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Constructing Instruments for Regressions With Measurement Error When No Additional Data Are Available: Comment

Timothy Erickson


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.