The Wiley Finance Series : Handbook of News Analytics in Finance

In a classicalt-test, thet-statistic is distributed according to Student’st-distribution, and
thusis large enough to be statistically significant if:

t > t^1 ð 1 = 2 ;Þð 3 : 11 Þ

whereð 1  Þ100% is the confidence level, andis the number of degrees of freedom,
calculated as

 

ð

^2 þ

w

^2 

w

Þ^2

^4 

w^2 ðw 1 Þ

þ

^4 þ

w^2 ðw 1 Þ

: ð 3 : 12 Þ

When measuring variables that are not completely independent (such as pre-event
returns/volatility), thet-statistic may not follow an exactt-distribution. To ensure that
the confidence levels we compute are accurate, we empirically determine the distribution
of thet-statistic under the null hypothesis as follows.
We construct random event studies by choosing, say, 500 random points in time and
declare these as ‘‘events’’. We then compute thet-statistic of this event study and repeat
this process 5,000 times to generate the finite sample null distribution. The resulting
samples give a reliable estimateDof the distribution of the variabletfor random
(insignificant) events. We can then compare thet-statistics obtained from our (non-
random) events and measure their significance according to the following formula:

sigðtÞ¼Pr

x D

½Šxt : ð 3 : 13 Þ

This yields a more robust significance measure for ourt-tests. The empirical values of
t-statistics are reported in Section 3.A.3 (see appendix on p. 102).

3.5.4 Levene’s Test for equality of variance

Another statistical test we apply to averaged event window samples is Levene’s Test,
which tests for a change in standard deviation before and after the event. This test is
most naturally applied to returns since a change in standard deviation would suggest a
change in volatility. Thus we begin with the averaged event window samples of log
returnsf^rrwþ 1 ;:::;rr^ 0 ;^rr 1 ;:::;^rrwg, and then compute the following quantities:

^rrmedianmedianfrr^wþ 1 ;::;^rr 0 g ; rr^medianþ  medianf^rr 1 ;::;^rrwgð 3 :14aÞ

Zjj^rrj^rrmedianj ; Zjþ jrr^jþrr^medianþjð 3 :14bÞ

Z

1

w

j 0 Zj ;Zþ 

1

w

j> 0 Zjþ ;Z 

1

2

ðZþZþÞð 3 :14cÞ

Finally, we compute:

Qwð 2 w 2 ÞððZZÞ^2 þðZþZÞ^2 Þð 3 :15aÞ

Rj 0 ðZjZÞ^2 þj> 0 ðZjþZþÞ^2 ð 3 :15bÞ

WQ=R ð 3 :15cÞ

Standard deviation changes withð 1  Þ100% confidence ifW>F^1 ð 1 ; 1 ;w 2 Þ.

88 Quantifying news: Alternative metrics