Inference on Regressions with Interval Data on a Regressor or Outcome

Abstract

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"-sub-0, "v"-sub-1), where "y" is an element of "R"-super-1, "x" is an element of "R-super-k", and the real variables ("v", "v"-sub-0, "v"-sub-1) satisfy "v"-sub-0?"v"<"v"-sub-1. Let a random sample be drawn from "P" and the realizations of ("y", "x", "v"-sub-0, "v"-sub-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: (I) "P"("v"-sub-0<"v"<"v"-sub-1)& equals;1; (M) "E"("y"|"x", "v") is monotone in "v"; (MI) "E"("y"|"x", "v", "v"-sub-0, "v"-sub-1)="E"("y"|"x", "v"). No restrictions are imposed on the distribution of the unobserved values of "v" within the observed intervals ["v"-sub-0, "v"-sub-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 (MMS)" 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 (MMD)" 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.