Instrumental Variable Estimation and Selection with Many Weak and Irrelevant Instruments

This paper proposes a new two stage least squares (2SLS) estimator which is consistent and asymptotically normal in the presence of many weak and irrelevant instruments and heteroskedasticity. In the first stage the estimator uses an adaptive absolute shrinkage and selection operator (LASSO) that selects the relevant instruments with high probability. However, the adaptive LASSO estimates have a considerable bias. To address this concern and exploit the instrument selection properties of the adaptive LASSO, an OLS regression with the selected instruments is performed in the first stage. The first stage can be constructed using the assumption of mean independence between the instruments and the structural random disturbance. This allows for the possibility of a nonparametric adaptive LASSO in the first stage. The second stage uses an OLS regression with the fitted values of the first stage. Simulation results suggest that the proposed methodology has better performance than the existing estimators when the levels of endogeneity and instrument weakness are high.