A. Colin Cameron 767There are several ways to secure identification ofβ. These include instrumental
variables methods (assuming availability of an instrumentzthat is correlated with
x∗but not with the model erroru), use of replicated data or validation sample
data to estimate key sample cross-moments, and use of additional distributional
assumptions, such as symmetry of the error. Bounds onβcan also be obtained
using reverse regression. Wansbeek and Meijer (2000) review many identification
methods. Few studies correct for measurement error, however, in part due to lack
of necessary data or reluctance to make strong assumptions about the nature of the
measurement error.
The preceding methods do not generalize easily and in a systematic way to
nonlinear models. Carroll, Ruppert and Stefanski (1995) summarize the statis-
tics literature and Hausman (2001) considers the econometrics literature. For
measurement error in regressors in nonlinear regression with additive error, an
early reference is Y. Amemiya (1985) and a more recent reference is Schennach
(2004). For nonlinear models with nonadditive errors, such as discrete outcome
and count models, measurement error in the dependent variable can also cause
problems. For example, Hausman, Abrevaya and Scott-Morton (1998) consider
mismeasurement in the dependent variable in binary outcome models, taking a
parametric approach with strong assumptions.
The classical measurement error model maintains that the measurement error
is i.i.d. Some work relaxes this. An early example is that, for a binary regressor,
the measurement error is necessarily correlated with the true value, since the only
way to mismeasure a value of 0 is as a 1, and vice versa. Mahajan (2006) gives a
quite general treatment for binary regressors. Kim and Solon (2005) consider stan-
dard linear panel estimators when measurement error in a regressor is negatively
correlated with the true value.
14.8 Conclusion
Microeconometricians are very ambitious in their desire to obtain marginal effects
that can be given a causative interpretation, permit individual heterogeneity
and are obtained under minimal assumptions. The associated statistical inference
should also rely on minimal assumptions. This has led to a literature and toolkit
that is quite advanced for an area of applied statistics.
This survey has of necessity been selective. The methods used in labor economics
and public economics have been emphasized. General approaches have been pre-
sented, with specialization usually to the linear model. For econometrics methods
for specific types of data – binary, multinomial, durations and counts – good start-
ing points are the specialized monographs by, respectively, Maddala (1983), Train
(2003), Lancaster (1990), and Cameron and Trivedi (1998), as well as the more gen-
eral texts cited in the introduction and Cameron and Trivedi (2008). The chapters
by Greene and Jones in this volume are also highly relevant.