From Many Models, One: Macroeconomic Forecasting with Reservoir Ensembles

TL;DR

The paper advances macroeconomic forecasting by building ensembles of Multi-Frequency Echo State Networks (MFESNs) and integrating online expert-aggregation methods. It develops regret- and concentration-type guarantees for Follow-the-Leader and Hedge variants (including Constant Hedge, Decreasing Hedge, and AdaHedge) under independent and φ-mixing losses, while applying these ideas to MFESN ensembles. Empirically, EN-MFESN-RP and EN-MFESN-αRP ensembles substantially improve GDP growth forecasts relative to individual MFESNs and standard benchmarks, with gains up to around 30–50% in relative MSFE and notable robustness to hyperparameter settings. The work demonstrates that ensemble approaches can exploit diverse reservoir dynamics and memory properties to adapt to changing macroeconomic regimes, reducing the need for extensive hyperparameter tuning.

Abstract

Model combination is a powerful approach for achieving superior performance compared to selecting a single model. We study both theoretically and empirically the effectiveness of ensembles of Multi-Frequency Echo State Networks (MFESNs), which have been shown to achieve state-of-the-art macroeconomic time series forecasting results (Ballarin et al., 2024a). The Hedge and Follow-the-Leader schemes are discussed, and their online learning guarantees are extended to settings with dependent data. In empirical applications, the proposed Ensemble Echo State Networks demonstrate significantly improved predictive performance relative to individual MFESN models.

Figures (5)

• Figure 1: EN-MFESN-RP ensembles constructed out of S-MFESN A (a), S-MFESN B (b), M-MFESN A (c), and M-MFESN B (d) models. Plots display MSFE relative to AR(1) baseline for ensemble aggregation methods (colored dashed lines), the median ensemble performance (black dotted line), specific draws of S-MFESN A/B and M-MFESN A/B from ballarin2022reservoir (black solid line), and empirical CDF of the relative MSFE across individual ensemble models (gray curve). Colored markers are included to distinguish overlapping lines.
• Figure 2: EN-MFESN-$\alpha$RP ensembles constructed out of S-MFESN A (a), S-MFESN B (b), M-MFESN A (c), and M-MFESN B (d) models. Plots display MSFE relative to AR(1) baseline for ensemble aggregation methods (colored dashed lines), the median ensemble performance (black dotted line), specific draws of S-MFESN A/B and M-MFESN A/B from ballarin2022reservoir (black solid line), and ECDF of the relative MSFE across individual ensemble models (gray curve). Colored markers are included to distinguish overlapping lines.
• Figure 3: EN-MFESN-$\alpha$RP ensemble: MSFE relative to AR(1). Plots provide the MSFE ECDF of the entire ensemble (black curve) and models grouped by leak rate $\alpha$ (colored).
• Figure G.4: EN-MFESN-$\alpha$RP ensemble: AdaHedge (left column) and FTL (right column) weights over the forecasting interval. For FTL, the index $\textsc{ftl}_t$ of the leader model is shown.
• Figure G.5: EN-MFESN-$\alpha$RP ensemble: AdaHedge (left column) and FTL (right column) weights over the forecasting interval. For FTL, the index $\textsc{ftl}_t$ of the leader model is shown.

Theorems & Definitions (32)

• Remark 2.1
• Lemma 3.1: rooijFollowLeaderIf2014, Lemma 10
• Lemma 3.2: rooijFollowLeaderIf2014, Corollary 9
• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Remark 3.2
• Proposition A.1: chernovPredictionExpertAdvice2010
• Lemma A.1
• Lemma A.2: mourtadaOptimalityHedgeAlgorithm2019, Lemma 13