It is well-known that the approximate factor models have the rotation indeterminacy. It has been considered that the principal component (PC) estimators estimate some rotations of the true factors and factor loadings, but the rotation matrix commonly used in the literature depends on the PC estimator itself. This raises a question: what does the PC estimator consistently estimate? This paper aims to explore the answer. We first show that, assuming a quite general weak factor model with the $r$ signal eigenvalues diverging possibly at different rates, there always exists a unique rotation matrix composed only of the true factors and loadings, such that it rotates the true model to the identifiable model satisfying the standard $r^2$ restrictions. We call the rotated factors and loadings the pseudo-true parameters. We next establish the consistency and asymptotic normality of the PC estimator for this pseudo-true parameter. The results give an answer for the question: the PC estimator consistently estimates the pseudo-true parameter. We also investigate similar problems in the factor augmented regression. Finite sample experiments confirm the excellent approximation of the theoretical results.

翻译：众所周知，近似因子模型存在旋转不可识别性。通常认为，主成分（PC）估计量能够估计真实因子和因子载荷的某些旋转结果，但文献中常用的旋转矩阵依赖于PC估计量本身。这引发了一个问题：PC估计量一致估计的是什么？本文旨在探讨该问题的答案。我们首先证明，在信号特征值以不同速率发散的相当一般的弱因子模型假设下，始终存在一个仅由真实因子和载荷构成的唯一旋转矩阵，使得该矩阵将真实模型旋转为满足标准$r^2$约束的可识别模型。我们将旋转后的因子和载荷称为伪真实参数。接着，我们建立了PC估计量对此伪真实参数的一致性和渐近正态性。这些结果为上述问题提供了答案：PC估计量一致估计伪真实参数。我们还研究了因子增强回归中的类似问题。有限样本实验证实了理论结果的优良近似性。