Abstract
This article extends the Imbens and Manski and Stoye confidence interval for a partially identified scalar parameter to a vector-valued parameter. The proposed method produces uniformly valid simultaneous confidence intervals for each dimension, or, equivalently, a rectangular confidence region that covers points in the identified set with a specified probability. The method applies when asymptotically normal estimates of upper and lower bounds for each dimension are available. The intervals are computationally simple and fast relative to methods based on test inversion or bootstrapped calibration, and do not suffer from the conservativity of projection-based approaches.
Acknowledgments
We thank Federico Bugni, Hiroaki Kaido, Francesca Molinari, Jörg Stoye, Matthew Thirkettle, and Tiemen Woutersen for helpful discussion. Thanks to Matthew Thirkettle for help with the software implementation of Kaido, Molinari, and Stoye (Citation2019).
Disclosure Statement
The authors report there are no competing interests to declare.
Notes
1 In the special case of a two-dimensional subvector, one could skirt the projection problem and directly plot the confidence region. For dimensions higher than three, as in the empirical application in Section 3, this option isn’t workable, at least in print.