A Nontrivial Upper Bound on the Out-of-Sample $R^2$ in Return Forecasting

Abstract

This study establishes a nontrivial upper bound on the out-of-sample $R^2$ ($R^2_{\text{OOS}}$) in return forecasting. In particular, we define a coin-flip oracle model that, under the same directional accuracy, theoretically outperforms practical models in terms of MSE. The $R^2_{\text{OOS}}$ of the oracle model, whose analytical expression is a quadratic function of directional accuracy, can therefore serve as a tractable upper bound on the actual $R^2_{\text{OOS}}$. Empirical analyses across multiple forecasting scenarios reveal that the $R^2_{\text{OOS}}$ values of common predictive models are fundamentally bounded by this quadratic function.