Estimation in linear errors-in-variables models with unknown error distribution

Parameter estimation in linear errors-in-variables models typically requires that the measurement error distribution be known or estimable from replicate data. A generalized method of moments approach can be used to estimate model parameters in the absence of knowledge of the error distributions, but it requires the existence of a large number of model moments. In this paper, parameter estimation based on the phase function, a normalized version of the characteristic function, is considered. This approach requires the model covariates to have asymmetric distributions, while the error distributions are symmetric. Parameters are estimated by minimizing a distance function between the empirical phase functions of the noisy covariates and the outcome variable. No knowledge of the measurement error distribution is needed to calculate this estimator. Both asymptotic and finite-sample properties of the estimator are studied. The connection between the phase function approach and method of moments is also discussed. The estimation of standard errors is considered and a modified bootstrap algorithm for fast computation is proposed. The newly proposed estimator is competitive with the generalized method of moments, despite making fewer model assumptions about the moment structure of the measurement error. Finally, the proposed method is applied to a real dataset containing measurements of air pollution levels.