WP41: main study results (2024), pairwise/subgroup/stress/economic analyses, phase 5 report, results section

Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>

Simon-Pierre Boucher committed 18 h ago (Nov 3, 2026) parent 015a1d5 Browse files

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++# PHASE 4 — Scale: profiling, feasible Base architecture, main ingestion, frozen test protocol (2026-11-02)
++
++## Profiling (`scripts/benchmark.py`, M3U96b, Apple M3 Ultra, torch 2.15.0 MPS, FP32, real pilot grids)
++Data loader (batch 64, memory-mapped grids, spawned workers, after warm-up): 3 971 ex/s (0 workers), 13 016 (4), 38 486 (8)
++→ the loader is never the bottleneck (training consumes < 800 ex/s).
++
++| Preset | Parameters (measured) | Batch | Steps/s | Examples/s | Peak MPS memory | Inference ms/origin (batched) |
++|---|---|---|---|---|---|---|
++| Small (d192, 4L) | 3 455 619 | 32 / 64 / 128 | 20.8 / 12.2 / 5.9 | 666 / 782 / 758 | 54 / 69 / 97 MB | 0.34 / 0.25 / 0.27 |
++| Spec (d256, 6L, the brief) | 7 677 955 | 32 / 64 / 128 | 11.5 / 6.3 / 3.0 | 369 / 402 / 380 | 114 / 119 / 147 MB | 0.59 / 0.48 / 0.53 |
++| Base (d384, 8L) | 20 718 339 | 32 / 64 / 128 | 6.1 / 3.2 / 1.5 | 195 / 207 / 193 | 275 / 284 / 315 MB | 5.0 / 1.6 / 1.8 |
++Batch-1 latency 4.3 / 5.1 / 5.9 ms. Losses finite in FP32 at every setting. Mixed precision was not enabled (no stability
++study yet). Memory is tiny relative to the 96 GB node; batch 64 is the throughput optimum for all presets.
++
++## Decision: Base = 20.7 M parameters (inside the 15–30 M target); Small = 3.5 M (inside 3–5 M)
++Budget at batch 64 for the frozen main protocol (12 000 steps = 768 000 examples ≈ 1.3 passes over the 600 k training
++origins drawn at 2 per instrument-session): Small ≈ 16 min, Spec ≈ 32 min, Base ≈ 62 min of pure training plus
++validation passes (60 batches every 1 000 steps) and prediction (≈ 1.45 M evaluation origins at stride 15 → ≈ 6 min
++per Small model, ≈ 40 min for Base at 1.6 ms/origin). The 50–100 M "Large" preset is not trained in v1.0
++(no scaling evidence yet; documented limitation).
++
++## Main ingestion (completed before this phase, `docs/PHASE_1_REPORT.md`)
++300 instruments × 2018–2025: 233 958 776 rows (220 142 835 regular hours), 3.5 GB Parquet, 414 s of exports, 19 s of
++grids; disk usage ≈ 8 GB of 423 GB free.
++
++## Frozen test protocol (`configs/main.yaml`, `configs/final.yaml`)
++- Train ≤ 2022-12-31, validation 2023, development test 2024 (model selection and ablations), final 2025 untouched
++  (run once with the main-study checkpoints, `--models-from results/metrics/main`).
++- Evaluation origins every 15 minutes from 10:00 (24 per normal session); 200 Monte Carlo draws per origin.
++- Held-out instruments (60) evaluated zero-shot with the pooled checkpoints.
++- Ablations (`configs/ablations.yaml`): Small scale, 4 000 steps for every variant **and** for its reference
++  (`mrpt_small_b4k`), seed 1 (+ seeds 2, 3 for the reference).
++- Statistics: day-cluster bootstrap, HAC Diebold–Mariano with lag = overlap × instruments, Holm/BH.
++
++## Go / no-go
++GO to P5 (`main_chain` launched on M3U96b 2026-11-02 18:5x ET: benchmark → `configs/main.yaml`).

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++# PHASE 5 — Main experiments (development test 2024) — 2026-11-02/03
++
++## Run (`configs/main.yaml`, M3U96b, launched 19:02 ET, finished 23:06 ET; `results/logs/main_chain.out`)
++- 240 pooled instruments, 2018-01-02 → 2025-12-31 grids; origins: train 586 202 (2 per instrument-session, 2018–2022),
++  validation 1 040 345 (2023, stride 15), **test 986 468 (2024, stride 15)**, final 1 008 425 (2025, untouched).
++- Baselines stage: origin frames 153 s (test) + train; fits + scoring 1 581 s total for 8 baselines.
++- Neural stage (batch 64, 12 000 steps = 768 000 examples, seed 1, early stopping on validation NLL every 1 000 steps, patience 4):
++
++| Model | Params | Train+predict wall | Best val NLL | Examples/s (train, incl. validation) |
++|---|---|---|---|---|
++| MRPT-S (A+B+C) | 3 455 619 | 2 372 s | −6.4535 | ≈ 450 |
++| MRPT-S (A only) | 3 200 259 | 1 509 s | −6.4520 | ≈ 600 |
++| MRPT-B (A+B+C) | 20 718 339 | 6 684 s | −6.4526 | ≈ 180 |
++| TCN (A) | 231 459 | 1 329 s | −6.4518 | |
++| MLP (A, last 60 bars) | 301 827 | 1 219 s | −6.4497 | |
++All five reached the full 12 000 steps (no early stop); validation NLL plateaued from ≈ 7 000 steps for MRPT-S.
++
++## Headline scores (scaled CRPS of the cumulative return; day-cluster 95 % CI; `leaderboard_test.csv`)
++| Model | H=5 | H=15 | H=30 | H=60 | Δ vs random walk (H=5 / 60) |
++|---|---|---|---|---|---|
++| Random walk | 0.4699 [0.4648, 0.4754] | 0.4575 | 0.4469 | 0.4347 [0.4271, 0.4441] | — |
++| EWMA-t | 0.4547 | 0.4425 | 0.4326 | 0.4214 | −3.65 % / −3.47 % |
++| GARCH(1,1)-t | 0.4558 | 0.4430 | 0.4333 | 0.4232 | −3.60 % / −3.40 % |
++| HAR-RV | 0.4615 | 0.4403 | 0.4295 | **0.4166** | −2.19 % / **−4.89 %** |
++| Ridge / elastic net | 0.4603 | 0.4405 | 0.4297 | 0.4167 | −2.88 % / −4.89 % |
++| LightGBM | 0.4591 | 0.4415 | 0.4293 | 0.4166 | −2.88 % / −4.85 % |
++| MLP (1m) | 0.4538 | 0.4421 | 0.4319 | 0.4197 | −3.94 % / −4.07 % |
++| TCN (1m) | 0.4538 | 0.4427 | 0.4327 | 0.4207 | −3.93 % / −3.73 % |
++| MRPT-S 1m only | 0.4533 | 0.4416 | 0.4314 | 0.4192 | −4.02 % / −4.17 % |
++| **MRPT-S multi-res.** | **0.4531** | **0.4414** | **0.4311** | 0.4187 | −4.05 % / −4.27 % |
++| MRPT-B multi-res. | 0.4532 | 0.4414 | 0.4312 | 0.4189 | −4.04 % / −4.23 % |
++Every model beats the random walk (DM t between −25 and −72, all p < 1e-10 after Holm).
++
++## Pre-specified contrasts (`pairwise_test.csv`; relative CRPS change, DM-HAC t, Holm-adjusted p)
++| Contrast | H=5 | H=15 | H=30 | H=60 |
++|---|---|---|---|---|
++| **H1** MRPT-S multi-res vs 1m-only | −0.04 % (t −5.3) | −0.04 % (t −4.5) | −0.05 % (t −4.5) | −0.10 % (t −5.3) |
++| MRPT-B vs MRPT-S | +0.01 % (t 1.6, p 0.35) | −0.01 % (t −1.3) | −0.01 % (t −0.8) | +0.04 % (t 2.3, p 0.10) |
++| MRPT-S vs HAR-RV | −1.91 % (t −58) | +0.34 % (t 12) | +0.53 % (t 12) | +0.65 % (t 10) |
++| MRPT-S vs GARCH-t | −0.47 % (t −16) | −0.27 % (t −9) | −0.37 % (t −11) | −0.90 % (t −19) |
++| MRPT-S vs EWMA-t | −0.42 % | −0.35 % | −0.47 % | −0.83 % |
++| MRPT-S vs LightGBM | −1.21 % (t −22) | +0.09 % (t 2.0, p 0.19) | +0.53 % (t 12) | +0.61 % (t 10) |
++| MRPT-S vs TCN | −0.13 % | −0.29 % | −0.39 % | −0.56 % |
++| MRPT-S vs MLP | −0.12 % | −0.11 % | −0.12 % | −0.21 % |
++Reading: H1 is statistically detectable but economically negligible (a tenth of a percent of CRPS at 60 minutes).
++Model size does not matter (Base ≈ Small). The neural family wins at 5 minutes; the feature-based volatility
++regressions (HAR-RV, ridge, LightGBM) win from 15 minutes on, by 0.3–0.7 %.
++
++## Calibration (`leaderboard_test.csv`, `stress_calibration_test.csv`)
++Neural models and GARCH/EWMA are over-dispersed in the centre (50 % coverage 0.55–0.59, PIT KS 0.03–0.05) and
++near nominal at 95 %; HAR-RV / ridge / LightGBM are close to nominal from 15 minutes (0.49 / 0.80 / 0.96 at 60 min,
++KS 0.008) but badly under-dispersed at 5 minutes (50 % coverage 0.37, 95 % coverage 0.90). On stress days (top
++decile of cross-sectional σ_t, 100 846 origins) MRPT-S keeps 0.57 / 0.85 / 0.97 coverage (normal days 0.58 / 0.85 / 0.96):
++calibration does not deteriorate under stress, but the regime-conditioning hypothesis (H3) is tested in the ablations.
++
++## Subgroups (MRPT-S vs HAR-RV at H=60, `subgroups_mrpt_small_vs_har_rv_test.csv`)
++HAR-RV's advantage concentrates in low-ADV stocks (+1.3 %), low-volatility instruments (+1.0 %), transport/utilities
++(+1.9 %), fixed-income ETFs (+1.8 %) and the "unknown"-sector group (+2.8 %); MRPT-S is better on broad-equity,
++sector and volatility ETFs (−0.3 to −1.1 %) and on high-ADV instruments (−0.03 %, n.s.). Afternoon origins favour HAR (+1.0 %).
++
++## Economic content (H5, exploratory; `economic_test.csv`)
++Correlation between the predictive median and the realised 60-minute return: 0.003 (MRPT-S), 0.025 (MRPT-B), 0.014 (TCN),
++0.007 (MLP); regression slopes between −1 and 1.2; the median's typical magnitude is 0.06–0.7 bp. With a 1 bp
++round-trip cost MRPT-S trades 17 % of origins at 60 minutes for a net −0.00 bp [−1.7, +1.8]; with 2.5–10 bp costs the
++traded share collapses below 4 % and every net return interval contains zero. **No evidence of economic content.**
++
++## Artifacts
++`results/metrics/main/`: `summary_*_test.json`, `per_origin_*_test.npz` (node + laptop, not committed), `leaderboard_test.csv`,
++`compare_{crps,wis,qlike}_vs_naive_rw_test.csv`, `pairwise_test.csv`, `subgroups_*.csv`, `stress_calibration_test.csv`,
++`economic_test.csv`, `<model>/train_log.jsonl`, `train_summary.json`, `best.pt` (node only). Tables and figures:
++`paper/tables/main_*.tex`, `paper/figures/fig_main_*`.

