Efficiency properties of weighted mixed regression estimation

This paper considers the estimation of the coe cient vector in a linear regression model subject to a set of stochastic linear restrictions binding the regression coe cients and presents the method of weighted mixed regression estimation which permits to assign possibly unequal weights to the prior information in relation to the sample information E ciency properties of this estimation procedure are analyzed when disturbances are not necessarily normally distributed Introduction When a set of stochastic linear constraints binding the regression coe cients in a linear regression model is available Theil and Goldberger have pro posed the method of mixed regression estimation see Srivastava for an annotated bibliography Their method typically assumes that the prior infor mation in the form of stochastic linear constraints and the sample information in the form of observations on the study variable and explanatory variables are equally important and therefore receive equal weights in the estimation pro cedure In practice situations may occur where this assumption may not be tenable For example one may conduct a statistical test for the compatibility of sample and prior information see Theil for instance If the statis tical test reveals that they are compatible we may combine the two kinds of information assigning equal weights and use the method of mixed estimation accordingly On the other hand if the statistical test is indicative of incom patibility the conventional procedure is to ignore the prior information This strategy of discarding the prior information outrightly is rather unappealing in comparison to the one which assigns unequal weights to the prior information in comparison to the sample information Some extraneous considerations may often be suggestive of giving unequal weights In such circumstances it may be imperative to assign not necessarily equal weights during the process of com bining the prior and sample information Appreciating this viewpoint Scha rin and Toutenburg have developed the method of weighted mixed regression Institut f ur Statistik Universit at M unchen Akademiestr M unchen Germany Department of Statistics Lucknow University Lucknow India Department of Civil and Environmental Engineering and Geodetic Science Ohio State University Columbus OH U S A estimation Such a method o ers considerable exibility in the sense that one can assign possibly di erent weights to sample information and prior informa tion depending upon the degree of belief Besides this the method provides a kind of uni ed treatment to traditional pure and mixed regression methods The purpose of this article is to analyze the e ciency properties of the weighted mixed regression method Section describes the model and the method of weighted mixed regression estimation proposed by Scha rin and Toutenburg A feasible version of it is developed when the disturbance variance is not known In Section we discuss the e ciency properties when disturbances are small but not necessarily normally distributed The results related to both bias vector and mean squared error matrix are derived in the Appendix Finally some remarks are o ered in Section Model Speci cation and Some Estimators Let us postulate the following linear regression model