Home  /  Resources & support  /  FAQs  /  Fitting ordered logistic and probit models with constraints

## 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

Consider a parameterization in which a constant is present, e.g., Greene’s formulation (Greene 2018, Chapter 18):

    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)

. 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   Coefficient  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 + [/]cut2 = 0

rep78   Coefficient  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 + [/]cut3 = 0

rep78   Coefficient  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. 2018.
Econometric Analysis. 8th ed. NJ: Prentice Hall.