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++\newcommand{\pwmainmrptsmallvstcnHsixzero}{-0.56}
++\newcommand{\tpwmainmrptsmallvstcnHsixzero}{-23.4}
++\newcommand{\ppwmainmrptsmallvstcnHsixzero}{0}
++\newcommand{\pwmainmrptsmallvsmlpHfive}{-0.12}
++\newcommand{\tpwmainmrptsmallvsmlpHfive}{-8.0}
++\newcommand{\ppwmainmrptsmallvsmlpHfive}{2.02e-14}
++\newcommand{\pwmainmrptsmallvsmlpHonefive}{-0.11}
++\newcommand{\tpwmainmrptsmallvsmlpHonefive}{-7.4}
++\newcommand{\ppwmainmrptsmallvsmlpHonefive}{1.73e-12}
++\newcommand{\pwmainmrptsmallvsmlpHthreezero}{-0.12}
++\newcommand{\tpwmainmrptsmallvsmlpHthreezero}{-6.0}
++\newcommand{\ppwmainmrptsmallvsmlpHthreezero}{1.48e-08}
++\newcommand{\pwmainmrptsmallvsmlpHsixzero}{-0.21}
++\newcommand{\tpwmainmrptsmallvsmlpHsixzero}{-7.0}
++\newcommand{\ppwmainmrptsmallvsmlpHsixzero}{3.27e-11}
++\newcommand{\pwmainharrvvsgarchtHfive}{+1.46}
++\newcommand{\tpwmainharrvvsgarchtHfive}{41.9}
++\newcommand{\ppwmainharrvvsgarchtHfive}{0}
++\newcommand{\pwmainharrvvsgarchtHonefive}{-0.61}
++\newcommand{\tpwmainharrvvsgarchtHonefive}{-22.0}
++\newcommand{\ppwmainharrvvsgarchtHonefive}{0}
++\newcommand{\pwmainharrvvsgarchtHthreezero}{-0.89}
++\newcommand{\tpwmainharrvvsgarchtHthreezero}{-21.4}
++\newcommand{\ppwmainharrvvsgarchtHthreezero}{0}
++\newcommand{\pwmainharrvvsgarchtHsixzero}{-1.54}
++\newcommand{\tpwmainharrvvsgarchtHsixzero}{-23.7}
++\newcommand{\ppwmainharrvvsgarchtHsixzero}{0}

