tailcor

Non-asymptotic, distribution-free decomposition of linear correlation vs non-linear tail contagion — a Python implementation of Ricci, Tristani & Vergote (2023) "TailCoR" (Banco de España WP 1227, Journal of Empirical Finance). Pure numpy + scipy, no C++ deps, no copula fitting, no asymptotic limits.

What it does

Given two paired time series (X, Y), TailCoR returns three numbers per window:

• Linear — always 1.0. Bivariate-Gaussian reference at the empirical Pearson correlation.
• Composite — empirical tail inter-quantile range divided by the Gaussian reference. Equals 1.0 under exact Gaussian, >1 when the joint tail is fatter than Gaussian predicts.
• Non-linear — composite − linear. The "pure tail-contagion" component that correlation cannot explain.

Why it matters: during the 2008 + 2011 crises, linear correlations between Euro-area sovereign bonds collapsed to ~1 then dropped, while tail contagion spiked. Classical EVT was too slow to catch it (extremes are scarce at acceleration); TailCoR tracked it in real time because it does not need asymptotic-limit samples.

Install

pip install tailcor

Requires Python ≥ 3.9, numpy ≥ 1.23, scipy ≥ 1.10.

Quickstart

import numpy as np
from tailcor import tailcor, rolling_tailcor

rng = np.random.default_rng(0)
x = rng.standard_normal(1000)
y = 0.5 * x + rng.standard_normal(1000)

result = tailcor(x, y, q=0.95)
print(result.composite)   # ~1.0 under Gaussian
print(result.nonlinear)   # ~0.0 under Gaussian
print(result.rho)         # empirical Pearson

# rolling time series:
r = rolling_tailcor(x, y, window=252, q=0.95)
print(r.composite.shape)  # (1000 - 252 + 1,)

The math (1-minute version)

For bivariate (X, Y) with empirical Pearson correlation ρ, project onto the sum axis standardised by the Gaussian variance:

Z = (X + Y) / √(2(1 + ρ))

Under bivariate standard Gaussian at any ρ, Z is N(0, 1) — so its inter-quantile range at level q equals Φ⁻¹(q) − Φ⁻¹(1−q) (the Gaussian reference IQR at that quantile).

Define

S(q) = IQR_q(Z) / ( Φ⁻¹(q) − Φ⁻¹(1−q) )

• S(q) ≈ 1 under exact bivariate Gaussian (any ρ)
• S(q) > 1 when the empirical tail is fatter than Gaussian (tail contagion)
• S(q) < 1 when the empirical tail is thinner (rare — anti-concentration)

The linear component is exactly 1.0 by construction (Gaussian reference). The non-linear component is S(q) − 1. "A few lines of basic coding", per the paper — and literally is (see tailcor/core.py).

Quantile profile

Fat-tailed joint distributions show TailCoR rising with q:

from tailcor import tailcor_profile
import numpy as np

rng = np.random.default_rng(1)
# Student-t copula with df=4 (canonical tail dependence)
g = rng.multivariate_normal([0, 0], [[1, 0], [0, 1]], size=15000).T
chi2 = rng.chisquare(4, size=15000)
t = g * np.sqrt(4 / chi2)
x, y = t[0], t[1]

for r in tailcor_profile(x, y):
    print(f"q={r.q:.2f}  composite={r.composite:.3f}  nonlinear={r.nonlinear:+.3f}")

Output: composite starts ~1 at q=0.6, grows past 1.5 at q=0.99 — the classic "tail contagion signature" that Ricci et al. use as a crisis early-warning.

Correctness

15 unit tests cover:

• Bivariate Gaussian at six ρ levels (−0.5 … 0.8): composite within [0.9, 1.1] of 1, non-linear near 0
• Student-t copula df=4: composite > 1.15 even at ρ=0 (tail contagion with zero Pearson)
• Student-t copula df=30: nearly Gaussian (|non-linear| < 0.15)
• ρ recovered to within 0.05 on n=10000 Gaussian samples
• NaN-row handling (10% injection preserves a valid result)
• q monotonicity on tail-dependent copulas (composite grows with q)
• Input-validation error paths
• Rolling: Gaussian → Student-t regime shift produces post-shift composite lift > 0.1

Use cases

• Crisis early warning on bivariate index pairs (e.g. equity vs sovereign, or two credit spreads)
• Risk decomposition: split pairwise CoVaR into what Gaussian explains and what it can't
• Stress-test input: feed the non-linear component into correlation-matrix generators (CorrGAN and similar) as a fat-tail signal
• Regime detection complementary to HMM / change-point methods: the non-linear component reacts to copula-shape changes that HMM trained on returns may miss

Roadmap

v0.2 (planned):

• Alternative rotations (principal-component, anti-diagonal) — rotation="sum"|"pc"|"anti"
• Tail-CoVaR helper — conditional-VaR-style wrapper on top of the core decomposition
• Block-bootstrap confidence intervals for composite + non-linear
• Multivariate extension via pairwise aggregation matrix

Authors

• Pierre Samson (@darw007d) — idea, use-case, design decisions
• Claude Opus (Anthropic) — implementation and tests

Originally motivated by the OMEGA Swarm project, Phase 19 Tier A #4. Sister package to phawkes (Hawkes processes) and fisherrao (information geometry). Same "small, tested, publishable" ethos.

Citation

If this library contributes to a published result, please cite the underlying paper:

• Ricci, L., Tristani, O. & Vergote, O. (2023). TailCoR. Banco de España Working Paper 1227. Journal of Empirical Finance.

License

MIT — see LICENSE.