On Nonlinear Measurement Error Errors-in-Variables Models

Abstract

A nonlinear structural errors-in-variables model with normal underlying distributions is investigated. The response variable has a density belonging to an exponential family. An asymptotic covariance matrix of the quasi-likelihood estimator [1] is computed, including the corrected terms which appear because in the score function the sample mean and the sample variance are plugged in. The convergence of the corresponding iteratively reweighted least squares procedure is shown. The corrected maximum likelihood estimator [2] and the naive maximum likelihood estimator are also considered. While the naive estimator is biased the other two estimators are consistent. The asymptotic covariance matrices of the three estimators are compared in border cases of small errors. A polynomial model and a Poisson model are explored in more detail as particular cases. The results concerning the polynomial model are contained in [3, 4]. References: 1.Carrol, R.J., Ruppert, D. and Stefanski, L.A. (1995). Measurement Error in Nonlinear Models. Chapman and Hall, London. 2.Thamerus, M. (1998). Different nonlinear regression models with incorrectly observed covariates. In R. Galata and H. Kuchenhoff (Eds.): Econometrics in Theory and Practice, Festschrift for Hans Schneeweiss. Physica. Heidelberg, New York. 3.Kukush, A. and Schneeweiss, H. (2000). A comparison of asymptotic covariance matrices of adjusted least squares and structural least squares in error ridden polynomial regression. Discussion Paper 218, Sonderforschungsbereich 386, University of Munich. 4.Kukush, A., Schneeweiss, H., and Wolf, R. (2001). Comparison of three estimators in a polynomial regression with measurement errors. Discussion Paper , Sonderforschungsbereich 386, University of Munich. (To appear).