Inference on regressions with interval data on a regressor or outcome

This paper examines inference on regressions when interval data are available on one variable, the other variables being measured precisely. Let a population be characterized by a distribution P(y, x, v, v(0), v(1)), where y is an element of R-1, x is an element of R-k, and the real variables (v, v(0), v(1)) satisfy v(0) less than or equal to v less than or equal to v(1). Let a random sample be drawn from P and the realizations of (y, x, v(0), v(1)) be observed, but not those of v. The problem of interest may be to infer E(y\x, v) or E(v\x). This analysis maintains Interval (I), Monotonicity (M), and Mean Independence (MI) assumptions: (1) P(v(0) less than or equal to v less than or equal to v(1)) = 1; (M) E(y\x, v) is monotone in v; (MI) E(y\x, v, v(0), v(1)) = E(y\x, v). No restrictions are imposed on the distribution of the unobserved values of v within the observed intervals [v(0), v(1)]. It is found that the IMMI Assumptions alone imply simple nonparametric bounds on E(y\x, v) and E(v\x). These assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score (MAIS) method. The IMMI assumptions combined with a parametric model for E(y\x, v) or E(v\x) yield an identification region that may be estimated consistently by a modified minimum-distance (MAID) method. Monte Carlo methods are used to characterize the finite-sample performance of these estimators. Empirical case studies are performed using interval wealth data in the Health and Retirement Study and interval income data in the Current Population Survey.