We consider quadratic regression models where the explanatory variable
is measured with error. The effect of classical measurement error is to
flatten the curvature of the estimated function. The effect on the
observed turning point depends on the location of the true turning point
relative to the population mean of the true predictor. Two methods for
adjusting parameter estimates for the measurement error are compared.
First, two versions of regression calibration estimation are considered.
This approximates the model between the observed variables using the
moments of the true explanatory variable given its surrogate
measurement. For certain models an expanded regression calibration
approximation is exact. The second approach uses moment-based methods
which require no assumptions about the distribution of the covariates
measured with error. The estimates are compared in a simulation study, and
used to examine the sensitivity to measurement error in models relating
income inequality to the level of economic development. The simulations
indicate that the expanded regression calibration estimator dominates
the other estimators when its distributional assumptions are satisfied.
When they fail, a small-sample modification of the method-of-moments
estimator performs best. Both estimators are sensitive to
misspecification of the measurement error model.