A. Colin Cameron 731it is convenient to denote theith observation bywi=(yi,xi)orwi=(yi,xi,zi).
The parameter vector in general is aq×1 vectorθ. In some cases this is specialized
to ak×1 parameter vectorβ. Combining allNobservations,yis theN×1 vector
of dependent variables, andXis theN×Kregressor matrix. The linear regression
model is written asyi=x′iβ+uiory=Xβ+u.
The reader should be aware that this is a methods survey, rather than a litera-
ture survey. It is not possible to cite more than a few relevant references for each
topic, leading to omission of the important contributions of many authors. More
complete references are given in the relevant texts by T. Amemiya (1985), Greene
(2003, first edition 1990), Davidson and MacKinnon (1993), Wooldridge (2008,
first edition 2002) and Cameron and Trivedi (2005), and in the lectures by Imbens
and Wooldridge (2007). The most recent references given in this chapter should
provide a useful start to the current literature.
14.2 Identification
Most applied econometrics studies use methods and models for which identifica-
tion is not an issue, with a notable exception being the need to have sufficient
instruments in linear regression with endogenous regressors. Identification does
come to the forefront when more complex models are estimated, or when models
are incompletely parameterized.
14.2.1 Point identification
Introductory treatments of econometrics focus on specifying a parametric model
for the conditional distributionf(y|x,θ), or for the conditional mean, E[y|x]=
g(x,β). Given a specification off(·)org(·)and a sampling process, such as ran-
dom sampling or an exogenous stratified sampling that provides no additional
complication, the emphasis is on estimation of the parametersθorβ, and on statis-
tical inference based on these parameter estimates. Identification, meaning unique
determination ofθorβ, is discussed briefly in the context of rank conditions to
ensure identification in linear simultaneous equations models.
For nonlinear parametric models, identification can be more challenging. A stan-
dard result is that in, for example, Newey and McFadden (1994, p. 2134), who state
that “the identification condition for consistency of an extremum estimator is that
the limit of the objective function has a unique maximum at the truth.”
For semiparametric modeling, the identification question is whether a model,
or key features of that model, can be estimated assuming an infinitely large sam-
ple is available and given the relevant sampling scheme. Only after identification
is secured can one move on to estimation and inference given a finite sample. An
example is a censored regression model, with observed datayi=yi∗ify∗i=x′iβ+ui≥
0 andyi=0 otherwise. The goal is to (uniquely) identifyβgiven assumptions on
the distribution ofuithat fall short of complete parameterization of the distribu-
tion ofui(such as assuming normality). In general there is no unified theory and
identification conditions vary with the model being considered and the sampling