A Multivariate Discrete Choice Models
The binary case. Choice among discrete alternatives is a deeply studied topic
in modern econometrics. Consider an individual facing a binary choice. This
situation may be modelled by assuming the existence of a latent continuous
variable, supposed to be an index of the propensity to undertake this decision.
A threshold model is a model in which the choice of the individual is supposed
to be caused by this index crossing a deterministic value.18 In the following we
will consider a threshold of 0, since no restriction is imposed in this setting.
If we indicate with Y* the latent continuous variable, we have:
Y=( 1, ifyi*>o
г ɪ 0, otherwise
or
5zi = l{γ,*>o}
where l{.j is the indicator function. If Y* has a cumulative distribution function
Fγ* (∙), we can express the probability of Yi taking a value 1 as:
Pr (Yi = 1) = E(yi) = E (l{y?>0}) = Pr (yi* > 0) = 1 - Fγ∙ (0)
Similarly the probability of having a value of 0 is:
Pr (Yi = 0) = 1 - Pr (Yi = 1) = Pr (У/ ≤ 0) = Fγ∙ (0)
A particular case of this general framework is the latent regression model, that
can be obtained when Y* is a (usually linear) function of observed and unob-
served characteristics of the individual and of the environment:
E(yi* I xi) = β'xi
Y*=β'xi+εi
Here Xi is a vector of observed covariates, while e⅛ represents the unobserved
ones. Assuming that e⅛ has a c.d.f. Fe (∙), the probability of the discrete binary
variable Yi is:
Pr (Yi = 0 I xi) = Pr (У/ ≤ 0) = Pr (β'xi + εi ≤ 0)
= Pr (ɛi ≤ -β'xi) = Fe (-β'xi)
Pr (Yi = 1 I xi) = 1 - Pr (Yi = 0) = 1 - Fe (-β,xi)
If the density function fe (∙) is symmetric around 0, then F (-y) = 1 — F (y)
and:
Pr (Yi = 1 I xi) = 1 - Fe (-β'xi) = Fe (β'xi)
18Stochastic threshold are commonly advocated in biometrical response study: here an
individual is treated with a deterministic quantity of a reagent; the reaction (death or survival)
is caused by the crossing of a stochastic threshold, function of observed (age, etc.) and
unobserved (frailty or proneness, etc.) characteristics of the individual. In economics, it
seems more natural to suppose the threshold deterministic.
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