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RE: st: Likelihood function of uniform distribution


From   "Verkuilen, Jay" <JVerkuilen@gc.cuny.edu>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: Likelihood function of uniform distribution
Date   Wed, 2 Apr 2008 18:38:37 -0400

Mostafa Beshkar wrote:


>>I think I should explain my question in more detail, since I think
there has been some misunderstanding.

Indeed.


>>I want to estimate the following probability model (this comes from my
game-theoretic model):

Pr(s=1|X)=Pr(p>BX)
Pr(s=0|X)=1-Pr(p>BX)

where s is a binary variable, X is the vector of observable variables, B
is the vector of parameters to be estimated, and p is an unobservable
random varibale that is distributed according to F on the interval
[0,1].<<

Unless I'm mistaken, this is just an ordinary binary regression. You
actually observe S = 0 or 1, you have a vector of predictors for
characteristics of each choice. The usual random utility formulation
sets BX on the real line and uses a link function generated by making
assumptions about the distribution of the disturbance in a random
utility model. Kenneth Train's most excellent book on discrete choice
(see http://elsa.berkeley.edu/books/choice2.html) explains things quite
well. 

Depending on your design, you will have dependency among observations
because you have observed choices for two players in the same game, you
are in a more complex situation requiring simultaneous equations with a
non-recursive model. There is a literature on econometrics in the
context of game theoretic models; I am aware of it but don't know much
about what's going on currently. I'm guessing that biprobit in Stata
(generalizes probit to two simultaneous equations) would be of help.
Googling and a trip to Jstor gives: Estimation of Econometric Models of
Some Discrete Games, Peter Kooreman,  Journal of Applied Econometrics,
Vol. 9, No. 3. (Jul. - Sep., 1994), pp. 255-268. 

I must confess this is getting out of my area.... I is just a poor, dumb
psychometrician. :) 

Jay

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