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st: Adding up constraints in a nonlinear system of share equations


From   Alex Olssen <Alex.Olssen@motu.org.nz>
To   "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject   st: Adding up constraints in a nonlinear system of share equations
Date   Thu, 24 Feb 2011 13:18:48 +1300

Hi Statalisters,

I am estimating a system of nonlinear share equations.  It is similar to an AIDS model but I want to make sure the predicted shares always lie in the [0,1] interval.  Am I correct in thinking that predicted vaules from AIDS and QUAIDS models can fall outside this interval?

In a linear system of share equations there are adding up constraints which make sure the sum of the shares is always one.  Such constraints correspond to the net marginal effect of changing a regressor being zero.

My question is whether the equivalent constraints for a nonlinear system provide for an estimable model.  The specific nonlinear model I am thinking results in assuming the shares follow a logit model - this assumption may be shaky but please bear with me.

In the logit case the marginal effect of changing x_k is well known to be  e^(v_i)/((1+e^(v_i))^2)*B_k.  Where v_i is the linear combination of regressors and coefficients and B_k is the estimated coefficient on x_k.  Following the logic from the linear system I translated the adding up constraint to say that the sum of these marginal effects over i must be equal to zero.
Is this correct?  Can it be estimated?  My problem with estimation is that the constraint must hold across all values of x_k as the marginal effects vary with x_k.  To estimate this, must I solve for parameters as functions of x_k?

Does anyone know of an easier solution?  I have yet to see an AIDS/QUAIDS paper that guarantees predicted values to fall in the [0,1] interval.  If you know of one I would be very grateful for the pointer!

Kind regards,

Alex Olssen
Research Analyst
Motu Economic & Public Policy Research
Ph 939 4250  Fax 939 4251
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