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Empirical researchers are increasingly faced with rich data sets containingmany controls or instrumental variables, making it essential to choose anappropriate approach to variable selection. In this paper, we provide resultsfor valid inference after post- or orthogonal $L_2$-Boosting is used forvariable selection. We consider treatment effects after selecting among manycontrol variables and instrumental variable models with potentially manyinstruments. To achieve this, we establish new results for the rate ofconvergence of iterated post-$L_2$-Boosting and orthogonal $L_2$-Boosting in ahigh-dimensional setting similar to Lasso, i.e., under approximate sparsitywithout assuming the beta-min condition. These results are extended to the 2SLSframework and valid inference is provided for treatment effect analysis. Wegive extensive simulation results for the proposed methods and compare themwith Lasso. In an empirical application, we construct efficient IVs with ourproposed methods to estimate the effect of pre-merger overlap of bank branchnetworks in the US on the post-merger stock returns of the acquirer bank.

Estimation and Inference of Treatment Effects with $L_2$-Boosting in High-Dimensional Settings