[Limdep Nlogit List] ordered tobit model

[William Greene]

Sami.
First, note that there is a methodological error in Bellemare and Barret (2006) that
you may or may not want to pay attention to.  In their specification, they explicitly
note that y(i,2) > 0 and y(i,3) > 0.  As such, the log likelihood on page 329 is not
correct. That function treats y(i,2) and y(i,3) as if they were determined by simple
linear regressions with normally distributed disturbances when they should be
truncated regressions, instead. That would make for a much more complicated function.
However, my hunch is that the net sales or purchases (y(i,2) and y(i,3) are not really
truncated normal variables - if so, they would be clustered near zero,and it does
not sound like that is the case, so we'll assume that linear regression models are OK
and as such, so is their (18).  In fact, the first and third parts of (18) are the
log likelihood functions for the familiar sample selection model.  (Note, by the way, 
although they call the model in (18) some kind of "tobit" model, it is not a tobit model.
There are no limit values for y(i,2) or y(i,3) in the log likelihood function.
Second, a simple note that they should have made and didn't.  In x(i,1), there is no
constant term.  But, there is in x(i,2) and x(i,3).  The reason there is no constant
term in x(i,1) is that the 3 outcome ordered probit model has two threshold parameters,
alpha(1) and alpha(2). If there were a constant term, alpha(1) would have to be fixed 
at zero, and a nonzero constant and a different value would have been reported for
alpha(2).
Third, it is claimed that the three epsilons are trivariate normally distributed. Strictly
speaking, this is true.  However, it should be noted that rho(2,3) must equal zero because
a family cannot be a net seller and a net buyer at the same time.

With all that in place, you can easily fit this model using the FIML estimator and the
MAXIMIZE command in LIMDEP. I suggest you use the olsen transformation. Rather than
estimating beta2, estimate gamma2 = beta2/sigma2 and estimate gamma3 = beta3/sigma3.
Also, reparameterizing the model in terms of theta2 = 1/sigma2 and theta3 = 1/sigma3 will
simplify the log likelihood function.  You can use the delta method when you are done
to recover beta2 and beta3 if you want to.

The authors' two step ordered probit method is actually a waste of effort.  It is not
necessary to fit the ordered probit model.  If you define the variable z(i,1) = 1 if y(i,1)
equals 2 and z(i,1) = zero if y(i,1) 0 or 1, then the garden variety famous Heckman model
can be applied to z(i,1) and y(i,3).  Be sure to include a constant term on the RHS of 
the probit model.  Likewise, if you define z(i,0) = 1 if y(i,1) = 0 and z(i,0) = 0 if y(i,1)
= 1 or 2, then the Heckman model applies to z(i,0) and y(i,2).   

Two more notes:
(A) the methodology in the appendix has a flaw in it in that the cvariance matrix used 
in A.11 does not account for the estimated threshold parameters. This would make a small
difference in the standard errors.
(B) The authors indicate that they used an ad hoc "instrumental variable" procedure on 
page 331.  I do not see anywhere in any of the "derivations" where they accounted for the
fact that they are using a predicted regressor in their model. This would have to be 
accounted for with a Murphy/Topel type of correction.  This is not an instrumental 
variable; it is a constructed regressor.

Sincerely,
Bill Greene



----- Original Message -----
From: "Sami Haile" <samex_zegr8 at yahoo.com>
To: limdep at limdep.itls.usyd.edu.au
Sent: Monday, July 7, 2008 2:13:55 PM (GMT-0500) America/Bogota
Subject: [Limdep Nlogit List] ordered tobit model

Hi,
I want to eastimate an ordered tobit model or ordered probit model followed by bivariate tobit model after heckman selection is applied, similar to that of Bellemare and Barret (2006). That is how can I eastimate an ordered heckit with limited information maximum likelihood, using Heckman's two step approach. How can I do this in Limdep, and with which version of limdep. My dependent variable is market participation which takes three forms, net buyers, Autarkic and Net sellers.  
Regards,
Sami



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