HMD_ES

Finite-Sample Precision Limits for Expected Shortfall Forecast Comparisons

Authors: Daniel Traian Pele and Miruna Mazurencu-Marinescu-Pele (Bucharest University of Economic Studies)

Overview

Replication code and data for "Finite-Sample Precision Limits for Expected Shortfall Forecast Comparisons".

The paper converts the known $(n\alpha)^{-1/2}$ information limit for Expected Shortfall estimation into an operational precision-audit framework for pairwise ES forecast comparison.

Results

Detrended SD vs. plug-in precision benchmark — each point is one (asset, forecaster, α) cell; the dashed line marks R = 1.

VaR-first diagnostic — Christoffersen conditional-coverage statistic vs. benchmark ratio R at α = 1%. Excess dispersion concentrates in poorly VaR-calibrated cells.

Required calibration window for 50 bp ES precision — the vertical line marks the 250-day default.

Window-length scaling — non-overlapping windows confirm the $(n\alpha)^{-1/2}$ rate.

Repository Structure

code/           Python analysis scripts
  pipeline.py           Main rolling-window FZ recalibration pipeline
  regen_all_figures.py  Regenerate all figures and tables
  figures.py            Figure generation routines
  v_next/               Robustness and diagnostic scripts
data/           Processed CSV results
  recalib_results.csv   Rolling recalibration results (24 assets × 4 forecasters × 3 alpha)
  rolling_estimates.csv Rolling-window ES correction estimates
tables/         Generated LaTeX tables
figures/        Generated PDF/PNG plots

Requirements

Python 3 with: numpy, pandas, scipy, matplotlib, statsmodels, arch (the last two are needed only for the 2026 revision scripts: ARCH-LM tests and the Model Confidence Set).

Usage

python code/pipeline.py            # Main analysis pipeline
python code/regen_all_figures.py   # Regenerate all figures and tables

Revision analyses (2026)

The reviewer revision added four analyses, reproducible from the committed data:

python code/v_next/revision_2026_reviewers.py   # bootstrap CI for the
                                                # precision-fragile share;
                                                # Engle ARCH-LM tests (24 assets);
                                                # CC-vs-R permutation + panel tests
                                                # within the well-calibrated subset
python code/v_next/revision_2026_dm_mcs.py      # Diebold-Mariano (HAC + HLN)
                                                # and Model Confidence Set
                                                # benchmarks for the screen

Outputs (tables/): fragile_bootstrap_ci.tex (share 20.1% [16.0, 25.0] at α=2.5%), arch_lm_test.tex (no-ARCH null rejected at 1% for all 24 assets), cc_R_subset_permutation.tex / cc_R_subset_panel.tex (VaR-first link does not survive asset fixed effects among well-calibrated models), dm_mcs_crosstab.tex (90% of precision-fragile pairs are also Diebold-Mariano-indistinguishable; mean MCS size 3.0/4), plus the supporting CSVs dm_mcs_results.csv and cc_R_all_alphas.csv.

Data

Daily return data are sourced from Yahoo Finance for 24 global assets (equities, bonds, commodities, cryptocurrencies, FX). Forecasters: GJR-GARCH-t, TimesFM 2.5, Chronos-Small, Moirai 2.0.

Citation

Pele, D. T. & Mazurencu-Marinescu-Pele, M. (2026). Finite-Sample Precision Limits for Expected Shortfall Forecast Comparisons. Mathematics (submitted).

Funding

This project has received funding from the Marie Skłodowska-Curie Actions under the European Union's Horizon Europe research and innovation program for the Industrial Doctoral Network on Digital Finance, acronym DIGITAL, Project No. 101119635; the project "IDA Institute of Digital Assets", CF166/15.11.2022, contract number CN760046/23.05.2023; the project "AI for Energy Finance (AI4EFin)", CF162/15.11.2022, contract number CN760048/23.05.2023; the project "Accountable Governance and Responsible Innovation in Artificial Intelligence", CF158/15.11.2022, contract number CN760047/23.05.2023, financed under Romania's National Recovery and Resilience Plan, Apel nr. PNRR-III-C9-2022-I8.

We acknowledge the support of the project "MA'AT — Autonomous Model for Textual Assistance", SMIS Code 2021+: 330941, funding contract no. 390090/11.11.2025, project co-financed by the European Regional Development Fund through the Smart Growth, Digitalisation and Financial Instruments Programme 2021–2027 (POCIDIF).