This paper considers aggregation in binary outcome models. Let p
= Pr(y=1|x,b) = F(x'b) where y takes value 1 or 0. The micro function is Y
= F(x'b). The corresponding macro function is E[Y] = Integral F(x'b)g(x'b)dx'b
where g(x'b) is the density of x'b.
The paper begins with a brief review of complete aggregation (Stoker (1984) and Kelijian (1980)).
It then considers obtaining the macro function from the micro function
for binary outcome models. In the special case that x ~ N[mu,S], so
that x'b ~ N[mu'b, b'Sb], and we consider the probit model so F(x'b) = PHI(x'b),
it can be shown that E[Y] = PHI(x'b/sqrt(1+s^2) where s^2 = 1+b'Sb. Otherwise
analytical results are hard to obtain.
The paper then considers macro prediction from a micro binary choice
model using macro data.