Bounding parameter estimates with nonclassical measurement error

The bias introduced by errors in the measurement of independent variables h as increasingly been a topic of interest among researchers estimating econo mic parameters. However, studies typically use the assumption of classical measurement error; that is, the variable of interest and its measurement er ror are uncorrelated, End the expected value of the mismeasured variable is equal to the expected value of the true measure. These assumptions often a rise from convenience rather than conviction. When a variable is bounded, i t is likely that the measurement error and the true value of the variable a re negatively correlated. We consider the case of a noisily measured variab le with a negative covariance between the measurement error and the true va lue of the variable. We show that, asymptotically, the parameter in a univa riate regression is bounded between the ordinary least squares (OLS)estimat or and an instrumental variables (IV) estimator. Further, we demonstrate th at the OLS bound can be improved in the case where there are two noisy repo rts 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 e stimates using a method-of-moments framework. We then extend our bounding r esults to simple multivariate models with measurement error. We provide emp irical applications of our analytical results using employer and employee r eports on health insurance coverage and wage growth, and reports of identic al twins on the level of schooling and wages. Using OLS, hearth insurance c overage 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 insuranc e coverage. We are able to improve the lower bound estimate to 8.2% using o ur bounding strategy and obtain a point estimate of 8.8% using the method-o f-moments framework. The estimates using the data for identical twins, thou gh not correcting for problems such as endogenous determination of the leve l of schooling, do illustrate the potential usefulness of correcting for me asurement error as a complement to other approaches. Using the multiple rep orts 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 percent age points.