Three Essay in Econometrics

Abstract

The development of statistical tools benefit economic modeling by relaxing the assumptions for identification. For example, kernel or SEIVE methods allow us to remove parametric assumptions on the functional form. Penalized estimators allow us to run a regression when the dimension of the data exceeds the number of observations. This dissertation presents new economic models and new estimators based on LASSO-type estimators and kernel estimators. I study their statistical properties and show how these new estimators can be applied to study specific economic problems. The first chapter, Heterogeneous Endogenous Effects in Networks, proposes a new method to identify leaders and followers in a network. Current literature models peer effects using spatial autoregression models (SARs). SARs implicitly assume that each individual in the network has the same endogenous effects on others and conclude the key player in the network to be the one with the highest centrality. However, when some individuals are more influential than others, centrality may fail to be a good measure. I develop a SAR model that allows for individual-specific endogenous effects and propose a two-stage LASSO (2SLSS) procedure to identify influential individuals in a network. My method allows me to identify leaders and followers through their observed behaviors on the network. Under an assumption of sparsity: only a subset of individuals (which can increase with sample size $n$) is influential, I show that my 2SLSS estimator is consistent and achieves asymptotic normality. I develop robust inference including uniformly valid confidence intervals. These results also carry through to scenarios where the influential individuals are not sparse. I extend the analysis to allow for multiple types of connections (multiple networks), and I show how to use the square-root sparse group LASSO to detect which of the multiple connection types is more influential. Simulation evidence shows that my estimator has good finite sample performance. I further applied my method to the data in Banerjee et al. (2013) and my proposed procedure is able to identify leaders and effective networks.\\ The second chapter, On Testing Continuity and the Detection of Failures, propose a new estimator to detect discontinuities in the functional form. This is coauthored work with Professor Matthew Backus. Estimation of discontinuities is pervasive in applied economics: from the study of sheepskin effects to prospect theory and the bunching" of reported income on tax returns. This salience of discontinuities makes the models that generate them empirically testable. However, detection and identification of those discontinuities typically relies on knowledge of their number, their type, their location, or their underlying functional form. We develop a nonparametric approach to the study of arbitrary discontinuities --point discontinuities as well as jump discontinuities in the nth derivative, where n = 0,1… that does not require ex ante knowledge of their number or location. Our approach exploits the recent development of false discovery rate control methods for LASSO regression as proposed by G'Sell et al. (2015). This framework affords us the ability to construct valid tests for both the null of continuity as well as the significance of any particular discontinuity without the computation of nonstandard distributions. We illustrate the method with a series of Monte Carlo examples and by replicating prior work, e.g. classical regression discontinuity election study Lee (2008), Card et al. (2008) and Backus et al. (2015). The third chapter, Local Regression Smoothers with Set Valued Outcome Data, provides statistical results on local linear regression smoothing when the outcome data is set valued and the regressors are exactly measured. This is coauthored work with Qiyu Li, Professor Ilya Molchanov and Professor Francesca Molinari. We derive the asymptotic properties of our estimator, propose a bias correction method, and adapt results from Beresteanu and Molinari (2008) to obtain point-wise confidence bands that asymptotically cover the functional of interest with probability 1-alpha. We demonstrate the usefulness of our approach using a novel dataset that follows 132 patients during anti-cancer treatment.