Bounding Parameter Estimates with Nonclassical Measurement Error

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Abstract
The bias introduced by errors in the measurement of independent variables has increasingly been a topic of interest among researchers estimating economic parameters. However, studies typically use the assumption of classical measurement error; that is, the variable of interest and its measurement error are uncorrelated, and the expected value of the mismeasured variable is equal to the expected value of the true measure. These assumptions often arise from convenience rather than conviction. When a variable is bounded, it is likely that the measurement error and the true value of the variable are negatively correlated. We consider the case of a noisily measured variable with a negative covariance between the measurement error and the true value of the variable. We show that, asymptotically, the parameter in a univariate regression is bounded between the ordinary least squares (OLS) estimator and an instrumental variables (IV) estimator. Further, we demonstrate that the OLS bound can be improved in the case where there are two noisy reports on the variable of interest. In the case of continuous variables, this lower-bound estimate is a consistent estimate of the parameter. In the case of binary or discrete noisily measured variables, we also identify point estimates using a method-of-moments framework. We then extend our bounding results to simple multivariate models with measurement error. We provide empirical applications of our analytical results using employer and employee reports on health insurance coverage and wage growth, and reports of identical twins on the level of schooling and wages. Using OLS, health insurance coverage is associated with a reduction in wage growth of 6.5–7.4%, whereas IV estimates suggest a 11.2–11.8% reduction associated with health insurance coverage. We are able to improve the lower bound estimate to 8.2% using our bounding strategy and obtain a point estimate of 8.8% using the method-of-moments framework. The estimates using the data for identical twins, though not correcting for problems such as endogenous determination of the level of schooling, do illustrate the potential usefulness of correcting for measurement error as a complement to other approaches. Using the multiple reports on the level of schooling and the our proposed estimators, we are able to tighten the spread between the upper- and lower-bound estimates of the returns to schooling from 7–10 percentage points to approximately 4 percentage points.

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