modified paper/sections/results.tex +95 −1

++\section{Results}
++\label{sec:results}
++
++\subsection{Pilot}
++The pilot (20 instruments, September--December 2022, test on the last ten sessions, \num{4659} origins) validated
++the pipeline from export to LaTeX table and gave a first, deliberately unweighted, reading: every volatility-aware
++model improved on the random walk by 2--3\,\% in CRPS and the multi-resolution MRPT, its one-minute-only variant, a
++TCN and an MLP were within 0.2\,\% of one another (Table~\ref{tab:pilot_crps_scaled}, Appendix). Nothing was tuned on it.
++
++\subsection{Development test (2024): all models beat the random walk, by about four percent}
++Table~\ref{tab:main_crps_scaled} reports the scaled CRPS of the cumulative return on the \num{986468} evaluation
++origins of 2024 (240 pooled instruments, 24 origins per session), with trading-day cluster-bootstrap intervals, and
++Table~\ref{tab:main_compare_crps} the relative change against the random walk with its Diebold--Mariano statistic.
++Figure~\ref{fig:main_crps} summarises both. Three facts stand out.
++
++First, the random walk with a trailing one-minute scale is beaten by every model at every horizon, by
++3.5--4.9\,\%, with DM statistics between $-25$ and $-72$: minute-level history carries real information about the
++\emph{dispersion} of the next hour, and the improvement is essentially a volatility improvement (directional
++accuracy of every location forecast stays at 0.50).
++
++Second, the neural family (MRPT-S, MRPT-B, TCN, MLP) is the best at five minutes
++(scaled CRPS \crpsmainmrptsmallHfive{} for MRPT-S against \crpsmainharrvHfive{} for HAR-RV, a difference of
++\pwmainmrptsmallvsharrvHfive{}\,\% with $t=\tpwmainmrptsmallvsharrvHfive{}$), whereas the feature-based volatility
++regressions---HAR-RV, ridge, elastic net and LightGBM---are the best from fifteen minutes on
++(\crpsmainharrvHsixzero{} against \crpsmainmrptsmallHsixzero{} at sixty minutes; MRPT-S is worse by
++\pwmainmrptsmallvsharrvHsixzero{}\,\%, $t=\tpwmainmrptsmallvsharrvHsixzero{}$). MRPT-S beats GARCH(1,1)-$t$, EWMA, the TCN
++and the MLP at every horizon (by 0.1--0.9\,\%, all Holm-adjusted $p<10^{-6}$).
++
++Third, and central to the paper, \textbf{Hypothesis~\ref{h1} is rejected in economic terms}: the multi-resolution
++MRPT-S improves on its one-minute-only twin by \pwmainmrptsmallvsmrptsmallonemonlyHfive{}\,\% at five minutes and
++\pwmainmrptsmallvsmrptsmallonemonlyHsixzero{}\,\% at sixty minutes. The differences are statistically detectable
++($t=\tpwmainmrptsmallvsmrptsmallonemonlyHsixzero{}$ at $H=60$, \num{986468} origins over 252 days) but they are an order
++of magnitude smaller than the gap between any neural model and the random walk, and smaller than the gap between
++the neural models and HAR-RV at sixty minutes. Five days of 5-minute aggregates and twenty days of 30-minute
++aggregates add almost nothing that the last 390 one-minute bars do not already contain. Size does not matter
++either: the 20.7-million-parameter Base model and the 3.5-million-parameter Small model are indistinguishable
++(\pwmainmrptbasevsmrptsmallHsixzero{}\,\% at $H=60$, Holm $p=\ppwmainmrptbasevsmrptsmallHsixzero{}$).
++
++
++\begin{figure}[htbp]\centering
++\includegraphics[width=\textwidth]{figures/fig_main_crps.pdf}
++\caption{Development test 2024 (240 instruments, \num{986468} origins). Panel A: scaled CRPS of the cumulative-return
++forecast by horizon. Panel B: relative change versus the random walk. Source: \texttt{results/metrics/main/leaderboard\_test.csv}.}
++\label{fig:main_crps}
++\end{figure}
++
++\begin{figure}[htbp]\centering
++\includegraphics[width=\textwidth]{figures/fig_main_calibration.pdf}
++\caption{Empirical coverage of the central 50\,\%, 80\,\% and 95\,\% prediction intervals by horizon (dotted line = nominal), development test 2024.}
++\label{fig:main_calibration}
++\end{figure}
++
++\begin{figure}[htbp]\centering
++\includegraphics[width=\textwidth]{figures/fig_main_pit.pdf}
++\caption{PIT histograms at $H=60$ for the random walk, GARCH(1,1)-$t$, LightGBM and MRPT-S (dashed line = uniform density).}
++\label{fig:main_pit}
++\end{figure}
++
++\begin{figure}[htbp]\centering
++\includegraphics[width=\textwidth]{figures/fig_main_training.pdf}
++\caption{Training (moving average) and validation negative log-likelihood per optimiser step, main study.}
++\label{fig:main_training}
++\end{figure}
++
++\subsection{Calibration}
++Table~\ref{tab:main_calibration} and Figure~\ref{fig:main_calibration} show empirical coverage of the 50/80/95\,\%
++central intervals and the PIT Kolmogorov--Smirnov statistic. The random walk is over-dispersed everywhere (50\,\%
++intervals cover \covfiftymainnaiverwHsixzero{}\,\% of outcomes at sixty minutes). The Student-$t$ neural models and
++GARCH/EWMA are nearly nominal at 95\,\% but over-dispersed in the centre (50\,\% coverage 0.55--0.59): their
++degrees of freedom and scale produce tails that are right and a body that is too wide. HAR-RV, ridge and LightGBM
++are the best-calibrated models from fifteen minutes on (0.49/0.80/0.96 at sixty minutes, KS 0.008) and the worst at
++five minutes (50\,\% coverage 0.37, 95\,\% coverage 0.90): a lognormal variance regression on 390-minute features
++cannot react to the last five minutes. PIT histograms (Figure~\ref{fig:main_pit}) make the same point visually.
++Scores and calibration thus tell one consistent story: at the shortest horizon the sequence models' ability to
++read the very last bars wins; at longer horizons a well-specified heterogeneous variance regression with a
++kurtosis-matched Student-$t$ is both sharper and better calibrated.
++
++\subsection{Subgroups}
++Splitting the MRPT-S versus HAR-RV contrast at sixty minutes by subgroup, HAR-RV's advantage is concentrated in
++low-dollar-volume stocks ($+1.3$\,\%), low-volatility instruments ($+1.0$\,\%), transport and utilities ($+1.9$\,\%),
++fixed-income ETFs ($+1.8$\,\%) and the stocks without a SIC code ($+2.8$\,\%); MRPT-S is better on broad-equity,
++sector and volatility-linked ETFs ($-0.3$ to $-1.1$\,\%) and statistically tied on high-dollar-volume instruments.
++Afternoon origins favour HAR-RV ($+1.0$\,\%), morning origins much less ($+0.2$\,\%). On stress days (top decile of
++cross-sectional $\sigma_t$, \num{100846} origins) the ordering is unchanged and every model's coverage moves by at
++most one percentage point: none of the models breaks under stress, and none gains a special advantage there.
++
++\subsection{Economic content (Hypothesis~\ref{h5})}
++The predictive medians carry no usable directional information. Their correlation with the realised sixty-minute
++return is 0.003 (MRPT-S), 0.025 (MRPT-B), 0.014 (TCN) and 0.007 (MLP), the typical magnitude of the median is below
++one basis point, and a sign rule that trades only when the median exceeds an assumed round-trip cost of 1 basis
++point trades 17\,\% of the origins for a net return of $0.0$ bp $[-1.7, +1.8]$; at 2.5--10 bp the traded share
++falls below 4\,\% and every interval contains zero. We report this null without further sensitivity analysis.
++
++\subsection{Zero-shot transfer and the untouched 2025 period}
++\IfFileExists{sections/results_heldout.tex}{\input{sections/results_heldout}}{\emph{(Held-out and final-period results are added when the runs complete.)}}

