Significance testing

The following section presents the significance testing methods performed in this paper.

First off, the reasoning behind using the KP-test and GRANK test will be laid out. Next, the KP-test and GRANK test will be explained in detail in their own sub-section.

This paper makes use of two tests to test the null hypotheses: The parametric KP-test and non-parametric GRANK test, which have both been presented in the literature review. This combination of tests follows Kolari and Pynnönen’s (2011) recommendations on performing

event studies on stock returns. This combination allows this paper to perform a powerful significance test for the presented hypotheses, with the GRANK-tests, while improving the robustness of the result by performing the KP-test.

These tests are performed with the statistical software STATA16, with the command estudy provided by Pacicco et al. (2018). This useful command has been published in the official The Stata Journal: Promoting Communications On Statistics And Stata. This command allows one to perform a variety of event studies, of which the two chosen by this paper are part of. The estudy command automates the execution of parametric and non-parametric tests. However, the functioning and concepts behind both the KP-test and GRANK will be presented.

3.5.1. KP-test

The KP-test is a parametric significance t-test that is applied in event studies. It tests if abnormal returns are significant within an event or post-event window (Kolari and Pynnönen, 2010). This section will present how it is performed.

The test requires a sample of abnormal returns, determined by an estimation model and estimation window, an event date, and a (post-)event window. In order for the t-test to determine the significance of the abnormal returns, the dependent variable should be normally distributed, which is not the case for stock return data (Friedman, 1937). Kolari and Pynnönen (2010) solved this issue by standardising the abnormal returns on an individual level across time, but also for cross-section correlation between stocks.

First, we must standardise the 𝐴𝑅_𝑖𝑡 from formula (6) on an individual level. The standardised abnormal return, 𝑆𝐴𝑅_𝑖𝑡, can be found by formula (10):

𝑆𝐴𝑅

_𝑖𝑡

=

^{𝐴𝑅𝑖𝑡}

𝑠𝐴𝑅_𝑖

(

10

)

With 𝑠𝐴𝑅_𝑖, the standard deviation of abnormal returns coming from formula (11) (Campbell et al., 1998). 𝑠𝐴𝑅_𝑖, is computed on the basis of data within the estimation window (T) of length N.

𝑠𝐴𝑅

_𝑖²

=

¹

𝑁−2

∑ (𝐴𝑅

^𝑇2_{𝑇1 𝑖𝑡}

)

²

(

11

)

This study wishes to test the significance of stock returns across time. It needs to adapt its standardised abnormal returns into cumulative standardised abnormal returns. This is done with formula (12) which borrows the 𝐶𝐴𝑅_𝑖 from formula (7).

𝑆𝐶𝐴𝑅

_𝑖τ

=

^{𝐶𝐴𝑅𝑖}

𝑠𝐶𝐴𝑅_𝑖

(

12

)

With 𝑠𝐶𝐴𝑅_𝑖 being the square root of 𝑠𝐶𝐴𝑅_𝑖² coming from formula (13):

𝑠𝐶𝐴𝑅

_𝑖²

= 𝑠𝐴𝑅

_𝑖²

∗ (𝑁) (

13

)

Next, one must take the 𝑆𝐶𝐴𝑅_𝑖𝑡 and re-standardise it with the cross-sectional standardised deviation. This enables the resulting 2𝑆𝐶𝐴𝑅_𝑖τ to account for possible event-induced volatility impacting the return of other firms in the sample. Formula (14) shows the necessary step to compute the 2𝑆𝐶𝐴𝑅_𝑖τ.

2𝑆𝐶𝐴𝑅

_𝑖,τ

=

^{𝑆𝐶𝐴𝑅𝑖,τ}

𝑠𝑆𝐶𝐴𝑅τ

(

14

)

With 𝑠𝑆𝐶𝐴𝑅_𝑖 being the cross-sectional standard deviation of 𝑆𝐶𝐴𝑅_𝑖 from formula (12).

𝑠𝑆𝐶𝐴𝑅_𝑖 is the square root of 𝑠𝑆𝐶𝐴𝑅_𝑖², which is computed from formula (15) with 𝑛 being the number of firms in the sample.

𝑠𝑆𝐶𝐴𝑅

_τ²

=

¹

𝑛−1

∑

^𝑛_𝑖=1

(𝑆𝐶𝐴𝑅

_𝑖

− 𝑆𝐶𝐴𝑅 ̅̅̅̅̅̅̅

_τ

)

²

(

15

)

With 𝑆𝐶𝐴𝑅̅̅̅̅̅̅̅ being the average of 𝑆𝐶𝐴𝑅_𝑖 within the event window, see formula (16):

𝑆𝐶𝐴𝑅

_τ

̅̅̅̅̅̅̅̅ =

¹

𝑛

∑

^𝑛_𝑖=1

𝑆𝐶𝐴𝑅

_𝑖τ

(

16

)

