Lit Synthesis: DFM Forecasting ∩ Sinkhorn-as-Loss

Classical DFM forecasting anchors

• Stock & Watson — dynamic factor models for large panels; diffusion-index / factor-augmented forecasting. Textbook reference for "few latent factors drive co-movement; forecast via estimated factors."
• Banbura, Giannone & Reichlin — nowcasting with mixed-frequency / ragged-edge data via state-space DFMs and Kalman filtering/smoothing; EM for parameters. The operational template for "publish forecasts as data arrive asymmetrically." Mechanism: observation equation \(y_t = \Lambda f_t + e_t\), factor dynamics (VAR), missing entries handled by Kalman with time-varying observation selection; typically Gaussian likelihood / quadratic loss.

These anchors define what a true intersection must touch: factors + missingness + real-time information sets — not merely OT somewhere near time series.

Sinkhorn divergence as loss

• Cuturi (2013) — entropic regularization makes OT differentiable/scalable via Sinkhorn–Knopp matrix scaling.
• Genevay / Feydy-style Sinkhorn divergence — debiased Sinkhorn loss \(S_\varepsilon(\mu,\nu) = OT_\varepsilon(\mu,\nu) - \tfrac12 OT_\varepsilon(\mu,\mu) - \tfrac12 OT_\varepsilon(\nu,\nu)\), usable as a training criterion between empirical measures (batches, predictive residual clouds, feature distributions).

In forecasting, Sinkhorn-as-loss usually means: match distributions (or feature measures), not just pointwise MSE — useful when phase/shape errors matter or when imputing under a distributional invariance prior (Spelta's "two batches, same law").

True intersection: Spelta et al. (JRSS-A 2025)

Adjacent clusters — what each is not

True ∩ (external to brief): Spelta et al. (2025) JRSS-A — Sinkhorn loss + DFM-EM for nowcasting/short-term forecasting.

Research recipe: DFM / state-space + Sinkhorn observation loss

Goal (generic financial / macro panel): Improve ragged-edge multi-series short-horizon forecasts by replacing or augmenting Gaussian observation likelihood with a Sinkhorn divergence between predicted and observed panel distributions, following the Spelta-style "distribution-preserving imputation" idea.

Assumptions

1. Approximate low-rank factor structure: \(y_t = \Lambda f_t + e_t\), \(f_t\) VAR / state-space.
2. Missingness is ragged-edge / publication lag, treated as missing-at-random conditional on the information set at forecast origin \(t_0\) (no peeking at same-period revisions released later).
3. Sinkhorn acts on empirical measures over cross-section and/or residual batches, not on a single scalar target only.
4. Entropic regularization \(\varepsilon\) chosen for stability; debiasing (Sinkhorn divergence) preferred over raw \(OT_\varepsilon\).

Protocol (leakage-safe)

• Walk-forward: expanding or rolling origin; at each origin, use only data that would have been available (vintage / pseudo-real-time mask).
• No same-period leakage: never train or evaluate with contemporaneous realizations that arrive after the decision time; hold out the full evaluation window from \(\varepsilon\) / factor-count tuning (nested walk-forward).
• Imputation vs prediction: if Sinkhorn fills missing entries inside EM, freeze filled values for out-of-sample scoring against eventual releases; score point (RMSE/MAE) and distributional (Sinkhorn / CRPS) metrics separately.

Candidate nulls / baselines

1. Classical Gaussian DFM + Kalman (MSE) — Banbura/Giannone/Reichlin-style.
2. Same DFM with Student-t / Huber observation loss (robustness without OT).
3. Sinkhorn-only nonparametric imputer without factors (Spelta stand-alone arm).
4. Factor model + MSE, then Sinkhorn only as evaluation metric (not train).
5. Adjacent deep baseline: Feature-aligned N-BEATS + Sinkhorn (2305.15196) on the same panel.

Success criteria (scientific, not commercial)

• Statistically better short-horizon scores vs (1) on held-out origins, with Diebold–Mariano or similar.
• Gains concentrated where raggedness is severe (ablate missingness rate).
• Ablation: remove Sinkhorn step → performance collapses toward MSE-DFM; remove factors → collapses toward nonparametric OT imputer.

Top citations

1. Stock & Watson — dynamic factor / diffusion-index forecasting (classical DFM).
2. Banbura, Giannone & Reichlin — DFM nowcasting with ragged edge / Kalman.
3. Cuturi (2013) + Genevay/Feydy Sinkhorn divergence — OT loss technology.
4. Spelta, Pagnottoni & Pecora (JRSS-A 2025) — DFM-EM + Sinkhorn loss nowcasting. DOI 10.1093/jrsssa/qnaf036.
5. Lee et al. 2305.15196 — strongest adjacent forecasting+Sinkhorn deep baseline.

Open questions

1. Is Sinkhorn best as an EM augmentation (Spelta), a direct observation loss in gradient-based state-space learning, or only an evaluation metric?
2. How to define the ground metric on heterogeneous financial panels (scales, signs, mixed frequencies)?
3. Computational scaling: Sinkhorn per EM iteration vs large \(N\) panels — when does it dominate Kalman cost?
4. Does distributional matching help tail events / crises, or only average nowcast RMSE?
5. Identification: can Sinkhorn over-smooth factor dynamics and hurt multi-step forecasts?
6. Replication: open re-implementation of Spelta DFM+Sinkhorn on a generic public panel (e.g. FRED-MD style) under strict vintage rules.

Tooling notes (this run)

• OpenAlex / feynman rank and raw arXiv API hit 429 earlier in the workflow; abs HTML cache + arxiv_hits.json + priority_brief.md were the reliable literature substrate.
• Jev Ultrafast successfully opened the Spelta JRSS-A landing page — critical because the true ∩ paper was journal-first, not in the arXiv brief.
• Feynman (GLM via NVIDIA NIM) was used for one-shot synthesis from the brief head.