A previous version of this working paper was originally published as Identification Through Sparsity in Factor Models in June 2020.
The key insight is that any rotation of a sparse loading vector will be less sparse. While a rotation criterion based on the ℓ0-norm of the loading matrix is infeasible, we prove that a rotation criterion based on the ℓ1-norm will consistently recover the individual loading vectors under sparsity in the loading matrix. Existing rotation criteria (e.g., the Varimax rotation, Kaiser ) lack such theoretical guarantees. We further show that the assumption of sparsity in the loading matrix is testable and develop such a test. In our simulations, the ℓ1-rotation performs better than existing rotation criteria, and we find strong evidence for the presence of local factors in two economic applications.