Logit, Probit and Tobit: Models for Categorical and Limited ...

Introduction
• With such variables variables, we can build models that
describe the response probabilities, say P(yi = 1), of
the dependent p variable y yi. i
– For a sample of N independently and identically distributed
observations i = 1, ... ,N and a (K+1)-dimensional vector x′ i
of f explanatory l t variables, i bl the th probability b bilit th that t y tk takes value l
1 is modeled as
P ( yi
= 1|
xi
) = F ( xi′
β ) = F ( zi
where β is a (K + 1)-dimensional column vector of
parameters.
• The transformation function F is crucial. It maps the
linear combination into [0,1] and satisfies in general
F(−∞) =0 = 0, F(+∞) = =1 1, and andδF(z)/δz δF(z)/δz > 0 [that is is, it is a
cumulative distribution function].
)