Improve convergence rate for eigenvalue approximation under heterogeneous weak loadings

Derive the exact limit of the matrix B_N t^{-1} \widehat{F}_t' F_t B_N^{-1} under Assumptions A.1–A.4 with heterogeneously weak factor loadings (1 ≥ α_1 ≥ ⋯ ≥ α_r > 1/2), and use this limit to improve the convergence rate in Lemma 5(b)(ii) for sup_{k_0 ≤ t ≤ T−1} ||(N B_N^{-2} D_{Nt,r}^2)^{-1} − Σ_Λ^{-1}|| beyond the currently established bound O_p(N/N^{α_r}·T^{-1/2}) + O_p(N^{-(1−2α_r)/2}).

Background

The paper develops asymptotic theory for forecast accuracy and encompassing tests when the alternative model uses principal component factors with potentially weak, including heterogeneously weak, loadings. Central to the theory are convergence rates for principal component quantities, including eigenvalue matrices and rotation matrices, under recursive estimation.

In the heterogeneous weak loading case, Lemma 5 provides improved rates for certain quantities but maintains a specific convergence rate for the difference between the inverse of the scaled eigenvalue matrix (N B_N^{-2} D_{Nt,r}^2)^{-1} and the population limit Σ_Λ^{-1}. The authors note that this rate may be sharpened if one can characterize the exact limit of B_N t^{-1} \widehat{F}_t' F_t B_N^{-1}.

Establishing this limit would allow a more precise analysis of eigenvalue behavior under heterogeneously weak loadings and could lead to stronger asymptotic results for the feasible test statistics.