David T. Jacho-Chávez and Pravin K. Trivedi 801smoothing parameter(s). Therefore, at least asymptotically, any sequence of
smoothing parameters is going to give a consistent estimator as long as it satis-
fies certain conditions. However, this observation does not necessarily imply that
such an estimator is going to be optimal in the mean squared error (MSE) sense
(see, e.g., Linton, 1995; Härdle, Hall and Ichimura, 1993).
Finally, the asymptotic theory for the above estimators requires trimming those
observations arbitrarily close to the boundary of their observed supports. We did
not address this in the above discussion, partly because of the lack of guidance on
how to select trimming parameters with fixed sample sizes, and partly because no
trimming amounts to assuming that the actual support of the variables involved
is larger than that observed in the data.
15.5 Modeling heterogeneity
In the introductory section we noted that heterogeneity is a pervasive feature in
microeconometrics. The most tractable way of handling observed heterogeneity
is to control for it by including sociodemographic variables such as family com-
position, age, gender, location, etc. The potential of this approach is limited by
the scope of the available data. There still remains variation that is induced by
unobserved factors, referred to in econometrics as unobserved heterogeneity.
Neglecting individual-level unobserved heterogeneity can have potentially seri-
ous consequences, analogous to those of omitted regressors. Even when hetero-
geneity is explicitly accommodated, the precise manner and assumptions under
which it enters the model can have important consequences. Hence, not surpris-
ingly, every proposed econometric specification is routinely scrutinized for the
manner in which it accommodates heterogeneous behavior.
Not only are models with heterogeneity more flexible, and hence generally fit the
data better, but they also lead to relaxation of strong constraints. For example, the
multinomial logit (MNL) discrete choice model is subject to the strong restriction
of independence of irrelevant alternatives (IIA), whereas the random parameter
version of the MNL does not have the IIA restriction.
There are a number of distinctive ways of allowing for unobserved heterogeneity.
- Treat heterogeneity either as an additive or a multiplicative random effect
(uncorrelated with included regressors) or as a fixed effect (potentially corre-
lated with included regressors). Within the class of random effects models,
heterogeneity distributions may be treated as continuous or discrete. Exam-
ples include a random intercept in linear regression and linear panel models,
fixed effects in linear and nonlinear panels, and multiplicative heterogeneity in
models of counts and durations. - Allow both intercept and slope parameters to vary randomly and parametrically.
Examples include random parameter discrete choice and outcome models and
finite mixture models. - Model heterogeneity explicitly in terms of both observed and unobserved
variables using mixed models, hierarchical models and/or models of clustering.