Partial Identification of Expectations with Interval Data

TL;DR

This paper introduces new nonparametric bounds and methods for analyzing conditional expectations with interval-censored data, enabling more informative inference in cases with limited or coarse data, and demonstrates applications in mortality and mobility studies.

Contribution

It develops three innovative approaches for partial identification of expectations with interval data, including bounds when the distribution is known, measures over fixed intervals, and curvature constraints.

Findings

Bounds improve inference in mortality estimation with censored education data

Interval-based mobility measures are tightly bounded, unlike traditional rank correlations

Method performs well in simulations and real-world applications

Abstract

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.