The canonical least squares estimation of large-scale simultaneous-equations models

The equation-by-equation ordinary least squares (OLS) estimation of simultaneous-equations models (SEMs) is known to be biased and even inconsistent. In addition, OLS is not normalization invariant. Yet, the OLS method is still widely used, especially for large-scale SEMs with many exogenous variables, because two-stage least squares, limited information maximum likelihood, and other related methods are not applicable. A canonical least squares (CANLS) method is developed here to overcome the normalization bias and to alleviate the simultaneity bias. The CANLS method contrasts one set of endogenous variables with the other set of exogenous variables in a structural equation through canonical correlation analysis. A linear combination of endogenous variables is then determined by the first canonical variable, which is subsequently regressed on the individual exogenous variables of the structural equation. The new method is shown to be more accurate than OLS through a series of Monte Carlo experiments.