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st: Test of ordered probit vs ordinary probits


From   "Schaffer, Mark E" <M.E.Schaffer@hw.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Test of ordered probit vs ordinary probits
Date   Wed, 31 Oct 2007 19:07:46 -0000

Hi everybody.  I have a question about ordered probits that follows on from the FAQ by Bill Gould about the role of the constant/cutoffs:

http://www.stata.com/support/faqs/stat/constant.html

At the end of the FAQ, Bill sets out a 3-outcome example:

        Pr(outcome==1) = Pr(        X*b + u <= /cut1)            
        Pr(outcome==2) = Pr(/cut1 < X*b + u <= /cut2)               (1)
        Pr(outcome==3) = Pr(/cut2 < X*b + u)

and then shows how these relate to two 2-outcome (basic probit) equations:

        Pr(outcome==3) = Pr(/cut2 < X*b + u)
                       = Pr(    0 < X*b + (−/cut2) + u)
                       = Pr( X*b + (−/cut2) + u > 0)                 (1.3)

        Pr(outcome>=2) = Pr(/cut1 < X*b + u)
                       = Pr(    0 < X*b + (−/cut1) + u)
                       = Pr( X*b + (−/cut1) + u > 0)                 (1.2)

Bill writes,

"Ordered probit amounts to estimating (1.3) and (1.2) simultaneously, and with the constraint, that b in (1.3) equals b in (1.2). Ergo, ordered probit amounts to estimating the standard binary probit models

        Pr(outcome==3) = Pr( X*b + (−/cut2) + u > 0)                 (1.3)

and

        Pr(outcome>=2) = Pr( X*b + (−/cut1) + u > 0)                 (1.2)

with the constraint that the cofficients, but not the INTERCEPTS, are equal."

My question: Say we estimate two probits as above and test the constraint that the coefficients are equal.  Can we interpret this test as a test of probit vs. ordered probit?  That is, if the test fails to reject the null that the coefficients are different, would estimated an ordered probit be a logical next step?

I suspect the answer is "not quite", because the ordered probit also has embodied in it the constraint that the intercepts (cutoffs) are ordered, but maybe I'm wrong.

A specific example, using suest:

sysuse auto, clear
drop if rep78==.
gen rep78_12=rep78>1
gen rep78_23=rep78>2
gen rep78_34=rep78>3
gen rep78_45=rep78>4

qui probit rep78_12 turn
est store probit12

qui probit rep78_23 turn
est store probit23

qui probit rep78_34 turn
est store probit34

qui probit rep78_45 turn
est store probit45

suest probit12 probit23 probit34 probit45

test [probit12]turn=[probit23]turn
test [probit23]turn=[probit34]turn, accum
test [probit34]turn=[probit45]turn, accum

oprobit rep78 turn, robust

Cheers,
Mark

Prof. Mark E. Schaffer
Director
Centre for Economic Reform and Transformation
Department of Economics
School of Management & Languages
Heriot-Watt University
Edinburgh EH14 4AS  UK
44-131-451-3494 direct
44-131-451-3296 fax
http://www.sml.hw.ac.uk/cert

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