Ordered probit

Suppose the underlying relationship to be characterized is[2]

${\displaystyle y^{*}=\mathbf {x} ^{\mathsf {T}}\beta +\epsilon }$,

where ${\displaystyle y^{*}}$ is the exact but unobserved dependent variable (perhaps the exact level of improvement by the patient); ${\displaystyle \mathbf {x} }$ is the vector of independent variables, and ${\displaystyle \beta }$ is the vector of regression coefficients which we wish to estimate. Further suppose that while we cannot observe ${\displaystyle y^{*}}$, we instead can only observe the categories of response:

${\displaystyle y={\begin{cases}0~~{\text{if}}~~y^{*}\leq 0,\\1~~{\text{if}}~~0<y^{*}\leq \mu _{1},\\2~~{\text{if}}~~\mu _{1}<y^{*}\leq \mu _{2}\\\vdots \\N~~{\text{if}}~~\mu _{N-1}<y^{*}.\end{cases}}}$

Then the ordered probit technique will use the observations on ${\displaystyle y}$, which are a form of censored data on ${\displaystyle y^{*}}$, to fit the parameter vector ${\displaystyle \beta }$.

The model cannot be consistently estimated using ordinary least squares; it is usually estimated using maximum likelihood. For details on how the equation is estimated, see the article Ordinal regression.

• Becker, William E.; Kennedy, Peter E. (1992). "A Graphical Exposition of the Ordered Probit". Econometric Theory. 8 (1): 127–131. doi:10.1017/S0266466600010781.