  Counterfactual distributions are important ingredients for policy analysis
and decomposition analysis in empirical economics. In this article we develop
modeling and inference tools for counterfactual distributions based on
regression methods. The counterfactual scenarios that we consider consist of
ceteris paribus changes in either the distribution of covariates related to the
outcome of interest or the conditional distribution of the outcome given
covariates. For either of these scenarios we derive joint functional central
limit theorems and bootstrap validity results for regression-based estimators
of the status quo and counterfactual outcome distributions. These results allow
us to construct simultaneous confidence sets for function-valued effects of the
counterfactual changes, including the effects on the entire distribution and
quantile functions of the outcome as well as on related functionals. These
confidence sets can be used to test functional hypotheses such as no-effect,
positive effect, or stochastic dominance. Our theory applies to general
counterfactual changes and covers the main regression methods including
classical, quantile, duration, and distribution regressions. We illustrate the
results with an empirical application to wage decompositions using data for the
United States.
  As a part of developing the main results, we introduce distribution
regression as a comprehensive and flexible tool for modeling and estimating the
\textit{entire} conditional distribution. We show that distribution regression
encompasses the Cox duration regression and represents a useful alternative to
quantile regression. We establish functional central limit theorems and
bootstrap validity results for the empirical distribution regression process
and various related functionals.
