Pervasive cross-section dependence is increasingly recognized as a characteristic of economic data and the approximate factor model provides a useful framework for analysis. Assuming a strong factor structure where $\Lop\Lo/N^\alpha$ is positive definite in the limit when $\alpha=1$, early work established convergence of the principal component estimates of the factors and loadings up to a rotation matrix. This paper shows that the estimates are still consistent and asymptotically normal when $\alpha\in(0,1]$ albeit at slower rates and under additional assumptions on the sample size. The results hold whether $\alpha$ is constant or varies across factor loadings. The framework developed for heterogeneous loadings and the simplified proofs that can be also used in strong factor analysis are of independent interest.

翻译：截面数据普遍存在的相互依赖关系日益被视为经济数据的特征，近似因子模型为此类分析提供了有效框架。早期研究在假设强度因子结构（当α=1时，$\Lop\Lo/N^\alpha$在极限条件下正定）的基础上，建立了因子与载荷的主成分估计在旋转变换下的收敛性。本文证明，当α∈(0,1]时，尽管收敛速度较慢且需额外样本量假设，该估计仍具有一致性和渐近正态性。上述结论在α为常数或随因子载荷变化时均成立。本文针对异质性载荷开发的分析框架及可应用于强因子分析的简化证明方法，其本身亦具有独立学术价值。