In ordered probit and logit, what are the cut points?
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Title
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Interpreting the cut points in ordered probit and logit
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Author
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William Gould, StataCorp
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Date
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January 1999
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Say we have a dataset where y takes on the values 0, 1, and 2 and we
estimate the following ordered probit model:
. oprobit y x1 x2
Iteration 0: Log Likelihood = -27.49743
Iteration 1: Log Likelihood =-12.965819
Iteration 2: Log Likelihood =-9.5150903
Iteration 3: Log Likelihood = -8.606356
Iteration 4: Log Likelihood =-8.4755449
Iteration 5: Log Likelihood =-8.4711766
Iteration 6: Log Likelihood =-8.4711702
Ordered probit estimates Number of obs = 40
LR chi2(2) = 38.05
Prob > chi2 = 0.0000
Log Likelihood = -8.4711702 Pseudo R2 = 0.6919
------------------------------------------------------------------------------
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | 1.494236 .5281424 2.829 0.005 .4590964 2.529377
x2 | -.6365205 .2387014 -2.667 0.008 -1.104367 -.1686744
---------+--------------------------------------------------------------------
_cut1 | -.4097024 .6693587 (Ancillary parameters)
_cut2 | 3.073797 1.155658
------------------------------------------------------------------------------
The cut points _cut1 and _cut2 are really just coefficients of
the model.
The interpretation of this model is
Pr(y=0) = Pr(Xb+u < _cut1) = Pr(u < _cut1-Xb) = F(_cut1-Xb)
Pr(y=1) = Pr(_cut1 < Xb + u < _cut2)
= Pr(_cut1-Xb < u < _cut2-Xb)
= F(_cut2-Xb) - F(_cut1-Xb)
Pr(y=2) = Pr( _cut2 < xb+u) = Pr(u > _cut2-xb) = 1 - F(_cut2-Xb)
where F() stands for the cumulative normal distribution.
This is confusing because different authors use different notations. Greene
(1993, 674) includes an intercept in his Xb term and we do not. So
Greene writes Pr(y=0) as F(-Xb). In our notation,
_cut1 is Greene's intercept, but with a reversed sign, and our Xb
does not have an intercept at all.
We did not design our notation to be complicated; it is just that we use
different notations than Greene, and it is confusing to go between them. Try
ours; it is really very easy.
* u ~ N(0,1)
** **
* * Define z = X*b + u with NO intercept
* *
* * Pr(y=0) = Pr(z < _cut1)
** **
*| ** Pr(y=1) = Pr(_cut1 < z < _cut2)
* | *
* | * Pr(y=2) = Pr(_cut2 < z)
** | **
** | |**
** | | **
* | | *
** y=0 | y=1 | y=2**
***** | | ******
---------------|------------|------------
_cut1 _cut2
References
- Greene, W. H. 1993.
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Econometric Analysis: 2d ed.
New York: Macmillan.
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