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st: Asymptotic covariance matrix for a structural probit equation in a selection model
i am running a selection model with endogenous switching and limited
dependent variables described in the maddala book (1983) explained on pages
236-238. this is also the type of a model that is equivalent to lee's union
wages model (1978).
to summarize, i have two regime functions
(1) y1 = B1'X + u1
(2) y2 = B2'X + u2
and a criterion function
(3) C = A'B + d(y1-y2) + u
so, the procedure is to estimate the reduced form of (3) and to get the
selectivity terms (inverse mills ratios).
(I) then, use the estimated selectivity ratios to predict y1^ and y2^.
(II) finally, i have to plug the estimated y1^ and y2^ (second-stage) into
the stuctural probit equation (3) to get the coefficient d.
firstly, to my understanding, procedure (I) does not provide the correct
standard errors because of the fact that selectivity terms (inverse mills
ratios) are estimated. in this case, standard errors are underestimated.
as far as i understand the problem, LIMDEP takes care of this problem and
provides consistent standard errors. my first question is whether there is a
procedure already developed in stata that takes care of this? or, does one
have to manually adjust the standard errors by constructing the correct
secondly, since y1 and y2 are estimated in the first stage, when i use them
in the second stage (II) to estimate the structural probit equation, i also
run into a problem of incorrect standard errors. maddala mentions that
standard errors are also underestimated in the second stage because y1^ and
y2^ are estimated in the first step. he mentions lee's (1978) example how
his standard errors are underestimated (p. 238 of maddala) and points to the
appendix (p. 255) where the derivation of the correct covariance matrix is
shown. so, my question is again, whether there is a command in stata that i
am overlooking or does one have to construct the correct covariance matrix
by hand? to my understanding, LIMDEP also does not take care of this
it might be that case that i just cannot find already existing solution to
this problem and i'd appreciate if someone could point me in the right
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