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：跨截面普遍依赖性日益被视为经济数据的一个特征，而近似因子模型为分析提供了有用的框架。在假设强因子结构下，即当$\alpha=1$时$\Lop\Lo/N^\alpha$在极限下为正定矩阵，早期工作确立了因子和载荷的主成分估计在旋转矩阵意义下的收敛性。本文表明，当$\alpha\in(0,1]$时，尽管估计的收敛速度较慢且需对样本量施加额外假设，但估计仍具有一致性和渐近正态性。该结果无论$\alpha$是常数还是随因子载荷变化均成立。针对异质性载荷开发的这一框架，以及可用于强因子分析的简化证明，本身也具有独立的研究价值。