Comparing Distribution and Quantile Regression

We study the sampling properties of two alternative approaches to estimating the conditional distribution of a continuous outcome Y given a vector X of regressors. One approach – distribution regression – is based on direct estimation of the conditional distribution function; the other approach – quantile regression – is instead based on direct estimation of the conditional quantile function. Indirect estimates of the conditional quantile function and the conditional distribution function may then be obtained by inverting the direct estimates obtained from either approach or, to guarantee monotonicity, their rearranged versions. We provide a systematic comparison of the asymptotic and finite sample performance of monotonic estimators obtained from the two approaches, considering both cases when the underlying linear-in-parameter models are correctly specified and several types of model misspecification of considerable practical relevance.