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How can I convert Stata’s parameterization of ordered probit and logistic models to one in which a constant is estimated?

Why is there no constant term reported in ologit and oprobit?

Title   Fitting ordered logistic and probit models with constraints
Author Mark Inlow, StataCorp
Ronna Cong, StataCorp
Date August 1999; minor revisions July 2009

Consider a parameterization in which a constant is present, e.g., Greene’s formulation (Greene 2003, 736):

    Pr(Y = 0) = F(−Xb)
    Pr(Y = 1) = F(u1 −Xb) − F(−Xb)
    Pr(Y = 2) = F(u2 −Xb) − F(u1 −Xb)
    ...

In the preceding, F is the cumulative distribution function (CDF), either the cumulative standard normal distribution for ordered probit regression or the cumulative logistic distribution for ordered logistic regression. Since Greene includes a constant in his Xb, we need to indicate this to make his notation and Stata’s ordered probit/logistic notation comparable:

    Pr(Y = 0) = F(−Xb − con)
    Pr(Y = 1) = F(u1 − Xb − con) − F(−Xb − con)
    Pr(Y = 1) = F(u2 − Xb − con) − F(u1 −Xb − con)
    ...

Now, compare this with Stata’s no-constant model:

    Pr(Y = 0) = F(/cut1 − Xb)
    Pr(Y = 1) = F(/cut2 − Xb) − F(/cut1 − Xb)
    Pr(Y = 2) = F(/cut3 − Xb) − F(/cut2 − Xb)
    ...

Examining the expressions for Pr(Y = 0), we see that

    −Xb − con = /cut1 − Xb 

so Greene’s constant equals –/cut1. Greene set the first cut point to zero, whereas Stata set the constant to zero.

Combining this observation with the expressions for Pr(Y = 1), we see that Greene’s u1 = /cut2 + con = /cut2 − /cut1. Doing the same for Pr(Y = 2), we see that u2 = /cut3 − /cut1. Thus to estimate Greene’s model using the coefficient estimates from Stata’s ordered probit/logistic regression commands we can use the following:

    Greene's intercept = −/cut1
    Greene's u1 = /cut2 − /cut1
    Greene's u2 = /cut3 − /cut1
    ...

After you fit your model using Stata, you can convert to Greene’s parameterization using lincom, which will provide both the coefficient estimate and the standard error as follows:

    ologit/oprobit ...
    lincom _b[/cut2] - _b[/cut1]
    lincom _b[/cut3] - _b[/cut1]
    ...

To make things concrete, consider the following example using the auto dataset, which is shipped with Stata.

. sysuse auto, clear
(1978 Automobile Data)

. replace rep78 = 2 if rep78 == 1 | missing(rep78)
(7 real changes made)

. tabulate rep78

Repair
Record 1978 Freq. Percent Cum.
2 15 20.27 20.27
3 30 40.54 60.81
4 18 24.32 85.14
5 11 14.86 100.00
Total 74 100.00
. oprobit rep78 price weight Iteration 0: log likelihood = -97.440603 Iteration 1: log likelihood = -91.088192 Iteration 2: log likelihood = -91.074229 Iteration 3: log likelihood = -91.074223 Ordered probit regression Number of obs = 74 LR chi2(2) = 12.73 Prob > chi2 = 0.0017 Log likelihood = -91.074223 Pseudo R2 = 0.0653
rep78 Coef. Std. Err. z P>|z| [95% Conf. Interval]
price .0000966 .0000515 1.88 0.061 -4.36e-06 .0001976
weight -.0007095 .0002013 -3.52 0.000 -.0011041 -.000315
/cut1 -2.468357 .5580629 -3.56214 -1.374573
/cut2 -1.276601 .5310947 -2.317528 -.2356748
/cut3 -.3720451 .5046055 -1.361054 .6169635

Thus the intercept (constant) is −/cut1 = 2.47, and now we compute the point estimate and standard error of u1:

. lincom _b[/cut2] - _b[/cut1]

 ( 1)  - [cut1]_cons + [cut2]_cons = 0

rep78 Coef. Std. Err. z P>|z| [95% Conf. Interval]
(1) 1.191755 .183964 6.48 0.000 .8311925 1.552318

Our estimate of u1 is 1.19 with a standard error of 0.18. Finally we estimate u2:

. lincom _b[/cut3] - _b[/cut1]

 ( 1)  - [cut1]_cons + [cut3]_cons = 0

rep78 Coef. Std. Err. z P>|z| [95% Conf. Interval]
(1) 2.096311 .2457135 8.53 0.000 1.614722 2.577901

Thus our estimate of u2 is 2.096 with a standard error of .246.

Reference

Greene, W. H. 2003.
Econometric Analysis. 5th ed. Upper Saddle River, NJ: Prentice Hall.
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