Germán Rodríguez

Generalized Linear Models
Princeton University
In this section I will describe an extension of the multinomial logit model that is particularly appropriate in models of choice behavior, where the explanatory variables may include attributes of the choice alternatives (for example cost) as well as characteristics of the individuals making the choices (such as income). To motivate the extension I will first reintroduce the multinomial logit model in terms of an underlying latent variable.

Suppose that \( Y_i \) represents a discrete choice among \( J \) alternatives.
Let \( U_{ij} \) represent the value or *utility* of the \( j \)-th choice to
the \( i \)-th individual. We will treat the \( U_{ij} \) as independent
random variables with a systematic component \( \eta_{ij} \) and a
random component \( \epsilon_{ij} \) such that

We assume that individuals act in a rational way, maximizing their
utility. Thus, subject \( i \) will choose alternative \( j \) if
\( U_{ij} \) is the largest of \( U_{i1}, \ldots, U_{iJ} \). Note that the
choice has a random component, since it depends on random utilities.
The *probability* that subject \( i \) will choose alternative \( j \) is

It can be shown that if the error terms \( \epsilon_{ij} \) have standard Type I extreme value distributions with density

\[\tag{6.11}f(\epsilon) = \exp\{ -\epsilon - \exp\{-\epsilon\}\}\]then (see for example Maddala, 1983, pp 60–61)

\[\tag{6.12}\pi_{ij} = \frac{\exp\{\eta_{ij}\}}{\sum\exp\{\eta_{ik}\}},\]which is the basic equation defining the multinomial logit model.

In the special case where \( J=2 \), individual \( i \) will choose the first alternative if \( U_{i1}-U_{i2} > 0 \). If the random utilities \( U_{ij} \) have independent extreme value distributions, their difference can be shown to have a logistic distribution, and we obtain the standard logistic regression model.

Luce (1959) derived Equation 6.12 starting from a simple requirement
that the odds of choosing alternative \( j \) over alternative \( k \) should be
independent of the choice set for all pairs \( j,k \). This property
is often referred to as the axiom of *independence from
irrelevant alternatives*. Whether or not this assumption is
reasonable (and other alternatives are indeed irrelevant) depends
very much on the nature of the choices.

A classical example where the multinomial logit model does not work well is the so-called “red/blue bus” problem. Suppose you have a choice of transportation between a train, a red bus and a blue bus. Suppose half the people take the train and half take the bus. Suppose further that people who take the bus are indifferent to the color, so they distribute themselves equally between the red and the blue buses. The choice probabilities of \( \pi = (.50, .25, .25) \) would be consistent with expected utilities of \( \eta =(\log 2, 0, 0) \).

Suppose now the blue bus service is discontinued. You might expect that all the people who used to take the blue bus would take the red bus instead, leading to a 1:1 split between train and bus. On the basis of the expected utilities of \( \log 2 \) and \( 0 \), however, the multinomial logit model would predict a 2:1 split.

Keep this caveat in mind as we consider modeling the expected utilities.

In the usual multinomial logit model, the expected utilities \( \eta_{ij} \) are modeled in terms of characteristics of the individuals, so that

\[ \eta_{ij} = \boldsymbol{x}_i'\boldsymbol{\beta}_j. \]Here the regression coefficients \( \boldsymbol{\beta}_j \) may be interpreted as reflecting the effects of the covariates on the odds of making a given choice (as we did in the previous section) or on the underlying utilities of the various choices.

A somewhat restrictive feature of the model is that the same attributes \( \boldsymbol{x}_i \) are used to model the utilities of all \( J \) choices.

McFadden (1973) proposed modeling the expected utilities \( \eta_{ij} \) in terms of characteristics of the alternatives rather than attributes of the individuals. If \( \boldsymbol{z}_j \) represents a vector of characteristics of the \( j \)-th alternative, then he postulated the model

\[ \eta_{ij} = \boldsymbol{z}_j'\boldsymbol{\gamma}. \]This model is called the *conditional logit* model,
and turns out to be equivalent to a log-linear model where
the main effect of the response is represented in terms of
the covariates \( \boldsymbol{z}_j \).

Note that with \( J \) response categories the response margin may be reproduced exactly using any \( J-1 \) linearly independent attributes of the choices. Generally one would want the dimensionality of \( z_j \) to be substantially less than \( J \). Consequently, conditional logit models are often used when the number of possible choices is large.

A more general model may be obtained by combining the multinomial and conditional logit formulations, so the underlying utilities \( \eta_{ij} \) depend on characteristics of the individuals as well as attributes of the choices, or even variables defined for combinations of individuals and choices (such as an individual’s perception of the value of a choice). The general model is usually written as

\[\tag{6.13}\eta_{ij} = \boldsymbol{x}_i'\boldsymbol{\beta}_j + \boldsymbol{z}_{ij}'\boldsymbol{\gamma}\]where \( \boldsymbol{x}_i \) represents characteristics of the individuals that are constant across choices, and \( \boldsymbol{z}_{ij} \) represents characteristics that vary across choices (whether they vary by individual or not).

Some statistical packages have procedures for fitting conditional logit models to datasets where each combination of individual and possible choice is treated as a separate observation. These models may also be fit using any package that does Poisson regression. If the last response category is used as the baseline or reference cell, so that \( \eta_{iJ} = 0 \) for all \( i \), then the \( \boldsymbol{z}_{ij} \) should be entered in the model as differences from the last category. In other words, you should use \( \boldsymbol{z}^*_{ij} = \boldsymbol{z}_{ij} - \boldsymbol{z}_{iJ} \) as the predictor.

Changing the distribution of the error term in Equation 6.9
leads to alternative models.
A popular alternative to the logit models considered so far
is to assume that the \( \epsilon_{ij} \) have independent
standard normal distributions for all \( i,j \).
The resulting model is called the multinomial/conditional
*probit* model,
and produces results very similar to the multinomial/conditional
logit model after standardization.

A more attractive alternative is to retain independence across
subjects but allow dependence across alternatives, assuming that
the vector \( \boldsymbol{\epsilon}_i =
(\epsilon_{i1}, \ldots, \epsilon_{iJ})' \)
has a *multivariate* normal distribution with mean vector **0**
and arbitrary correlation matrix **R.** (As usual with latent
variable formulations of binary or discrete response models, the variance
of the error term cannot be separated from the regression coefficients.
Setting the variances to one means that we work with a correlation
matrix rather than a covariance matrix.)

The main advantage of this model is that it allows correlation between the utilities that an individual assigns to the various alternatives. The main difficulty is that fitting the model requires evaluating probabilities given by multidimensional normal integrals, a limitation that effectively restricts routine practical application of the model to problems involving no more than three or four alternatives.

For further details on discrete choice models see Chapter 3 in Maddala (1983).