This paper studies optimal estimation of large-dimensional nonlinear factor
models. The key challenge is that the observed variables are possibly nonlinear
functions of some latent variables where the functional forms are left
unspecified. A local principal component analysis method is proposed to
estimate the factor structure and recover information on latent variables and
latent functions, which combines K-nearest neighbors matching and principal
component analysis. Large-sample properties are established, including a sharp
bound on the matching discrepancy of nearest neighbors, sup-norm error bounds
for estimated local factors and factor loadings, and the uniform convergence
rate of the factor structure estimator. Under mild conditions our estimator of
the latent factor structure can achieve the optimal rate of uniform convergence
for nonparametric regression. The method is illustrated with a Monte Carlo
experiment and an empirical application studying the effect of tax cuts on
economic growth.Comment: arXiv admin note: text overlap with arXiv:2008.1365