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Logit models of individual choices


Logit models of individual choices
Thierry Magnac
Université de Toulouse
First version: September 2005.
(prepared for the New Palgrave)
The logit function is the reciprocal function to the sigmoid logistic function. It maps the interval
[0,1] into the real line and is written as:
logit(p) = ln(p=(1 􀀀 p))
Two traditions are involved in the modern theory of logit models of individual choices. The …rst
one concerns curve …tting as exposed by Berkson (1944) who coined the term logit after its close
competitor probit which is derived from the normal distribution. Both models are by far the most
popular econometric methods used in applied work to estimate models for binary variables, even
though the development of semiparametric and non parametric alternatives over the last 30 years
has been intensive (Horowitz and Savin, 2001).
In the second strand of literature, models of discrete variables and discrete choices as originally
set up by Thurstone (1927) in psychometrics have been known as random utility models (RUM)
since Marschak (1960) introduced them to economists. As the availability of individual databases
and the need for tools to forecast aggregate demands derived from discrete choices were increasing
from the 1960s onwards, di¤erent waves of innovations, fostered by McFadden (see his Nobel lecture,
2001) elaborated more and more sophisticated and ‡exible logit models. The use of these models
and of simulation methods triggered burgeoning applied research in demand analysis in recent years.
Those who wish to study the subject in greater detail are referred to Gouriéroux (2000), McFad-
den (2001) or Train (2003) where references to applications in economics and marketing can also
be found.
Measurement models
As Berkson (1951) put it, logit (or probit) models may be seen as “merely a convenient way of
graphically representing and …tting a function”. It is used for any empirical phenomena delivering a
1
binary random variable Yi; taking values 0 and 1, to be analyzed. In a logit model, it is postulated
that its probability distribution conditional on a vector of covariates Xi is given by:
Pr(Yi = 1 j Xi) =
exp(Xi )
1 + exp(Xi )
where is a vector of parameters. This model can also be derived from more general frameworks
in statistical mechanics or spatial statistics (Strauss, 1992).
Using cross-sectional samples, the parameter of interest is estimated using maximum likelihood
or by GLM methods where the link function is logit (McCullagh and Nelder, 1989). Under the
maintained assumption that it is the true model and other standard assumptions, the Maximum
Likelihood Estimator (MLE) is consistent, asymptotically normal and e¢ cient (Amemiya, 1985).
Nevertheless, the MLE may fail to exist, or more exactly be at the bounds of the parameter space,
when the samples are uniformly composed by 0s or 1s for instance (Berkson, 1955).
When repeated observations are available, the method of Berkson delivers an estimator close
to MLE since they are asymptotically equivalent. Observe …rst that the logit function of the true
probability obeys the linear equation:
logit(Pr(Yc = 1 j Xc)) = Xc :
where the covariates Xc now takes a discrete number of values de…ning each cell, c. Second use the
observed frequencies in each cell, ^pc; and contrast it with theoretical probabilities, pc, as:
logit(^pc) = Xc + (logit(^pc) 􀀀 logit(pc))
= Xc + “c:
The random term “c properly scaled by the square root of the number of observations in cell c is
asymptotically normally distributed with variance equal to 1=(pc(1 􀀀 pc)). The method of Berkson
then consists in using minimum chi-square, i.e. a method of moments, to estimate ; an instance of
what is know as minimum distance or asymptotic least squares (Gouriéroux, Monfort and Trognon,
1985).
When measurements for a single individual are repeated, Rasch (1960) suspected that individual
e¤ects might be important and proposed to write:
logit(Pr(Yit = 1 j Xit)) = Xit + i
2
where t indexes the di¤erent items that are measured and i is an individual speci…c intercept
or …xed e¤ect. Items can be di¤erent questions in performance tests or di¤erent periods. In the
original Rasch formulation, parameters were allowed to be di¤erent across items, t, and there were
no covariates.
Given that the number of items is small, it is well known that the estimation of such a model
runs into the problem of incidental parameters (see Lancaster, 2000). As the number of parameters
i increases with the cross-section dimension, the MLE is inconsistent (Chamberlain, 1984). Nev-
ertheless, the nuisance parameters i can be di¤erenced out using conditional likelihood methods
(Andersen, 1971) because:
logit(Pr(Yit = 1 j Xit; Yit + Yit0 = 1)) = (Xit 􀀀 Xit0) :
The conditional likelihood estimator of is consistent and root n asymptotically normal but it is
not e¢ cient although no e¢ cient estimator is known. Furthermore, when binary variables Yit are
independent, conditionally on Xi, the only model where a root n consistent estimator exists is a logit
model (Chamberlain, 1992). Extensions of Rasch rely on the fact that root n consistent estimators
exist if and only if Yit + Yit0 is a su¢ cient statistic for the nuisance parameters i (Magnac, 2004).
