A note on the double k-class estimator in simultaneous equations

Dwivedi and Srivastava (DS) (J. Econometrics 25 (1984) 263) studied the exact finite sample properties of Nagar's (Internat. Econom. Review 3 (1962) 168) double k-class estimator as continuous functions of its two characterizing scalars k₁ and k₂, and provided guidelines for their choice in empirical work. In this note we show that the empirical guidelines provided by DS are not entirely valid since they did not explore the complete range of the relevant parameter space in their numerical evaluations. We find that the optimal values of k₂ leading to unbiased and mean squared error (MSE) minimizing double k-class estimators are not symmetric with respect to the sign of the product ρω₁₂, where ρ is the correlation coefficient between the structural and reduced form errors, and ω₁₂ is the covariance between the unrestricted reduced form errors. Specifically, when ρω₁₂ is positive, the optimal value of k₂ is generally positive and greater than k₁, which partly explains the superior performance of Zellner's (J. Econometrics 83 (1998) 185) Bayesian Method of Moments (BMOM) and Extended MELO estimators reported in Tsurumi (in: Geisser, Hodges, Press, Zellner (Eds.), Bayesian and Likelihood Methods in Statistics and Econometrics, North-Holland, Amsterdam, 1990).