added paper/tables/benchmark.tex +21 −0

++\begin{table}[htbp]\centering\small
++\caption{Measured training throughput and memory on the compute node (Apple M3 Ultra{,} m3u96b{,} torch 2.15.0{,} mps; FP32)}
++\label{tab:benchmark}
++\begin{tabular}{lrrrrrr}
++\toprule
++Preset & Batch & Parameters & Steps/s & Examples/s & Peak memory (MB) & Latency/origin (ms) \\
++\midrule
++small & 32 & 3{,}455{,}619 & 20.81 & 666 & 54 & 0.34 \\
++small & 64 & 3{,}455{,}619 & 12.21 & 782 & 69 & 0.25 \\
++small & 128 & 3{,}455{,}619 & 5.93 & 758 & 97 & 0.27 \\
++spec & 32 & 7{,}677{,}955 & 11.54 & 369 & 114 & 0.59 \\
++spec & 64 & 7{,}677{,}955 & 6.29 & 402 & 119 & 0.48 \\
++spec & 128 & 7{,}677{,}955 & 2.97 & 380 & 147 & 0.53 \\
++base & 32 & 20{,}718{,}339 & 6.11 & 195 & 275 & 5.04 \\
++base & 64 & 20{,}718{,}339 & 3.24 & 207 & 284 & 1.59 \\
++base & 128 & 20{,}718{,}339 & 1.51 & 193 & 315 & 1.76 \\
++\midrule
++Data loader (64/batch) & 0 workers: 3971 ex/s; 4 workers: 13016 ex/s; 8 workers: 38486 ex/s & & & & & \\
++\bottomrule
++\end{tabular}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_calibration.tex +24 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: empirical 50/80/95\% coverage and PIT KS statistic}
++\label{tab:main_calib}
++\begin{tabular}{lcccccccccccccccc}
++\toprule
++Model & \multicolumn{4}{c}{$H=5$} & \multicolumn{4}{c}{$H=15$} & \multicolumn{4}{c}{$H=30$} & \multicolumn{4}{c}{$H=60$} \\
++ & 50 & 80 & 95 & KS & 50 & 80 & 95 & KS & 50 & 80 & 95 & KS & 50 & 80 & 95 & KS \\
++\midrule
++Random walk (trailing RMS) & 0.65 & 0.88 & 0.97 & 0.078 & 0.66 & 0.89 & 0.97 & 0.079 & 0.67 & 0.90 & 0.97 & 0.083 & 0.68 & 0.91 & 0.98 & 0.089 \\
++Seasonal volatility & 0.56 & 0.82 & 0.94 & 0.031 & 0.56 & 0.83 & 0.94 & 0.031 & 0.56 & 0.83 & 0.95 & 0.033 & 0.57 & 0.83 & 0.95 & 0.034 \\
++EWMA-$t$ & 0.54 & 0.83 & 0.97 & 0.023 & 0.54 & 0.84 & 0.97 & 0.024 & 0.55 & 0.85 & 0.97 & 0.028 & 0.56 & 0.86 & 0.97 & 0.034 \\
++HAR-RV & 0.60 & 0.89 & 0.99 & 0.058 & 0.50 & 0.81 & 0.96 & 0.008 & 0.48 & 0.80 & 0.96 & 0.009 & 0.49 & 0.80 & 0.96 & 0.008 \\
++GARCH(1,1)-$t$ & 0.53 & 0.83 & 0.97 & 0.021 & 0.55 & 0.85 & 0.97 & 0.027 & 0.56 & 0.86 & 0.98 & 0.035 & 0.59 & 0.88 & 0.98 & 0.047 \\
++Ridge & 0.37 & 0.67 & 0.90 & 0.076 & 0.47 & 0.77 & 0.95 & 0.018 & 0.48 & 0.78 & 0.95 & 0.015 & 0.49 & 0.80 & 0.96 & 0.009 \\
++Elastic net & 0.37 & 0.67 & 0.90 & 0.076 & 0.47 & 0.77 & 0.95 & 0.018 & 0.48 & 0.78 & 0.95 & 0.014 & 0.49 & 0.80 & 0.96 & 0.009 \\
++LightGBM & 0.38 & 0.67 & 0.90 & 0.073 & 0.44 & 0.75 & 0.94 & 0.036 & 0.49 & 0.79 & 0.95 & 0.010 & 0.50 & 0.80 & 0.96 & 0.008 \\
++MLP (1m) & 0.55 & 0.83 & 0.96 & 0.029 & 0.57 & 0.84 & 0.96 & 0.034 & 0.58 & 0.85 & 0.96 & 0.039 & 0.58 & 0.86 & 0.96 & 0.043 \\
++TCN (1m) & 0.56 & 0.84 & 0.96 & 0.032 & 0.57 & 0.85 & 0.96 & 0.037 & 0.58 & 0.85 & 0.96 & 0.042 & 0.59 & 0.86 & 0.96 & 0.046 \\
++MRPT-S (1m+5m+30m) & 0.55 & 0.83 & 0.96 & 0.029 & 0.56 & 0.84 & 0.96 & 0.033 & 0.57 & 0.85 & 0.96 & 0.037 & 0.58 & 0.85 & 0.96 & 0.040 \\
++MRPT-S (1m only) & 0.55 & 0.83 & 0.96 & 0.030 & 0.56 & 0.84 & 0.96 & 0.033 & 0.57 & 0.85 & 0.96 & 0.037 & 0.58 & 0.85 & 0.96 & 0.041 \\
++MRPT-B (1m+5m+30m) & 0.55 & 0.83 & 0.96 & 0.029 & 0.56 & 0.84 & 0.96 & 0.033 & 0.57 & 0.85 & 0.96 & 0.037 & 0.58 & 0.85 & 0.96 & 0.041 \\
++\bottomrule
++\end{tabular}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_compare_crps.tex +23 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: CRPS relative to Random walk (trailing RMS)}
++\label{tab:main_cmp_crps}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Seasonal volatility & -2.89\% (-48.8) & -3.09\% (-49.9) & -3.22\% (-39.3) & -3.44\% (-30.7) \\
++EWMA-$t$ & -3.65\% (-57.2) & -3.69\% (-53.4) & -3.60\% (-37.3) & -3.47\% (-25.5) \\
++HAR-RV & -2.19\% (-35.7) & -4.36\% (-64.1) & -4.56\% (-47.4) & -4.89\% (-36.6) \\
++GARCH(1,1)-$t$ & -3.60\% (-57.1) & -3.77\% (-56.0) & -3.70\% (-40.5) & -3.40\% (-27.9) \\
++Ridge & -2.88\% (-33.2) & -4.35\% (-58.3) & -4.55\% (-45.9) & -4.89\% (-36.2) \\
++Elastic net & -2.88\% (-33.2) & -4.35\% (-58.2) & -4.56\% (-46.1) & -4.89\% (-36.2) \\
++LightGBM & -2.88\% (-33.4) & -4.12\% (-51.1) & -4.56\% (-47.4) & -4.85\% (-37.6) \\
++MLP (1m) & -3.94\% (-71.6) & -3.93\% (-68.6) & -3.94\% (-51.5) & -4.07\% (-38.7) \\
++TCN (1m) & -3.93\% (-69.2) & -3.75\% (-65.1) & -3.68\% (-48.7) & -3.73\% (-37.3) \\
++MRPT-S (1m+5m+30m) & -4.05\% (-69.3) & -4.03\% (-67.0) & -4.05\% (-50.6) & -4.27\% (-39.7) \\
++MRPT-S (1m only) & -4.02\% (-69.2) & -4.00\% (-66.9) & -4.00\% (-50.4) & -4.17\% (-39.6) \\
++MRPT-B (1m+5m+30m) & -4.04\% (-69.5) & -4.04\% (-67.2) & -4.06\% (-51.3) & -4.23\% (-39.8) \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Relative change in mean CRPS versus the reference (negative = better); Diebold--Mariano $t$-statistic with HAC variance in parentheses.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_compare_qlike.tex +23 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: QLIKE relative to Random walk (trailing RMS)}
++\label{tab:main_cmp_qlike}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Seasonal volatility & -8472593190631.71\% (-26.5) & -8305269577503.39\% (-23.8) & -7325382995932.99\% (-14.1) & -6019653909459.67\% (-7.4) \\
++EWMA-$t$ & -12797135612615.71\% (-41.9) & -10996225269347.05\% (-30.9) & -8956891977333.71\% (-18.6) & -6156613654049.44\% (-9.7) \\
