Validity of Assumption af-2 in observed-factor covariance estimation

Determine whether Assumption af-2 in the observed-factor model covariance estimation framework of Fan, Liao, and Mincheva (2011)—which posits that, uniformly over assets and time, OLS residuals \(\hat{u}_{it}\) closely approximate the true idiosyncratic errors \(u_{it}\) via high-probability bounds on both the average squared difference and the maximum absolute difference—actually holds across the high-dimensional financial time series settings considered for portfolio construction when \(p>T\).

Background

The observed-factor approach of Fan et al. (2011) constructs a high-dimensional covariance estimator by combining OLS-estimated factor loadings with a thresholded covariance matrix of idiosyncratic errors. A critical ingredient is Assumption af-2, which requires that OLS residuals provide sufficiently accurate proxies for the unobserved errors uniformly over assets and time, controlled by sequences and probability bounds.

The paper flags Assumption af-2 as high-level and questions its applicability across cases, highlighting a gap between theoretical conditions and practical financial datasets. Establishing the empirical or theoretical regimes under which Assumption af-2 is satisfied is necessary to justify the consistency results of the observed-factor precision matrix estimator in realistic portfolio settings.