AI 中文总结
本文通过精确分解和不可识别性定理，阐明信号相关性与PnL相关性的差异，指出二者在无约束时互不界定，并给出最优组合规则一致性条件。
AI 中文摘要
信号相关性与 PnL 相关性是不同指标集上的相关性——前者是每个日期上跨资产的相关性，后者是标量收益跨日期的相关性——实践者常将前者视为后者的代理。我们给出了一个精确分解，展示该代理所捕捉与所忽略的内容。我们回顾到，在每个日期，归一化信号在已实现去均值收益方向上的投影即为已实现横截面 Pearson IC，因此固定信号的 PnL 等于收益离散度乘以 IC（Qian 和 Hua，2004），并识别出两个信号在正交补空间中的归一化相似度为其在控制已实现收益后的偏相关；剩余的旋转自由度是一个正交规范，其不变量为横向 Gram 矩阵。因此，信号相关性等于非中心化 IC 交叉矩加上横向相似度，而 Pearson PnL 相关性仅对 IC 序列进行中心化和离散度加权。我们的主要结果是一个不可识别性定理：在缺乏横向几何约束的情况下，两种相关性既不能相互界定也不能相互排序，这将 Sorensen、Qian、Schoen 和 Hua（2004）的模拟发现精炼为精确陈述。对于归一化系综，横向 Gram 矩阵通过归一化分母重新进入；在欧几里得和一般协方差风险度量下的事后最优组合是经典多重相关界，我们证明当且仅当风险度量在信号张成空间上是各向同性时，两种规则对每个纵向暴露都一致。经典的在各向同性下偏相关的精确零分布、弱 IC 尺度下的合成机制示例，以及时间依赖下的推断指导，共同完成了这一处理。
英文摘要
Signal correlation and PnL correlation are correlations over different index sets - across assets at each date versus across dates for scalar payoffs - and practitioners often treat the first as a proxy for the second. We give an exact decomposition that shows what that proxy sees and what it discards. We recall that at each date a normalized signal's projection onto the realized demeaned return direction is its realized cross-sectional Pearson IC, so that fixed-signal PnL is return dispersion times IC (Qian and Hua, 2004), and we identify the normalized similarity of two signals in the orthogonal complement as their partial correlation controlling for realized returns; the residual rotational freedom is an orthogonal gauge whose invariants are the transverse Gram matrix. Signal correlation therefore equals an uncentered IC cross-moment plus transverse similarity, while Pearson PnL correlation centers and dispersion-weights the IC series alone. Our main result is a non-identifiability theorem: absent constraints on transverse geometry, neither correlation bounds or orders the other, which sharpens the simulation finding of Sorensen, Qian, Schoen, and Hua (2004) into an exact statement. For normalized ensembles the transverse Gram matrix re-enters through the normalization denominator; the ex post optimal combination under Euclidean and general covariance-risk metrics is the classical multiple-correlation bound, and we show that the two rules agree for every longitudinal exposure if and only if the risk metric is isotropic on the signal span. The classical exact null law of partial correlation under isotropy, a synthetic mechanism illustration at weak-IC scales, and inference guidance under temporal dependence complete the treatment.
Comments24 pages, 2 figures, 3 tables. Reproduction code for the synthetic study is included as an appendix
