[Limdep Nlogit List] A selection qeustion--a quick follow-up

[William Greene]

Fred (and all).  The Terza JAE (2002) paper is a FIML estimator for
the MNL model with selection that was queried in the first email. The
use of the Taylor series in this paper was to describe an alternative
approach by Mullahy and Sindelar.  I am current working on an implementation
of the model in LIMDEP, but am not ready to say it is complete. The
Terza and Kenkel (2001) paper is a formal sample selection model for
counts that derives the appropriate conditional mean function for the
Poisson variable under selection, and estimates it by nonlinear least
squares. It does not use an IMR approach.  Terza and I developed the
FIML estimator for this model simultaneously (to some extent, jointly)
in the early 1990s.  It is currently fully implemented in LIMDEP; however, 
being a model for counts, it is not helpful to this discussion.
  The Boyes, Hoffman and Low paper describes the bivariate probit model
with sample selection that has been fully implemented in LIMDEP since 
about 1992.  Likewise the Poirier model on partial observability, which
is one of several forms of this model.  The LIMDEP manual contains 
extensive documentation on all of these models.  FIML estimators for all
of them are incorporated in LIMDEP.  They do not use IMRs.  On the other
hand, neither are these helpful; they are not the model that was queried
about.  As Fred points out, the modeling of the joint distribution of the
disturbances that is used in the probit and Heckman linear models does not
carry over to the multinomial logit model, so the problem remains.
   The business of the standard errors of MLEs in these models relying
on suspect asymptotic results is a canard.  The idea that the Bayesian 
estimators somehow get everything right and the MLE doesn't has two flaws, 
notwithstanding all the other problems listed: (1) If the sample is 
at all large, and that does not have to be excessively so, though the 
typical application does use a big sample, then the asymptotics for these 
models are fine, and, in fact, they are very well behaved.  I and many
others have been using them for a couple decades.  Moreover, even in a
moderately sized sample, unless one has a very strong prior - and that
would taint the results - the Bayesian estimates are generally close
to identical to the MLEs (a theorem of Bernstein and von Mises is at work).
Indeed, it is a signal that something is wrong when the Bayesian
estimate deviates too much from the MLE.  After all, if the sample is
relatively large, the log likelihood is nearly symmetric and unimodal
(central limit theorem), so the mode (MLE) and the mean (posterior mean)
coincide.
(2) If the sample is so
small that these criticisms would have any relevance -- and we would be
talking about small 2 digit sample sizes, then the fact that the Bayesian
estimators have exact small sample properties is a flaw not a virtue. The
results are posterior to just the sample data, and can't be extended beyond
the small sample in hand. But, it is a heroic (and very classical) assumption
that somehow this small sample tells you the story about the whole 
population you are trying to characterize.

/Bill Greene


----- Original Message -----
From: "Fred Feinberg" <feinf at umich.edu>
To: "Limdep and Nlogit Mailing List" <limdep at limdep.itls.usyd.edu.au>
Sent: Thursday, October 18, 2007 1:55:53 PM (GMT-0500) America/New_York
Subject: Re: [Limdep Nlogit List] A selection qeustion--a quick follow-up

William Greene wrote:

> Christer.  There is a paper by Joe Terza in the Journal of Applied
> Econometrics, 2004, I believe, that specifically discusses this
> application.  Your suggestion does not work - the IMR approach is
> only appropriate for linear models.

Prof. Greene is of course correct about Inverse Mills Ratios; you need an
actual residual to apply them, and logit/probit models don't supply those
(unless you use some form of multiple imputation or Bayesian methods to sample
over the unobserved "utility"; but then you wouldn't bother with IMLs in the
first place.)

There are two papers by Terza that deal with this:

"The Effect of Physician Advice on Alcohol Consumption: Count Regression with
an Endogenous Treatment," DS Kenkel, JV Terza - Journal of Applied
Econometrics, 2001.

"Alcohol abuse and employment: a second look," JV Terza - Journal of Applied
Econometrics, 2002.

Both use estimators based on approximations (in one case, something like the
Mills Ratio; in the other, a Taylor series). These may be expedient, but there
is no guarantee that they look like the real likelihood, particularly when the
error correlations intrinsic to the Heckman set-up are far from zero.
Generally speaking, there are three ways to go about it; putting each
casually:

1) Two-Step Estimators: quick, easy to use, but you have no idea if the answer
is correct
2) MLEs: higher-dimensonal search, provides correct answer for unimodal
likelihoods, but standard errors of coefficients based on asymptotics that may
not hold
3) Bayesian: correct everything, but you have to code it yourself, wait
forever for the sampling, and you can't be sure it's really converged (least
noxious possibility: use WinBugs; at least you don't have to calculate full
conditional densities)

Dr. Bilgic recommended two other papers:

Boyes, W. J., D.L. Hoffman, and S. A. Low. 1989. An Econometric analysis of
the bank credit scoring problem, Journal of Econometrics 40, 3-14.

Poirier D. J. 1980. Partial observability in bivariate probit models, Journal
of Econometrics 12, 210-217.

The Low paper cites Poirier, and writes the likelihood out explicitly, which
it then claims to maximize... somehow (also calculate standard errors). It
seems to be a "Heckman's Probit" model -- with a binary selection and
prediction -- and as such is directly estimable by Limdep. I don't think it
shows how to estimate the model you're after, but I could be wrong. That would
be covered by the Lee paper in my prior e-mail, as well as the other paper of
mine I linked to.

FF

=====

Fred Feinberg
Hallman Fellow and Professor of Management
Stephen M. Ross School of Business
University of Michigan
feinf at umich.edu


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Professor William Greene
Department of Economics
Stern School of Business
New York University
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