++HAR-RV & +1922904424583.55\% (4.0) & -11677912373599.43\% (-9.2) & -11280891155603.60\% (-11.2) & -11508236846123.03\% (-17.7) \\
++GARCH(1,1)-$t$ & -3792274777371.91\% (-4.2) & -2936683613506.50\% (-4.8) & -1014981889598.51\% (-1.7) & +2553431567866.42\% (4.3) \\
++Ridge & +5434479946610.18\% (7.4) & -12177565285940.69\% (-21.4) & -12517051234684.73\% (-20.7) & -12281264246708.48\% (-17.5) \\
++Elastic net & +5514663565784.96\% (7.5) & -12129129818422.78\% (-21.2) & -12468574596082.83\% (-20.4) & -12234566499031.67\% (-17.2) \\
++LightGBM & +1814267205302873.75\% (4.8) & +11471871421175.42\% (0.8) & -13278787159454.87\% (-25.8) & -13466213447329.12\% (-26.1) \\
++MLP (1m) & -15818212313424.51\% (-41.8) & -14508869311135.06\% (-36.7) & -13489944655719.03\% (-28.8) & -12445594328523.75\% (-19.9) \\
++TCN (1m) & -16924042856514.05\% (-63.3) & -15020597397889.10\% (-35.3) & -13794157215647.81\% (-23.8) & -13055303736382.31\% (-25.8) \\
++MRPT-S (1m+5m+30m) & -17518582839370.88\% (-61.1) & -16395858212306.31\% (-59.2) & -15469510096077.54\% (-47.9) & -14813941021870.59\% (-38.0) \\
++MRPT-S (1m only) & -17184931027568.06\% (-60.5) & -15902869282831.41\% (-45.2) & -14857414727344.76\% (-34.7) & -14011901777049.01\% (-28.0) \\
++MRPT-B (1m+5m+30m) & -17153971110963.66\% (-60.7) & -16037277205437.76\% (-48.1) & -15081771195336.03\% (-37.9) & -14340084012486.90\% (-30.2) \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Relative change in mean QLIKE versus the reference (negative = better); Diebold--Mariano $t$-statistic with HAC variance in parentheses.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_compare_wis.tex +23 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: WIS relative to Random walk (trailing RMS)}
++\label{tab:main_cmp_wis}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Seasonal volatility & -3.00\% (-46.4) & -3.20\% (-47.8) & -3.33\% (-37.8) & -3.54\% (-29.1) \\
++EWMA-$t$ & -3.75\% (-54.5) & -3.75\% (-50.4) & -3.60\% (-34.8) & -3.40\% (-23.3) \\
++HAR-RV & -2.00\% (-29.7) & -4.49\% (-62.7) & -4.67\% (-46.4) & -4.99\% (-35.5) \\
++GARCH(1,1)-$t$ & -3.66\% (-54.1) & -3.79\% (-52.8) & -3.67\% (-37.7) & -3.29\% (-25.3) \\
++Ridge & -2.83\% (-29.9) & -4.50\% (-56.7) & -4.68\% (-44.7) & -5.02\% (-35.0) \\
++Elastic net & -2.83\% (-29.8) & -4.49\% (-56.5) & -4.69\% (-44.9) & -5.02\% (-35.0) \\
++LightGBM & -2.79\% (-30.1) & -4.26\% (-50.5) & -4.72\% (-46.8) & -5.01\% (-36.9) \\
++MLP (1m) & -4.17\% (-71.5) & -4.15\% (-68.5) & -4.15\% (-51.3) & -4.28\% (-38.1) \\
++TCN (1m) & -4.16\% (-68.4) & -3.96\% (-64.5) & -3.88\% (-48.4) & -3.92\% (-36.8) \\
++MRPT-S (1m+5m+30m) & -4.30\% (-68.4) & -4.28\% (-66.6) & -4.28\% (-50.4) & -4.50\% (-39.2) \\
++MRPT-S (1m only) & -4.26\% (-68.4) & -4.23\% (-66.3) & -4.22\% (-50.0) & -4.38\% (-38.8) \\
++MRPT-B (1m+5m+30m) & -4.28\% (-68.8) & -4.28\% (-66.7) & -4.29\% (-51.0) & -4.44\% (-39.3) \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Relative change in mean WIS versus the reference (negative = better); Diebold--Mariano $t$-statistic with HAC variance in parentheses.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_crps_scaled.tex +24 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: mean scaled CRPS of the cumulative-return forecast by horizon (minutes)}
++\label{tab:main_crps_scaled}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Random walk (trailing RMS) & 0.4699 {\scriptsize[0.4648, 0.4754]} & 0.4575 {\scriptsize[0.4515, 0.4639]} & 0.4469 {\scriptsize[0.4412, 0.4538]} & 0.4347 {\scriptsize[0.4271, 0.4441]} \\
++Seasonal volatility & 0.4582 {\scriptsize[0.4527, 0.4641]} & 0.4453 {\scriptsize[0.4390, 0.4519]} & 0.4347 {\scriptsize[0.4285, 0.4418]} & 0.4221 {\scriptsize[0.4139, 0.4320]} \\
++EWMA-$t$ & 0.4547 {\scriptsize[0.4492, 0.4606]} & 0.4425 {\scriptsize[0.4363, 0.4491]} & 0.4326 {\scriptsize[0.4264, 0.4396]} & 0.4214 {\scriptsize[0.4133, 0.4314]} \\
++HAR-RV & 0.4615 {\scriptsize[0.4563, 0.4671]} & 0.4403 {\scriptsize[0.4338, 0.4471]} & 0.4295 {\scriptsize[0.4230, 0.4369]} & 0.4166 {\scriptsize[0.4080, 0.4270]} \\
++GARCH(1,1)-$t$ & 0.4558 {\scriptsize[0.4502, 0.4616]} & 0.4430 {\scriptsize[0.4367, 0.4497]} & 0.4333 {\scriptsize[0.4271, 0.4403]} & 0.4232 {\scriptsize[0.4153, 0.4328]} \\
++Ridge & 0.4603 {\scriptsize[0.4542, 0.4667]} & 0.4405 {\scriptsize[0.4339, 0.4476]} & 0.4297 {\scriptsize[0.4231, 0.4371]} & 0.4167 {\scriptsize[0.4081, 0.4272]} \\
++Elastic net & 0.4603 {\scriptsize[0.4543, 0.4668]} & 0.4406 {\scriptsize[0.4339, 0.4476]} & 0.4296 {\scriptsize[0.4231, 0.4370]} & 0.4167 {\scriptsize[0.4081, 0.4272]} \\
++LightGBM & 0.4591 {\scriptsize[0.4531, 0.4655]} & 0.4415 {\scriptsize[0.4347, 0.4484]} & 0.4293 {\scriptsize[0.4228, 0.4366]} & 0.4166 {\scriptsize[0.4080, 0.4269]} \\
++MLP (1m) & 0.4538 {\scriptsize[0.4484, 0.4598]} & 0.4421 {\scriptsize[0.4357, 0.4486]} & 0.4319 {\scriptsize[0.4258, 0.4388]} & 0.4197 {\scriptsize[0.4116, 0.4291]} \\
++TCN (1m) & 0.4538 {\scriptsize[0.4484, 0.4597]} & 0.4427 {\scriptsize[0.4364, 0.4491]} & 0.4327 {\scriptsize[0.4265, 0.4396]} & 0.4207 {\scriptsize[0.4126, 0.4303]} \\
++MRPT-S (1m+5m+30m) & 0.4531 {\scriptsize[0.4476, 0.4590]} & 0.4414 {\scriptsize[0.4351, 0.4479]} & 0.4311 {\scriptsize[0.4249, 0.4381]} & 0.4187 {\scriptsize[0.4105, 0.4284]} \\
