Bayesian Econometrics Software

6.5 Multinomial Probit

Mathematical representation

\begin{matrix} p_{m i} = Prob (y_{i} = m | x_{i}^{'} β_{m}), & m = 1, \dots, M \\ p_{0 i} = 1 - \sum_{m = 1}^{M} p_{m i}, & with additional structure on each p_{m i} \end{matrix}

(6.9)

the model is estimated using $N$ observations
$y_{i}$ is the value of the dependent variable for observation $i$ and it can take $M + 1$ values: 0, 1, …, M
$x_{i}$ is a $K \times 1$ vector that stores the values of the $K$ independent variables for observation $i$
$β_{m}$ is a $K \times 1$ vector of parameters for alternative $m$ , $m = 1, 2, \dots, M$

The latent-variable representation of the multinomial Probit is:

\begin{matrix} y_{1 i}^{*} & = & x_{i}^{'} β_{1} & + & 𝜀_{1 i} \\ y_{2 i}^{*} & = & x_{i}^{'} β_{2} & + & 𝜀_{2 i} \\ ⋮ & ⋮ & ⋮ \\ y_{M i}^{*} & = & x_{i}^{'} β_{M} & + & 𝜀_{M i} \end{matrix} y_{i} = \{\begin{matrix} 0 & if & max_{j} \{y_{j i}^{*}\} \leq 0 \\ 1 & if & max_{j} \{y_{j i}^{*}\} = y_{1 i}^{*} & and & y_{1 i}^{*} > 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ M & if & max_{j} \{y_{j i}^{*}\} = y_{M i}^{*} & and & y_{M i}^{*} > 0 \end{matrix}

(6.10)

Let:

\underset{(M K) \times 1}{β} = [\begin{matrix} β_{1} \\ β_{2} \\ ⋮ \\ β_{M} \end{matrix}] and \underset{M \times 1}{𝜀_{i}} = [\begin{matrix} 𝜀_{1 i} \\ 𝜀_{2 i} \\ ⋮ \\ 𝜀_{M i} \end{matrix}]

$𝜀_{i}$ is assumed to follow a multivariate-Normal distribution: $𝜀 \sim N (0, Ω^{- 1})$ . For identiﬁcation purposes the precision matrix is restricted such that $tr (Ω) = M$ .

Priors


Parameter	Probability density function	Default hyperparameters

$β$	$p (β) = \frac{\| P \|^{1 ∕ 2}}{{(2 π)}^{M K ∕ 2}} exp \{- \frac{1}{2} {(β - m)}^{'} P (β - m)\}$	$m = 0_{M K}$ , $P = 0.001 \cdot I_{M K}$
$Ω$	$p (Ω) = \frac{\| Ω \|^{\frac{n - M - 1}{2}} \| V^{- 1} \|^{n ∕ 2}}{2^{n M ∕ 2} Γ_{M} (\frac{n}{2})} exp \{- \frac{1}{2} tr (V^{- 1} Ω)\}$	$n = M^{2}$ , $V = \frac{100}{M} \cdot I_{M}$
Following Burgette & Nordheim (2012), the prior for $Ω$ is transformed such that $tr (Ω) = M$

Syntax

[

<model name> =

]

mnprobit( y ~ x1 x2 … xK

[

, <options>

]

);

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 0 (base category). 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} = 1

), then an error is produced.

The optional arguments for the multinomial Probit model are:⁵

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 $β$ ( $M K \times 1$ vector); the default value is $0_{M K}$
"P"	precision matrix of the prior for $β$ ( $M K \times M K$ symmetric and positive-deﬁnite matrix); the default value is $0.001 \cdot I_{M K}$
"V"	scale matrix of the prior for $Ω$ ( $M \times M$ symmetric and positive-deﬁnite matrix); the default value is $\frac{100}{M} \cdot I_{M}$
"n"	degrees-of-freedom parameter of the prior for $Ω$ (real number greater than or equal to $M$ ); the default value is $M^{2}$
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


$β$	variable_name	vector of parameters associated with the independent variables; these are broken into groups according to the alternative, $m$ , the parameter is associated with

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a multinomial 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 $β$ (across all alternatives, starting from the ﬁrst one) and the unique elements of $Ω$
y_m$x1, $\dots$ , y_m$xK	vectors containing the draws from the posterior of the parameters associated with variables x1, $\dots$ ,xK, for $m = 1, 2, \dots, M$ (the names of these vectors are the names of the variables that were included in the right-hand side of the model, prepended by y_m$, where y_m is the name of the dependent variable followed by an underscore and the index of the alternative; this is done so that the samples on the parameters associated with the same independent variable but for diﬀerent alternatives can be distinguished)
Omega_i_j	vectors containing the draws from the posterior of the unique elements of $Ω$ ; because $Ω$ is symmetric, only $\frac{(M - 1) M}{2} + M$ of its elements are stored (instead of all $M^{2}$ elements); i and j index the row and column of $Ω$ , respectively, at which the corresponding element is located
Omega	$M \times M$ matrix that stores the posterior mean of $Ω$ ; $Ω$ is restricted such that $tr (Ω^{- 1}) = M$
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 multinomial Probit model uses the mfx() function to calculate and report the marginal eﬀects of the independent variables on the probability of each outcome, $m$ , occurring. 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 multinomial Probit model is:

mfx(

[

"type"=1

]

[

, "point"=<point of calculation>

]

[

, "model"=<model name>

]

);

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 multinomial 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 multinomial Probit model uses the predict() function to generate predictions of the probability each of the $M + 1$ outcomes occuring. 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 multinomial Probit model is:

[

]

= predict(

[

"type"=1

]

[

, "point"=<point of calculation>

]

[

,"model"=<model name>

]

[

, "stats"=true|false

]

[

, "prefix"=<prefix for new variable name>

]

);

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 multinomial 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

myData = import("$BayESHOME/Datasets/dataset7.csv");
myData.constant = ones(rows(myData), 1);

mnprobit( y ~ constant x1 x2 x3 x4 );

Example 2

myData = import("$BayESHOME/Datasets/dataset7.csv");
myData.constant = ones(rows(myData), 1);

myModel = mnprobit( y ~ constant x1 x2 x3 x4,
    "m"=ones(2*5,1), "P"=0.1*eye(2*5,2*5),
    "burnin"=20000, "draws"=40000, "thin"=4, "chains"=2,
    "logML_CJ" = true );

diagnostics("model"=myModel);

kden(myModel.y_2$x3, "title" = "∖beta3 for the 2nd alternative");

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

predict();

⁵Optional arguments are always given in option-value pairs (eg. "chains"=3).

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