Extend the latent-factor randomized alpha test to RP-PCA and Projected PCA

Prove that the asymptotic validity of the max-type randomized alpha test for latent factor models—specifically, that the suitably normalized test statistic converges to a Gumbel distribution under the null of zero pricing errors and diverges under the alternative—continues to hold when the K latent factors and loadings are estimated using the Risk-Premium PCA (RP-PCA) method of Lettau and Pelger and the Projected PCA method of Fan, Liao, and Wang, possibly after minor modifications to the assumptions on loadings, factor processes, and idiosyncratic errors. Precisely formulate the required assumptions for RP-PCA and Projected PCA and establish the corresponding limiting behavior of the randomized test statistics constructed from these estimators.

Background

The paper introduces a randomized max-type test for the joint null of zero pricing errors ("zero alpha") in linear factor asset pricing models and develops extensions to non-tradable and latent factors. For latent factors, the authors estimate factors and loadings via standard principal component analysis (PCA) and prove that the randomized test statistic based on PCA retains the same asymptotic properties as in the observable-factor case: a Gumbel limit under the null and divergence under the alternative.

In Section 4.2 they suggest that similar theoretical results should carry over when latent factors and loadings are estimated using alternative PCA-based methods tailored to asset pricing—Risk-Premium PCA (RP-PCA) and Projected PCA—because these approaches rely on conditions closely related to those used in their PCA analysis (e.g., assumptions akin to Bai, 2003). However, they do not provide proofs, leaving the extension to RP-PCA and Projected PCA as a conjecture.