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

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  • Cheng Zhang

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.

Suggested Citation

  • Cheng Zhang, 2026. "A Quadratic Link between Out-of-Sample $R^2$ and Directional Accuracy," Papers 2602.07841, arXiv.org, revised Feb 2026.
    • Handle: RePEc:arx:papers:2602.07841
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