Bootstrap inference for penalized GMM estimators with oracle properties

We study the validity of bootstrap methods in approximating the sampling distribution of penalized GMM estimators with oracle properties. More precisely, we focus on bridge estimators with Lq penalty for 0<q<1, and adaptive lasso estimators. We show that the nonparametric bootstrap with recentered moment conditions provides a valid method for approximating the distribution of these estimators. Furthermore, using the bootstrap approach, we also propose a data-driven method for the selection of tuning parameters in the penalization terms. Monte Carlo simulations confirm the reliability and accuracy of the bootstrap procedure. The empirical coverages for the active variables implied by the nonparametric bootstrap are always very close to the nominal coverage probabilities.