The HPQLIM Procedure

Ordinal Discrete Choice Modeling

Subsections:

Binary Probit and Logit Model
Ordinal Probit/Logit

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 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 $j$ th 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 User's Guide.