The HPQLIM Procedure

Ordinal Discrete Choice Modeling

Subsections:

Binary Probit and Logit Model

The binary choice model is

\[ y^{*}_{i} = \mathbf{x}_{i}’\bbeta + \epsilon _{i} \]

where the value of the latent dependent variable, $y^{*}_{i}$, is observed only as follows:

\begin{eqnarray*} y_{i} & = 1 & \hbox{if } y^{*}_{i}>0 \\ & = 0 & \hbox{otherwise} \end{eqnarray*}

The disturbance, $\epsilon _{i}$, of the probit model has a standard normal distribution with the distribution function (CDF)

\[ \Phi (x)=\int _{-\infty }^{x}\frac{1}{\sqrt {2\pi }}\exp (-t^2/2)dt \]

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

\[ \Lambda (x)=\frac{\exp (x)}{1+\exp (x)} = \frac{1}{1+\exp (-x)} \]

The binary discrete choice model has the following probability that the event $\{ y_{i}=1\} $ occurs:

\[ P(y_{i}=1) = F(\mathbf{x}_{i}’\bbeta ) = \left\{ \begin{array}{ll} \Phi (\mathbf{x}_{i}’\bbeta ) & \mr{(probit)} \\ \Lambda (\mathbf{x}_{i}’\bbeta ) & \mr{(logit)} \end{array} \right. \]

For more information, see the section Ordinal Discrete Choice Modeling in SAS/ETS 13.2 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

\[ y_{i}^{*} = \mathbf{x}_{i}’\bbeta + \epsilon _{i} \]

where error disturbances, $\epsilon _{i}$, have the distribution function $F$. The unobserved continuous random variable, $y_{i}^{*}$, is identified as M categories. Suppose there are $M+1$ real numbers, $\mu _{0},\ldots ,\mu _{M}$, where $\mu _{0}=-\infty $, $\mu _{1}=0$, $\mu _{M}=\infty $, and $\mu _{0} \leq \mu _{1} \leq \cdots \leq \mu _{M}$. Define

\[ R_{i,j} = \mu _{j} - \mathbf{x}_{i}’\bbeta \]

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

\[ P[\mu _{j-1}< y_{i}^{*} \leq \mu _{j}] = F(R_{i,j}) - F(R_{i,j-1}) \]

For more information, see the section Ordinal Discrete Choice Modeling in SAS/ETS 13.2 User's Guide.