---
title: "6. Multinomial probit"
output: rmarkdown::html_vignette
bibliography: ../inst/REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{6. The multinomial probit model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r label = setup, include = FALSE}
knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE, widtht = 65)
options(width = 65)
```

## The model

The multinomial probit is obtained with the same modeling that we used
while presenting the random utility model. The utility of an
alternative is still the sum of two components : $U_j = V_j +
\epsilon_j$.

but the joint distribution of the error terms is now a multivariate
normal with mean 0 and with a matrix of covariance denoted
$\Omega$^[see [@HAUS:WISE:78] and [@DAGA:79]].

Alternative $l$ is chosen if : 
$$
\left\{
\begin{array}{rcl}
U_1-U_l&=&(V_1-V_l)+(\epsilon_1-\epsilon_l)<0\\
U_2-U_l&=&(V_2-V_l)+(\epsilon_2-\epsilon_l)<0\\
 & \vdots &  \\
U_J-U_l&=&(V_J-V_l)+(\epsilon_J-\epsilon_l)<0\\
\end{array}
\right.
$$

wich implies, denoting $V^l_j=V_j-V_l$ :

$$
\left\{
\begin{array}{rclrcl}
  \epsilon^l_1 &=& (\epsilon_1-\epsilon_l) &<& - V^l_1\\
  \epsilon^l_2 &=& (\epsilon_2-\epsilon_l) &<& - V^l_2\\
  &\vdots & & \vdots &  \\
  \epsilon^l_J &=& (\epsilon_J-\epsilon_l) &<& - V^l_J\\
\end{array}
\right.
$$

The initial vector of errors $\epsilon$ are transformed using the
following transformation :

$$\epsilon^l = M^l \epsilon$$

where the transformation matrix $M^l$ is a $(J-1) \times J$ matrix
obtained by inserting in an identity matrix a $l^{\mbox{th}}$ column
of $-1$. For example, if $J = 4$ and $l = 3$ :

$$
M^3 = 
\left(
\begin{array}{cccc}
1 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & -1 & 1 \\
\end{array}
\right)
$$

The covariance matrix of the error differences is obtained using the
following matrix :

$$
\mbox{V}\left(\epsilon^l\right)=\mbox{V}\left(M^l\epsilon\right)
=
M^l\mbox{V}\left(\epsilon\right){M^l}^{\top}
=
M^l\Omega{M^l}^{\top}
$$

The probability of choosing $l$ is then :

\begin{equation}
P_l =\mbox{P}(\epsilon^l_1<-V_1^l \;\&\; \epsilon^l_2<-V_2^l \;\&\; ... \; \epsilon^l_J<-V_J^l)
\end{equation}

with the hypothesis of distribution, this writes :

\begin{equation}
P_l = \int_{-\infty}^{-V_1^l}\int_{-\infty}^{-V_2^l}...\int_{-\infty}^{-V_J^l}\phi(\epsilon^l)
d\epsilon^l_1 d\epsilon^l_2... d^l_J
\end{equation}

with :

\begin{equation}
\phi\left(\epsilon^l\right)=\frac{1}{(2\pi)^{(J-1)/2}\mid\Omega^l\mid^{1/2}}
e^{-\frac{1}{2}\epsilon^l{\Omega^l}^{-1}\epsilon^l}
\end{equation}

Two problems arise with this model :

- the first one is that the identified parameters are the elements
  of $\Omega^l$ and not of $\Omega$. We must then carefully investigate
  the meanings of these elements.
- the second one is that the probability is a $J-1$ integral,
  which should be numerically computed. The relevant strategy in this
  context is to use simulations.

## Identification

The meaning-full parameters are those of the covariance matrix of the
error $\Omega$. For example, with $J = 3$ :

$$
\Omega =
\left(
\begin{array}{ccc}
\sigma_{11} & \sigma_{12} & \sigma_{13}  \\
\sigma_{21} & \sigma_{22} & \sigma_{23} \\
\sigma_{31} & \sigma_{32} & \sigma_{33} \\
\end{array}
\right)
$$

$$
\Omega^1 = M^1 \Omega {M^1}^{\top}=
\left(
\begin{array}{cc}
\sigma_{11}+\sigma_{22}-2\sigma_{12} & \sigma_{11} + \sigma_{23} - \sigma_{12} -\sigma_{13} \\
\sigma_{11}+\sigma_{23}- \sigma_{12} - \sigma_{13} & \sigma_{11} + \sigma_{33} - 2 \sigma_{13} \\
\end{array}
\right)
$$

The overall scale of utility being unidentified, one has to impose the
value of one of the variance, for example the first one is fixed to
1. We then have :

$$ \Omega^1 = \left( \begin{array}{cc} 1 & \frac{\sigma_{11}+
\sigma_{23} - \sigma_{12}
-\sigma_{13}}{\sigma_{11}+\sigma_{22}-2\sigma_{12}} \\
\frac{\sigma_{11}+\sigma_{23}- \sigma_{12} -
\sigma_{13}}{\sigma_{11}+\sigma_{22}-2\sigma_{12}} &
\frac{\sigma_{11} + \sigma_{33} - 2
\sigma_{13}}{\sigma_{11}+\sigma_{22}-2\sigma_{12}} \\ \end{array}
\right) 
$$

Therefore, out the 6 structural parameters of the covariance matrix,
only 3 can be identified. Moreover, it's almost impossible to
interpret these parameters.

