The iterative instrumental variables method and the full information maximum likelihood method for estimating interdependent systems

The "iterative instrumental variables" (IIV) method for estimating interdependent systems, originally referred to as a symmetric counterpart to the "fix-point" (FP) method, shares its symmetry properties with Durbin's iterative method for performing the "full information maximum likelihood" (FIML) estimation. Classical interdependent systems are considered and identities may occur among the structural equations. Alternative symmetric procedures for obtaining FIML estimates are also dealt with, including the sequential maximization of the likelihood function with respect to the coefficients of one structural equation at a time. Two recent estimation methods developed by Brundy and Jorgenson (1971, Review of Economics and Statistics 53, 207-224) as well as Dhrymes (1971, Austral. J. Statist. 13, 168-175) can be considered the second approximation of the IIV method and Durbin's method respectively with the first approximation obtained by the "ordinary instrumental variables" (OIV) method. In practice the second approximation depends heavily on the choice of initial instrumental variables, although the asymptotic distribution is not changed by the continued iteration.