  This paper develops a general causal inference method for treatment effects
models with noisily measured confounders. The key feature is that a large set
of noisy measurements are linked with the underlying latent confounders through
an unknown, possibly nonlinear factor structure. The main building block is a
local principal subspace approximation procedure that combines $K$-nearest
neighbors matching and principal component analysis. Estimators of many causal
parameters, including average treatment effects and counterfactual
distributions, are constructed based on doubly-robust score functions.
Large-sample properties of these estimators are established, which only require
relatively mild conditions on the principal subspace approximation. The results
are illustrated with an empirical application studying the effect of political
connections on stock returns of financial firms, and a Monte Carlo experiment.
The main technical and methodological results regarding the general local
principal subspace approximation method may be of independent interest.
