The binary choice model is
where value of the latent dependent variable, , is observed only as follows:
The disturbance, , of the probit model has standard normal distribution with the distribution function (CDF)
The disturbance of the logit model has standard logistic distribution with the CDF
The binary discrete choice model has the following probability that the event occurs:
The log-likelihood function is
where the CDF is defined as for the probit model while for logit. The first order derivatives of the logit model are
The probit model has more complicated derivatives
where
Note that the logit maximum likelihood estimates are times greater than probit maximum likelihood estimates, since the probit parameter estimates, , are standardized, and the error term with logistic distribution has a variance of .
When the dependent variable is observed in sequence with categories, binary discrete choice modeling is not appropriate for data analysis. McKelvey and Zavoina (1975) proposed the ordinal (or ordered) probit model.
Consider the following regression equation:
where error disturbances, , have the distribution function . The unobserved continuous random variable, , is identified as categories. Suppose there are real numbers, , where , , , and . Define
The probability that the unobserved dependent variable is contained in the jth category can be written as
The log-likelihood function is
where
The first derivatives are written as
where and if , and otherwise. When the ordinal probit is estimated, it is assumed that . The ordinal logit model is estimated if . The first threshold parameter, , is estimated when the LIMIT1=VARYING option is specified. By default (LIMIT1=ZERO), so that threshold parameters () are estimated.
The ordered probit models are analyzed by Aitchison and Silvey (1957), and Cox (1970) discussed ordered response data by using the logit model. They defined the probability that belongs to jth category as
where and . Therefore, the ordered response model analyzed by Aitchison and Silvey can be estimated if the LIMIT1=VARYING option is specified. Note that .
The goodness-of-fit measures discussed in this section apply only to discrete dependent variable models.
McFadden (1974) suggested a likelihood ratio index that is analogous to the in the linear regression model:
where is the value of the maximum likelihood function and is the value of a likelihood function when regression coefficients except an intercept term are zero. It can be shown that can be written as
where is the number of responses in category j.
Estrella (1998) proposes the following requirements for a goodness-of-fit measure to be desirable in discrete choice modeling:
The measure must take values in , where 0 represents no fit and 1 corresponds to perfect fit.
The measure should be directly related to the valid test statistic for significance of all slope coefficients.
The derivative of the measure with respect to the test statistic should comply with corresponding derivatives in a linear regression.
Estrella’s (1998) measure is written
An alternative measure suggested by Estrella (1998) is
where is computed with null slope parameter values, is the number observations used, and represents the number of estimated parameters.
Other goodness-of-fit measures are summarized as follows:
where and .