A Quadratic Link between Out-of-Sample $R^2$ and Directional Accuracy

Cheng Zhang

Papers from arXiv.org

Abstract: This study provides a novel perspective on the metric disconnect phenomenon in financial time series forecasting through an analytical link that reconciles the out-of-sample $R^2$ ($R^2_{\text{OOS}}$) and directional accuracy (DA). In particular, using the random walk model as a baseline and assuming that sign correctness is independent of the realized magnitude, we show that these two metrics exhibit a quadratic relationship for MSE-optimal point forecasts. For point forecasts with modest DAs, the theoretical value of $R^2_{\text{OOS}}$ is intrinsically negligible. Thus, a negative empirical $R^2_{\text{OOS}}$ is expected if the model is suboptimal or affected by finite sample noise.

Date: 2026-02, Revised 2026-02
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