A Note on the Relationship of Minimum Expected Loss (MELO) and Other Structural Coefficient Estimates - Zellner, Arnold (The Review of Economics and Statistics, 1980)
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).