Parametric Test

Parametric Test
Remarks [begin]
The parameters used in this example are:
*
ESTIMATION PERIOD (EP) 6 MONTHS (122 DAYS);
*
ESTIMATION WINDOW (L1) 117 DAYS;
*
EVENT WINDOW (L2) 5 DAYS START FROM THE DAY BEFORE THE
EVENT DATE UNTILL THE THIRD DAY AFTER THE EVENT DATE;
*
N is the number of firms/assets;
*
Possible combinations of CAR in L2(-1;+3) are: CAR(-1;0), CAR(-1;+1), CAR(-
1;+2),CAR(1;+3),CAR(0;+1),CAR(0;+2),CAR(0;+3),CAR(+1;+2),CAR(+1;+3),CAR(+2;+3.

If you change any of the above parameters be careful to modify the following computer code.
[end]
Starting from the following matrices:


...
_
_

...
. . .
_
_

...

;
. . .
;

;
...
;

The above matrices comes from the previous post "Coding AR and CAR". For entirety these
matrices refers to an estimation period of 6 months (columns length 122 days in the
AR_est_window) and event window of 5 days: AR_event_window is an (N x 5) matrix and
CAR is an (N x 10) due to the different 10 combinations.

---

In this section, we provide the instrument to implement statistic inference on the (C)AR's
mean. In the ES (Event Study) the simplest hypothesis system is:

:

:

Where the
is the null hypothesis of "no event effect" in the sample across the event
window against the alternative hypothesis
:"presence of mean abnormal performance" in
the sample.
When performing a parametric test the researcher kwon the distribution or assume it as a
Normal and AR are distributed as follow:
~
0,


Where the expected value is 0 and the variance is
.1

The simplest parametric test is the t-Student test (plain vanilla). The test statistic is :






Where
is the mean of sample (C)AR and i started from the day (-1) to day (+3) (is
the L2 event window observations );
is the mean of the population that in our context is
0 under the null hypothesis; is the unbiased estimator of and finally N is the number of
firm in the sample.2
First of all generate a matrix where each columns are the follow variables respectively: AR(-
1), AR(0), AR(+1), AR(+2), AR(+3), CAR(-1;0), CAR(-1;+1), CAR(-1;+2), CAR(-1;+3),
CAR(0;+1), CAR(0;+2), CAR(0;+3), CAR(+1;+2), CAR(+1;+3), CAR(+2;+3). All the
columns are 16, 5 for AR in the estimation window and 10 for all possible combinations of
CAR, that correspond to the variables above (if you change the length of event window the
number 16 shift).





1The same consideration apply to CAR so :
,
~
0,
,
.
2More in detail:

.

---

Coding in MatLab:

%% t-TEST %%

Generate matrix X merge the columns of AR_event_window and the columns of the matrix
CAR with the next command line: X = [AR_event_window CAR];

Since MatLab implement the t-Student test write the command as follows:
[h,p,ci,stats] = ttest(X);


We present here two additional parametric test statistic J1 and J2 that can be helpful in the
ES.3 The hypothesis system it is the same shows before and under the null hypothesis as said
before the expectations of (Cumulative) Abnormal Returns is zero.
Before showing the test statistic J1 and J2 we define the following equation:

;
1


The expression above shows a proportional relation between the variance of AR and the
variance of CAR.
is calculate from estimation window observation contained into
AR_est_window.


Coding in MatLab:
% Standard deviation: AR %

MatLab provide a command std(X,flag,dim) that calculate automatically the standard
deviation.

Within parentheses are considered respectively: the standard deviation is calculate on X (in
our context AR_est_window matrix); flag takes the value of 0 or 1 depending on whether you
want to calculate s define above or that is the traditional standard deviation (in our context
the flag is 0). Finally dim computes the standard deviations along the dimension of X. dim
assume value 1 (standard deviation calculate on the columns) or 2 (standard deviation
calculate on the rows) in our context the flag is 1 and std return a vector of dimension (Nx1).
The command line is the following: d_st = std(AR_est_window,0,1)';