When the number of items or periods becomes large, pro…le likelihood methods where individual
e¤ects are treated as parameters seem to be accurate in Monte-Carlo experiments as soon as the
number of periods is 4 or 5 (Arellano, 2003).
Multinomial logit (or in disuse “conditional logit”) is to binary logit what is a multinomial to
a binomial distribution (Theil, 1969). Given a vector Yi consisting of K elements which are binary
random variables and lie in the RK􀀀simplex (their sum is equal to 1), it is postulated that:
Pr(Y (k)
i = 1 j Xi) =
exp(Xi (k))
1 +
XK
k=2
exp(Xi (k))
where by normalization, (1) = 0. Ordered logit has a di¤erent ‡avor since it applies to rank-ordered
data such as education levels (Gouriéroux, 2000).
As probits, logit models are very tightly speci…ed parametric models and can be substantially
generalized. Much e¤ort has been exerted to relax parametric and conditional independence as-
sumptions starting with Manski (1975). Manski (1988) analyzes the identifying restrictions in
3
binary models and Horowitz (1998) reviews estimation methods. In some cases, Lewbel (2000) and
Matzkin (1992) o¤er alternatives.
Random utility models
The theory of discrete choice is directly set-up in a multiple alternative framework. A choice of
an alternative k belonging to a set C is assumed to be probabilistic either because preferences are
stochastic, heterogenous or because choices are perturbed in a random way. By de…nition, choice
probability functions map each alternative and choice sets into the simplex of RK.
A strong restriction on choices is the axiom of Independence of Irrelevant Alternatives (IIA,
Luce, 1959). The axiom states that the choice between two alternatives is independent of any other
alternative in the choice set. The version that allows for zero probabilities (McFadden, 2001) states
that for any pair of choice set C;C0 such that fk; k0g 2 C and C  C0:
Pr(k is chosen in C0) = Pr(k is chosen in C): Pr(An element of C is chosen in C0):
Under this axiom, choice probabilities take a multinomial generalized logit form.
Moreover, assume that choices are associated with utility functions, fu(k)gk that depend on
determinants Xi and random shocks:
u(k) = X (k) + “(k);
and that the actual choice of the decision maker yields maximum utility to her. Then, the IIA
axiom is veri…ed if and only if “(k) are independent and extreme value distributed (McFadden,
1974). Extensions of decision theory under IIA were proposed in the continuous case (Resnick and
Roy, 1991) or in an intertemporal context (Dagsvik, 2002).
The IIA axiom is a strong restriction as in the famous red and blue bus example where if IIA is
assumed, the existence of di¤erent colours a¤ect choices of transport between bus and other modes
while introspection suggests that colours should indeed be irrelevant. Several generalizations which
procede from logit were proposed to bypass IIA. Hierarchical or tree structures were the …rst to
be used. At the upper level, the choice set consists in broad groups of alternatives. In each of
these groups, there are various alternatives which can consist themselves in subsets of alternatives
etc. The most well known model is the two-level nested logit where alternatives are grouped by
4
similarities. For instance, the …rst level is the choice of the type of the car, the second level being
the make of the car. The formula of choice probabilities for nested logit:
p(k) =
exp(X (k)=Bs)
P
j2Bs exp(X (j)=Bs)
Bs􀀀1
PT
t=1
P
j2Bt
exp(X (j)=Bs)
Bt
;
where alternative k belongs to Bs; is not illuminating but the logic of construction is clear. Choices
at each level are modelled as multinomial logit (Train, 2003).
General Extreme Value distributions (McFadden, 1984) provide more extensions although they
do not generate all con…gurations of choice probabilities. In contrast, mixed logit does, as shown
by McFadden & Train (2000). Instead of considering that parameters are deterministic, make them
random or heterogeneous across agents. The resulting model is a mixture model where individual
probabilities of choice are obtained by integrating out the random elements as in
p(k) =
Z
p(k)( )f( )d :
Integrals are computed using simulation methods (MacFadden, 2001). The same principle is used
by Berry, Levinsohn & Pakes (1995) with a view to generalize the aggregate logit choice models
using market data. Logit models are still very much in use in applied settings in demand analysis
and marketing and are equivalent to a representative consumer model (Anderson, de Palma and
Thisse, 1992). Mixed logits permit much more general patterns of substitution between alternatives
and should probably become the standard tool in the near future.
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