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Inference on Regressions with Interval Data on a Regressor or Outcome
Author(s) -
Manski Charles F.,
Tamer Elie
Publication year - 2002
Publication title -
econometrica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 16.7
H-Index - 199
eISSN - 1468-0262
pISSN - 0012-9682
DOI - 10.1111/1468-0262.00294
Subject(s) - mathematics , estimator , nonparametric statistics , statistics , interval (graph theory) , population , monotonic function , binary number , distribution (mathematics) , combinatorics , mathematical analysis , demography , arithmetic , sociology
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 ∈ R 1 , x ∈ R k , and the real variables ( v , v 0 , v 1 ) satisfy v 0 ≤ v ≤ 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: (I) P ( v 0 ≤ v ≤ 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 (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.