Forecasting on the Accuracy-Timeliness Frontier: Two Novel `Look Ahead' Predictors

TL;DR

This work re-examine the traditional Mean-Squared Error (MSE) forecasting paradigm by formally integrating an accuracy-timeliness trade-off, and derives a universal upper bound on lead over MSE for any linear predictor under a consistency constraint and proves that their methods hit this ceiling.

Abstract

We re-examine the traditional Mean-Squared Error (MSE) forecasting paradigm by formally integrating an accuracy-timeliness trade-off: accuracy is defined by MSE (or target correlation) and timeliness by advancement (or phase excess). While MSE-optimized predictors are accurate in tracking levels, they sacrifice dynamic lead, causing them to lag behind changing targets. To address this, we introduce two `look-ahead' frameworks--Decoupling-from-Present (DFP) and Peak-Correlation-Shifting (PCS)--and provide closed-form solutions for their optimization. Notably, the classical MSE predictor is shown to be a special case within these frameworks. Dually, our methods achieve maximum advancement for any given accuracy level, so our approach reveals the complete efficient frontier of the accuracy-timeliness trade-off, whereas MSE represents only a single point. We also derive a universal upper bound on lead over MSE for any linear predictor under a consistency constraint and prove that our methods hit this ceiling. We validate this approach through applications in forecasting and real-time signal extraction, introducing a leading-indicator criterion and tailored linear benchmarks.

Figures (10)

• Figure 1: Top graph: CCF of MSE 5-step ahead forecast filter and MA(9) process at leads $9\geq\delta> 0$ and lags $-4\leq \delta\leq 0$. Bottom graph: a comparison of realizations of the process (black line) and of the optimal 5-step ahead forecast filter (green line)
• Figure 2: Geometry of the DFP predictor. The solution $\mathbf{{b}}=\lambda_1\boldsymbol{\gamma}_h+\lambda_2\boldsymbol{\gamma}_0$ lies at the intersection of the plan spanned by $(\boldsymbol{\gamma}_0,\boldsymbol{\gamma}_h)$, the circular cone with semi-angle $\theta_{0b}$ (red), and the unit-sphere (blue). The cone has axis $\boldsymbol{\gamma}_0$ and half-angle $\theta_{0b}=\arccos(\alpha_0)$. The figure depicts one of the two solutions of the quadratic in $\lambda_2$ associated with $+\theta_{0b}$ (with $\lambda_1>0,\lambda_2<0$); the mirrored solution at $-\theta_{0b}$ (with $\lambda_1<0,\lambda_2>0$) does not maximize the objective $\boldsymbol{\gamma}_h'\mathbf{b}$ and corresponds to a lag, so it is omitted. The quantity $\theta_{0b}-\theta_{0h}>0$ represents the phase excess of the DFP relative to the MSE predictor $\boldsymbol{\gamma}_h$ and is associated with its advancement (lead), see Section \ref{['time_shift']}.
• Figure 3: Side lengths $a$, $b$ and $c$ (red) as well as angles $\alpha$, $\beta$ and $\gamma$ of the DFP-triangle spanned by the vectors $\boldsymbol{\gamma}_h$ and $\mathbf{{b}}=\boldsymbol{\gamma}_h+\lambda\boldsymbol{\gamma}_0$ (black), with $\lambda<0$.
• Figure 4: Geometry of the PCS predictor. Two solutions $\mathbf{b}_1$ (red) with $\beta_h=\mathbf{b}_1'\boldsymbol{\gamma}_{h-1}-\mathbf{b}_1'\boldsymbol{\gamma}_{h}=0$ and $\mathbf{b}_2$ (blue) with $\beta_h=\mathbf{b}_2'\boldsymbol{\gamma}_{h-1}-\mathbf{b}_2'\boldsymbol{\gamma}_{h}<0$, are shown in the plane spanned by $(\boldsymbol{\gamma}_{h-1},\boldsymbol{\gamma}_h)$. The positive angles $\theta_{hb1}$ and $\theta_{hb2}$ denote the phase excesses of the respective PCS predictors relative to the MSE predictor $\boldsymbol{\gamma}_h$.
• Figure 5: MSE (green) and DFP predictors under weak decoupling (red) and complete decoupling (blue). Top-left: forecast weights. Top-right: CCF across leads (the CCF under complete decoupling is zero at $\delta= 0$). Bottom: standardized forecasts. Increasing decoupling produces a leftward shift (advancement) of the DFP forecasts relative to the MSE benchmark.

Theorems & Definitions (10)

• Theorem 1
• Corollary 1
• Proposition 1
• Corollary 2
• Proposition 2
• Theorem 2
• Corollary 3
• Proposition 3
• Corollary 4
• Proposition 4