Aggregation in Discrete Choice Models: An Illustration of Nonlinear Aggregation

A. Colin Cameron

"Aggregation in Discrete Choice Models: An Illustration of Nonlinear Aggregation"

in T.S. Barker and H.M. Pesaran eds., Disaggregation in Economic Modelling, 1990, pp. 206-234, Routledge, London.

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.