Dear Achim,

As you probably saw, STATA estimates a function of rho, atanh rho
=1/2ln(1+rho/1-rho), instead of rho itself. A rho of precisely 1 would
make the denominator zero, and so make atanh rho undefined.

If you have a rho of near one but not precisely one, there are some
things you can do to investigate further. As you mention, the Heckman
estimator often produces strange results when you don't have enough
information to estimate. So, the obvious first step is to think about
whether you have a good exclusion restriction (and whether you otherwise
like your specification), and get a better exclusion restriction if you
can (easier to say than do, of course). Second, take a look at the
confidence interval around rho. This is often more reliable in small
samples than looking at the standard error of rho. (STATA calculates the
standard error of rho using the delta method, but it calculates the
confidence interval without approximation, by calculating the confidence
interval around atanh rho and transforming the end points back. In the
limit, these methods give the same answer, but in small samples, the
approximation doesn't always seem very good.) Third, if the CI tells you
that rho could be almost anything, you can perform some sensitivity
analyses. Try fixing rho at some other reasonable levels and see if
changing rho changes your substantive conclusions about your independent
variables. If so, you are back at step one. You also could gather more
data. Of course, it's always possible that your true rho is high (you
might believe it if the estimate is high but not 1, you are confident in
your specification, and the CI isn't wide), but I think you're smart to
investigate further given what you say in your message.

The procedure that I developed, that others mentioned, isn't going to
help you here. It assumes a rho of 1, and is useful in some situations
*when you have prior theoretical reason to believe that the true rho is
near 1* and you have dichotomous dependent variables in the selection
and the outcome equations. (See Political Analysis 11:2 for a discussion
of when that might be.) Here, you didn't say anything about a theory
that leads you to believe rho is 1; STATA just estimated it as 1. Since
the Heckman estimates of rho are often either inaccurate (see CIs) or
quite misleading in small samples, estimating a high rho doesn't mean
that the true rho is high. 

Best, Anne

Anne Sartori
Assistant Professor and
Charles G. Osgood Preceptor
Princeton University

-----Original Message-----
From: Political Methodology Society [mailto:[log in to unmask]] On
Behalf Of Achim Kemmerling
Sent: Thursday, February 08, 2007 11:32 AM
To: [log in to unmask]
Subject: [POLMETH] Heckman

Dear Polmeth-members:

I have a problem with Heckman models. I have found several applications
of this model for comparative political economy purposes. In some of
these rho is close to or equal to 1. In the STATA reference manual I
have found that such a rho should lead you to reject the Heckman
approach. Since I am also writing on a paper using a Heckman model and I
am facing similar problems I would like to get some opinion on this. I
use some 120 European regions of which only 30-60 are observed on the
(final) dependent variable. My guess is that a rho equal to 1 - implying
that the selection and the regression process are equal - reveal two
problems: either it shows that sample size is not large enough for an
iterative estimation procedure, or that specification of the selection
process is poor.

Can anyone help?

Best

Achim

--
Dr. Achim Kemmerling
Research Fellow
Social Science Research Centre Berlin (WZB) Reichpietschufer 50
10785 Berlin/ Germany
Tel.: +49-(0)30-25491-150
Fax.: +49-(0)30-25491-222
Email: [log in to unmask]