Partial Identification in Monotone Binary Models: Discrete Regressors and Interval Data

Abstract

We investigate identification in semi-parametric binary regression models, y = 1(xβ + v + e> 0) when v is either discrete or measured within intervals. The error term e is assumed to be uncorrelated with a set of instruments z, e is independent of v conditionally on x and z, and the support of −(xβ + e) is finite. We provide a sharp characterization of the set of observationally equivalent parameters β. When there are as many instruments z as variables x, the bounds of the identified intervals of the different scalar components βk of parameter β can be expressed as simple moments of the data. Also, in the case of interval data, we show that additional information on the distribution of v within intervals shrinks the identified set. Specifically, the closer the conditional distribution of v given z is to uniformity, the smaller is the identified set. Point identified is achieved if and only if v is uniform within intervals.