This paper establishes central limit theorems (CLTs) and proposes how to perform valid inference in factor models. We consider a setting where many counties/regions/assets are observed for many time periods, and when estimation of a global parameter includes aggregation of a cross-section of heterogeneous microparameters estimated separately for each entity. The CLT applies for quantities involving both cross-sectional and time series aggregation, as well as for quadratic forms in time-aggregated errors. This paper studies the conditions when one can consistently estimate the asymptotic variance, and proposes a bootstrap scheme for cases when one cannot. A small simulation study illustrates performance of the asymptotic and bootstrap procedures. The results are useful for making inferences in two-step estimation procedures related to factor models, as well as in other related contexts. Our treatment avoids structural modeling of cross-sectional dependence but imposes time-series independence.

中文翻译：

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因子模型的极限定理

本文建立了中心极限定理 (CLT)，并提出了如何在因子模型中进行有效推理。我们考虑在许多时间段内观察到许多县/地区/资产的设置，并且当全局参数的估计包括为每个实体分别估计的异质微参数的横截面的聚合时。CLT 适用于涉及横截面和时间序列聚合的量，以及时间聚合误差中的二次形式。本文研究了可以一致地估计渐近方差的条件，并针对不能一致的情况提出了一种引导方案。一项小型模拟研究说明了渐近和引导程序的性能。结果对于在与因子模型相关的两步估计过程中进行推断很有用，以及在其他相关情况下。我们的处理避免了横截面依赖的结构建模，但强加了时间序列独立性。