Determine the estimation error of the signal’s second moment without additional assumptions

Ascertain the estimation error of the population second moment E[f_t^2] from out-of-sample data, specifically the quantity E[f_t^2] − E_oos[f^2], under only Assumption 1 for the predictive model y_{t+1} = f_t + ε_{t+1}, without imposing additional technical conditions on the predictable component f_t. This would enable direct inference on the population R^2 rather than relying on the modified object Ř^2.

Background

The paper derives a confidence interval for a modified infeasible R^2 (Ř^2) because the authors cannot, under their minimal assumptions, characterize the estimation error associated with the signal’s second moment. This limitation motivates the use of Ř^2 instead of the true population R^2.

Resolving this issue—determining the estimation error of E[f_t^2] from finite samples without additional structure on f_t—would strengthen inference by enabling direct confidence bounds for R^2 under the paper’s general environment.