A METHOD OF UNDETERMINED COEFFICIENT IN ESTIMATING TWO SEEMINGLY UNRELATED REGRESSION EQUATIONS

For the system of two seemingly unrelated regression equations: Y_i=X_iβ_i+δ_i（i=1,2）, a new method of estimating β_i’s is introduced in this paper. Theestimator of β₁ is given as\beta_1^*(K)=\left(X_1^\prime X_1\right)^-1 X_1^\prime Y_1-\frac\sigma_12\sigma_22\left(X_1^\prime X_1\right)^-1 X_1 N_2 Y_2-K \frac\sigma_12^2\sigma_11 \sigma_22\times\left(X_1^\prime X_1\right)^-1 X_1^\prime P_2 Y_1, where K is an arbitrary constant. The unrestricted two-step estimator, which is the feasible counterpart to β₁^*（K）, is denoted asβ₁^*（K,T）. In particular, β₁^*（1）=\widetilde\beta_1, the covariance improved estimator introduced in 1, and β₁^*（1）=\widetilde\beta_1, a biased estimator introduced in 2. It is shown that choosing a reasonable K, the estimator β₁^*（K） may work better than\widetilde\beta_1, and β₁^*（K,T） may perform better than \widetilde\beta_1(T), with respect to the mean square error matrix （MSEM） criterion. How to choose the optimal value of K is also discussed.