pycointbreak 0.1.0

Tests for breaks in fractional cointegration (Hassler & Breitung 2006; Rodrigues, Sibbertsen & Voges 2019).

pycointbreak

Tests for breaks in fractional cointegration — a Python implementation of the Hassler & Breitung (2006) residual-based LM test and the Rodrigues, Sibbertsen & Voges (2019) supremum tests for breaks in the cointegrating relationship.

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Author

Dr. Merwan Roudane Email: [e-mail] GitHub: https://github.com/merwanroudane/pycointbreak

If you use this library in academic work, please cite both the underlying papers and this package — see pycointbreak.cite() for ready-made BibTeX entries.

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What does it do?

Given a scalar series y and a (possibly multivariate) series x, each integrated of order d, pycointbreak lets you test whether

y_t = β' x_t + z_t,        z_t ~ I(d − b),   b ≥ 0,

is

1. Cointegrated at all (b > 0) — using the Hassler & Breitung (2006) LM-type test.
2. Cointegrated only on part of the sample (segmented fractional cointegration) — using the Rodrigues, Sibbertsen & Voges (2019) sup-tests over split, forward incremental, backward incremental, and rolling sub-samples.

When the answer is "yes, breaks", the library can also estimate the break date via the consistent estimator of RSV (2019, eq. 19).

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Installation

git clone https://github.com/merwanroudane/pycointbreak.git
cd pycointbreak
pip install -e .

Dependencies: numpy, pandas, scipy, statsmodels, matplotlib, seaborn. Optional: yfinance (for the real-data example).

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Quick start

import pycointbreak as pcb

# 1. Generate (or load) two I(d) series.
y, x = pcb.simulate_segmented_cointegration(
    T=500, d=1.0, b=0.4, break_frac=0.5,
    regime="coint_then_spurious", seed=2024,
)

# 2. Full-sample Hassler-Breitung test.
hb = pcb.hassler_breitung_test(y, x, d=1.0, p=0)
print(hb.summary())

# 3. RSV (2019) sup-test battery.
battery = pcb.rsv_battery(y, x, d=1.0, lam0=0.5, n_grid=120)
for k, r in battery.items():
    print(r.summary())

# 4. Estimate the break date.
bp = pcb.estimate_break_point(y, x, d=1.0, direction="auto")
print(bp.summary())

# 5. Paper-style summary table.
df = pcb.battery_to_dataframe(battery, hb=hb, label="my pair")
print(pcb.render_table(df, fmt="text",
                       title="Tests for segmented cointegration"))

# 6. Plots.
import matplotlib.pyplot as plt
pcb.plot_series_with_breaks({"y": y, "x": x}, {"break": bp.obs})
pcb.plot_rsv_battery(battery)
pcb.plot_breakpoint_objective(bp)
plt.show()

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Mathematical compatibility with the papers

The implementation follows the papers' notation closely. The key ingredients are exposed as building blocks:

Symbol	Function
Δ⁺ᵈ x_t — truncated (Type II) fractional difference	pcb.fdiff(x, d)
π_j(d) — coefficients of (1−L)^d	pcb.frac_coefs(d, n)
x*_{t−1} = Σ_{j=1}^{t−1} x_{t−j}/j — harmonic-weighted sum (HB eq. 3)	pcb.harmonic_running_sum(x)
(1−L)^{−d} x_t — fractional integration	pcb.frac_integrate(x, d)
Geweke & Porter-Hudak d̂	pcb.gph(x)
HB test (eqs. 12–14, iid case)	pcb.hassler_breitung_test(..., p=0)
HB test with AR(p) augmentation (eqs. 17–18)	pcb.hassler_breitung_test(..., p=p)
RSV sub-sample t-stat (eq. 5)	internal _hb_subsample_tstat
Sup-statistics 𝒯_S, 𝒯_S*, 𝒯_If, 𝒯_Ib, 𝒯_R (eqs. 10–15)	pcb.rsv_sup_test(..., kind=...)
RSV break-point estimator (eq. 19)	pcb.estimate_break_point(...)
Critical values from Table 1 of RSV2019	pcb.rsv_critical_value(test, T, alpha)

On the eq.-(5) √n factor

Equation (5) of RSV (2019) literally reads

t(ê(λ₁,λ₂)) = √(⌊λ₂T⌋ − ⌊λ₁T⌋) · Σ ê_t · ê*_{t−1}
              / [√Σ ê*²_{t−1} · √((T−1)⁻¹ · Σ ê²_t)]

With the full-sample variance 1/(T−1) and the extra √n factor, under H0 this statistic is O_p(√T) rather than asymptotically N(0,1), which would contradict Theorem 2 (sup χ²₁ limit) and the critical values of Table 1 (5%-cv ≈ 4–6 for T = 500). We therefore interpret the formula as the standard HB statistic (eq. 14 of HB2006) applied to the sub-sample, with sub-sample variance 1/(n−1) — the same convention used by Davidson & Monticini (2010). This yields t → N(0, 1) per Proposition 3 of HB2006 and T_K → sup χ²₁ consistent with Theorem 2 of RSV2019.

