Quantile regression, as introduced in Koenker and Bassett (1978), is gradually evolving into a comprehensive approach to the analysis of statistical models for univariate response. By supplementing the exclusive focus of least-squares-based methods on the estimation of conditional mean functions with a general approach to estimating conditional quantile functions, it expands the scope of both parametric and non-parametric statistical methods.
Broader Impact of the Proposed Research: Since Quetelets work in the 19th century social science has iconified the average man, that hypothetical man without qualities who is comfortable with his head in the oven, and his feet in a bucket of ice. Conventional statistical methods, since Gauss, seek to estimate the effects of policy treatments on this 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 in schools may improve academic performance for good students, but not help weaker ones. Quantile regression methods helps to explore these heterogeneous effects
Intellectual Merit of the Proposed Research: Koenker (2005a) offers a comprehensive view of the current state of play, but inevitably any such monograph constitutes no more than an interim progress report. This proposal sketches several prospective research topics that the PI would like to explore over the next three years Among these include: 1. Penalty methods play an increasingly important role in non-parametric statistics. With the notable exception of some recent work in image processing, virtually all of this work employs L2 roughness penalties and related Hilbertian machinery. But a compelling case can be made for complementary methods based on total variation roughness penalties for functions in spaces of bounded variation. Such !1 penalties are better suited to estimating functions with sharply defined features. In addition to continuing work on nonparametric regression, the PI anticipates a major effort directed toward density estimation. 2. Panel data methods in econometrics are still predominantly the provence of Gaussian random effects models, however there is often a strong motivation in applications for also estimating conditional quantile models. Growth curve data is a leading example 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 a penalty approach to quantile regression estimation for longitudinal data. 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. With Zhijie Xiao, the PI has begun to investigate more general specifications, focusing initially on a class of models that exhibit some features of persistent random-walk behavior, while also exhibiting a sporadic form of mean reversion. The PI believes that these models offer considerable potential for broadening the scope of applied time series analysis. 4. Estimation of mean effects in linear structural models with endogonous covariates lies at the foundation of econometrics as a discipline. Significant progress has been made in recent years to expand the scope of such models to incorporate heterogeneous treatment effects. Building on work of Chesher (2003), Koenker and Ma (2005) compare two approaches to quantile regression for recursive structural models; expanding this inquiry from both theoretical and empirical perspectives would be highly desirable.
