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st: MLE est. linear form with two sets of parameters


From   Denis Kalugin <denis.kalugin@gmail.com>
To   statalist <statalist@hsphsun2.harvard.edu>
Subject   st: MLE est. linear form with two sets of parameters
Date   Sun, 30 May 2010 23:57:36 +0400

Dear all,

I'm currently facing a problem of estimating the following setup:
The goal is to estimate two latent variables, let's say L1 and L2. The
variables are linked to variable y in the following manner:
y=0 if L1<z<L2
y=-1 if z>L2
y=1 if z<L1

L1=exp(x*b1+error)/(1+exp(x*b1+error)
L2=min[exp(x*b1+error)/(1+exp(x*b1+error)
, exp(x*b2+error)/(1+exp(x*b2+error)

This is estimated via slightly altered probit (as far as I understand).

P(y=-1)=norm[(log(z/(1-z)) - xb1)/sigma(error1)]
P(y=1)=(1-norm[(log(z/(1-z)) - xb1)/sigma(error1)])*(1-norm[(log(z/(1-z)) -
xb2)/sigma(error2)])
P(y=0)=1-p(y=-1)-p(y=1)

However, my problem right now is how to feed TWO sets of parameters (b1 and
b2) to be estimated. If i write this as a linear form, then stata would
generate only one vector. in "ml model" function there is no option to
define the parameters manually (as there is in non-linear least squares for
example).

A related question is how to estimate these dependencies with non-linear
least squares method? The original estimation routine is from Dudley (2007),
a paper on Tests of Dynamic Trade-off theory of Capital Structure. The
authors just says "estimation is done by NNLS", but i don't get a clue as to
how he goes about it.

Could you please help me out on this coz i'm utterly stuck.

Many thanks in advance!

Best regards, Denis.
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