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Re: st: Nonlinear regression and constraints

From   "Erik �. S�rensen" <[email protected]>
To   [email protected]
Subject   Re: st: Nonlinear regression and constraints
Date   Wed, 29 Jun 2005 09:11:34 +0200

On 29. jun. 2005, at 08.49, Daniel Schneider wrote:
Without going to much into detail: my parameters are percentages. They
can only range from 0 to 1. There may be a better solution (i.e. a
solution that better fits the data) beyond 1, BUT, as I said, by
definition they cannot be above 1 (or below 0). So the best solution
that is possible has to be between 0 and 1.
I think the easiest way to do this is to define a function g:R-> (0,1),
and then where you want to estimate a parameter \theta \in (0,1),
you instead make the transformation of variables

\theat = exp(x)/(1+exp(x)).

So stata optimises to find an x \in R, but you know that the estimate of your parameter is \theta^*=exp(x^*)/(1+exp(x^*)). You can use asymptotic theory to find the standard errors of this transformation. Or you can estimate first with this method and then reestimate with starting points close to the estimates and hope that it converges to the same estimates instead of going off out of the allowed parameter space.

I'm not sure it is a good idea to try to adopt to the possibility that \theta=0 or \theta=1, at least all standard asymptotic distribution theory would break down at the limits of the parameter space.

best regards,

Erik �. S�rensen, dept of Econ., Norwegian School of Economics

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