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.

翻译：本文研究大规模非线性因子模型的最优估计问题。核心挑战在于观测变量可能是某些潜变量的非线性函数，且函数形式未加限定。我们提出一种局部主成分分析方法，该方法结合K近邻匹配与主成分分析，用于估计因子结构并恢复潜变量及潜函数的信息。本文建立了大样本性质，包括近邻匹配差异的尖锐界、局部因子与因子载荷估计的极大模范差界，以及因子结构估计量的一致收敛速率。在温和条件下，潜变量因子结构估计量可达到非参数回归的最优一致收敛速率。通过蒙特卡洛实验及一项关于减税对经济增长影响的实证应用，验证了该方法的有效性。