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
*
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