A traditional t-test can now be performed on 2𝑆𝐶𝐴𝑅_𝑖τ in order to determine if the standardised cumulative abnormal return is significantly different to zero. The hypotheses for this test are:

𝑁𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 ∶ 2𝑆𝐶𝐴𝑅 = 0 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 ∶ 2𝑆𝐶𝐴𝑅 = 0

Similarly, to formula (9) the SSCAAR can be tested cumulatively on the sample with:

𝑁𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 ∶ 2𝑆𝐶𝐴𝐴𝑅 = 0 𝐴𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 ∶ 2𝑆𝐶𝐴𝐴𝑅 = 0

3.5.2. GRANK-test

Transforming the AR into the 2SCAR has made it a variable with a mean of zero and variance of 1 (Kolari and Pynnönen, 2011). The previously computed 2SCAR and SAR are the

starting points for creating the generalised standardized abnormal return, GSAR. The GSAR is found through formula (17) and (18)

Within the event window 𝐺𝑆𝐴𝑅_𝑖𝑡 = 2𝑆𝐶𝐴𝑅_𝑖𝑇

(

17

)

Outside of the event window 𝐺𝑆𝐴𝑅_𝑖𝑡 = 𝑆𝐴𝑅_𝑖𝑡

(

18

)

During the event window, τ, GSAR behaves like any other standardized return under the null hypothesis but begins to deviate under the alternative hypothesis. Kolari and Pynnönen (2011) use the GRANK test to take advantage of this and test for possible deviation under the alternative hypothesis.

To derive their rank test, they redefine the time indexing in order to squeeze the whole CAR-period length into τ (Cowan, 1992). This is defined as the cumulative event date. Kolari and Pynnönen (2011) note the T is considered to be one time point in which the abnormal return (GSAR) equals the re-standardized 2SCAR from formula (14). For other time points, GSAR equals the usual standardized abnormal return SAR from formula (10). This means that the new number of observation is L1+1. L1 are the days in the estimation period and the one additional observation is the cumulative event day.

Next, we define the demeaned standardized abnormal ranks of the GSAR in formula (18) 𝑈_𝑖𝑡 denotes the demeaned standardized abnormal rank of each cumulative abnormal return.

Under the null hypothesis we expect the value of 𝑈_𝑖𝑡 to be 0:

𝑈

_𝑖𝑡

=

^{𝑅𝑎𝑛𝑘(𝐺𝑆𝐴𝑅𝑖𝑡)}

𝑇+1

−

¹

2

(

18

)

According to the Kolari and Pynnönen (2011): For i=1,…,n where t ∈ T = {T-1+1,…,T0,0}

is the set of time indexes including the estimation period t = T0 + 1, …, T1 and the CAR at t = 0.

Where the dates T0 + 1 and T1 corresponding to the first and last observations in the estimation window, where T = L1 + 1 = T1 − T0 + 1 is the total number of observations with L1=T1 − T0 or the number of estimation period returns. 𝑅𝑎𝑛𝑘(𝐺𝑆𝐴𝑅_𝑖𝑡) substitutes 𝐺𝑆𝐴𝑅_𝑖𝑡 by its respective rank number, between 1 and T.

The authors of the model define the following null hypothesis of no mean effect on abnormal returns by the event:

𝑁𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠 ∶ μ_τ = 0

(

19

)

With, μ_τ = E[CARτ], is the expected value of the cumulative abnormal returns over the period of length τ.

The generalized rank t-statistic (GRANK-T), with T-2 degrees of freedom, for testing hypothesis (19) is defined as:

𝑡

_{𝑔𝑟𝑎𝑛𝑘}

= 𝑍(

^𝑇−2

𝑇−1−𝑍²

)

^1⁄2

(

20

)

Where,

𝑍 =

^𝑈̅0

𝑆_𝑈̅

(

21

)

With,

𝑆

_𝑈̅

= √

¹

𝑇

∑

^𝑛𝑡

𝑛

𝑈 ̅

_𝑡²

𝑡∈

(

22

)

&

𝑈 ̅

_𝑡

=

¹

𝑛_𝑡

∑

^𝑛_𝑖=1^𝑡

𝑈

_𝑖𝑡

(

23

)

𝑛_𝑡is the number of valid 𝐺𝑆𝐴𝑅_𝑖𝑡, available at time point 𝑡, with 𝑡 ∈ T, T = {T0+1,…,T1,0}.

T=T1-T0+1 equals to the numbers of observations (estimation window length +1 observation).

𝑈 ̅

is the mean

𝑈 ̅

_𝑡 at time t= 0 (the cumulative abnormal return).

This testing methodology allows the GRANK procedure to outperform previous rank tests of CARs and was robust to abnormal return serial correlation and event-induced

volatility (Kolari & Pynnonen, 2011). Additionally, the GRANK procedure exhibits superior empirical power relative to popular parametric tests.

In document TERM RETURNS OF STOCKS . A N EVENT STUDY ON SHORT SQUEEZES AND THEIR EFFECT ON THE LONG - (pagina 30-34)