I've been working on a paper for random effects modeling of
conditionally beta random variables which I am going to be diving into
again in the next few weeks (finally!). We don't have code for Stata but
I can send you SAS code (or winBUGS scripts for Bayesian versions). Also
see:
Z. Qiu, P. Song, & M. Tan (2009). Simplex Mixed-Effects Models for
Longitudinal Proportional Data. Scandanavian J. Stat. 35(4): 577-596.
This paper proposes using the simplex distribution for the error
density, with the multivariate Gaussian as the mixture. (The simplex
distribution is to the inverse Gaussian what the beta is to the gamma.)
There are many other such distributions and at this point I really don't
have a clue as to what distribution on a bounded interval is better.
Anyhow, my feeling is that if the Tobit approach is working and giving
sensible answers, and if you're comfortable with the linearity
assumptions inherent in it as opposed to the diminishing returns near
the boundary assumption inherent in the nonlinear models, I'd stick with
it. If you feel there should be diminishing returns as you get close to
the boundary the nonlinear model makes sense. For many models, you can
"cheat" them back into the unit interval quite safely, e.g., by
rescaling all the observations as:
Ynew = eps/2 + (1-eps/2)*Yold
without disturbing estimation and inference much. (However, the
logit-transformed normal is not one of these models.)
JV
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