Identification, Estimation and Inference Based on Constructed Instruments

Abstract

We propose a structural projection (SP) method for identifying and estimating structural parameters in regression models associated with endogenous regressors. The method projects the structural error onto the sigma-field generated by the regressors, decomposing it into a conditional mean component and an orthogonal residual. This decomposition yields an internally constructed instrument that is valid whenever the conditional mean lies in a known subspace, avoiding the need for external instruments. We establish identification conditions, derive consistency and asymptotic normality for the resulting estimator, and propose a LASSO-based selection procedure for choosing the subspace when no natural candidate is available. The SP method applies to a broad class of linear, nonlinear, and non-separable models. Simulations show that SP substantially outperforms least-squares-based benchmarks under endogeneity while remaining competitive when it is absent, and we illustrate the method using data on the returns to schooling.