Chapter on Discrete Choice Models Linear Probability Model ...

<strong>Chapter</strong> on Discrete Choice Models
Linear Probability Model: Sample observations on (I i ,x i )whereI i =0, 1 is a dichotomous indicator,
I i = x i β + ² i ,
i =1, ···,n
where E(² i |x) = 0. This is a linear regression speciÞcation.
As E(² i |x) = 0, it implies that E(I i |x i )=x i β. Because E(I i |x i )=P (I i =1|x i ) by the expectation
formula of a discrete r.v., x i β can be interpreted as the choice probability of I i =1givenx i .
The limitation of the linear probability model is that there is no restriction on the possible value of xβ
between 0 and 1. So, for certain values of x and β, the predicated values of xβ can be less than 0 and greater
than 1.
Furthermore, the variance of ² i is not a constant. The variances of ²s areheterokedastic. Asvar(²|x) =
var(I|x) =P (I =1|x)[1 − P (I =1|x)] = xβ(1 − xβ), the variance of ²s depends on x and β.
Probit and Logit models: To impose probability structure, specify P (I 1 =1|x) =F (xβ), where F
is a distribution function. A Probit model speciÞes that F (x) =Φ(x) whereΦ(x) is a standard normal
distribution. A Logit model speciÞes that F (x) = e x /(1 + e x ). The linear probability corresponds to
F (x) =x — a uniform distribution (if x is restricted to lie between 0 and 1).
The logistic distribution has a zero mean and a variance = π 2 /3. The standard logistic distribution
e λx /(1 + e λx )withλ = π/ √ 3 has slightly heavier tails than the standard normal distribution.
These models can be estimated by the method of maximum likelihood. With an independent sample
for I 0 s, the log likelihood function is
ln L =
nX
[I i ln F (x i β)+(1− I i )ln(1− F (x i β))].
i=1
The log likelihood functions of the logit and probit models are concave in parameters.
The Conditional Logit Model
The classical consumer demand model assumes that consumers are rational in the sense that they
make choices maximizing their perceived utility subject to constraints on expenditure. Under this classical
framework, McFadden (1974) provides microeconomic justiÞcation of the speciÞcation of popular probability
choice models.
Suppose there are m choice alternatives for a consumer. To allow unobservable attributes, unobservable
characteristics or imperfect perception, the utility function is assumed to be a random function from the
econometrician’s point of view:
y ∗ j = V (x j,z,θ)+²,
j =1,...,m.
Let I j be a dichotomous indicator that the jth alternative is chosen. Under the utility maximization
hypothesis, I j =1ify ∗ j = max(y ∗ 1 , ···,y∗ m)andI j = 0, otherwise. In practice, one may assume that
V (x j ,z,θ) =x j β + zα j . With stochastic distributions for the disturbances ², choice probabilities can be derived.
The choice probabilities may not have, in general, closed expressions. But for some speciÞc parametric
distributions, choice probabilities can be derived analytically with closed form expressions.
Suppose that ² j , j =1,...,m are i.i.d. with the Gumbel type I extreme-value distribution:
The corresponding density function is
F (² j

where V j denotes V (x j ,z) for simplicity. This can be shown as follows. Since yj ∗ =max(y∗ 1 , ···,y∗ m )is
equivalent to ² k

The Hessian matrix is non-positive deÞnite and the log likelihood function is concave.
The conditional logit model has a restrictive property known as the independence of irrelevant alternative
property (IIA): The probability odds ratio for the jth and the kth choices is the same irrespective of the
total number m of choices. This is so, because
Prob(I j =1|{1, 2,...,m})
Prob(I k =1|{1, 2,...,m}) = eV j
e V k
= Prob(I j =1|{j, k})
Prob(I k =1|{j, k}) .
The IIA property is inappropriate for some situations. An example is the so-called blue bus and red bus
paradox. Suppose that each consumer has three alternatives in choosing the red bus, the blue bus or
his/her own automobile to work. Let x 1 (red bus), x 2 (blue bus) and x 3 (car) be the attributes of the
three transportation modes. Suppose that consumers treat the two buses indifferent and are also indifferent
between the automobile mode and the bus mode. In such situation, Prob(I 1 =1|x 1 ,x 2 )=Prob(I 1 =
1|x 1 ,x 3 )=Prob(I 2 =1|x 2 ,x 3 )=1/2 andProb(I 1 =1|x 1 ,x 2 ,x 3 )=Prob(I 2 =1|x 1 ,x 2 ,x 3 )=1/4. It
follows that Prob(I1=1|x1,x2,x3)
Prob(I1=1|x1,x3)
Prob(I 3=1|x 1,x 2,x 3)
=1/2 and
Prob(I 3=1|x 1,x 3)
=1. TheIIApropertyisnotsatisÞed in this
situation. The inconsistency occurs because the two bus modes are perceived as similar alternatives rather
than independent alternatives by the individual.
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