**Publications**

Smoothed GMM in Quantile Models,

**, 2019 (with Luciano de Castro, Antonio F. Galvao, and David Kaplan)**

*Journal of Econometrics*

**Working papers**

I propose a quantile-based random coefficient panel data framework to study heterogeneous causal effects. The heterogeneity depends on unobservables, as opposed to heterogeneity for which we can add interaction terms. This connects it to other structural quantile models. My approach uses panel data to address ``endogeneity,'' meaning dependence between the explanatory variables and the random coefficients. The random coefficient vector depends on an unobserved, scalar, time-invariant ``rank'' variable, in which outcomes are monotonic at a particular point. I develop the theory first in a simplified model and then extend results to a more general model. First, I establish identification and uniformly consistent estimation. Second, I use a Dirichlet approach to establish small-n confidence sets or uniform confidence bands of the slope function. Third, I establish asymptotic normality of the slope estimator in the simplified model by applying the functional delta method to the empirical process. This facilitates a bootstrap confidence interval for the slope estimator at each specific rank. Finally, I illustrate the proposed methods by examining the causal effect of a country's oil wealth on its political violence and military defense spending.

This paper proposes averaging estimation methods to improve the finite-sample efficiency of the instrumental variables quantile regression (IVQR) estimator. First, I apply Cheng, Liao, and Shi's (2019) averaging GMM framework to the IVQR model. I propose using the usual quantile regression moments for averaging to take advantage of cases when endogeneity is not too strong. I also propose using two-stage least squares slope moments to take advantage of cases when heterogeneity is not too strong. The empirical optimal weight formula of Cheng et al. (2019) helps optimize the bias-variance tradeoff, ensuring uniformly better (asymptotic) risk of the averaging estimator over the standard IVQR estimator under certain conditions. My implementation involves many computational considerations and builds on recent developments in the quantile literature. Second, I propose a bootstrap method that directly averages among IVQR, quantile regression, and two-stage least squares estimators. More specifically, I find the optimal weights in the bootstrap world and then apply the bootstrap-optimal weights to the original sample. The bootstrap method is simpler to compute and generally performs better in simulations, but it lacks the formal uniform dominance results of Cheng et al. (2019). Simulation results demonstrate that in the multiple-regressors/instruments case, both the GMM averaging and bootstrap estimators have uniformly smaller risk than the IVQR estimator across data-generating processes (DGPs) with all kinds of combinations of different endogeneity levels and heterogeneity levels. In DGPs with a single endogenous regressor and instrument, where averaging estimation is known to have the least opportunity for improvement, the proposed averaging estimators outperform the IVQR estimator in some cases but not others.

**Work in progress**

Smoothed GMM Inference for Quantile Models

The k-Class Estimator for Instrumental Variables Quantile Regression

Partial Identification in Panel Quantile Models with Time-Invariant Rank, (with David Kaplan)

IVQR Bootstrap Averaging, (Stata command/article)