  A conditional expectation function (CEF) can at best be partially identified
when the conditioning variable is interval censored. When the number of bins is
small, existing methods often yield minimally informative bounds. We propose
three innovations that make meaningful inference possible in interval data
contexts. First, we prove novel nonparametric bounds for contexts where the
distribution of the censored variable is known. Second, we show that a class of
measures that describe the conditional mean across a fixed interval of the
conditioning space can often be bounded tightly even when the CEF itself
cannot. Third, we show that a constraint on CEF curvature can either tighten
bounds or can substitute for the monotonicity assumption often made in interval
data applications. We derive analytical bounds that use the first two
innovations, and develop a numerical method to calculate bounds under the
third. We show the performance of the method in simulations and then present
two applications. First, we resolve a known problem in the estimation of
mortality as a function of education: because individuals with high school or
less are a smaller and thus more negatively selected group over time, estimates
of their mortality change are likely to be biased. Our method makes it possible
to hold education rank bins constant over time, revealing that current
estimates of rising mortality for less educated women are biased upward in some
cases by a factor of three. Second, we apply the method to the estimation of
intergenerational mobility, where researchers frequently use coarsely measured
education data in the many contexts where matched parent-child income data are
unavailable. Conventional measures like the rank-rank correlation may be
uninformative once interval censoring is taken into account; CEF interval-based
measures of mobility are bounded tightly.
