This project extends the domain of applicability of quantile regressions to three important and fast growing areas of economic research---survival analysis, models with endogeneity, and portfolio theory. Besides theoretical extensions, the research project will also areas continue working on problems broadly relating to quantile regression and its applications. This research project relies on the penalty method. Quantile regression, as introduced in Koenker and Bassett (1978), is gradually evolving into a comprehensive approach to the statistical analysis of linear and nonlinear response models. It is capable of greatly expanding the flexibility of both parametric and non-parametric statistical methods. Research is proposed on a broad range of problems related to quantile regression. Panel, or longitudinal, data analysis is still predominately the province of Gaussian random effects models, however there is often a strong motivation in applications for estimating conditional quantile models. Exploiting the close relationship between random effects estimation and penalty methods research is proposed on a L1 approach for quantile regression random effects estimation.
Considerable progress has been made on inference in quantile regression over the course of the PI's current (2000-02) grant. The objective of the 1999 proposal was to extend the theoretical developments in Koenker and Machado (1999) for inference based on the quantile regression, extending the results to composite null hypotheses involving unknown nuisance parameters. Koenker and Xiao (2002) describes a general approach to such problems based on an ingenious suggestion by Khmaladze (1981). The classical Doob-Meyer decomposition is employed to transform a parametric version of the quantile regression process, rendering it asymptotically distribution free. In related work on survival analysis, Koenker and Geling (2001) and Koenker and Bilias (2001) attention is focused on more general hypotheses about covariate effects on the scale and tail behavior of the response, motivated by the Lehmann (1974) quantile treatment effect. Related software has been made available on the web.
Although there is already quite an extensive literature on quantile regression methods for time-series, most of the attention has focused on iid error, pure location shift model. In collaboration with Zhijie Xiao, work will be done on more general specifications, focusing initially on a class of models that exhibit some features of persistent "unit-root" behavior, while also exhibiting a sporadic form of mean reversion. These models pose serious technical challenges, but offer considerable potential for broadening the scope of applied time series analysis. The results of this research could have enormous impact on time series and panel data analyses.
