Theory for cross-validated PGMM penalty selection

Develop theoretical guarantees for the K-fold cross-validated penalized GMM (PGMM) procedure used to select the penalty multiplier for estimating the Riesz representer, including conditions ensuring reliable penalty selection and preservation of the convergence rate of the PGMM estimator.

Background

The paper proposes a K-fold cross-validation criterion to choose the penalty multiplier in the PGMM estimator of the Riesz representer, using an out-of-sample GMM objective constructed from fold-specific moments and weight matrices.

While empirically effective, this cross-validated penalty selection currently lacks formal guarantees. Establishing such results would clarify when the chosen penalty yields near-optimal performance and maintains the oracle-type convergence rates derived for the PGMM estimator.