Event Study: negative CAAR but positive test statistic (Wilcoxon rank test)

[Phil Bach]

Dear Forum,

I recently conducted an event study using eventstudy2 by Thomas Kaspareit. The resulting CAAR exhibit a barely negative result of -0.0003448 which corresponds to a negative abnormal return of -0.03%:

However, the resulting Wilxocon signed-rank test shows a positive sign. In fact, all non-parametric tests show a positive sign for this CAAR-result:

"NCAARE" refers to the test-statistic and "PCAAR" to the corresponding p-value of the above illustrated non-parametric tests. The yellow box indicates the Wilcoxon test, which is of particular interest to me.
I searched for possible explanations but I couldn't find anything about this issue.

I am happy for every hint or help!

Thank you very much in advance.

Best regards,

[Carlo Lazzaro]

Phil:
given the limited details you provided, a tentative reply rests on which one of the two groups is coded 1:

. use "https://www.stata-press.com/data/r17/fuel2.dta"

. g treat_slave=0 if treat==1

. replace treat_slave=1 if treat==0

. ranksum mpg, by(treat)

Two-sample Wilcoxon rank-sum (Mann–Whitney) test

       treat |      Obs    Rank sum    Expected
-------------+---------------------------------
   Untreated |       12         128         150
     Treated |       12         172         150
-------------+---------------------------------
    Combined |       24         300         300

Unadjusted variance      300.00
Adjustment for ties       -4.04
                     ----------
Adjusted variance        295.96

H0: mpg(treat==Untreated) = mpg(treat==Treated)
         z = -1.279
Prob > |z| = 0.2010
Exact prob = 0.2117

. ranksum mpg, by( treat_slave )

Two-sample Wilcoxon rank-sum (Mann–Whitney) test

 treat_slave |      Obs    Rank sum    Expected
-------------+---------------------------------
           0 |       12         172         150
           1 |       12         128         150
-------------+---------------------------------
    Combined |       24         300         300

Unadjusted variance      300.00
Adjustment for ties       -4.04
                     ----------
Adjusted variance        295.96

H0: mpg(treat_~e==0) = mpg(treat_~e==1)
         z =  1.279
Prob > |z| = 0.2010
Exact prob = 0.2117

.

[Phil Bach]

Hello Carlo,

thanks for your reply. Sorry, I should have elaborate my question more precisely.

I was wondering why the Wilcoxon test exhibits a positive value, even though the CAAR is negative. As far as I know, these two values should share the same sign, or is it possible, under certain circumstances, that test statistic and coefficient have opposite signs?
Thanks a lot in advance.

Kind regards,
Phil

[Carlo Lazzaro]

Phil:
this is not necessarily so.
See Bland M. An introduction to medical statsistics. 3rd ed. Oxford: OUP, 2000: 217-20.