### 6.1 Binary Probit

Mathematical representation

 $Prob\left({y}_{i}=1\right)=\Phi \left({\mathbf{x}}_{i}^{\prime }\beta \right)$ (6.1)
• the model is estimated using $N$ observations
• ${y}_{i}$ is the value of the dependent variable for observation $i$ and it can take two values: 0 and 1
• ${\mathbf{x}}_{i}$ is a $K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector that stores the values of the $K$ independent variables for observation $i$
• $\beta$ is a $K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector of parameters
• $\Phi \left(\cdot \right)$ is the standard-Normal cdf

An equivalent representation uses the latent variable ${y}_{i}^{\ast }$:

 $\begin{array}{cc}\begin{array}{rl}{y}_{i}^{\ast }& ={\mathbf{x}}_{i}^{\prime }\beta +{𝜀}_{i},\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{𝜀}_{i}\sim \mathrm{N}\left(0,1\right)\\ {y}_{i}& =\left\{\begin{array}{ccc}1\hfill & \hfill \mathrm{if}\hfill & \hfill {y}_{i}^{\ast }>0\\ 0\hfill & \hfill \mathrm{if}\hfill & \hfill {y}_{i}^{\ast }\le 0\end{array}\right\\end{array}& \end{array}$ (6.2)

Priors

 Parameter Probability density function Default hyperparameters $\beta$ $p\left(\beta \right)=\frac{|\mathbf{P}{|}^{1∕2}}{{\left(2\pi \right)}^{K∕2}}exp\left\{-\frac{1}{2}{\left(\beta -\mathbf{m}\right)}^{\prime }\mathbf{P}\left(\beta -\mathbf{m}\right)\right\}$ $\mathbf{m}={\mathbf{0}}_{K}$, $\mathbf{P}=0.001\cdot {\mathbf{I}}_{K}$

Syntax

$\left[$<model name> = $\right]$ probit( y ~ x1 x2  xK $\left[$, <options> $\right]$ );

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 two values: 0 and 1 (with 1 indicating “success"). Observations with missing values in y 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 binary Probit model are:1

 Gibbs parameters "chains" number of chains to run in parallel (positive integer); the default value is 1 "burnin" number of burn-in draws per chain (positive integer); the default value is 10000 "draws" number of retained draws per chain (positive integer); the default value is 20000 "thin" value of the thinning parameter (positive integer); the default value is 1 "seed" value of the seed for the random-number generator (positive integer); the default value is 42 Hyperparameters "m" mean vector of the prior for $\beta$ ($K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector); the default value is ${\mathbf{0}}_{K}$ "P" precision matrix of the prior for $\beta$ ($K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}K$ symmetric and positive-deﬁnite matrix); the default value is $0.001\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}{\mathbf{I}}_{K}$ Dataset and log-marginal likelihood "dataset" the id value of the dataset that will be used for estimation; the default value is the ﬁrst dataset in memory (in alphabetical order) "logML_CJ" boolean indicating whether the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood should be calculated (true$|$false); the default value is false

Reported Parameters

 $\beta$ variable_name vector of parameters associated with the independent variables

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

 Samples a matrix containing the draws from the posterior of $\beta$ x1,$\dots$,xK vectors containing the draws from the posterior of the parameters associated with variables x1,$\dots$,xK (the names of these vectors are the names of the variables that were included in the right-hand side of the model) logML the Lewis & Raftery (1997) approximation of the log-marginal likelihood logML_CJ the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood; this is available only if the model was estimated with the "logML_CJ"=true option nchains the number of chains that were used to estimate the model nburnin the number of burn-in draws per chain that were used when estimating the model ndraws the total number of retained draws from the posterior ($=$chains $\cdot$ draws) nthin value of the thinning parameter that was used when estimating the model nseed value of the seed for the random-number generator that was used when estimating the model

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

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

The binary Probit model uses the mfx() function to calculate and report the marginal eﬀects of the independent variables on the probability of success. 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 a binary Probit model is:

mfx( $\left[$"type"=1$\right]$ $\left[$, "point"=<point of calculation>$\right]$ $\left[$, "model"=<model name>$\right]$ );

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

The binary Probit model uses the predict() function to generate predictions of the probability of success. Because the model generates only one type of predictions, the only valid value for the "type" option is 1. The generic syntax for a statement involving the predict() function after estimation of a binary Probit model is:

$\left[$<id value>$\right]$ = predict( $\left[$"type"=1$\right]$ $\left[$, "point"=<point of calculation>$\right]$ $\left[$,"model"=<model name>$\right]$ $\left[$, "stats"=true|false$\right]$ $\left[$, "prefix"=<prefix for new variable name>$\right]$ );

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

Examples

Example 1

myData = import("$BayESHOME/Datasets/dataset4.csv"); myData.constant = ones(rows(myData), 1); probit( y ~ constant x1 x2 x3 x4 ); Example 2 myData = import("$BayESHOME/Datasets/dataset4.csv");
myData.constant = ones(rows(myData), 1);

myModel = probit( y ~ constant x1 x2 x3 x4,
"m"=ones(5,1), "P"=0.1*eye(5,5),
"burnin"=10000, "draws"=40000, "thin"=4, "chains"=2,
"logML_CJ" = true );

diagnostics("model"=myModel);

kden(myModel.x3, "title" = "beta3 from the Probit model");

margeff_mean = mfx("point"="mean","model"=myModel);
margeff_median = mfx("point"="median","model"=myModel);
margeff_eachpoint = mfx("point"="x_i","model"=myModel);
margeff_atx = mfx("point"=[1,1,0.5,2,0],"model"=myModel);

predict();

1Optional arguments are always given in option-value pairs (eg. "chains"=3).