---
output: rmarkdown::html_vignette
bibliography: ../inst/REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{mlogit}
  %\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)
```

Random utility models is the reference approach in economics when one
wants to analyze the choice by a decision maker of one among a set of
mutually exclusive alternatives. Since the seminal works of Daniel Mc
Fadden [@MCFAD:74, MCFAD:78] who won the Nobel prize in economics "for
his development of theory and methods for analyzing discrete choice",
a large amount of theoretical and empirical literature have been
developed in this field.^[For a presentation of this literature, see
@TRAI:09 ; the theoretical parts of this paper draw heavily on this
book.]

Among the numerous applications of such models, we can cite the
following: @HEAD:MAYE:04 investigate the determinants of the choice by
Japanese firms of an European region for implementing a new production
unit, @FOWL:10 analyse the choice of a NO$_x$ emissions reduction
technology by electricity production plants, @KLIN:THOM:96 and
@HERR:KLIN:99 consider how the choice of a fishing mode can be
explained by the price and the catch expectency, @DIPA:JAIN:94
investigate the brand choice for yogurts, @BHAT:95 analyse transport
mode choice for the Montreal-Toronto corridor.

These models rely on the hypothesis that the decision maker is able to
rank the different alternatives by an order of preference represented
by a utility function, the chosen alternative being the one which is
associated with the highest level of utility. They are called random
utility models because part of the utility is unobserved and is
modelised as the realisation of a random deviate. 

Different hypothesis on the distribution of this random deviate lead
to different flavors of random utility models. Early developments of
these models were based on the hypothesis of identically and
independent errors following a Gumbel distribution,^[This
  distribution has the distinctive advantage that it leads to a
  probability which can be written has an integral which has a closed
  form.] leading to the multinomial logit model (**MNL**). More
general models have since been proposed, either based on less
restrictive distribution hypothesis or by introducing individual
heterogeneity.

Maintaining the Gumbel distribution hypothesis but relaxing the iid
hypothesis leads to more general logit models (the heteroscedastic and
the nested logit models). Relaxing the Gumbel distribution hypothesis
and using a normal distribution instead leads to the multinomial
probit model which can deal with heteroscedasticity and correlation of
the errors.

Individual heterogeneity can be introduced in the parameters
associated with the covariates entering the observable part of the
utility or in the variance of the errors. This leads respectively to
the mixed effect models (**MXL**) and the scale heterogeneity
model (**S-MNL**).


The first version of `mlogit` was posted in 2008, it was the first `R`
package allowing the estimation of random utility models. Since then,
other package have emerged [see @SARR:DAZI:17 page 4 for a survey of
revelant R pakages].  `mlogit` still provides the widests set of
estimators for random utility models and, moreover, its syntax has
been adopted by other `R` packages. Those packages provide usefull
additions to `mlogit`:

- `mnlogit` enables efficient estimation of **MNL** for
  large data sets,
- `gmnl` estimates **MXL** and **S-MNL**, but also the
  so called generalized multinomial logit model **G-MNL** which
  nests them,
- latent-class multinomial logit models (**LC-MNL**), for
  which the heterogeneity is due to the fact that individuals belong to
  different classes and mixed-mixed models (**MM-MNL**) which are a
  mixture of **LC-MNL** and **MXL** can also be estimated
  using the `gmnl` package,
- bayesian estimators for multinomial models are provided by the
  `bayesm`, `MNP` and `RSGHB` packages.

The article is organized as follow. Vignette
[formula/data](./c2.formula.data.html) explains how the usual
formula-data and testing interface can be extended in order to
describes in a very natural way the model to be estimated. Vignette
[random utility models](./c3.rum.html) describe the landmark
multinomial logit model. Vignette [relaxing the iid
hypothesis](./c4.relaxiid.html), [mixed logit model](./c5.mxl.html) and
[multinomial probit model](./c6.mprobit.html) present three important
extensions of this basic model: Vignette [relaxing the iid
hypothesis](./c4.relaxiid.html) presents models that relax the iid
Gumbel hypothesis, [mixed logit model](./c5.mxl.html) introduces slope
heterogeneity by considering some parameters as random and
[multinomial probit model](./c6.mprobit.html) relax the Gumbel
distribution hypothesis by assuming a multivariate norm
distribution.

### Bibliography