event study that uses the Theil’s estimator see Nicolau (2001) and Saleh (2007).
47The step procedure follows closely the methodology outlined in Dombrow et al. (2000).
48In case of an odd numbered interval we drop the median observation.
1.7.1 Testing for statistical signi…cance
1.7.1.1 Parametric test
The parametric test builds upon the OLS estimators to derive the statistical properties of abnormal returns. Under the assumption that asset returns are jointly multivariate normal, independently and identically distributed through time, it can be showed that the OLS model estimated in (1.6) is a consistent and e¢ cient estimator for the market model parameters.
From the OLS model it is then possible to derive the statistical properties of the abnormal returns, under the null hypothesis of zero abnormal returns (Campbell et al. (1997)):
E[ dARijRmt] = 0 (1.10)
Vi = I 2+ Xi(Xi0Xi) 1Xi0 2 (1.11) where Vi is the variance covariance matrix of abnormal returns. Xi and Xi are respec-tively a ((T3 T2) X 2) and a ((T1 T0)X 2) matrices of regressors (market return, Rm;and a constant) at the event window and at the estimation window. 2 is the variance of the errors estimated from the OLS estimation of the market model.
From this we can then estimate:
V = 1 N2
XN i=1
Vi (1.12)
that is the aggregate variance matrix of the average daily abnormal returns.
From the above results we can then construct the three test for the above statistics 1.3, 1.4, and 1.5. These are respectively:
[
V AR(DAARt) = vtt (1.13)
where vttis the (t; t) element of the variance covariance matrix V:
[
V AR(CAAR) = 0V (1.14)
where is a vector of 1s of dimension (T3 T2).
[
V AR(CARi) = 0Vi (1.15)
From which we can derive the appropriate three tests as:
JtDAAR = DAARt
where the distributional results are for large samples and not exact because the estimator of the variance appear in the denominator.
Non Parametric Test
For the nonparametric estimates, derived by estimating the market model using Theil’s method, we also use a nonparametric test statistics. Hence we follow the advice of Dombrow et al. (2000) and perform what they call a complete nonparametric event study. The non-parametric test we use is known as the rank test and was outlined by Corrado (1989). The test is developed as follow. First, for each case, compute abnormal returns for all the days considered both in the estimation and event window. Then for every case i convert all the daily abnormal returns into their rank within the distribution of abnormal returns of that case.
Kit = rank(dARit) (1.19)
Higher values of rank K denote an higher abnormal return. This transformation turns the distribution of the abnormal returns into a uniform distribution of the possible ranks.
Under the null hypothesis of zero abnormal returns the expected rank is just one plus half the number of days considered (if we run the analysis for 250 days the expected rank is 125,5). Then two tests, depending on the level of aggregation, are computed as follow:
StDAAR =
The test in 1.20 refers to daily average abnormal return estimates and T represents the sum of days both in the event and estimation window (i.e. T = T1 T0+ T3 T2+ 2):
When we aggregate the daily average abnormal return to construct the CAR measure we then use the following test proposed by Cowan (1992) in which he extends the original test proposed by Corrado (1989) to multi day event window assuming that the daily return ranks within the window are independent.
Under the null hypothesis of zero abnormal returns on the event window and using the result that the asymptotic null distribution of test 1.20 and test 1.21 are standard normal we can then test the null hypothesis.