** My apologies if this comes through twice. **
Several people have written about the random effects probit estimator
in Stata. There are several issues here.
(*) Limdep uses BHHH in fitting this model (and uses it in many
of the models it fits; check the documentation). That
is why the standard errors do not match between the
packages. Stata uses the usual or observed Hessian
(inverse matrix of second derivatives).
(*) Stata fits a constant only random effects probit estimator
at the first stage. It keeps this log-likelihood for a
future test of the fully specified model.
(*) When Stata starts to fit the full model, it takes the
values for a regular probit model and then calculates
the log-likelihood of the random effects probit model
using those coefficients with rho=0.0, 0.1, 0.2, ... It
continues to fit these models until the log-likelihood
fails to improve. It then starts the optimization using
the (regular) probit estimates along with the "best"
value of rho it found in this simpleminded search.
So , several have asked why Stata doesn't use the
regular probit answers as the starting value. The
answer is that Stata does use those values, but
immediately tries to find a better starting value for
rho before starting the optimization process. I believe
that all of the random-effects models in Stata that use
Gauss-Hermite quadrature take this same sequence of
steps.
It is easy to verify these starting values by specifying
the -trace- option on the command (and running the
regular probit command separately honoring any sample
selection that occurs in the xtprobit command).
(*) Typically, Stata has no problems fitting this model except
when rho is very small (it could have problems when rho
is close to one, but this is not very common). When rho
is very small, the parameterization of the correlation
using the arctan wants to march off to negative
infinity. The calculation of the second derivatives
becomes numerically difficult and results with missing
standard errors are not uncommon.
(*) Users can specify their own starting values, but must
remember to specify a value for the correlation
parameter in the arctangent metric.
(*) As has been shown in a very nice article in The Stata Journal,
the model may be fit much more efficiently (and
numerically stable) using adaptive quadrature. So, use
the -gllamm- command instead.
(*) Whether one would want to use BHHH calculated standard errors
or use this for the optimization step-size calculations
is open for debate. Personally, I want the standard
errors that Stata reports. The xtprobit command is
mostly internal calculations. It was written at a time
that internal commands used a different (but very
similar) collection of optimization routines from the
-ml- collection. Nowadays, I think that they may use
the same routines. Coding your own estimator using the
-d2- form of the -ml- command allows you to use whatever
method you want to form the stepsize matrix for the
optimization.
(*) You can affect the answers that you get using several of the
options on the -xtprobit- command. You can specify a
less conservative tolerance for the log-likelihood. You
can specify a small number of maximum iterations. You
can specify a different number of quadrature points,
though as mentioned earlier you are likely better off
using a command that offers adaptive quadrature in the
calculation. To be clear, the quadrature is used to
calculate the log-likelihood, the gradient, and the
hessian matrix. You can specify different starting
values -- at the very least this will assist you in
determining if values from another package result in a
better log-likelihood than that found at the last step
in Stata.
(*) Running -quadchk- is a must. You should always do this
and the manuals give this advice. With the availability
of gllamm, I would run that command instead specifying
the adapt option. At the very least I would use the
gllamm with adapt to verify the answers in xtprobit.
Best,
James
----------------------------------------------------------------------
James W. Hardin, Ph.D., Lecturer [email protected]
Department of Statistics, Blocker 416G 979-845-3141 (phone)
Texas A&M University Mail Stop-3143 979-845-3144 (fax)
College Station, TX 77843-3143 http://stat.tamu.edu/~jhardin
----------------------------------------------------------------------
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