The assumption of symmetry is particularly well suited when there is no natural
order in the responses, i.e. when 1 and 0 are logically interchangeable.19 In
particular, when εi is a normal r.v.
εi ~ N (θ,σ2)
we recover the standard probit model:
Pr(Ki = 0 I xi) = Φ (-i3⅛d),σ2)
Pr(yi = l∣a⅛) = Φ(∕3⅛0,σ2)
or, compactly, for y ∈ {0,1}:
Pr (Yi = y∖ xi) = Φ ((2 ∙ y - 1) ∙ β'xi-0, σ2)
However, in this model β and σ2 are not separately identified and this can be
easily shown considering a new latent continuous random variable Y ° = c∙Yi* =
c ■ (tβxi + ɛi) = rγxi + ŋi, with c > 0. In fact, in this case:
Pr (Yi = 1 I xi) = Pr (Yi° > 0) = Φ (7aιi; 0, (c ∙ σ)2}
^fXi
= f √⅛7exp(-⅛⅛)d≡'
— ∞
7Xi βxi
= I √⅛ eχP (-⅛) d © = / √⅛exp (-⅛) dz
— ∞ —∞
= Pr(yi*>0)
To avoid this problem σ2 is customarily set at the arbitrarily value of 1, and β
is evaluated under this condition.
A multivariate extension. The latent regression approach can be generalized
to two or more binary decisions.20 Dropping the i subscript for notational ease,
we can suppose that every decision j can be expressed as a function of a latent
continuous regression:
У/ = β'jX + Ξj = Vj + Ξj, j = 1, ..., n
Yj = ^17>0}
19This happens in most cases. However, suppose a group of individuals in a controlled
trial is observed on a window [0,T]: assuming the DGP to be a Cox (1972) Proportional
Hazards Model, survival probability in T is the c.d.f. of an Extreme Value Type I or Gumbel
distribution. Clearly, it is a skewed distribution.
20The first attempt in this direction is due to Ashford and Sowden (1970); several papers
have followed this one, expanding the subject and proposing estimation metods different
from maximum likelihood. We refer to Amemiya (1981) for a survey of multivariate binary
regression models.
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