Dear Statalist,
While doing an extension of the code weibhet_glfa I
found what I believe is a mistake in the
log-likelihood expression of this code. Consider that
the mixed survivor function is given by
Sm(t)= [ 1 � theta*ln{S(t)}]^(-1/tetha)
(1)
Where S(t) is the survivor function corresponding to
the case in which there is no unobserved
heterogeneity; and theta is the variance of the gamma
distribution that it has been assumed for the
unobserved characteristic. Now, using the fact that
-dln(Sm)/dt = hm(t) I obtain the mixed
hazard function:
hm (t)= - {S�(t)/S(t)}*{1-theta*ln{S(t)}}^(-1)
(2)
Using (1) and (2) I have written down the
log-likelihood obtaining (in an intermediate
expression that helps comparison) the expression:
Logl = ln{1+theta*exp[-x�b*p]*t^(p)}^(-(1/theta + d))
+
+ ln{exp[-x�b*p]*p*t^(p-1)}
(3)
Where x is the vector of observed characteristics and
b is its corresponding vector of coefficients. In the
weibhet_glfa code the log-likelihood is written as
Logl = ln{1+theta*exp[-x�b*p]*t^(p)}^(-(1/theta + d))+
+ ln {exp[-x�b*p]*p*t^(p)}
(4)
Notice that while in (3) we have a term t^(p-1), in
(4) the corresponding term is t^(p) (look in the
second ln(.)). After checking the consistency of (3)
with the log-likelihood that Lancaster (1979) would
write using his own expressions for Sm(t) and hm(t), I
have become to believe that (4) is incorrect and that
the code of weibhet_glfa requires a slight
correction.
Could you double-check my calculations so that I have
a second opinion on the issue?.
Thank you,
Alfonso Miranda
Research Student
Economics Department, University of Warwick.
References
1. Stata maunal pp. 343-375 (on the st streg code)
2. Lancaster (1979) Econometric Methods for the
Duration of Unemployment. Econometrica 47 (4), pp.
939-956.
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