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 \to \infty 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.
We develop new procedures to quantify the statistical uncertainty of data-driven clustering algorithms. In our panel setting, each unit belongs to one of a finite number of latent groups with group-specific regression curves. We propose methods for computing unit-wise and joint confidence sets for group membership. The unit-wise sets give possible group memberships for a given unit and the joint sets give possible vectors of group memberships for all units. We also propose an algorithm that can improve the power of our procedures by detecting units that are easy to classify . The confidence sets invert a test for group membership that is based on a characterization of the true group memberships by a system of moment inequalities. To construct the joint confidence, we solve a high-dimensional testing problem that tests group membership simultaneously for all units. We justify this procedure under N, T \to \infty asymptotics where we allow T to be much smaller than N. As part of our theoretical arguments, we develop new simultaneous anti-concentration inequalities for the MAX and the QLR statistics . Monte Carlo results indicate that our confidence sets have adequate coverage and are informative. We illustrate the practical relevance of our confidence sets in two applications.
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