A Note on the Relationship of Minimum Expected Loss (MELO) and Other Structural Coefficient Estimates
证明在线性联立方程模型中，最小期望损失（MELO）估计量可表示为直接最小二乘（DLS）和两阶段最小二乘（2SLS）估计量的矩阵加权平均，并讨论了该加权平均的性质。
In previous work (Zellner, 1978) structural coefficient estimates that minimize the posterior expectation of a generalized quadratic loss function were derived. On computing these minimum expected loss (MELO) estimates for coefficients of Klein's Model I and the Girschick-Haavelmo supply and demand model for food, it was found that the MELO coefficient estimates have values between corresponding direct least squares (DLS) and two stage least squares (2SLS) estimates (see Zellner and Park, 1979). This note explores the relation of MELO, DLS and 2SLS estimates. Below it is shown,that the MELO estimate of a vector of coefficients of endogenous variables in an equation of a linear, interdependent econometric model can be expressed as a matrix weighted average of the DLS and 2SLS estimates. In the scalar case, the MELO estimate can be expressed as a simple weighted average of the DLS and 2SLS estimates. Some properties of these matrix weighted averages are discussed using the results in Chamberlain and Leamer (1974).