Thursday, March 16, 2006
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 covariance matrix?
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 automatically.
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 direction.
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