Threshold-Dependent Dominance in Tail Risk Approximation — Replication

Replication package for:

Agbeyegbe, T. D. (2026). Threshold-Dependent Dominance in Tail Risk Approximation. Econometrics (MDPI), forthcoming. https://doi.org/10.3390/econometrics1010000

The paper compares the fourth-order Edgeworth expansion and the four-cumulant saddlepoint approximation for tail-risk measurement in rare-disaster and heavy-tailed financial settings. It derives a closed-form admissibility envelope for the fourth-order Edgeworth density in the (γ₁, γ₂) plane, shows that 87.5% of 648 calibrated rare-disaster moment configurations fall outside it, and documents that Edgeworth's mean Expected-Shortfall error rises from 6.20% inside the envelope to 47.04% outside it, while the saddlepoint error stays near 13.64%.

Running the notebook end-to-end reproduces every table (Tables 1–11), both manuscript figures, the three Supplementary Figures (S1–S3), and every headline statistic in the published version from a single deterministic pass. Notebook table numbering matches the manuscript one-to-one.

Contents

Path	Purpose
EdgeworthSaddlepoint_replication.ipynb	Main notebook. Builds the 648-cell rare-disaster grid; derives the Edgeworth validity envelope in closed form (Proposition 1); computes Edgeworth and saddlepoint tail probabilities and Expected Shortfall; runs the Student-t / Hansen skewed-t / GED / two-sided-jump / generalised-hyperbolic robustness sweeps; classifies rolling empirical moments against the envelope; computes the per-ticker kurtosis-exceedance and plug-in exceedance-error tables; assembles the criterion-dominance summary; runs the Monte Carlo validation; and emits all tables and figures.
sample_data/sp500_returns.csv	Frozen daily log-return panel (percent) for the S&P 500 and the XLE / XLF / XLK / XLV sector ETFs, 1990–2026 (retrieved 8 May 2026). Used by the empirical-anchoring section. See sample_data/README.md.
sample_data/fetch_data.py	Provenance: (re)creates the panel from Yahoo Finance via yfinance (executed with yfinance 0.2.66). Not needed for replication — the frozen CSV ships with the repo.
outputs/	Reference outputs produced by the notebook: table1–table11 CSVs matching the manuscript tables one-to-one, one auxiliary diagnostic CSV (auxiliary_empirical_regime.csv, not a manuscript table), the robustness CSVs, the closed-form envelope, and the two manuscript figures. Provided so reviewers can diff against a fresh run.
supplementary/	Supplementary Figures S1–S3 as published with the article (cited in Sections 4.5 and 4.7 of the paper) and their description.

How to run

Locally

git clone https://github.com/ProfTemi/threshold-risk-replication.git
cd threshold-risk-replication
pip install -r requirements.txt
jupyter notebook EdgeworthSaddlepoint_replication.ipynb   # Kernel → Restart & Run All

A full run takes ≈ 4–5 minutes on a laptop CPU. All outputs are written to outputs/, and the two manuscript figures (figure1_preview.pdf, figure2_preview.pdf) are also written to the working directory so they can be picked up directly by the LaTeX source.

Google Colab

Open the notebook in Colab, upload sample_data/sp500_returns.csv to a sample_data/ folder, then Runtime → Run all.

Reproducibility

• The committed notebook was executed under Python 3.12.13; any Python ≥ 3.10 with the pinned requirements reproduces the same numbers.
• The notebook sets np.random.seed(456), so the Monte Carlo cells are deterministic (true tail probability 0.011109, MC mean 0.011067, Edgeworth error 194.44%, saddlepoint error 88.08%; manuscript Table 9).
• The empirical section reads the frozen sample_data/sp500_returns.csv rather than calling yfinance live, so the rolling-moment and per-ticker tables (manuscript Tables 7 and 8) are bit-reproducible and do not drift with Yahoo Finance's retroactive adjustments.
• The lateral peak of the Edgeworth envelope is found by continuous optimization (scipy.optimize.minimize_scalar), reproducing |γ₁*| ≈ 0.6846 at γ₂ ≈ 2.533 independent of plotting resolution.

Note on the four-cumulant saddlepoint

The truncated four-cumulant cumulant-generating function is non-convex when κ₄ < 0 (sub-Gaussian / thin-tailed distributions), so the saddlepoint is mathematically undefined there. The implementation flags such cells and records them as missing (NaN) rather than substituting a value; they are excluded from the saddlepoint error averages. This affects only the two sub-Gaussian GED cells (ν = 3, 5) in Table 6; every other family and the entire rare-disaster grid are well-posed.

Requirements

Python ≥ 3.10 with numpy, scipy, pandas, matplotlib (see requirements.txt).

Acknowledgments

The author thanks Toneukarin G. Agbeyegbe for research assistance and helpful discussions throughout the development of this package.