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Re: st: Fixed effects ordinal probit regression


From   wgould@stata.com (William Gould, Stata)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Fixed effects ordinal probit regression
Date   Fri, 05 Sep 2003 09:10:21 -0500

James Shaw <shaw@pharmacy.arizona.edu> wrote, 

> I was wondering if there is such a thing as fixed effects ordinal probit
> regression.  If so, could one simply add dummy variables for the panel
> indicator (e.g., subject id) to the ordinal probit model to obtain fixed
> effects estimates?  Also, when estimating a fixed effects regression model
> with a subject-level effect, how problematic is it if there are missing
> observations on the dependent variable for some subjects (i.e., unbalanced
> panels)?

We the undersigned do *NOT* recommend using -oprobit- with dummies to fit a
fixed-effects ordinal probit model unless you have a large number of
observations within each group (a number to be made precise below).  Bo
Honore, an econometrician at Princeton, has derived a semiparametric estimator
for fixed-effects probit models, but we do not know of any generalization of
that work to ordered probit.  (We suspect that ordered logit would be easier
because of a property of the logit likelihood function that makes the
intercepts separable from the other coefficients.  It is that property that
makes the -clogit- estimator so "easy" -- the individual intercepts literally
drop out of the estimating equation.)

It is true that, in the linear-regression case, one can estimate fixed-effects
models by including the dummies, but that is a unique feature of the
linear regression estimator that does not carry over to nonlinear estimators.
The statistical problem is that, as the number of groups tends to infinity,
the number of estimated parameters increases at the same rate.  The estimates
are not consistent.

The argument sounds arcane.  How bad would it be, we wondered, if we ignorred
the theoretical problem and included the dummy variables anyway?  
One of us (Vince Wiggins) did simulations, using -logit- with dummies as a way
to fit fixed-effects logit models.  The results were awful.  Not only were
standard errors biased, but so were the coefficients and increasing the number
of observations by increasing the number of groups did nothing to eliminate
the problem.  Only when he held the number of groups constant and increased
the number of observations within group did he obtain reasonable results.  In
his case, he needed about 50 observations per group.  This should not have
surprised us.

One way to think about fixed-effects models is as fitting separate models
for each group, with the added constraint that the coefficients other than 
the intercepts be equal.  I.e., 

           Pr(outcome) = G(a_1 + b_1*X)          for group 1 

           Pr(outcome) = G(a_2 + b_2*X)          for group 2 

           ...

           Pr(outcome) = G(a_G + b_G*X)          for group G 

We then constraint b_1 == b_2 == ... == b_G.

Here is an infaliable rule you can use for when you can include the dummies to
estimate fixed-effects:  When you have enough observations in each group that
you would be willing to use the estimator on each group BY ITSELF and publish
those results in isolation.  There is a little efficiency to be gained by the
imposition of the constraint, so in fact you can even have a few groups with
fewer observations, but at that point, the rule is not infaliable and you
ought to do a simulation to be certain.

Jim also asked, 

> Also, when estimating a fixed effects regression model with a subject-level
> effect, how problematic is it if there are missing observations on the
> dependent variable for some subjects (i.e., unbalanced panels)?

It is not problamatic at all, either in the case of linear regression or 
the nonlinear estimators.  What you have heard about unbalanced panels has 
to do with the fact the that computations are more difficult for the 
random-effects estimator, not the fixed-effects estimator.

-- Bill                  -- Vince                  -- David
   wgould@stata.com         vwiggins@stata.com        ddrukker@stata.com
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