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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 2003, 736):

Pr(Y = 0) = F(−Xb) Pr(Y = 1) = F(u_{1}−Xb) − F(−Xb) Pr(Y = 2) = F(u_{2}−Xb) − F(u_{1}−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(u_{1}− Xb − con) − F(−Xb − con) Pr(Y = 1) = F(u_{2}− Xb − con) − F(u_{1}−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 u_{1} = /cut2 + con = /cut2 − /cut1. Doing the same
for Pr(Y = 2), we see that u_{2} = /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 u_{1}= /cut2 − /cut1 Greene's u_{2}= /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 |

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 u_{1}:

. 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 u_{1} is 1.19 with a standard error of 0.18.
Finally we estimate u_{2}:

. 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 u_{2} is 2.096 with a standard error of .246.

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