We develop inferential tools for latent factor analysis in short panels. The pseudo maximum likelihood setting under a large cross-sectional dimension $n$ and a fixed time series dimension $T$ relies on a diagonal $T \times T$ covariance matrix of the errors without imposing sphericity or Gaussianity. We outline the asymptotic distributions of the latent factor and error covariance estimates as well as of an asymptotically uniformly most powerful invariant (AUMPI) test based on the likelihood ratio statistic for tests of the number of factors. We derive the AUMPI characterization from inequalities ensuring the monotone likelihood ratio property for positive definite quadratic forms in normal variables. An empirical application to a large panel of monthly U.S. stock returns separates date after date systematic and idiosyncratic risks in short subperiods of bear vs. bull market based on the selected number of factors. We observe an uptrend in idiosyncratic volatility while the systematic risk explains a large part of the cross-sectional total variance in bear markets but is not driven by a single factor. We also find that observed factors, scaled or not, struggle spanning latent factors. Rank tests reveal that observed factors struggle spanning latent factors with a discrepancy between the dimension of the two factor spaces decreasing over time.

翻译：我们开发了短面板中潜在因子分析的推断工具。在大横截面维度 $n$ 和固定时间序列维度 $T$ 的设定下，伪极大似然方法依赖于误差项的对角化 $T \times T$ 协方差矩阵，且不要求球面性或高斯性假设。我们刻画了潜在因子与误差协方差估计的渐近分布，并基于似然比统计量提出了用于检验因子数量的渐近一致最大功效不变（AUMPI）检验。通过确保正态变量中正定二次型的单调似然比性质的不等式，我们推导出AUMPI的特征。实证分析应用于美国月度股票收益的大面板数据，根据所选因子数量在熊市与牛市的短子周期中分离逐日系统性风险与异质性风险。我们观察到异质性波动呈上升趋势，而系统性风险在熊市中解释了横截面总方差的大部分，但并非由单一因子驱动。我们还发现，无论是否经尺度调整，观测因子均难以涵盖潜在因子。秩检验表明，观测因子与潜在因子之间存在维度不一致，且两个因子空间的维度差异随时间递减。