A Generalization of the Nested Logit Model

Abstract

Amemiya and Shimono (1989) estimated the parameters of various nested logit models to explain the choice of four alternatives facing those who have retired from their regular employment at age 55. The four alternatives are 1 to retire (to be denoted by the symbol “R”), 2 to be self-employed (“S”), 3 to be employed full-time (“F”), and 4 to be employed part-time (“P”). They estimated four types of nested logit models differing by the way four alternatives are nested. Here we mention only what they called Type I, which is characterized by the following joint distribution of the stochastic terms of the utilities associated with the four alternatives:

$$ F({\varepsilon _1},{\varepsilon _2},{\varepsilon _3},{\varepsilon _4}) = \exp \left\{ { - {{\left[ {{{\left( {{e^{\frac{{ - {\varepsilon _1}}}{\rho }}} + {e^{\frac{{ - {\varepsilon _2}}}{\rho }}}} \right)}^{\frac{\rho }{\sigma }}} + {e^{\frac{{ - {\varepsilon _3}}}{\sigma }}}]} \right]}^\sigma } - {e^{ - {\varepsilon _4}}}} \right\}$$

(1.1)

for 0 < σ, ρ ≤ 1. Here σ and ρ are called the association parameters, and the bounds on them are required to make F a proper distribution function.