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

Cheng Zhang

arXiv 8 Feb 2026 · Econometrics

arXiv:2602.07841 · PDF · DOI · OpenAlex · Extracted main text

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 $R^2$ ($R^2_{OOS}$) 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 $R^2_{OOS}$ is intrinsically negligible. Thus, a negative empirical $R^2_{OOS}$ is expected if the model is suboptimal or affected by finite sample noise.

Citation extraction

18
references
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in-text mentions
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distinct cited
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main-text words

appendix boundary found by appendix_command · 87% of the source is main text. Read the extracted text to check this.

Most heavily cited references

The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.

ReferenceIntensityMentionsSectionsMain text
1Christoffersen, P.F., Diebold, F.X (2006) Financial asset returns, direction-of-change forecasting, and volatility dynamics0.64422100%
2Gu, S., Kelly, B., Xiu, D (2020) Empirical asset pricing via machine learning0.64422100%
3Welch, I., Goyal, A (2008) A comprehensive look at the empirical performance of equity premium prediction0.64422100%
4Petropoulos, F., Apiletti, D., Assimakopoulos, V., Babai, M.Z., Barr… (2022) Forecasting: theory and practice0.51121100%
5Anatolyev, S., Gospodinov, N (2010) Modeling financial return dynamics via decomposition0.40511100%
6Bollerslev, T (2023) Reprint of: Generalized autoregressive conditional heteroskedasticity0.40511100%
7Campbell, J.Y., Thompson, S.B (2008) Predicting excess stock returns out of sample: Can anything beat the historical average?0.40511100%
8Ellwanger, Snudden (2023) Forecasts of the real price of oil revisited: Do they beat the random walk?0.40511100%
9Engle, R.F., Gallo, G.M (2006) A multiple indicators model for volatility using intra-daily data0.40511100%
10Farmer, L., Schmidt, L., Timmermann, A (2023) Pockets of predictability0.40511100%

Showing the top 10 of 18 scored citations.