Two-step GMM estimation of the errors-in-variables model using high-order moments

We consider a multiple mismeasured regressor errors-in-variables model where the measurement and equation errors are independent and have moments of every order but otherwise are arbitrarily distributed. We present parsimonious two-step generalized method of moments (GMM) estimators that exploit overidentifying information contained in the high-order moments of residuals obtained by "partialling out" perfectly measured regressors. Using high-order moments requires that the GMM covariance matrices be adjusted to account for the use of estimated residuals instead of true residuals defined by population projections. This adjustment is also needed to determine the optimal GMM estimator. The estimators perform well in Monte Carlo simulations and in some cases minimize mean absolute error by using moments up to seventh order. We also determine the distributions for functions that depend an both a GMM estimate and a statistic not jointly estimated with the GMM estimate.