Abstract
Timothy Erickson (1999) "Constructing Instruments for
Regressions with Measurement Error When No Additional Variables Are Available:
Comment."
Lewbel (1997) has ingeniously shown that linear instrumental variables estimators for
the errors-in-variables model can be constructed using functions of the dependent
variable, proxy, and perfectly measured regressors as instruments. He proves consistency
for the estimator and then asserts that "standard limiting distribution theory for
TSLS can now be applied." In this note I assume that "standard theory" is
given by White (1982), the source of the standard errors used by Professor Lewbel in his
empirical application. I show that when White's formulas are applied to Lewbel's
instruments, they give an inefficient estimator, an incorrect asymptotic covariance
matrix, and an inconsistent covariance matrix estimator. These results stem from a subtle
violation of the familiar instrumental variable orthogonality condition. Specifically,
only one of Lewbel's instruments can be measured from an arbitrary origin and satisfy the
orthogonality condition; the remaining instruments satisfy orthogonality only if measured
as deviations from their population means. The substitution of sample means therefore
generates a nonstandard asymptotic covariance matrix of the type described by Newey and
McFadden (1994) in their discussion of ``plug-in'' estimators. I apply the theory for such
estimators to Lewbel's instruments to obtain an efficient estimator, the correct
asymptotic covariance matrix, and a consistent covariance matrix estimator.
Last Modified Date: July 19, 2008
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