Before revision:  In the panel case, observations from different periods can be used as proxy controls and our key identifying assumptions follow from restrictions on the serial dependence  of the data . Our estimation problem is (under our assumptions) `well-posed' and our estimator is straight-forward to implement. The key step is a penalized sieve minimum distance estimator. We derive simple convergence rates under high-level assumptions .
After revision:  In the panel case, observations from different periods can be used as proxy controls and our key identifying assumptions follow from restrictions on the serial dependence of the data and latent variables. We apply our methodology to two separate empirical settings that respectively showcase the cross-sectional and the panel implementations of our approach. We estimate causal effects of grade retention on performance in a range of cognitive tests using cross-sectional variation in the data and we estimate a structural Engel curve for food using panel data .