Conditional Density Estimation

Conditional Density Estimation: An Approximate Bayesian Method
Jungmo Yoon

First draft: November 2006, This version: July 2008.
previously circulated under the title, Bayesian Conditional Density Estimation.

We develop a nonparametric Bayesian method for conditional density function estimation. Applications include the density of income conditioned on individual specific characteristics, and the predictive density of returns conditioned on the current state of the market. To model the unknown conditional density, we use location mixtures where the mixing distribution of location parameters plays an important role for inference. The key idea of this paper is to use the conditional quantile function as a mixing distribution. We construct the posterior distribution on the space of mixing distribution, i.e. on the space of conditional quantiles, and derive the posterior on the conditional density. We establish the posterior consistency of the proposed method and obtain the rate of convergence. In our simulation exercises where we estimate predictive densities of multi-factor diffusion processes, we find that our Bayes estimator performs better than the well-known double kernel method of Fan, Yao, and Tong (1996).

The paper is available upon request.

: presented at the 2007 North American Summer meeting of the Econometric Society (Duke University) and Econometrics lunch seminar (UIUC, Fall 2006).