Estimating the Conditional Forecast-Revision Scale in Sequential Models: Local-Smoothing Limits, Matched Models, and Cost--Accuracy Trade-offs

Abstract

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.

32 pages, 3 figures