Moment conditions for the quadratic regression model with measurement error

We consider a new estimator for the quadratic errors-in-variables model that exploits higher-order moment conditions under the assumption that the distribution of the measurement error is symmetric and free of excess kurtosis. Our approach contributes to the literature by not requiring any side information and by straightforwardly allowing for one or more error-free control variables. We propose a Wald-type statistical test, based on an auxiliary method-of-moments estimator, to verify a necessary condition for our estimator's consistency. We derive the asymptotic properties of the estimator and the statistical test and illustrate their finite-sample properties by means of a simulation study and an empirical application to existing data from the literature. Our simulations show that the method-of-moments estimator performs well in terms of bias and variance and even exhibits a certain degree of robustness to the distributional assumptions about the measurement error. In the simulation experiments where such robustness is not present, our statistical test already has high power for relatively small samples.