6.9 Multivariate Probit

Mathematical representation

$$\begin{array}{cc}\begin{array}{rl}& \begin{array}{ccccc}\hfill y_{1i}^{\ast }\hfill & \hfill =\hfill & \hfill {\mathbf{x}}_{1i}^{\prime }{\beta }_1\hfill & \hfill +\hfill & \hfill {𝜀}_{1i}\hfill \\ \hfill y_{2i}^{\ast }\hfill & \hfill =\hfill & \hfill {\mathbf{x}}_{2i}^{\prime }{\beta }_2\hfill & \hfill +\hfill & \hfill {𝜀}_{2i}\hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill y_{Mi}^{\ast }\hfill & \hfill =\hfill & \hfill {\mathbf{x}}_{Mi}^{\prime }{\beta }_M\hfill & \hfill +\hfill & \hfill {𝜀}_{Mi}\hfill \\ \hfill \hfill \end{array}\\ & y_{mi}=\left\{\begin{array}{ccc}1\hfill & \hfill \mathrm{if}\hfill & \hfill y_{mi}^{\ast }>0\\ 0\hfill & \hfill \mathrm{if}\hfill & \hfill y_{mi}^{\ast }\le 0\end{array}\right.\forall m=1,2.\dots ,M\end{array}& \end{array}$$

(6.17)

where the $y_{mi}^{\ast }$’s are not observed, but, similarly to binary Probit and Logit models, their signs are determined by the corresponding observed $y_{mi}$s.

• the model is estimated using $N$ observations
• $y_{mi}$ is the value of equation $m$’s dependent variable for observation $i$ and it can take two values: 0 and 1, $\forall m=1,2,\dots ,M$
• ${\mathbf{x}}_{mi}$ is a $K_m\times 1$ vector that stores the values of the $K_m$ independent variables for observation $i$, as they appear in equation $m$
• the same independent variable can appear in multiple equations, associated with different coefficients
• ${\beta }_m$ is a $K_m\times 1$ vector of parameters associated with equation $m$’s independent variables
• in total, there are $K=\sum _{m=1}^M{K}_m$ $\beta$ parameters to be estimated
• the $M$ error terms jointly follow a multivariate Normal distribution with mean $\mathbf{0}$ and covariance matrix $\Sigma$

A more compact representation of the model is:

where:

and $\Sigma$ being the covariance matrix of ${𝜀}_i$: $\Sigma \equiv {\Omega }^{-1}$. For identification purposes, $\Sigma$ is normalized such that its diagonal elements are equal to one. Thus, $\Sigma$ is, in fact, a correlation matrix.

Priors

An inverse-Wishart prior is used for a positive-definite, but otherwise unrestricted, matrix, $\breve{\Sigma }$. This prior is then internally transformed to a prior for $\Sigma =\mathbf{D}\breve{\Sigma }\mathbf{D}$, where $\mathbf{D}$ is an $M\times M$ diagonal matrix constructed by taking the inverse of the square root of the diagonal elements of $\breve{\Sigma }$: $\mathbf{D}=diag\left(\left[\begin{array}{cccc}\hfill {\breve{\sigma }}_{11}^{-1∕2}\hfill & \hfill {\breve{\sigma }}_{22}^{-1∕2}\hfill & \hfill \cdots \hfill & \hfill {\breve{\sigma }}_{MM}^{-1∕2}\hfill \end{array}\right]\right)$.

The magnitude of the elements of the scale matrix, $\mathbf{S}$, in the prior for $\breve{\Sigma }$ affects the prior for $\Sigma$ only if $\mathbf{S}$ is not diagonal. The degrees-of-freedom parameter, $n$, in the prior for $\breve{\Sigma }$ can be used to control the dispersion of the implied prior density of $\Sigma$ around the prior expected value: $E\left(\Sigma \right)={\mathbf{I}}_M$ if $\mathbf{S}$ is diagonal $E\left(\Sigma \right)=\mathbf{D}\mathbf{S}\mathbf{D}$ if $\mathbf{S}$ is not diagonal In both cases smaller values of $n$ allow larger deviations form $E\left(\Sigma \right)$.

Due to an inverse-Wishart distribution being used as the prior for $\breve{\Sigma }$, the prior expected value of $\breve{\Sigma }$ is $\frac{1}{n-M-1}\mathbf{S}$. Although the prior for $\breve{\Sigma }$ is transformed to a prior for $\Sigma$ such that the latter is a correlation matrix, due to simulation-based integration taking place in this transformation when $\mathbf{S}$ is not diagonal, hyperparameter values that result into an expected value of $\breve{\Sigma }$ that is far from satisfying the conditions necessary for it to be a correlation matrix may lead to numerical unstability issues when calculating the log-marginal likelihood of the model. It is, therefore, advised that $n$ is used to control the dispersion of $\Sigma$ around its expected value and $\mathbf{S}$ is subsequently defined such that $\frac{1}{n-M-1}\mathbf{S}$ is close to being a proper correlation matrix (positive definite with values equal to one on the main diagonal).

Syntax

where:

• y1, y2, …, yM are the dependent variable names, as they appear in the dataset used for estimation
• xm1 xm2 $\dots$xmK${}_m$ is a list of the $K_m$ independent variable names for equation $m=1,2,\dots ,M$, as they appear in the dataset used for estimation; when a constant term is to be included in an equation, this must be requested explicitly; $M$ such lists must be provided

All dependent variables, y1, y2, …, yM, in the dataset used for estimation must contain only two values: 0 and 1 (with 1 indicating “success" for the respective outcome). Observations with missing values in any dependent variable are dropped during estimation, but if a numerical value other than 0 and 1 is encountered, then an error is produced.

The optional arguments for the multivarite Probit model are:⁹

Reported Parameters

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a multivariate 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()
• predict()

The multivariate Probit model uses the mfx() function to calculate and report the marginal effects of the independent variables on the probability of success for each of $M$ dependent variables. There are two types of marginal effects which can be requested by setting the "type" argument of the mfx() function equal to 1 or 2:

The generic syntax for a statement involving the mfx() function after estimation of a multivariate Probit model is:

and:

for calculating these two types of marginal effects. The default value of the "type" option is 1. See the general documentation of the mfx() function (section B.14) for details on the other optional arguments.

Although BayES can calculate marginal effects for the multivariate Probit model at each observation, the calculations may take an excessive amount of time to complete. This is because the GHK simulator needs to be invoked at every observed data point, each time using all draws from the posterior, thus leading to an immense number of computations.

The multivariate Probit model uses the predict() function to generate predictions of the probability of success for each of $M$ dependent variables. There are two types of predictions which can be requested by setting the "type" argument of the predict() function equal to 1 or 2:

The generic syntax for a statement involving the predict() function after estimation of a random-effects binary Probit model is:

and:

for generating these two types of predictions effects. The default value of the "type" option is 1. See the general documentation of the predict() function (section B.14) for details on the other optional arguments.

Although BayES can generate summary statistics of the predictions for the multivariate Probit model at each observation, the calculations may take an excessive amount of time to complete. This is because the GHK simulator needs to be invoked at every observed data point, each time using all draws from the posterior, thus leading to an immense number of computations.

Examples

Example 1

Example 2

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