A. Colin Cameron 733

14.3 Estimation

Most applied microeconometric studies include estimation of parametric models
or the conditional mean, E

[

yi|xi

]
, or the conditional densityf(yi|xi). The specific
models used vary with the type of outcomeythat is being modeled. The com-
monly used estimation methods are ML and quasi-ML (where appropriate) for fully
parameterized models, and LS and IV for linear conditional mean models.
In the simplest caseyis continuous on(−∞,∞)and the linear model E

[

yi|xi

]

x′iβis used. But often the outcome is restricted in some way, leading to various non-
linear models. In the most extreme caseycan take only one of two values, such
as whether or not employed. Then the distribution is necessarily a Bernoulli (the
binomial with one trial) and different models for the probability parameter corre-
spond to logit and probit models. When there are only a few possible categorical
outcomes a wide range of multinomial models exist, including multinomial and
ordered logit and probit. For count outcomeythat takes only non-negative integer
values, such as number of doctor visits, the standard parametric models are Poisson
and negative binomial with E

[

yi|xi

]
=exp(x′iβ). For duration outcomey, such as
length of employment spell, the standard parametric models are exponential and
Weibull.
Econometrics packages for cross-section data provide estimators for these
models, as well as for the standard corrections for the common complications
of truncation and censoring. We do not detail these models and their standard
estimators, though the appropriate statistical inference is detailed in section 14.4.
This section instead reviews more advanced estimation methods that permit esti-
mation of more flexible parametric models, when models are fully parameterized,
as well as methods that permit estimation when models are not fully parameter-
ized. The most important of these methods is GMM, which provides a very general
framework for estimation of nonlinear models that nests OLS, IV and ML esti-
mation. Empirical likelihood is an adaptation of GMM that has different finite
sample properties. Simulation methods permit classical and Bayesian methods to
be applied to a much wider range of parametric models. Quantile regression and
semiparametric methods place less structure on the data generating process.
Among these methods only GMM, quantile regression and nonparametric regres-
sion (with single regressor) appear in one or more standard econometrics software
packages, so the methods are currently not as widely used as they might be.

14.3.1 Generalized method of moments

The starting point for GMM is the moment condition:

E[h(wi,θ)]= 0 , (14.1)

whereh(·)is anr×1 vector.
The analogy principle, emphasized by Manski (1988) who attributes it to
Goldberger, proposes estimation using the sample analog of the population con-
dition (14.1). In the just-identified case this leads to a method of moments

(MM) estimator̂θMMthat solvesN−^1

∑

h(wi,θ)= 0. A simple example is that