Chapter 8
Stochastic regressors and measurement errors
Overview
Until this point it has been assumed that the only random element in a regression model is the disturbance term.
This chapter extends the analysis to the case where the variables themselves have random components.
The
initial analysis shows that in general OLS estimators retain their desirable properties.
A random component
attributable to measurement error, the subject of the rest of the chapter, is however another matter.
While
measurement error in the dependent variable merely inflates the variances of the regression coefficients,
measurement error in the explanatory variables causes OLS estimates of the coefficients to be biased and
invalidates standard errors,
t
tests, and
F
tests.
The analysis is illustrated with reference to the Friedman
permanent income hypothesis, the most celebrated application of measurement error analysis in the economic
literature.
The chapter then introduces instrumental variables (IV) estimation and gives an example of its use to
fit the Friedman model. The chapter concludes with a description of the Durbin–Wu–Hausman test for
investigating whether measurement errors are serious enough to warrant using IV instead of OLS.
Learning outcomes
After working through the corresponding chapter in the text, studying the corresponding slideshows, and doing
the starred exercises in the text and the additional exercises in this guide, you should be able to:
•
•
•
•
•
•
•
•
IV
explain the conditions under which OLS estimators remain unbiased when the variables in the regression
model possess random components
derive the large-sample expression for the bias in the slope coefficient in a simple regression model with
measurement error in the explanatory variable
demonstrate, within the context of the same model, that measurement error in the dependent variable does not
cause the regression coefficients to be biased but does increase their standard errors
describe the Friedman permanent income hypothesis and explain why OLS estimates of a conventional
consumption function will be biased if it is correct
explain what is meant by an instrumental variables estimator and state the conditions required for its use
demonstrate that the IV estimator of the slope coefficient in a simple regression model is consistent, provided
that the conditions required for its use are satisfied
explain the factors responsible for the population variance of the IV estimator of the slope coefficient in a
simple regression model
perform the Durbin–Wu–Hausman test in the context of measurement error.
Additional exercises
A8.1
A researcher believes that a variable
Y
is determined by the simple regression model
Y
=
β
1
+
β
2
X
+
u
She thinks that
X
is not distributed independently of
u
but thinks that another variable,
Z
, would be a
suitable instrument.
The instrumental estimator of the intercept,
, is given by
1
b
,
IV
2
IV
1
X
b
Y
b
−
=
October 2007

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