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Econometrics Procedures
 

CQLIM Procedure

Ordinal Discrete Choice Modeling

Binary Probit and Logit Model

The binary choice model is

y Subscript i Superscript asterisk Baseline equals bold x prime Subscript i Baseline bold-italic beta plus epsilon Subscript i

where the value of the latent dependent variable, y Subscript i Superscript asterisk, is observed only as follows:

StartLayout 1st Row 1st Column y Subscript i 2nd Column equals 1 3rd Column if y Subscript i Superscript asterisk Baseline greater-than 0 2nd Row 1st Column Blank 2nd Column equals 0 3rd Column otherwise EndLayout

The disturbance, epsilon Subscript i, of the probit model has a standard normal distribution with the cumulative distribution function (CDF)

normal upper Phi left-parenthesis x right-parenthesis equals integral Subscript negative normal infinity Superscript x Baseline StartFraction 1 Over StartRoot 2 pi EndRoot EndFraction exp left-parenthesis minus t squared slash 2 right-parenthesis d t

The disturbance of the logit model has a standard logistic distribution with the CDF

normal upper Lamda left-parenthesis x right-parenthesis equals StartFraction exp left-parenthesis x right-parenthesis Over 1 plus exp left-parenthesis x right-parenthesis EndFraction equals StartFraction 1 Over 1 plus exp left-parenthesis negative x right-parenthesis EndFraction

The binary discrete choice model has the following probability that the event left-brace y Subscript i Baseline equals 1 right-brace occurs:

upper P left-parenthesis y Subscript i Baseline equals 1 right-parenthesis equals upper F left-parenthesis bold x prime Subscript i Baseline bold-italic beta right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column normal upper Phi left-parenthesis bold x prime Subscript i Baseline bold-italic beta right-parenthesis 2nd Column left-parenthesis normal p normal r normal o normal b normal i normal t right-parenthesis 2nd Row 1st Column normal upper Lamda left-parenthesis bold x prime Subscript i Baseline bold-italic beta right-parenthesis 2nd Column left-parenthesis normal l normal o normal g normal i normal t right-parenthesis EndLayout

For more information, see SAS/ETS User's Guide.

Ordinal Probit/Logit

When the dependent variable is observed in sequence with M categories, binary discrete choice modeling is not appropriate for data analysis. McKelvey and Zavoina (1975) propose the ordinal (or ordered) probit model.

Consider the regression equation

y Subscript i Superscript asterisk Baseline equals bold x prime Subscript i Baseline bold-italic beta plus epsilon Subscript i

where error disturbances, epsilon Subscript i, have the distribution function F. The unobserved continuous random variable, y Subscript i Superscript asterisk, is identified as M categories. Suppose there are upper M plus 1 real numbers, mu 0 comma ellipsis comma mu Subscript upper M Baseline, where mu 0 equals negative normal infinity, mu 1 equals 0, mu Subscript upper M Baseline equals normal infinity, and mu 0 less-than-or-equal-to mu 1 less-than-or-equal-to midline-horizontal-ellipsis less-than-or-equal-to mu Subscript upper M. Define

upper R Subscript i comma j Baseline equals mu Subscript j Baseline minus bold x prime Subscript i Baseline bold-italic beta

The probability that the unobserved dependent variable is contained in the jth category can be written as

upper P left-bracket mu Subscript j minus 1 Baseline less-than y Subscript i Superscript asterisk Baseline less-than-or-equal-to mu Subscript j Baseline right-bracket equals upper F left-parenthesis upper R Subscript i comma j Baseline right-parenthesis minus upper F left-parenthesis upper R Subscript i comma j minus 1 Baseline right-parenthesis

For more information, see SAS/ETS User's Guide.

Last updated: July 09, 2026