This project considers new methods for (i) analyzing the bootstrap in a general class of estimators and (ii) estimating semiparametric models without smoothing-parameter choices. The importance of using the bootstrap for semiparametric estimators arises since the usual methods for asymptotic inference (e.g., consistent estimation of asymptotic standard errors) are often very difficult and sometimes infeasible to implement for these estimators. In ongoing work, a new framework for analyzing the consistency of the bootstrap within the class of M-estimators is being developed. Using this framework, the bootstrap has been found to be inconsistent for a class of "cube-root estimators" (which includes the maximum score estimator), thus solving a longstanding problem in the econometrics literature. For single-parameter estimators within this class, the results suggest an indirect method of asymptotic inference using the bootstrap (even though the bootstrap itself is inconsistent). The theoretical framework will be extended to the class of (non-smooth) U-estimators. Many semiparametric estimators in the economics literature fall within this class, and as of yet, there has been no proof that the bootstrap is applicable for these estimators. The project also considers estimation of flexible conditional quantile models. The asymptotic distribution for the isotonic quantile regression estimator (an estimator introduced in 1976) will be derived. A new bootstrap-based method for smoothing isotonic regression estimators is proposed. In addition, a new estimator for a semiparametric linear-index conditional quantile is proposed. This estimator combines isotonic quantile regression with traditional least-absolutedeviations (LAD) techniques. This estimator avoids smoothing-parameter choice and satisfies the underlying monotonicity condition of the model. The application of the bootstrap concepts in this project (and other subsampling ideas from the statistics literature) to this and related estimators is considered.
Intellectual merit of the proposed activity: This theoretical framework for the bootstrap differs markedly from previous approaches for analyzing consistency of the bootstrap. By utilizing empirical process theory, the approach is able to deal with estimators for which no bootstrap theory has been available. The proposed semiparametric quantile estimator (and associated subsampling strategy) falls within an appealing class of estimators for which smoothing-parameter choice is not required.
Broader impacts resulting from the proposed activity: Since this framework for analyzing the bootstrap differs from previous approaches, it will likely result in applications to other bootstrap problems beyond those considered in this project. The framework may also be useful as a pedagogical tool in teaching doctoral students about the bootstrap. The proposed estimator and associated resampling techniques should be of interest to those researchers employing quantile regression methods. This research is particularly timely given the many recent applications of quantile regression. Finally, all computer programs associated with this project will be made publicly available to researchers upon completion.
