Partial identification in monotone binary models: Discrete regressors and interval data

We investigate identification in semi-parametric binary regression models, y = 1(x beta + upsilon + epsilon > 0) when upsilon is either discrete or measured within intervals. The error term epsilon is assumed to be uncorrelated with a set of instruments z, epsilon is independent of upsilon conditionally on x and z, and the support of - (x beta + epsilon) is finite. We provide a sharp characterization of the set of observationally equivalent parameters beta. When there are as many instruments z as variables x, the bounds of the identified intervals of the different scalar components beta(k) of parameter beta 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 upsilon within intervals shrinks the identified set. Specifically, the closer the conditional distribution of upsilon given z is to uniformity, the smaller is the identified set. Point identified is achieved if and only if upsilon is uniform within intervals.