More generally, with $J$ alternatives, the number of the parameters of
the covariance matrix is $(J+1)\times J/2$ and the number of identified
parameters is $J\times(J-1)/2-1$.

## Simulations

Let $L^l$ be the Choleski decomposition of the covariance matrix of
the error differences :

$$
\Omega^l=L^l {L^l}^{\top}
$$

This matrix is a lower triangular matrix of dimension $(J-1)$ :

$$
L^l=
\left(
\begin{array}{ccccc}
l_{11} & 0 & 0 &... & 0 \\
l_{21} & l_{22} & 0 & ... & 0 \\
l_{31} & l_{32} & l_{33} & ... & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
l_{(J-1)1} & l_{(J-1)2} & l_{(J-1)3} & ... & l_{(J-1)(J-1)} \\
\end{array}
\right)
$$

Let $\eta$ be a vector of standard normal deviates :

$$
\eta \sim N(0, I)
$$

Therefore, we have :

$$
\mbox{V}\left(L^l\eta\right)=L^lV(\eta){L^l}^{\top}=L^lI{L^l}^{\top}=\Omega^l
$$

Therefore, if we draw a vector of standard normal deviates $\eta$ and
apply to it this transformation, we get a realization of $\epsilon^l$.

This joint probability can be written as a product of conditional and
marginal probabilities :

$$
\begin{array}{rcl}
  P_l &=& \mbox{P}(\epsilon^l_1<- V_1^l \;\&\; \epsilon^l_2<-V_2^l \;\&\; ... \;\&\; \epsilon^l_J<-V_J^l))\\
  &=& \mbox{P}(\epsilon^l_1<- V_1^l))\\
  &\times&\mbox{P}(\epsilon^l_2<-V_2^l \mid \epsilon^l_1<-V_1^l) \\
  &\times&\mbox{P}(\epsilon^l_3<-V_3^l \mid \epsilon^l_1<-V_1^l \;\&\; \epsilon^l_2<-V_2^l) \\
  & \vdots & \\
  &\times&\mbox{P}(\epsilon^l_J<-V_J^l \mid \epsilon^l_1<-V_1^l \;\&\; ... \;\&\; \epsilon^l_{J-1}<-V_{J-1}^l)) \\
\end{array}
$$

The vector of error differences deviates is :

$$
\left(
\begin{array}{c}
  \epsilon^l_1 \\ \epsilon^l_2 \\ \epsilon^l_3 \\ \vdots \\ \epsilon^l_J
\end{array}
\right)
=
\left(
\begin{array}{ccccc}
l_{11} & 0 & 0 &... & 0 \\
l_{21} & l_{22} & 0 & ... & 0 \\
l_{31} & l_{32} & l_{33} & ... & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
l_{(J-1)1} & l_{(J-1)2} & l_{(J-1)3} & ... & l_{(J-1)(J-1)} \\
\end{array}
\right)
\times
\left(
\begin{array}{c}
\eta_1 \\ \eta_2 \\ \eta_3 \\ \vdots \\ \eta_J
\end{array}
\right)
$$


$$
\left(
\begin{array}{c}
  \epsilon^l_1 \\ \epsilon^l_2 \\ \epsilon^l_3 \\ \vdots \\ \epsilon^l_J
\end{array}
\right)
=
\left(
\begin{array}{l}
l_{11}\eta_1 \\ 
l_{21}\eta_1+l_{22}\eta_2 \\ 
l_{31}\eta_1+l_{32}\eta_2 + l_{33}\eta_3\\ 
\vdots \\ 
l_{(J-1)1}\eta_1+l_{(J-1)2}\eta_2+...+l_{(J-1)(J-1)}\eta_{J-1}
\end{array}
\right)
$$

Let's now investigate the marginal and conditional probabilities : 

- the first one is simply the marginal probability for a standard
  normal deviates, therefore we have :
  $\mbox{P}(\epsilon^l_1<-V_1^l) = \Phi\left(-\frac{V_1^l}{l_{11}}\right)$
- the second one is, for a given value of $\eta_1$ equal to
  $\Phi\left(-\frac{V^l_2+l_{21}\eta_1}{l_{22}}\right)$. We then have to compute the
  mean of this expression for any value of $\eta_1$ lower than
  $-\frac{V^l_1}{l_{11}}$. We then have, denoting $\bar{\phi}_1$ the truncated
  normal density :
  $$\mbox{P}(\epsilon^l_2<-V_2^l)=\int_{-\infty}^{-\frac{V^l_1}{l_{11}}}\Phi\left(-\frac{V^l_2+l_{21}\eta_1}{l_{22}}\right)
  \bar{\phi}_1(\eta_1)d\eta_1$$
- the third one is, for given values of $\eta_1$ and $\eta_2$
  equal to :
  $\Phi\left(-\frac{V^l_3+l_{31}\eta_1+l_{32}\eta_2}{l_{33}}\right)$. We
  then have :
  $$\mbox{P}(\epsilon^l_3<-V_3^l)=\int_{-\infty}^{-\frac{V^l_1}{l_{11}}}\int_{-\infty}^{-\frac{V^l_2+l_{21}\eta_1}{l_{22}}}
  \Phi\left(-\frac{V^l_3+l_{31}\eta_1+l_{32}\eta_2}{l_{33}}\right)\bar{\phi}_1(\eta_1)\bar{\phi}_2(\eta_2)d\eta_1d\eta_2$$
- and so on. 


