AI 中文总结
该研究针对序贯模型的条件预测修正尺度，对比多种估计量的精度与成本，给出匹配结构选合适估计量的实用规则。
AI 中文摘要
条件预测修正尺度$\tilde{\tau} = \text{Var}(\text{E}[X_{t+1} \rvert \f_t] \rvert \f_{t-1})^{1/2}$衡量观测$X_t$所引发的、依赖于历史的预测更新规模。由于它是由两个未知条件均值构成的条件二阶矩，无法直接观测。我们研究在不同结构假设和计算预算下应使用哪种$\tilde{\tau}$的估计量，比较对象包括分块自助法、条件方差模型、拟合的状态空间模型、两个$O(1)$流式平滑器以及已训练循环网络的遗忘门。误差分解将一步预测误差与条件二阶矩跟踪误差分离。我们证明，当$\tilde{\tau}$在采样尺度上变化时，外部调优的仅滞后平滑器不一致，但在缓慢变化下达到常规的$T^{-2/3}$均方误差率（$\tilde{\tau}$为$T^{-1/3}$）；正确设定的状态空间估计量通过使用当前状态突破该极限。在波动驱动设计中，廉价的条件方差模型作为点估计比分块自助法更准确且成本低百倍以上，分块自助法的价值在于其提供的采样分布而非点跟踪；在状态驱动设计中，仅结构匹配滤波器能恢复快速变化。直接来看，训练好的网络的遗忘门不跟踪$\tilde{\tau}$，但对完整门向量的监督线性探针可以，故$\tilde{\tau}$可线性解码但无法免费获取。这些结果给出实用规则：确定条件二阶矩结构，匹配估计量，再选择成本最低的合适方法。
英文摘要
The \emph{conditional forecast-revision scale} $\It=\{\Var(\E[X_{t+1}\mid\F_t]\mid\F_{t-1})\}^{1/2}$ measures the history-specific size of the forecast update induced by observing $X_t$. Because it is a conditional second moment built from two unknown conditional means, it is not directly observed. We study which estimator of $\It$ should be used under different structural assumptions and computational budgets. The comparison includes a block bootstrap, a conditional-variance model, a fitted state-space model, two $O(1)$ streaming smoothers, and the forget gate of an already-trained recurrent network. An error decomposition separates one-step-prediction error from conditional-second-moment tracking error. We show that externally tuned lag-only smoothers can be inconsistent when $\It$ changes at the sampling scale, although they attain the usual $T^{-2/3}$ mean-squared-error rate ($T^{-1/3}$ for $\It$) under slow variation; a correctly specified state-space estimator escapes this limit by using the current state. In volatility-driven designs, a cheap conditional-variance model is more accurate and over one hundred times cheaper \emph{as a point estimator} than the implemented block bootstrap, whose value lies in the sampling distribution it provides rather than in point tracking. In state-driven designs, only the structurally matched filter recovers the fast variation. Read directly, a trained network's forget gate does not track $\It$ --- though a supervised linear probe on the full gate vector does, so $\It$ is linearly decodable but not available for free. These results yield a practical rule: identify the conditional-second-moment structure, match the estimator to it, and then choose the least costly adequate method.
Comments32 pages, 3 figures