On the critical values

The tabulated critical values in pcb.RSV_TABLE1 reproduce Table 1 of RSV (2019) for T = 250 and T = 500 with λ₀ = 0.5. They are the result of a Monte Carlo simulation in the paper averaged over several values of d. For configurations far from (T ∈ {250, 500}, λ₀ = 0.5, d ≈ 1), use the bootstrap helper

cv = pcb.bootstrap_rsv_cv(
    T=1000, d=1.0, kind="T_If", n_sim=2000,
)
print(cv)   # {0.10: ..., 0.05: ..., 0.01: ...}

which returns critical values calibrated to your exact sample size, grid, and d.

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Modules

Module	Contents
pycointbreak.fracdiff	Truncated (Type II) fractional difference operators
pycointbreak.gph	GPH log-periodogram estimator of d
pycointbreak.hassler_breitung	HB (2006) LM-type test
pycointbreak.rsv_tests	RSV (2019) sup-tests for breaks
pycointbreak.breakpoint	Break-fraction estimator (RSV eq. 19)
pycointbreak.critical_values	Tabulated and bootstrap critical values
pycointbreak.simulate	DGPs from HB2006 §5 and RSV2019 §4 (Experiments 1–3)
pycointbreak.reporting	Paper-style tables (text/markdown/LaTeX/HTML)
pycointbreak.plots	Publication-quality figures (matplotlib + seaborn)

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API at a glance

Tests

hassler_breitung_test(y1, y2, d=1.0, p=0, include_const=True)
    → HBResult(t_stat, p_value, reject, ...)

rsv_sup_test(y1, y2, d=1.0, kind="T_If", lam0=0.5, n_grid=200, ...)
    → RSVResult(statistic, argmax, t2_path, ...)

rsv_battery(y1, y2, d=1.0, tests=("T_S_star", "T_If", "T_Ib", "T_R"), ...)
    → dict[str, RSVResult]

estimate_break_point(y1, y2, d=1.0, delta=0.05, direction="auto", ...)
    → BreakPointResult(tau_hat, obs, date, objective, ...)

Reporting

df = battery_to_dataframe(battery, hb=hb, label="series")
print(render_table(df, fmt="text"))      # ASCII
print(render_table(df, fmt="markdown"))   # for GitHub / Jupyter
print(render_table(df, fmt="latex"))      # for papers
print(render_table(df, fmt="html"))       # for web

Plots

plot_series_with_breaks(series_dict, breaks_dict)
plot_rsv_profile(rsv_result)            # one sup-test
plot_rsv_battery(battery_dict)          # 2×2 panel of all four
plot_breakpoint_objective(bp_result)
plot_residual_diagnostics(residuals)
plot_split_heatmap(y, x, d=1.0, ...)    # 2-D (λ₁, λ₂) heatmap

Simulation

simulate_rsv_dgp(T=500, d=1.0, b=0.0)
    # Experiment 1 — constant cointegration over the whole sample

simulate_segmented_cointegration(
    T=500, d=1.0, b=0.4, break_frac=0.5,
    regime="coint_then_spurious",
)
    # Experiments 2 & 3 — one break in the cointegrating relationship

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Examples

The examples/ folder contains three runnable scripts:

• example_basic.py — minimal end-to-end demo on simulated data.
• example_real_data.py — applies the full pipeline to real stock prices (Coca-Cola / Pepsi by default via yfinance; falls back to simulated data when the network is unavailable).
• example_simulation_study.py — small Monte Carlo reproducing a slice of Table 2 of RSV (2019).

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References

• Hassler, U. and Breitung, J. (2006). A Residual-Based LM Type Test Against Fractional Cointegration. Econometric Theory, 22(6), 1091–1111.
• Rodrigues, P. M. M., Sibbertsen, P. and Voges, M. (2019). Testing for breaks in the cointegrating relationship: On the stability of government bond markets' equilibrium. Discussion Paper 656.
• Davidson, J. and Monticini, A. (2010). Tests for cointegration with structural breaks based on subsamples. Computational Statistics & Data Analysis, 54(11), 2498–2511.
• Robinson, P. M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics, 47, 67–84.
• Breitung, J. and Hassler, U. (2002). Inference on the cointegration rank in fractionally integrated processes. Journal of Econometrics, 110, 167–185.
• Demetrescu, M., Kuzin, V. and Hassler, U. (2008). Long memory testing in the time domain. Econometric Theory, 24(1), 176–215.
• Geweke, J. and Porter-Hudak, S. (1983). The Estimation and Application of Long Memory Time Series Models. Journal of Time Series Analysis, 4, 221–238.

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License

MIT