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In ordered probit and logit, what are the cut points?

Title   Interpreting the cut points in ordered probit and logit
Author William Gould, StataCorp
Date January 1999

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.
Econometric Analysis: 2d ed. New York: Macmillan.
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