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, v0, v1), where y∈R1, x∈Rk, and the real variables (v, v0, v1) satisfy v0≤v≤v1. Let a random sample be drawn from P and the realizations of (y, x, v0, v1) 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(v0≤v≤v1)=1; (M) E(y|x, v) is monotone in v; (MI) E(y|x, v, v0, v1)=E(y|x, v). No restrictions are imposed on the distribution of the unobserved values of v within the observed intervals [v0, v1]. 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.

Identification, Data Combination and the Risk of Disclosure

“…Manski (2003Manski ( , 2007, and Horowitz and Manski (1995) note that data errors or data modifications pose identification problems and generally result in only set identification of the parameter of interest. Manski and Tamer (2002) and Magnac and Maurin (2008) give examples where-for confidentiality or anonymity reasons-the data may be transformed into interval data or some attributes may be suppressed, leading to the loss of point identification of the parameters of interest. Consideration of the general setup in Molinari (2008) allows one to assess the impact of some data anonymization as a general misclassification problem.…”

Sharp identification regions in models with convex moment predictions

“…In this paper, we provide a complementary approach based on tools of random sets theory. We characterize I avoiding altogether the need to deal with ; thereby contributing to a stream of previous literature which has provided tractable characterizations of sharp identi…cation regions without making any reference to the selection mechanism or the selected prediction (see, e.g., Manski (2003) and Manski and Tamer (2002)). …”

“…Examples of sharp identi…cation regions for parameters of incomplete models are given in Manski (1989Manski ( , 2003, Manski and Tamer (2002), and Molinari (2008), among others. In some cases, researchers are only able to characterize a region in the parameter space that includes all the parameter values that may have generated the observables, but may include other (infeasible) parameter values as well.…”

Sharp identification regions in games

“…In this case, the model is partially identified. The analyses in Manski (1989Manski ( , 2003, Manski and Tamer (2002), Haile and Tamer (2003), Ciliberto and Tamer (2004) and Andrews, Berry, and Jia (2004) are examples of research studying the identified features of incomplete econometric models.…”

“…The region in the parameter space which includes all possible parameter values that could generate the same distribution of observables for some data generation process consistent with the maintained modeling assumptions, and no other parameter value, is called the sharp identification region. Examples of sharp identification regions for parameters of incomplete models are given in Manski (2003) and Manski and Tamer (2002), among others. In some cases, researchers are only able to characterize a region in the parameter space that includes all the parameter values that may have generated the observables, but may include other (infeasible) parameter values as well.…”