This paper considers the effect of a continuous treatment on the entire distribution of outcomes after adjusting for differences in the distribution of covariates across different levels of the treatment. Our methodology encompasses dose‐response functions, counterfactual distributions, and ‘distributional policy effects’ depending on the assumptions invoked by the researcher. We propose a three‐step estimator that consists of (i) estimating the distribution of the outcome conditional on the treatment and other covariates using quantile regression; (ii) for each value of the treatment, averaging over a counterfactual distribution of the covariates holding the treatment fixed; (iii) converting the resulting counterfactual distribution into parameters of interest that are easy to interpret. We show that our estimators converge uniformly to Gaussian processes and that the empirical bootstrap can be used to conduct uniformly valid inference across a range of values of the treatment. We use our method to study intergenerational income mobility where we consider effects of parents’ income on features of their child's income distribution such as (i) the fraction of children with income below the poverty line; (ii) the variance of child's income; and (iii) the inter‐quantile range of child's income–all as a function of parents’ income.
The intergenerational elasticity (IGE) is the most common parameter reported in the intergenerational mobility literature. This paper proposes a "local" intergenerational mobility parameter (LIGE) that allows the effect of parents' income to vary across different values of parents' income. We also extend this result to an "adjusted" local intergenerational elasticity (ALIGE) which adjusts for differences in the distribution of observed characteristics at different values of parents' income. We develop the asymptotic properties of the LIGE and ALIGE, and apply them to study intergenerational mobility using data from the PSID. We find that the intergenerational elasticity is much larger for low values of parents' income (indicating less mobility) relative to high values of parents' income; adjusting for differences in characteristics reduces the local IGE at all values of parents' income as well as flattening it across different values of parents' income.
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