3 J1 and J2 parametric tests verify the significance of our CARs. Under suspicion of a CAR variance bias due to AR
aggregation the use of J2 is more appropriate since the J2 correction factor gives high weight to low variance observations.
As shown in Campbell et al. (1997) even if the two test are both related to the distortion in the variance of the aggregate
parameters (such as CAR); J2 checks the significance of Cumulative abnormal returns (CARs) when we are not sure about
the robustness of their variance; The correction factor in J2 test, gives more weight to low variance securities with respect to
the J1 test.
Quoting Campbell et al.: "If the true abnormal return is constant across securities then the better choice will give more
weight to securities with the lower abnormal return variance as J2 does. If the true value of abnormal return is larger for
securities with higher variance, then the better choice will give equal weight to the realized CAR of each securities as J1
does."
Campbell, Lo and MacKinlay (1997) "The Econometrics of Financial Markets", Princeton University Press, New Jersey.

---

% Standard deviation: CAR %

Using the expression of variance proportionality between AR and CAR shows above we
calculate the standard deviations of CAR as follow:

dev_CAR_10 = d_st *(2)^0.5;
dev_CAR_11 = d_st *(3)^0.5;
dev_CAR_12 = d_st *(4)^0.5;
dev_CAR_13 = d_st *(5)^0.5;
dev_CAR01 = d_st *(2)^0.5;
dev_CAR02 = d_st *(3)^0.5;
dev_CAR03 = d_st *(4)^0.5;
dev_CAR12 = d_st *(2)^0.5;
dev_CAR13 = d_st *(3)^0.5;
dev_CAR23 = d_st *(2)^0.5;

Each one correspond to all the different combinations of CAR.

Then for aggregate the outcomes we generate a matrix called std as follow:

std=
[dev_CAR_10,dev_CAR_11,dev_CAR_12,dev_CAR_13,dev_CAR01,dev_CAR
02,dev_CAR03,dev_CAR12,dev_CAR13,dev_CAR23];

Finally we eliminate those variables which are not needed anymore as follows:

clear dev_CAR_10 dev_CAR_11 dev_CAR_12 dev_CAR_13 dev_CAR01
dev_CAR02 dev_CAR03 dev_CAR12 dev_CAR13 dev_CAR23


The test statistic of J1 is

;

0,1
;

Where at the numerator there is the mean of the CAR in a range
that in our context is
comprise from the day (-1) and the day (+3). At the denominator there is the mean of
estimate of the variance of CAR shows above.
This distribution result is for large sample (for N large) and is not exact because an estimator
of variance appears in the denominator. The test statistic J2 uses a new method of aggregation
of the abnormal measures that called SCAR. The SCAR is the standardized CAR:
;
;
that for large sample is normally distributed with mean 0 and
;
variance
. Then define the
;
the simple mean of
;
.

---

Next we introduce the test statistic J2 as follows:

4

;
0,1
2



is a correction factor where N is the number of firms in the sample,
is the
length of the estimation window (in our context is 117 days for 6 months estimation period)
and
is define above. As can be seen both statistic test J1 and J2 can only be
implemented for CAR measures of abnormal performance.

Coding in MatLab:


First of all we resume the matrix X and consider the sub matrix that contain only CAR. In our
case we consider the columns ranging from the 6 to the 16. (All possible combinations of
CAR). The command line is: X1 = X(:,6:end);

% J1 & J2 %

In this case is better used a loop.

for j = 1:10

First of all calculate SCAR SCAR(:,j) = X1(:,j)./std(:,j);
It is important linger on ./ . This symbol means the matrix division element by element.

% J1 on the overall sample

Consider the test statistic J1 shows above:

J1(:,j) = mean(X1(:,j)) / mean(std(:,j));

The function mean (...) in MatLab by default make the mean of the columns present in both
in the numerator and the denominator.

Finally calculate J2 test where the important role play the function size. We want find the
number of observations in L1 without write the number 117 to generalized the test
statistic.

size(ar_event_window,1) With this command line for example we are asking to
MatLab to write the number of rows in the matrix ar_event_window because there is 1
that correspond to the length of L1 (If you want to know the number of columns that
correspond to the number of firms you write 2 instead of 1).

---

Consider the test statistic J2 shows above:

J2(:,j) =
mean(SCAR(:,j))*(((size(ar_event_window,1))*(size(ar_est_windo
w,2)-4)/(size(ar_est_window,2)-2))^0.5);

end;

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