A Note on Estimation in Seemingly Unrelated Semi-Parametric Regression Models

In this paper a system of two seemingly unrelated semi-parametric regression models is considered, in which, following the partial residual procedure, we first show that the weighted least squares estimator (WLSE) of the regression parameter from the system can be expressed as a matrix series. Then this estimator is shown to be the limit of the covariance-adjusted estimator sequence of the regression parameter. Furthermore, based on the matrix series, we prove that the WLSE actually has only one unique simpler form, which exactly equals to the one-step covariance-adjusted estimator of the regression parameter. We also show that when the variance-covariance matrix of disturbances is unknown, the corresponding two-stage WLSE too has exactly one simpler form, and for any finite k ≥ 2, the k-step covariance-adjusted estimator degenerates to the one-step covariance-adjusted estimator. Finally, we generalize our above conclusions to the system of m(m ≥ 3) seemingly unrelated semi-parametric regressions and point out that the conclusions presented in this paper include the system of m(m ≥ 2) seemingly unrelated linear regressions as its special case.