7.1 Ordered Probit model

Mathematical representation

$$\begin{array}{cc}\begin{array}{rl}{y}_i^{\ast }& ={\mathbf{x}}_i^{\prime }\beta +{𝜀}_i,{𝜀}_i\sim \mathrm{N}\left(0,1\right)\\ y_i& =\left\{\begin{array}{ccc}1\hfill & \hfill \mathrm{if}\hfill & \hfill {\gamma }_0<y_i^{\ast }\le {\gamma }_1\hfill \\ 2\hfill & \hfill \mathrm{if}\hfill & \hfill {\gamma }_1<y_i^{\ast }\le {\gamma }_2\hfill \\ ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ M\hfill & \hfill \mathrm{if}\hfill & \hfill {\gamma }_{M-1}<y_i^{\ast }\le {\gamma }_M\hfill \\ \hfill \end{array}\right.\end{array}& \end{array}$$

(7.1)

• the model is estimated using $N$ observations
• $y_i$ is the value of the dependent variable for observation $i$ and it can assume integer values in the range $1,\dots ,M$
• ${\mathbf{x}}_i$ is a $K\times 1$ vector that stores the values of the $K$ independent variables for observation $i$
• $\beta$ is a $K\times 1$ vector of parameters
• the $\gamma$s are parameters that represent the cutoff points between categories and they satisfy the relation ${\gamma }_0<{\gamma }_1<\cdots <{\gamma }_M$, with ${\gamma }_0=-\infty$, ${\gamma }_M=\infty$ and, for identification purposes, ${\gamma }_1=0$; there are $M-2$ $\gamma$s to be estimated
• following Albert & Chib (2001), to impose the inequality constraints on the $\gamma$s the problem is re-parameterized using the 1-1 mapping:
  $$\begin{array}{ccc}\hfill {\delta }_2\hfill & \hfill =\hfill & \hfill log\left({\gamma }_2-{\gamma }_1\right)\hfill \\ \hfill {\delta }_3\hfill & \hfill =\hfill & \hfill log\left({\gamma }_3-{\gamma }_2\right)\hfill \\ \hfill ⋮\hfill & \hfill =\hfill & \hfill ⋮\hfill \\ \hfill {\delta }_{M-1}\hfill & \hfill =\hfill & \hfill log\left({\gamma }_{M-1}-{\gamma }_{M-2}\right)\hfill \end{array}$$

  there are $M-2$ $\delta$s to be estimated and they are collected in an $\left(M-2\right)\times 1$ vector $\delta$

Priors

Syntax

where:

• y is the dependent variable name, as it appears in the dataset used for estimation
• x1 x2 $\dots$xK is a list of the $K$ independent variable names, as they appear in the dataset used for estimation; when a constant term is to be included in the model, this must be requested explicitly

The dependent variable, y, in the dataset used for estimation must contain only consecutive integer values, with the numbering starting at 1. Observations with missing values in y are dropped during estimation, but if a non-integer numerical value is encountered or if the integer values are not consecutive (for example there are no observations for which $y_i=2$), then an error is produced.

The optional arguments for the ordered Probit model are:¹

Reported Parameters

Stored values and post-estimation analysis
If a left-hand-side id value is provided when an ordered Probit model is created, then the following results are saved in the model item and are accessible via the ‘.’ operator:

Additionally, the following functions are available for post-estimation analysis (see section B.14):

• diagnostics()
• test()
• pmp()
• mfx()

The ordered Probit model uses the mfx() function to calculate and report the marginal effects of the independent variables on the probability of the response variable being in each one of the $M$ categories: $Prob\left(y=m|\mathbf{x}\right)$, for $m=1,2,\dots ,M$. Because the model calculates only one type of marginal effects, the only valid value for the "type" option is 1. The generic syntax for a statement involving the mfx() function after estimation of an ordered Probit model is:

See the general documentation of the mfx() function (section B.14) for details on the other optional arguments.

Examples

Example 1

Example 2

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© 2016–20 Grigorios Emvalomatis