++MRPT-S (1m only) & 0.4533 {\scriptsize[0.4478, 0.4592]} & 0.4416 {\scriptsize[0.4352, 0.4481]} & 0.4314 {\scriptsize[0.4251, 0.4382]} & 0.4192 {\scriptsize[0.4110, 0.4290]} \\
++MRPT-B (1m+5m+30m) & 0.4532 {\scriptsize[0.4478, 0.4591]} & 0.4414 {\scriptsize[0.4351, 0.4479]} & 0.4312 {\scriptsize[0.4249, 0.4381]} & 0.4189 {\scriptsize[0.4107, 0.4287]} \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Cluster-bootstrap (trading-day) 95\% intervals in brackets.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_logscore.tex +24 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: mean log score of the cumulative-return forecast by horizon (minutes)}
++\label{tab:main_logscore}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Random walk (trailing RMS) & nan & nan & nan & nan \\
++Seasonal volatility & nan & nan & nan & nan \\
++EWMA-$t$ & -5.3800 {\scriptsize[-5.3983, -5.3611]} & -4.8516 {\scriptsize[-4.8704, -4.8300]} & -4.5201 {\scriptsize[-4.5406, -4.4981]} & -4.1899 {\scriptsize[-4.2130, -4.1628]} \\
++HAR-RV & -5.3419 {\scriptsize[-5.3591, -5.3235]} & -4.8627 {\scriptsize[-4.8820, -4.8408]} & -4.5354 {\scriptsize[-4.5574, -4.5123]} & -4.2140 {\scriptsize[-4.2385, -4.1858]} \\
++GARCH(1,1)-$t$ & -5.3754 {\scriptsize[-5.3940, -5.3561]} & -4.8509 {\scriptsize[-4.8693, -4.8293]} & -4.5188 {\scriptsize[-4.5389, -4.4974]} & -4.1823 {\scriptsize[-4.2040, -4.1573]} \\
++Ridge & -5.3260 {\scriptsize[-5.3484, -5.3035]} & -4.8632 {\scriptsize[-4.8832, -4.8409]} & -4.5367 {\scriptsize[-4.5587, -4.5128]} & -4.2155 {\scriptsize[-4.2404, -4.1866]} \\
++Elastic net & -5.3261 {\scriptsize[-5.3486, -5.3037]} & -4.8631 {\scriptsize[-4.8831, -4.8407]} & -4.5368 {\scriptsize[-4.5588, -4.5128]} & -4.2156 {\scriptsize[-4.2404, -4.1867]} \\
++LightGBM & -5.3289 {\scriptsize[-5.3506, -5.3063]} & -4.8551 {\scriptsize[-4.8753, -4.8326]} & -4.5411 {\scriptsize[-4.5633, -4.5180]} & -4.2191 {\scriptsize[-4.2434, -4.1914]} \\
++MLP (1m) & nan & nan & nan & nan \\
++TCN (1m) & nan & nan & nan & nan \\
++MRPT-S (1m+5m+30m) & nan & nan & nan & nan \\
++MRPT-S (1m only) & nan & nan & nan & nan \\
++MRPT-B (1m+5m+30m) & nan & nan & nan & nan \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Cluster-bootstrap (trading-day) 95\% intervals in brackets.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_qlike.tex +24 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: mean QLIKE of the cumulative-return forecast by horizon (minutes)}
++\label{tab:main_qlike}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Random walk (trailing RMS) & -12.3011 {\scriptsize[-12.3357, -12.2647]} & -11.2438 {\scriptsize[-11.2774, -11.2053]} & -10.5819 {\scriptsize[-10.6160, -10.5427]} & -9.9314 {\scriptsize[-9.9665, -9.8897]} \\
++Seasonal volatility & -12.3858 {\scriptsize[-12.4188, -12.3489]} & -11.3268 {\scriptsize[-11.3639, -11.2834]} & -10.6551 {\scriptsize[-10.6952, -10.6061]} & -9.9916 {\scriptsize[-10.0361, -9.9323]} \\
++EWMA-$t$ & -12.4291 {\scriptsize[-12.4603, -12.3927]} & -11.3537 {\scriptsize[-11.3907, -11.3114]} & -10.6715 {\scriptsize[-10.7094, -10.6265]} & -9.9930 {\scriptsize[-10.0339, -9.9413]} \\
++HAR-RV & -12.2819 {\scriptsize[-12.3137, -12.2475]} & -11.3605 {\scriptsize[-11.4055, -11.3071]} & -10.6947 {\scriptsize[-10.7386, -10.6459]} & -10.0465 {\scriptsize[-10.0884, -9.9945]} \\
++GARCH(1,1)-$t$ & -12.3390 {\scriptsize[-12.3766, -12.2971]} & -11.2731 {\scriptsize[-11.3120, -11.2289]} & -10.5920 {\scriptsize[-10.6309, -10.5470]} & -9.9059 {\scriptsize[-9.9462, -9.8580]} \\
++Ridge & -12.2468 {\scriptsize[-12.2897, -12.2033]} & -11.3655 {\scriptsize[-11.4057, -11.3170]} & -10.7071 {\scriptsize[-10.7479, -10.6570]} & -10.0542 {\scriptsize[-10.0979, -9.9984]} \\
++Elastic net & -12.2460 {\scriptsize[-12.2890, -12.2026]} & -11.3651 {\scriptsize[-11.4051, -11.3165]} & -10.7066 {\scriptsize[-10.7475, -10.6563]} & -10.0537 {\scriptsize[-10.0974, -9.9978]} \\
++LightGBM & 5.8416 {\scriptsize[-1.1419, 14.3811]} & -11.1290 {\scriptsize[-11.3556, -10.8016]} & -10.7147 {\scriptsize[-10.7553, -10.6689]} & -10.0661 {\scriptsize[-10.1063, -10.0165]} \\
++MLP (1m) & -12.4593 {\scriptsize[-12.4939, -12.4208]} & -11.3889 {\scriptsize[-11.4249, -11.3434]} & -10.7168 {\scriptsize[-10.7542, -10.6698]} & -10.0558 {\scriptsize[-10.0975, -10.0073]} \\
++TCN (1m) & -12.4704 {\scriptsize[-12.5018, -12.4332]} & -11.3940 {\scriptsize[-11.4310, -11.3510]} & -10.7198 {\scriptsize[-10.7598, -10.6741]} & -10.0619 {\scriptsize[-10.1020, -10.0151]} \\
++MRPT-S (1m+5m+30m) & -12.4763 {\scriptsize[-12.5080, -12.4395]} & -11.4077 {\scriptsize[-11.4437, -11.3669]} & -10.7366 {\scriptsize[-10.7742, -10.6941]} & -10.0795 {\scriptsize[-10.1178, -10.0350]} \\
++MRPT-S (1m only) & -12.4730 {\scriptsize[-12.5048, -12.4358]} & -11.4028 {\scriptsize[-11.4405, -11.3594]} & -10.7305 {\scriptsize[-10.7695, -10.6850]} & -10.0715 {\scriptsize[-10.1112, -10.0212]} \\
++MRPT-B (1m+5m+30m) & -12.4727 {\scriptsize[-12.5044, -12.4354]} & -11.4041 {\scriptsize[-11.4413, -11.3610]} & -10.7327 {\scriptsize[-10.7716, -10.6880]} & -10.0748 {\scriptsize[-10.1147, -10.0255]} \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Cluster-bootstrap (trading-day) 95\% intervals in brackets.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/main_wis.tex +24 −0

