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 realized magnitude, we show that these two metrics exhibit a quadratic relationship for MSE-optimal point forecasts. For point forecasts with modest DA, 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.

翻译：本研究通过建立一种分析性联系，调和了金融时间序列预测中的样本外$R^2$ ($R^2_{OOS}$) 与方向准确性 (DA) 两种度量，从而为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，这两种度量之间存在二次关系。对于方向准确性一般的点预测，其$R^2_{OOS}$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，则负的经验$R^2_{OOS}$是预期之中的。