Contributions to the theory and methodology of quantile models with endogeneity
Authors:   , Liu Xin 
Published: August 2021

In the first chapter, I develop smoothed GMM estimation and inference for overidentified instrumental variables quantile regression (IVQR) and related quantile models. Previous approaches have been limited to exact identification, small models (due to computational burden), and/or iid sampling; we allow overidentification, large models, and non-iid sampling. I establish consistency and asymptotic normality of a GMM estimator with a general weighting matrix. The GMM framework provides the usual advantages: efficient two-step estimation, "distance metric" type inference, and overidentification testing. The new methods are illustrated in an application to consumption Euler equations under quantile utility maximization. Code is provided for all methods and applications. In the second chapter I propose two averaging estimation methods to improve the finite-sample efficiency of the instrumental variables quantile regression (IVQR) estimator. I propose using the usual quantile regression for averaging to take advantage of cases when endogeneity is not too strong. I also propose using two-stage least squares to take advantage of cases when heterogeneity is not too strong. The first averaging method is to apply Cheng, Liao, and Shi's (2019) averaging GMM framework to the IVQR model based on this proposed intuition. My implementation involves many computational considerations and builds on recent developments in the quantile literature. The second averaging method is a new bootstrap model averaging 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, Liao, and Shi (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 with all kinds of combinations of different endogeneity levels and heterogeneity levels. In the third chapter, 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 coefficient function. Third, I establish asymptotic normality of the coefficient estimator in the simplified model by applying the functional delta method to the empirical process. This facilitates a bootstrap confidence interval for the coefficient 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.