++\begin{table}[htbp]\centering\small
++\caption{Main: mean WIS of the cumulative-return forecast by horizon (minutes)}
++\label{tab:main_wis}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Random walk (trailing RMS) & 0.0007 {\scriptsize[0.0007, 0.0008]} & 0.0013 {\scriptsize[0.0012, 0.0013]} & 0.0017 {\scriptsize[0.0017, 0.0018]} & 0.0024 {\scriptsize[0.0023, 0.0024]} \\
++Seasonal volatility & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0024]} \\
++EWMA-$t$ & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0024]} \\
++HAR-RV & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++GARCH(1,1)-$t$ & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0024]} \\
++Ridge & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++Elastic net & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++LightGBM & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++MLP (1m) & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++TCN (1m) & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++MRPT-S (1m+5m+30m) & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++MRPT-S (1m only) & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++MRPT-B (1m+5m+30m) & 0.0007 {\scriptsize[0.0007, 0.0007]} & 0.0012 {\scriptsize[0.0012, 0.0012]} & 0.0017 {\scriptsize[0.0016, 0.0017]} & 0.0023 {\scriptsize[0.0022, 0.0023]} \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Cluster-bootstrap (trading-day) 95\% intervals in brackets.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/pilot_calibration.tex +23 −0

++\begin{table}[htbp]\centering\small
++\caption{Pilot: empirical 50/80/95\% coverage and PIT KS statistic}
++\label{tab:pilot_calib}
++\begin{tabular}{lcccccccccccccccc}
++\toprule
++Model & \multicolumn{4}{c}{$H=5$} & \multicolumn{4}{c}{$H=15$} & \multicolumn{4}{c}{$H=30$} & \multicolumn{4}{c}{$H=60$} \\
++ & 50 & 80 & 95 & KS & 50 & 80 & 95 & KS & 50 & 80 & 95 & KS & 50 & 80 & 95 & KS \\
++\midrule
++Random walk (trailing RMS) & 0.62 & 0.87 & 0.96 & 0.070 & 0.61 & 0.87 & 0.96 & 0.063 & 0.60 & 0.86 & 0.96 & 0.068 & 0.61 & 0.87 & 0.97 & 0.075 \\
++Seasonal volatility & 0.53 & 0.80 & 0.94 & 0.029 & 0.53 & 0.80 & 0.94 & 0.025 & 0.52 & 0.79 & 0.93 & 0.026 & 0.52 & 0.79 & 0.93 & 0.039 \\
++EWMA-$t$ & 0.51 & 0.81 & 0.97 & 0.017 & 0.50 & 0.80 & 0.96 & 0.012 & 0.49 & 0.80 & 0.96 & 0.027 & 0.51 & 0.80 & 0.96 & 0.034 \\
++HAR-RV & 0.63 & 0.92 & 0.99 & 0.082 & 0.48 & 0.78 & 0.96 & 0.021 & 0.46 & 0.77 & 0.95 & 0.039 & 0.48 & 0.77 & 0.95 & 0.033 \\
++GARCH(1,1)-$t$ & 0.54 & 0.85 & 0.97 & 0.035 & 0.56 & 0.85 & 0.98 & 0.037 & 0.56 & 0.85 & 0.98 & 0.043 & 0.59 & 0.87 & 0.98 & 0.065 \\
++Ridge & 0.41 & 0.70 & 0.91 & 0.074 & 0.44 & 0.74 & 0.94 & 0.042 & 0.44 & 0.75 & 0.94 & 0.052 & 0.46 & 0.75 & 0.94 & 0.040 \\
++Elastic net & 0.41 & 0.71 & 0.91 & 0.075 & 0.44 & 0.74 & 0.94 & 0.041 & 0.44 & 0.74 & 0.94 & 0.053 & 0.46 & 0.75 & 0.94 & 0.038 \\
++LightGBM & 0.35 & 0.64 & 0.89 & 0.110 & 0.41 & 0.71 & 0.92 & 0.063 & 0.46 & 0.74 & 0.92 & 0.044 & 0.45 & 0.75 & 0.92 & 0.042 \\
++MLP (1m) & 0.51 & 0.80 & 0.95 & 0.021 & 0.53 & 0.81 & 0.95 & 0.024 & 0.51 & 0.80 & 0.94 & 0.030 & 0.53 & 0.80 & 0.94 & 0.043 \\
++TCN (1m) & 0.54 & 0.82 & 0.96 & 0.027 & 0.55 & 0.82 & 0.96 & 0.034 & 0.52 & 0.81 & 0.95 & 0.032 & 0.54 & 0.81 & 0.95 & 0.050 \\
++MRPT-S (1m+5m+30m) & 0.54 & 0.83 & 0.96 & 0.035 & 0.55 & 0.83 & 0.96 & 0.034 & 0.53 & 0.82 & 0.95 & 0.037 & 0.54 & 0.81 & 0.95 & 0.053 \\
++MRPT-S (1m only) & 0.55 & 0.83 & 0.96 & 0.039 & 0.56 & 0.83 & 0.96 & 0.038 & 0.53 & 0.82 & 0.95 & 0.040 & 0.54 & 0.81 & 0.95 & 0.053 \\
++\bottomrule
++\end{tabular}
++\end{table}
+ \ No newline at end of file

added paper/tables/pilot_compare_crps.tex +22 −0

++\begin{table}[htbp]\centering\small
++\caption{Pilot: CRPS relative to Random walk (trailing RMS)}
++\label{tab:pilot_cmp_crps}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Seasonal volatility & -1.98\% (-5.7) & -2.51\% (-6.6) & -2.69\% (-5.8) & -2.12\% (-3.4) \\
++EWMA-$t$ & -2.83\% (-8.1) & -2.79\% (-6.7) & -2.62\% (-5.2) & -1.77\% (-2.3) \\
++HAR-RV & -0.24\% (-0.7) & -3.20\% (-8.1) & -3.25\% (-7.0) & -3.04\% (-4.6) \\
++GARCH(1,1)-$t$ & -2.58\% (-8.9) & -2.63\% (-8.0) & -2.45\% (-7.2) & -1.60\% (-3.7) \\
++Ridge & -2.06\% (-3.7) & -3.07\% (-5.7) & -3.29\% (-5.5) & -2.96\% (-4.1) \\
++Elastic net & -2.14\% (-3.8) & -3.10\% (-5.8) & -3.31\% (-5.6) & -2.91\% (-3.9) \\
++LightGBM & +3.20\% (3.7) & +1.62\% (2.0) & +1.59\% (1.5) & +2.49\% (1.8) \\
++MLP (1m) & -3.08\% (-9.1) & -2.96\% (-8.3) & -3.01\% (-7.2) & -2.47\% (-4.2) \\
++TCN (1m) & -3.11\% (-9.8) & -2.94\% (-7.8) & -3.16\% (-7.3) & -2.50\% (-4.0) \\
++MRPT-S (1m+5m+30m) & -2.96\% (-8.8) & -3.07\% (-7.9) & -3.36\% (-7.2) & -2.62\% (-4.2) \\
++MRPT-S (1m only) & -3.00\% (-8.9) & -2.99\% (-7.8) & -3.16\% (-6.8) & -2.42\% (-3.9) \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Relative change in mean CRPS versus the reference (negative = better); Diebold--Mariano $t$-statistic with HAC variance in parentheses.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/pilot_compare_qlike.tex +22 −0

++\begin{table}[htbp]\centering\small
++\caption{Pilot: QLIKE relative to Random walk (trailing RMS)}
++\label{tab:pilot_cmp_qlike}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Seasonal volatility & -6631047621301.96\% (-3.7) & -9798545943930.17\% (-6.7) & -8922777889662.27\% (-5.8) & -8074565519238.53\% (-4.6) \\
++EWMA-$t$ & -12554922968119.03\% (-8.8) & -11610326741044.22\% (-8.6) & -9962403507408.67\% (-6.7) & -7537309379639.13\% (-3.7) \\
++HAR-RV & +7877866372961.78\% (4.6) & -14099293226631.73\% (-13.0) & -13394448504861.81\% (-10.4) & -12560029461854.98\% (-8.0) \\
++GARCH(1,1)-$t$ & -6845837520747.71\% (-5.6) & -4710411807615.16\% (-4.2) & -1671497472176.46\% (-1.3) & +4026232839171.85\% (2.9) \\
++Ridge & -2945713870141.00\% (-1.4) & -14078646030114.02\% (-11.7) & -13850072137900.00\% (-9.9) & -13521968596908.26\% (-8.3) \\
++Elastic net & -2906922453727.77\% (-1.4) & -14052357953211.58\% (-11.7) & -13800380130330.72\% (-9.9) & -13480669954535.19\% (-8.2) \\
++LightGBM & +80780260086423.78\% (1.4) & -12012831272926.51\% (-7.3) & -12181626304741.97\% (-7.1) & -11572312577415.51\% (-6.0) \\
++MLP (1m) & -13608727182964.57\% (-13.2) & -13379373428123.78\% (-13.6) & -13042336995340.29\% (-11.2) & -12126414938724.40\% (-8.1) \\
++TCN (1m) & -13969630443261.25\% (-13.3) & -13939960177831.86\% (-13.9) & -13400649669390.51\% (-12.0) & -12492078176605.08\% (-8.7) \\
++MRPT-S (1m+5m+30m) & -14426818050181.94\% (-11.8) & -14161537101518.33\% (-13.4) & -13449231005232.20\% (-12.1) & -12662443344111.06\% (-9.2) \\
++MRPT-S (1m only) & -13769052086011.91\% (-11.5) & -13594106578801.23\% (-13.2) & -12903146151897.46\% (-11.4) & -12091461273483.05\% (-8.3) \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Relative change in mean QLIKE versus the reference (negative = better); Diebold--Mariano $t$-statistic with HAC variance in parentheses.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/pilot_compare_wis.tex +22 −0