This probabilities can easily be simulated by drawing numbers from a
truncated normal distribution.

This so called GHK algorithm^[see for example
  [@GEWE:KEAN:RUNK:94].] (for Geweke, Hajivassiliou and Keane who
developed this algorithm) can be described as follow :

\begin{enumerate}
- compute $\Phi\left(-\frac{V_1^l}{l_{11}}\right)$
- draw a number called $\eta_1^r$ from a standard normal
  distribution upper-truncated at $-\frac{V_1^l}{l_{11}}$ and compute
  $\Phi\left(-\frac{V^l_2+l_{21}\eta_1^r}{l_{22}}\right)$
- draw a number called $\eta_2^r$ from a standard normal
  distribution upper-truncated at
  $-\frac{V^l_2+l_{21}\eta_1^r}{l_{22}}$ and compute
  $\Phi\left(-\frac{V^l_3+l_{31}\eta_1^r+l_{32}\eta_2^r}{l_{33}}\right)$
- $...$ draw a number called $\eta_{J-1}^r$ from a standard
  normal distribution upper-truncated at $-\frac{V^l_{J-1}+l_{(J-1)1}\eta_1^r+... V^l_{J-1}+l_{(J-1)(J-2)}\eta_{J-2}^r}{l_{(J-1)(J-1)}}$
- multiply all these probabilities and get a realization of the
  probability called $P^r_l$.
- repeat all these steps many times and average all these
  probabilities ; this average is an estimation of the probability :
  $\bar{P}_l = \sum_{r=1}^R P^r_l/R$.
\end{enumerate}

Several points should be noted concerning this algorithm :

- the utility differences should be computed respective to the
  chosen alternative for each individual,
- the Choleski decomposition used should relies on the same
  covariance matrix of the errors. One method to attained this goal is
  to start from a given difference, *e.g.* the difference
  respective with the first alternative. The vector of error
  difference is then $\epsilon^1$ and its covariance matrix is
  $\Omega^1=L^1{L^1}^{\top}$. To apply a difference respective with an
  other alternative $l$, we construct a matrix called $S^l$ which is
  obtained by using a $J-2$ identity matrix, adding a first row of 0
  and inserting a column of $-1$ at the $l-1^{\mbox{th}}$
  position. For example, with 4 alternatives and $l=3$, we have :
  $$S^3=
  \left(
    \begin{array}{ccc}
      0 & -1 & 0 \\
      1 & -1 & 0 \\
      0 & -1 & 1 \\
    \end{array}
  \right)
  $$
  The elements of the choleski decomposition of the covariance matrix
  is then obtained as follow :
  $$
  \Omega^l = S^l \Omega^1 {S^l}^{\top}=L^l {L^l}^{\top}
  $$
- to compute draws from a normal distribution truncated at $a$,
  the following trick is used : take a draw $\mu$ from a uniform
  distribution (between 0 and 1) ; then $\eta = \Phi^{-1}\left(\mu
    \Phi(a)\right)$ is a draw from a normal distribution truncated at
  $a$

## Applications

We use again the `Fishing` data frame, with only a subset of three
alternatives used. The multinomial probit model is estimated using
`mlogit` with the `probit` argument equal to `TRUE`.

```{r }
library("mlogit")
data("Fishing", package = "mlogit")
Fish <- dfidx(Fishing, varying = 2:9, choice = "mode", idnames = c("chid", "alt"))
``` 


```{r }
Fish.mprobit <- mlogit(mode~price | income | catch, Fish, probit = TRUE, alt.subset=c('beach', 'boat','pier'))
```

```{r }
summary(Fish.mprobit)
``` 

## Bibliography
