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.

翻译：本研究通过建立样本外$R^2$（$R^2_{\text{OOS}}$）与方向准确率（DA）之间的解析联系，为金融时间序列预测中的度量脱节现象提供了一个新颖的视角。具体而言，以随机游走模型为基准，并假设符号正确性与已实现幅度相互独立，我们证明了对于均方误差最优的点预测，这两个度量指标呈现出二次关系。对于方向准确率较低的点预测，$R^2_{\text{OOS}}$的理论值本质上是可忽略的。因此，如果模型是次优的或受到有限样本噪声的影响，负的经验$R^2_{\text{OOS}}$是预期之内的。