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

 From "Brian P. Poi" To statalist@hsphsun2.harvard.edu Subject Re: st: Adding up constraints in a nonlinear system of share equations Date Wed, 23 Feb 2011 21:02:49 -0500

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On 2/23/2011 7:18 PM, Alex Olssen wrote:
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```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,

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The predicted expenditure shares from an AIDS model will sum to one by virtue of the fact that the parameters were constrained during estimation to ensure that happens.
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The more common issue that arises in demand system estimation is how to deal with expenditure shares that equal zero (a "corner" solution). One strand of literature started by Wales and Woodland (1983) is based on the Kuhn-Tucker conditions for constrained maximization, while another strand tries to tackle the problem from a multivariate generalization of the tobit model; Heien and Wessells (1990 and 1993) are well-known papers in this area. The American Journal of Agricultural Economics would be a good place to browse for more recent developments. I've never tried fitting any of those models, so I can't be of more help.
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-- Brian Poi
-- brian@poiholdings.com

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Heien, D. and Wessells, C. R. Demand systems estimation with microdata: A censored regression approach. J. Business and Economic Statistics 8:365-71.
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Wales, T. J. and Woodland, A. D. Estimation of consumer demand systems with binding non-negativity constraints. J. Econometrics 21: 263-85.
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