  A recent literature in econometrics models unobserved cross-sectional
heterogeneity in panel data by assigning each cross-sectional unit a
one-dimensional, discrete latent type. Such models have been shown to allow
estimation and inference by regression clustering methods. This paper is
motivated by the finding that the clustered heterogeneity models studied in
this literature can be badly misspecified, even when the panel has significant
discrete cross-sectional structure. To address this issue, we generalize
previous approaches to discrete unobserved heterogeneity by allowing each unit
to have multiple, imperfectly-correlated latent variables that describe its
response-type to different covariates. We give inference results for a k-means
style estimator of our model and develop information criteria to jointly select
the number clusters for each latent variable. Monte Carlo simulations confirm
our theoretical results and give intuition about the finite-sample performance
of estimation and model selection. We also contribute to the theory of
clustering with an over-specified number of clusters and derive new convergence
rates for this setting. Our results suggest that over-fitting can be severe in
k-means style estimators when the number of clusters is over-specified.
