802 Computational Considerations in Microeconometrics
Which approach one takes to modeling heterogeneity and how it is combined
with other assumptions, e.g., functional forms, has important consequences for
computation. One researcher may impose strong functional forms but achieve
modeling flexibility by allowing for behavior heterogeneity. Another may make
very flexible functional form assumptions but not allow explicitly for unobserved
heterogeneity. Currently both strategies coexist in the literature.
We now consider computational aspects of the different modeling strategies.
- Under the fixed effects specification, the most popular approach is to treat het-
erogeneity in the nuisance parameter framework and eliminate it by a suitable
transformation such as: within transformation; first-difference transformation;
differences-in-differences transformation. In some parametric models with fixed
effects, the nuisance parameters can be eliminated by applying the conditional
likelihood approach that replaces the fixed effect by a sufficient statistic, but
such a statistic is not always available. - Dummy variables can be introduced to capture individual-specific heterogene-
ity. This approach is familiar in linear panel models, but due to the incidental
parameter problem, this formulation may not lead to consistent estimates. In
panel models with a large cross-section dimension (n), the estimation dimen-
sion also increases and in nonlinear models this may lead to computational
challenges, though there are examples where ingenious algorithms can address
the issues (e.g., Greene, 2004a, 2004b). - In random effects models the standard and time-honored approach involves
integrating out the distribution of unobserved heterogeneity, as was discussed
in section 15.3.3. Such integration is often implemented numerically, leading to
nontrivial computational challenges, especially if parameter dimension is large.
Random parameter multinomial logit and multinomial probit models are two
outstanding examples of this approach (see Train, 2003).
All the foregoing discussion has been carried out by reference to individual level
heterogeneity. Heterogeneity may also exist at the level of groups, which leads
to issues of clustering and interdependence that create additional computational
challenges. A group can be a geographical, social, ethnic, or merely a sampling
unit that exhibits some form of interdependent behavior induced by common
environment or culture. Group membership may be modeled as an observable or a
latent variable, analogous to fixed and random effects formulations in panel data.
In the remainder of this section we illustrate, using two data-based examples,
how modeling heterogeneity can be computationally demanding but empirically
informative.
15.5.1 Example: quantile regression
Whereas the standard linear regression is a useful tool for summarizing the average
relationship between the outcome variable of interest and a set of regressors, i.e.,
the conditional mean function,E[y|x], QR can provide a more complete picture