Thanks to Kit Baum a new package -propcnsreg- is available on ssc.
-propcnsreg- fits by maximum likelihood a linear regression with a
proportionality constraint. Consider a model where the explained
variable (y) depends on two explanatory variables (x1 and x2).
Furthermore suppose that the effects of x1 and x2 differ for different
values of a third variable (x3), i.e. the model contains interaction
effects between x1 and x3 and between x2 and x3. The proportionality
constraint imposes the constraint that the effects of x1 and x2 change
by the same proportion as x3 changes. So if the effect of x1 doubles
for a unit change in x3, than so does the effect of x2.
One of the nice characteristics of this model is that the constrained
variables can be interpreted as together measuring a latent variable,
whereby their parameters measure how strong they load on that latent
variable. It is an implementation of a so called MIMIC model (Hauser
and Goldberger, 1971).
The difficulty with this model is that the parameters are highly
correlated, thus making it hard for the standard maximization
algorithms to find the maximum of the likelihood function. To overcome
this issue an EM algorithm is first used to find suitable starting
values. The EM algorithm breaks the correlation by first treating the
loadings of the constrained parameters as fixed and estimate the
effect of the latent variable, and than treat the effect of the latent
variable as fixed and estimate the loadings. This is iterated till
convergence. Although this algorithm is likely to find the maximum in
the end, it can end up taking a large number very small steps towards
that maximum. To "lively this algorithm up" every fifth iteration will
consist of two iterations from the standard -ml- algorithm for the
entire likelihood. Once the EM algorithm has converged, the parameter
estimates are fed into the standard -ml- algorithm to get the correct
standard errors.
I will give a presentation on this package at the UK Stata Users' Group
Meeting on Monday, September 10.
Maarten
Hauser, Robert M. and Arthur S. Goldberger. 1971. "The Treatment of
Unobservable Variables in Path Analysis." Sociological Methodology 3:
81-117.
-----------------------------------------
Maarten L. Buis
Department of Social Research Methodology
Vrije Universiteit Amsterdam
Boelelaan 1081
1081 HV Amsterdam
The Netherlands
visiting address:
Buitenveldertselaan 3 (Metropolitan), room Z434
+31 20 5986715
http://home.fsw.vu.nl/m.buis/
-----------------------------------------
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