Estimated generalized least squares estimation for the heterogeneous measurement error model

Abstract

<p>The measurement error model of interest is (UNFORMATTED TABLE OR EQUATION FOLLOWS)\eqalign y[subscript]t &= [beta][subscript]0 + x[subscript]t[beta][subscript]1 + q[subscript]t, Y[subscript]t &= y[subscript]t + w[subscript]t, X[subscript]t &= x[subscript]t + u[subscript]t, (TABLE/EQUATION ENDS)where Z[subscript]t = ( Y[subscript]t, X[subscript]t) is the observed p-dimensional vector, z[subscript]t = (y[subscript]t, x[subscript]t) is the true unknown random vector, q[subscript]t is the equation error, and the measurement errors a[subscript]t = (w[subscript]t, u[subscript]t) are distributed with mean zero and known variance [sigma][subscript]aatt. An estimated generalized least squares estimator of the mean and variance of z[subscript]t, denoted by [mu] and [sigma][subscript]zz respectively, is shown to have a limiting normal distribution under mild regularity conditions. An estimator of [beta][superscript]' = ([beta][subscript]0, [beta][subscript]sp1') based upon the proposed estimator of [mu] and [sigma][subscript]zz is constructed and shown to have a limiting normal distribution. The variances of the limiting distribution are less than or equal to the corresponding variances for other estimators that have been suggested for the heterogeneous error model. The estimated generalized least squares estimator also displayed smaller mean square error than other estimators in a Monte Carlo study. A program to implement the proposed estimators is developed. The iterated estimated generalized least squares estimators of the measurement error model are investigated. The limit of a modified iteration procedure is shown to be the maximum likelihood estimator for the normal distribution;Estimated generalized least squares estimation is considered for the general linear model, Y = X[beta] + u, where the variance of u is denoted by V[subscript]uu and the elements of X[superscript]' V[subscript]spuu-1 X may increase at different rates. Sufficient conditions are given for the estimated generalized least squares estimator to be consistent and asymptotically equivalent to the generalized least squares estimator constructed with known V[subscript]uu. Consistent estimators of the normalizing matrix are developed, and the asymptotic distribution of a linear combination of the elements of the estimator is considered. The model where V[subscript]uu is a function of a fixed, finite number of parameters and the use of ordinary least squares residuals to estimate the parameters of V[subscript]uu are examined. Applications of the results to the trend model with first order autoregressive errors and to the measurement error model are given.</p>