Instrumental Variables

Working in moment space, we develop instrumental-variables estimation via duality, in a coordinate-free way. In a structural equation, the explanatory variable causes the dependent variable. The unexplained component is the dependent variable minus the explanatory variable. An estimate of the unknown explanatory variable is a choice from the explanatory variable subspace. If one believes that the explanatory variable and the unexplained component are correlated, then estimation poses a problem, because least-squares linear regression of the dependent variable on the explanatory variable subspace will incorrectly estimate the explanatory variable. To overcome this problem, one approach is instrumental-variables estimation. One assumes that, in the population moment space, the subspace of instrumental random variables is orthogonal to the unexplained component. In the sample moment space, the estimation chooses the explanatory variable to achieve this orthogonality as well as possible. We first set up the instrumental-variables framework. The solution to a linear equation determines necessary and sufficient conditions for identification. The analysis revolves around the cross-moment transformation linking the explanatory- and instrumental-variable subspaces. For an identified equation, the explanatory variable is the dependent variable transformed by the generalized inverse of the cross-moment transformation. One also obtains the analog estimate as the solution to minimization problems. A quadratic minimization implements the generalized method of moments approach to estimating the explanatory variable. Solving this problem and its dual finds that the primal solution is the analog estimate. The two-stage least-squares estimate is also the analog estimate.