Mixed linear regression with equi-cross-correlated errors

Abstract

This paper investigates an one-step procedure for the General Least Squares Estimation with an estimated covariance matrix (GLSE) in the Linear Regression Model with stochastic linear restrictions on the regression coefficients. The covariance matrix of the errors is assumed to be known up to the correlations between the errors of the observations and the errors of the restrictions. It is assumed that all of these correlations are equal. After transforming the model into its canonical form we show, that the GLSE coincides with the Ordinary Least Squares Estimation (OLS) of the complete model, with OLS after deleting one appropriate observation or with the OLS after replacing two appropriate observations by their arithmetic average. Furthermore, we calculate the relative efficiencies of these estimators with respect to the BLUE using the generalized variance criterion. The resulting estimator may be better than the OLS-estimator, what depends essentially on the diagonal elements of the hat matrix. An example demonstrating this fact is given. Additionally, a new test is proposed for the independence of the disturbances.