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st: NLLS w/ normal distribution - some parameters not estimated


From   Denis Kalugin <denis.kalugin@gmail.com>
To   statalist <statalist@hsphsun2.harvard.edu>
Subject   st: NLLS w/ normal distribution - some parameters not estimated
Date   Sun, 13 Jun 2010 15:19:34 +0400

Dear All,

I'm trying to do a non-linear least squares estimation to nail down a
latent dependence. The latent variables can be regressed on a linear
form and look like
L1=xb1+error1
L2=xb2+error2

The probabilities of observing a value of y is given by
P(y=-1)=P(L1<z)=norm( (z-xb1)/error1 ),
P(y=1)=P(z<min(L1,L2))=( 1- norm( (z-xb1)/error1 ) )*( 1- norm(
(z-xb2)/error2 ) )
P(y=0) = 1 - P(y=1) - P(y=-1)
The expected value of the observable dependent variable is given by
P(y1=1| b1,b2) - P(y1=-1| b1,b2) so the expected value becomes
E(y_i)=( 1- normal ( (z-xb1)/error1) )*( 1- normal ( (z-xb2)/error2) )
- normal ( (z-xb1)/error1) ).

The problem is that when I estimate this equation, all the b1
parameters are not estimates, and are reported as if they were
constants. How can i address this problem?
Furthermore, I don't know how to include individual error terms in the
regression. Including them as parameters does not do the trick, they
need to be drawn from the original L1 and L2 equations somehow. Is
there a way to do this?

Thanks in advance!

Best regards, Denis.
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