Linear Regression with a Large Number of Weak Instruments using a Post-l1-Penalized Estimator

This paper proposes a new two stage least squares (2SLS) estimator which is consistent and asymptotically normal in the presence of many weak instruments and heteroskedasticity. The first stage consists of two components: first, an adaptive absolute shrinkage and selection operator (LASSO) that selects the instruments and second, an OLS regression with the selected regressors. This procedure is a post-l1-penalized estimator as proposed by Belloni and Chernozhukov (2010). The second stage uses an OLS regression with the fitted values from the post-l1-penalized regression of the first stage. The methodology exploits the model selection benefits of the adaptive LASSO and reduces the post-IV selection bias. More importantly, it provides a consistent and asymptotically normal estimator in a 2SLS framework in the presence of many weak instruments and heteroskedasticity, which is infeasible for the conventional 2SLS in this context. These results are driven by the fact that after the instrument selection stage the growth rate of the concentration parameter is higher than the growth rate of the number of instruments.