810 Computational Considerations in Microeconometrics
We consider computational issues of full information estimation in a model
where the outcome of interest is a non-negative count which depends on a set of
variables that includes dummy variables generated by a multinomial choice model
(see Deb and Trivedi, 2006a, 2006b). The following section formally describes the
nature of dependence.
15.6.1 Model specification
Consider a selectivity model in which individualichooses a treatment from a
set of three or more choices. This implies a multinomial choice model with a
benchmark choice.E[V∗ij]denotes the indirect utility of selecting thejth treatment,
j=0, 1, 2,...,J, and:
E[V∗ij]=z$iαj+δjlij+ηij, (15.36)wherezidenotes exogenous covariates, with associated parametersαj, andηijare
i.i.d. error terms. In addition,E[V∗ij]includes a latent factorlijwhich incorporates
unobserved characteristics common to individuali’s treatment choice and out-
come. Thelijare assumed to be independent ofηij. Without loss of generality, let
j=0 denote the control group andE[V∗i 0 ]=0.
Letdjbe binary (dummy) variables representing the observed treatment choice
anddi = (di 1 ,di 2 ,...,diJ)$. In addition, letli = (li 1 ,li 2 ,...,liJ)$. Then the
probability of treatment can be represented as:
Pr(di|zi,li)=g(z$iα 1 +δ 1 li 1 ,z$iα 2 +δ 2 li 2 ,...,z$iαJ+δJliJ), (15.37)wheregis an appropriate multinomial probability distribution. Specifically, we
specify a mixed multinomial logit structure (MMNL) defined as:
Pr(di|zi,li)=exp(z$iαj+δjlij)
1 +∑J
k= 1 exp(z$
iαk+δklik). (15.38)
For the count variable the expected outcome equation is:E(yi|di,xi,li)=x$iβ+∑J
j= 1 γjdij+∑J
j= 1 λjlij, (15.39)wherexiis a set of exogenous covariates with associated parameter vectorβand the
γjdenote the treatment effects relative to the control.E(yi|di,xi,li)also depends on
latent factorslij, i.e., the outcome is affected by unobserved characteristics that also
affect selection into treatment. Whenλj, the factor loading parameter, is positive
(negative), treatment and outcome are positively (negatively) correlated through
unobserved characteristics, i.e., there is positive (negative) selection, withγandλ
the associated parameter vectors respectively.
Assume thatfis the negative binomial-2 density:
f(yi|di,xi,li)=
(yi+ψ)
(ψ)(yi+ 1 )(
ψ
μi+ψ)ψ(
μi
μi+ψ)yi
, (15.40)