Advanced Econometrics Takeshi Amemiya

Sequential Probit and Logit Models

When the choice decision is made sequentially, the estimation of multinomial models can be reduced to the successive estimation of models with fewer responses, and this results in computational economy. We shall illustrate this with a three-response sequential probit model. Suppose an individual deter­mines whether у = 2 or у Ф 2 and then, given уф 2, determines whether у — 1 or 0. Assuming that each choice is made according to a binary probit model, we can specify a sequential probit model by

Р2 = Ф(х^) (9.3.75)

and

Л = [ 1 - Ф(х^2)]Ф(хі A)- (9.3.76)

The likelihood function of this model can be maximized by maximizing the likelihood function of two binary probit models.

A sequential logit model can be analogously defined. Kahn and Morimune

(1979) used such a model to explain the number of employment spells a worker experienced in 1966 by independent variables such as the number of grades completed, a health dummy, a marriage dummy, the number of chil­dren, a part-time employment dummy, and experience. The dependent vari­able у і is assumed to take one of the four values (0, 1,2, and 3) corresponding to the number of spells experienced by the fth worker, except that y, = 3 means “greater than or equal to 3 spells.” Kahn and Morimune specified probabilities sequentially as

р(Уі=0) = Л{х'А),

Р(Уі = 1 Уі * 0) = А(х'А),

and

РІУі = 2|у, Ф 0, yt Ф) = Л(х'&). (9.3.79)

Note that Kahn and Morimune could have used an ordered logit model with their data because we can conjecture that a continuous unobservable variable yf (interpreted as a measure of the tendency for unemployment) exists that affects the discrete outcome. Specifying yf = x'lfi + e, would lead to one of the ordered models discussed in Section 9.3.2.