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Re: st: Cutoff estimation by MLE

From   Richard Williams <[email protected]>
To   [email protected], [email protected]
Subject   Re: st: Cutoff estimation by MLE
Date   Thu, 06 Dec 2007 09:23:44 -0500

At 09:22 AM 12/4/2007, Bradley Chen wrote:
Dear Statlisters,

I have a problem with estimating the ordered probit cutoff, and I
realized that there is probably no precoded command in STATA doing
what I want so I need to do the maximum likelihood estimation

This is what I want to estimate:
I have a dependent variable, which is a four category ordered
response Yi={1,2,3,4}, whose value depends on a latent variable Y*=Xb+e
The observed Yi=j if C(j-1)<Y*<Cj with C0=minus infinity. And the
most important part for me is that the cutoff level C is also a
function of the observed characteristics: Cj=X*b'

So the the log likelihood equation I want to maximize is
L={Yi=1}*log(cnorm(-C1))+{Yi=2}*log(cnorm(C2-C1)-cnorm(-C1))++{Yi=3} *log(cnorm(C3-C1)-cnorm(-21))+{Yi=4}*log(cnorm(C1-C3)),
to attain the coefficients b' in the cutoff equation

Could someone kindly teach me how to write the STATA command?
I am wondering now if this is a problem gologit2 can handle. Suppose you have gender as an indpendent var. The coefficient for gender might reflect (a) a real effect of gender on the underlying latent variable, or (b) differences in the cutpoints for men and women, e.g. two men and women could have the same underlying value but their observed responses are different because women use different cutpoints than men. So, instead of using the effect of gender to compute y*, you use it to come up with the cutpoints for women as opposed to men. I think you have to choose (a) or (b) though; there is no way to estimate both effects, although I suppose you could have some kind of theory that would let you say there is a 50-50 split or something like that.

The terminology cut-point shift and index shift are sometimes used. If the difference in cutpoints between men and women is always the same (index shift), then oprobit is fine. If the difference varies (e.g. first 2 cutpoints are the same but the third differs) then you have cut-point shift and use gologit2 with link probit.

Put another way, you might estimate something like

oprobit y x1 x2 x3 male

You could then view the effect of male as the difference in cut points for men and women; and in this case you would be saying the difference was the same for every cutpoint. Or, if you did

gologit2 y x1 x2 x3 male, npl(male) link(probit)

you could again be saying the cut points for men and women differ, but the difference need not be the same at each cut point.

Just speculation here; maybe you have something very different in mind.

Richard Williams, Notre Dame Dept of Sociology
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