++\begin{table}[htbp]\centering\small
++\caption{Pilot: WIS relative to Random walk (trailing RMS)}
++\label{tab:pilot_cmp_wis}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Seasonal volatility & -1.98\% (-5.1) & -2.60\% (-6.2) & -2.77\% (-5.3) & -2.12\% (-3.1) \\
++EWMA-$t$ & -2.82\% (-7.5) & -2.86\% (-6.4) & -2.69\% (-5.1) & -1.70\% (-2.1) \\
++HAR-RV & +0.12\% (0.3) & -3.34\% (-8.1) & -3.44\% (-7.1) & -3.16\% (-4.5) \\
++GARCH(1,1)-$t$ & -2.56\% (-8.4) & -2.59\% (-7.4) & -2.45\% (-6.7) & -1.43\% (-3.1) \\
++Ridge & -2.06\% (-3.4) & -3.26\% (-5.8) & -3.55\% (-5.7) & -3.14\% (-4.2) \\
++Elastic net & -2.12\% (-3.5) & -3.30\% (-6.0) & -3.55\% (-5.7) & -3.07\% (-4.0) \\
++LightGBM & +3.37\% (3.7) & +1.49\% (1.8) & +1.39\% (1.3) & +2.43\% (1.7) \\
++MLP (1m) & -3.24\% (-9.0) & -3.15\% (-8.3) & -3.23\% (-7.3) & -2.59\% (-4.1) \\
++TCN (1m) & -3.25\% (-9.7) & -3.15\% (-7.9) & -3.41\% (-7.4) & -2.64\% (-3.9) \\
++MRPT-S (1m+5m+30m) & -3.13\% (-8.7) & -3.29\% (-8.1) & -3.57\% (-7.3) & -2.78\% (-4.2) \\
++MRPT-S (1m only) & -3.13\% (-8.7) & -3.20\% (-7.9) & -3.39\% (-6.9) & -2.53\% (-3.8) \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Relative change in mean WIS versus the reference (negative = better); Diebold--Mariano $t$-statistic with HAC variance in parentheses.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/pilot_crps_scaled.tex +23 −0

++\begin{table}[htbp]\centering\small
++\caption{Pilot: mean scaled CRPS of the cumulative-return forecast by horizon (minutes)}
++\label{tab:pilot_crps_scaled}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Random walk (trailing RMS) & 0.4803 {\scriptsize[0.4618, 0.5012]} & 0.4874 {\scriptsize[0.4641, 0.5093]} & 0.4952 {\scriptsize[0.4688, 0.5225]} & 0.4819 {\scriptsize[0.4381, 0.5395]} \\
++Seasonal volatility & 0.4746 {\scriptsize[0.4545, 0.4957]} & 0.4793 {\scriptsize[0.4526, 0.5047]} & 0.4870 {\scriptsize[0.4569, 0.5190]} & 0.4767 {\scriptsize[0.4268, 0.5419]} \\
++EWMA-$t$ & 0.4699 {\scriptsize[0.4502, 0.4912]} & 0.4775 {\scriptsize[0.4517, 0.5023]} & 0.4868 {\scriptsize[0.4561, 0.5196]} & 0.4773 {\scriptsize[0.4276, 0.5428]} \\
++HAR-RV & 0.4833 {\scriptsize[0.4648, 0.5032]} & 0.4761 {\scriptsize[0.4492, 0.5019]} & 0.4845 {\scriptsize[0.4540, 0.5168]} & 0.4728 {\scriptsize[0.4223, 0.5391]} \\
++GARCH(1,1)-$t$ & 0.4714 {\scriptsize[0.4522, 0.4932]} & 0.4783 {\scriptsize[0.4545, 0.5018]} & 0.4876 {\scriptsize[0.4602, 0.5167]} & 0.4789 {\scriptsize[0.4331, 0.5388]} \\
++Ridge & 0.4734 {\scriptsize[0.4518, 0.4968]} & 0.4765 {\scriptsize[0.4494, 0.5028]} & 0.4842 {\scriptsize[0.4533, 0.5170]} & 0.4723 {\scriptsize[0.4216, 0.5368]} \\
++Elastic net & 0.4732 {\scriptsize[0.4516, 0.4967]} & 0.4764 {\scriptsize[0.4494, 0.5026]} & 0.4841 {\scriptsize[0.4532, 0.5169]} & 0.4725 {\scriptsize[0.4217, 0.5371]} \\
++LightGBM & 0.5090 {\scriptsize[0.4871, 0.5325]} & 0.5034 {\scriptsize[0.4756, 0.5308]} & 0.5116 {\scriptsize[0.4812, 0.5471]} & 0.5019 {\scriptsize[0.4484, 0.5765]} \\
++MLP (1m) & 0.4689 {\scriptsize[0.4497, 0.4908]} & 0.4769 {\scriptsize[0.4522, 0.5015]} & 0.4847 {\scriptsize[0.4559, 0.5157]} & 0.4743 {\scriptsize[0.4250, 0.5390]} \\
++TCN (1m) & 0.4687 {\scriptsize[0.4497, 0.4896]} & 0.4767 {\scriptsize[0.4526, 0.5008]} & 0.4838 {\scriptsize[0.4552, 0.5144]} & 0.4743 {\scriptsize[0.4252, 0.5388]} \\
++MRPT-S (1m+5m+30m) & 0.4697 {\scriptsize[0.4509, 0.4903]} & 0.4763 {\scriptsize[0.4526, 0.5001]} & 0.4831 {\scriptsize[0.4553, 0.5135]} & 0.4736 {\scriptsize[0.4238, 0.5381]} \\
++MRPT-S (1m only) & 0.4693 {\scriptsize[0.4506, 0.4899]} & 0.4765 {\scriptsize[0.4527, 0.5006]} & 0.4842 {\scriptsize[0.4558, 0.5148]} & 0.4745 {\scriptsize[0.4245, 0.5395]} \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Cluster-bootstrap (trading-day) 95\% intervals in brackets.\end{minipage}
++\end{table}
+ \ No newline at end of file

added paper/tables/pilot_logscore.tex +23 −0

++\begin{table}[htbp]\centering\small
++\caption{Pilot: mean log score of the cumulative-return forecast by horizon (minutes)}
++\label{tab:pilot_logscore}
++\begin{tabular}{lcccc}
++\toprule
++Model & $H=5$ & $H=15$ & $H=30$ & $H=60$ \\
++\midrule
++Random walk (trailing RMS) & nan & nan & nan & nan \\
++Seasonal volatility & nan & nan & nan & nan \\
++EWMA-$t$ & -5.2979 {\scriptsize[-5.3501, -5.2489]} & -4.7338 {\scriptsize[-4.7970, -4.6663]} & -4.3653 {\scriptsize[-4.4434, -4.2821]} & -4.0215 {\scriptsize[-4.1344, -3.8696]} \\
++HAR-RV & -5.2318 {\scriptsize[-5.2750, -5.1939]} & -4.7437 {\scriptsize[-4.8070, -4.6722]} & -4.3804 {\scriptsize[-4.4598, -4.2939]} & -4.0439 {\scriptsize[-4.1592, -3.8939]} \\
++GARCH(1,1)-$t$ & -5.2893 {\scriptsize[-5.3330, -5.2481]} & -4.7274 {\scriptsize[-4.7788, -4.6722]} & -4.3585 {\scriptsize[-4.4205, -4.2937]} & -4.0114 {\scriptsize[-4.0946, -3.8983]} \\
++Ridge & -5.2731 {\scriptsize[-5.3350, -5.2135]} & -4.7400 {\scriptsize[-4.8076, -4.6674]} & -4.3819 {\scriptsize[-4.4641, -4.2950]} & -4.0459 {\scriptsize[-4.1620, -3.8925]} \\
++Elastic net & -5.2733 {\scriptsize[-5.3355, -5.2137]} & -4.7402 {\scriptsize[-4.8078, -4.6676]} & -4.3817 {\scriptsize[-4.4639, -4.2949]} & -4.0457 {\scriptsize[-4.1620, -3.8920]} \\
++LightGBM & -5.1540 {\scriptsize[-5.2276, -5.0829]} & -4.6712 {\scriptsize[-4.7450, -4.5915]} & -4.3235 {\scriptsize[-4.4092, -4.2250]} & -3.9773 {\scriptsize[-4.1001, -3.8019]} \\
++MLP (1m) & nan & nan & nan & nan \\
++TCN (1m) & nan & nan & nan & nan \\
++MRPT-S (1m+5m+30m) & nan & nan & nan & nan \\
++MRPT-S (1m only) & nan & nan & nan & nan \\
++\bottomrule
++\end{tabular}
++\begin{minipage}{0.95\textwidth}\scriptsize Cluster-bootstrap (trading-day) 95\% intervals in brackets.\end{minipage}
++\end{table}
+ \ No newline at end of file

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