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st: RE: one tailed F statistics from TEST


From   "Yulia Marchenko" <ymarchenko@stata.com>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: one tailed F statistics from TEST
Date   Tue, 31 Aug 2004 11:26:41 -0500

On Tuesday, John wrote:

>Dear all,
>I want to compare the difference between two coefficients from a linear
>regression. The STATA command TEST gives me a F statistic from a Wald test.
>If I have a direction prediction of the two coefficients, can I convert the
>p-vlaue from the F statistics from two tailed to one tailed, despite the F
>distribution is not symmetric? Or I have to convert the F into a t
>statistics? Thanks.

Since your null hypothesis has only one restriction, i.e. difference between
two coefficients, the F-statistic has one degree of freedom for numerator.
In this case it is a squared t-statistic. Thus, you can obtained the p-value
for one-sided test using this relationship and the symmetry of the
t-distribution.

 pval=P(F_1,m > F)=P((t_m)^2 > F)=P(|t_m|>sqrt(F))=2*P(t_m>sqrt(F))

 P-value for the H1: beta1-beta2>0 is P(t_m>sqrt(F))=pval/2 where pval is
what you obtained from F-test.

 P-value for the corresponding lower one-sided test of H1: beta1-beta2<0 is
1-P(t_m>sqrt(F))

The support of F-distribution has positive values so the probability of
getting a negative value is zero. But the difference can be negative.
Therefore, using t-statistic seems quite reasonable.

--Yulia
ymarchenko@stata.com


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