Heteroscedastic errors in a linear functional relationship
暂无分享,去创建一个
SUMMARY We consider the estimation of structural parameters in a multivariate linear func- tional relationship when the error variances and covariances are not necessarily homogeneous. Estimators are obtained as roots of a system of nonlinear equations and an iterative algorithm is briefly described for finding numerical solutions of the system. Asymptotic properties of the estimators are studied and in particular an estimate of the asymptotic covariance matrix of the estimators is derived. The estimation of linear functional relationships has been extensively investigated in the literature. Yet many of the models studied assumed that the errors are indepen- dently and identically distributed. Barnett (1970) discussed the application of functional relationship models to a medical example in which the assumption of homogeneous variances is unrealistic. In the present paper the following model will be considered. Suppose that two unobservable nonstochastic variables m and i of dimensions re- spectively p and q are connected by an underlying linear relationship m = a + B. The parameters a and B are unknown and to be estimated based on n independent pairs of observations (xi, yr) for i = 1, ..., n, with Xi = ~i31, yi=q+e +~+i where (5k, si) has zero mean and covariance matrix f2i. Here the f2i are assumed to be either all known or they are given functions of the same unknown parameter In ? 2, it is noted that different approaches, including Morton's (1981) generalized likelihood procedure, lead to the same system of equations for estimating the structural parameters a, B and 0. Consistency and asymptotic normality of the estimators which are roots of the derived equations are established under certain conditions and an estimate for the asymptotic covariance matrix of the estimators is also obtained in ? 3. These conditions are not satisfied when the corresponding structural relationship model is nonidentifiable. An algorithm is also proposed for solving the system of equations numerically.
[1] N. N. Chan,et al. Estimation of multivariate linear functional relationships , 1983 .
[2] On Sprent's Generalized Least-squares Estimator , 1983 .
[3] Robert V. Foutz,et al. On the Unique Consistent Solution to the Likelihood Equations , 1977 .
[4] V. D. Barnett,et al. Fitting Straight Lines—The Linear Functional Relationship with Replicated Observations , 1970 .
[5] R. Morton,et al. Efficiency of estimating equations and the use of pivots , 1981 .