Confidence Set for Group Membership
We develop new procedures to quantify the statistical uncertainty from sorting units in panel data into groups using data-driven clustering algorithms. In our setting, each unit belongs to one of a ﬁnite number of latent groups and its regression curve is determined by which group it belongs to. Our main contribution is a new joint conﬁdence set for group membership. Each element of the joint conﬁdence set is a vector of possible group assignments for all units. The vector of true group memberships is contained in the conﬁdence set with a pre-speciﬁed probability. The conﬁdence set inverts a test for group membership. This test exploits a characterization of the true group memberships by a system of moment inequalities. Our procedure solves a high-dimensional one-sided testing problem and tests group membership simultaneously for all units. We also propose a procedure for identifying units for which group membership is obviously determined. These units can be ignored when computing critical values. We justify the joint conﬁdence set under N, T → ∞ asymptotics where we allow T to be much smaller than N. Our arguments rely on the theory of self-normalized sums and high-dimensional central limit theorems. We contribute new theoretical results for testing problems with a large number of moment inequalities, including an anti-concentration inequality for the quasi-likelihood ratio (QLR) statistic. Monte Carlo results indicate that our conﬁdence set has adequate coverage and is informative. We illustrate the practical relevance of our conﬁdence set in two applications.
JEL: C23, C33, C38
joint one-sided tests
anti-concentration for QLR
Working Papers in Economics