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 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. Parameter estimation is then based on 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 required to calculate this estimator. Both the asymptotic and finite sample properties of the estimator are considered. The connection between the phase function approach and method of moments is also discussed. The estimation of standard errors is also considered and a modified bootstrap algorithm is proposed for fast computation. The newly proposed estimator is competitive when compared to generalized method of moments, even while making fewer model assumptions on the measurement error. Finally, the proposed method is applied to a real dataset concerning the measurement of air pollution.