Efficient Estimation of the Reduced Form from Incomplete Econometric Models

Abstract

For certain applications of estimated econometric models, such as forecasting, interest centres on the reduced form of the model concerned rather than on the structural form.2 In such cases it appears desirable to estimate the reduced form by the most efficient means available. A fairly obvious approach is to estimate the reduced form directly by least-squares. This method is computationally easy, but makes minimal use of information provided by the structural specification, and so may suffer from a substantial loss of efficiency. Another approach is to algebraically derive the reduced form from the consistently estimated structural equations. This is a method which has been considered in detail in an article by Goldberger, Nagar and Odeh [2], and makes use of the full structural specification. But, ipso facto, it requires estimation of the complete structural model, and this is something which an econometric researcher may not always be willing or able to achieve. Reasons why only some of the equations of a structural model may be estimated include limitations of data, computational difficulties, doubts regarding the correct specification, inadequacies of theory and the existence of certain nonlinearities. However, even an incomplete structural specification in general implies certain restrictions on the reduced form of the model,3 and an efficient method of reduced form estimation should take any such restrictions into account. A suitable method of efficiently utilizing restrictions implied by some equations in the estimation of other equations is the three-stage least-squares of Zellner and Theil [6].4 In the present context, use of this method requires writing the structural equations of interest together with any number of reduced form equations and estimating all equations jointly.5 This procedure is shown to produce exactly the same structural estimates as if the reduced form equations were ignored, but a new reduced form estimator results which is in general asymptotically more efficient than the direct least-squares estimator. When all structural equations are used this procedure gives identical results to reduced form estimates derived from the three-stage least-squares estimated structural model. When no structural equations are used (alternatively when the structural equations used are just-identified) the approach coincides with direct least-squares.