Quantile Regression and L1 Regularization

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Quantile regression is gradually evolving into a comprehensive strategy for the analysis of statistical models with univariate response, complementing the exclusive focus of least-squares-based methods with a general approach to estimating conditional quantile functions. This project would continue the development of quantile regression methods, and intensify research on penalty methods for density estimation. In addition to continued work on survival/duration models and models with endogonous covariates, this research includes: 1. Penalty methods play an increasingly important role throughout statistics. Penalties are increasingly recognized as a critical tool of modern data analysis. The investigator and Ivan Mizero have been exploring such methods for both quantile regression and density estimation. For densities, total variation roughness penalties and related shape-constrained estimators seem especially promising. A major thrust of this proposal is to broaden the scope of this inquiry, producing improved algorithms, inferential capabilities, better understanding of asymptotic behavior and extending the flexibility of the available penalties. 2. Panel data methods in econometrics are still predominately the province of Gaussian random effects models, however there is often a strong motivation in applications for also estimating conditional quantile models. Growth curve data offers leading examples in biostatistics, and program evaluation offers numerous examples in econometrics. Expanding upon the close relationship between random effects estimation and penalty methods in Gaussian settings, research is proposed on penalty approaches to quantile regression estimation for longitudinal data, particularly focused on dynamic panel models. 3. Although there is already quite an extensive literature on quantile regression methods for time-series, most of the attention has focused on models in which lagged response exert a pure location shift effect on the distribution of the current response. The investigator and Zhijie Xiao have been investigating more general specifications, focusing initially on a class of models that exhibit some features of persistent random-walk behavior, while also exhibiting an episodic form of mean reversion. These models offer considerable potential for broadening the scope of applied time series analysis. Broader Impact: Conventional statistical methods, since Quetelet, have sought to estimate the effects of policy treatments on the average man. But such effects are often quite diverse: medical treatments may improve life expectancy, but also impose serious short term risks; reducing class sizes may improve performance of good students, but not help weaker ones. Quantile regression methods help to explore these heterogeneous effects, as do improved methods for density estimation. These methods have broad applicability throughout science.