  This paper considers identification and estimation of distributional effect
parameters that depend on the joint distribution of an outcome and another
variable of interest ("treatment") in a setting with "two-sided" measurement
error -- that is, where both variables are possibly measured with error.
Examples of these parameters in the context of intergenerational income
mobility include transition matrices, rank-rank correlations, and the poverty
rate of children as a function of their parents' income, among others. Building
on recent work on quantile regression (QR) with measurement error in the
outcome (particularly, Hausman, Liu, Luo, and Palmer (2021)), we show that,
given (i) two linear QR models separately for the outcome and treatment
conditional on other observed covariates and (ii) assumptions about the
measurement error for each variable, one can recover the joint distribution of
the outcome and the treatment. Besides these conditions, our approach does not
require an instrument, repeated measurements, or distributional assumptions
about the measurement error. Using recent data from the 1997 National
Longitudinal Study of Youth, we find that accounting for measurement error
notably reduces several estimates of intergenerational mobility parameters.
