multicoint — Panel Cointegration & Multiple Long-Run Relations

A comprehensive Python library for estimating long-run equilibrium relationships in heterogeneous panel data.

Implements 7 state-of-the-art estimators from three foundational papers in modern panel econometrics, with publication-quality tables and visualizations.

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📖 Table of Contents

• Installation
• Quick Start
• Estimators
  • SPMG — System Pooled Mean Group
  • PMG — Pooled Mean Group
  • Breitung — Two-Step Parametric
  • PDOLS — Panel Dynamic OLS
  • MGMW — Müller-Watson Mean Group
  • MGDL — Mean Group Distributed Lag
  • PME — Pooled Minimum Eigenvalue
• Diagnostics
• Visualization
• Comparison Tables
• Simulated Datasets
• Full API Reference
• Mathematical Background
• References
• Author

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📦 Installation

# From PyPI (once published)
pip install multicoint

# From GitHub
pip install git+https://github.com/merwanroudane/multicointt.git

# Local development install
git clone https://github.com/merwanroudane/multicointt.git
cd multicointt
pip install -e .

Dependencies

Package	Version	Purpose
numpy	≥ 1.24	Core numerical computation
scipy	≥ 1.10	Statistical distributions, linalg
pandas	≥ 2.0	Data handling and export
matplotlib	≥ 3.7	Publication-quality plots
seaborn	≥ 0.12	Statistical visualization
rich	≥ 13.0	Beautiful terminal tables
tabulate	≥ 0.9	Markdown/LaTeX table export
openpyxl	≥ 3.1	Excel export

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🚀 Quick Start

import numpy as np
import multicoint as mc

# Simulate panel data: consumption/GDP for 15 countries, 143 years
Y, X = mc.datasets.simulate_great_ratios(n=15, T=143, theta=1.0, seed=42)

# Estimate with all five Great Ratios estimators
spmg = mc.SPMG(p=2, bootstrap_reps=500).fit(Y, X, var_names=["Consumption", "GDP"])
pmg  = mc.PMG(p=2, bootstrap_reps=500).fit(Y, X)
breit = mc.Breitung(p=2).fit(Y, X)
pdols = mc.PDOLS(leads_lags=4).fit(Y, X)
mgmw  = mc.MGMW(q=5).fit(Y, X)

# Print comparison table
mc.comparison_table([spmg, pmg, breit, pdols, mgmw],
                    title="Great Ratios: Consumption/GDP")

# Forest plot of confidence intervals
mc.plot_confidence_intervals([spmg, pmg, breit, pdols, mgmw], theta0=1.0)

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📐 Estimators

1. SPMG — System Pooled Mean Group

The flagship estimator. Handles two-way long-run causality, non-cointegrating units, and cross-sectional dependence.

Paper: Chudik, Pesaran & Smith (2023), "Revisiting the Great Ratios Hypothesis", Fed Dallas GI WP 415.

Model:

$$\Delta w_{it} = -\phi_i \beta' w_{i,t-1} + \Upsilon_i q_{it} + u_{it}$$

where $w_{it} = (y_{it}, x_{it})'$, $\beta = (1, -\theta)'$, $\phi_i = (\phi_{yi}, \phi_{xi})'$ is a 2×1 vector (both equations), and $\Sigma_i = \text{Var}(u_{it})$ is 2×2.

Key advantages over PMG:

• φ_i is bivariate → captures two-way long-run causality
• Units with φ_i ≈ 0 contribute negligibly → robust to non-cointegration
• Normalization invariant: θ̂_{y.x} · θ̂_{x.y} = 1

Syntax

from multicoint import SPMG

estimator = SPMG(
    p=2,                  # Lag order (ARDL order, default 2)
    max_iter=500,         # Maximum iterations for MLE convergence
    precision=1e-4,       # Convergence criterion |θ^(k) - θ^(k-1)|
    bootstrap_reps=2000,  # Bootstrap replications for robust CI
    seed=1234,            # Random seed for reproducibility
)

result = estimator.fit(
    Y,                    # np.ndarray, shape (T, n) — dependent variable
    X,                    # np.ndarray, shape (T, n) — independent variable
    var_names=["y", "x"],        # Optional variable names
    unit_names=["US", "UK", …],  # Optional unit names
    bootstrap=True,              # Compute bootstrap CI (default True)
)

Result Attributes

result.theta_hat        # float — Estimated long-run coefficient θ̂
result.std_error        # float — Asymptotic standard error
result.t_ratio          # float — t-ratio for H₀: θ = 1
result.p_value          # float — Two-sided p-value
result.ci_95            # ConfidenceInterval — 95% asymptotic CI
result.ci_99            # ConfidenceInterval — 99% asymptotic CI
result.ci_bootstrap     # ConfidenceInterval — 95% bootstrap CI (if computed)
result.n_units          # int — Number of cross-section units
result.n_periods        # int — Number of time periods
result.n_iterations     # int — Iterations to convergence
result.converged        # bool — Whether MLE converged
result.phi_hat          # np.ndarray (n, 2) — Unit-specific φ̂_i = (φ̂_{yi}, φ̂_{xi})
result.sigma_hat        # np.ndarray (n, 2, 2) — Unit-specific Σ̂_i
result.is_unit_coefficient  # bool — Whether θ=1 is within 95% CI
result.summary()        # str — Rich-formatted summary table
result.to_dict()        # dict — Export as dictionary
result.to_series()      # pd.Series — Export as pandas Series

Example

spmg = SPMG(p=2, bootstrap_reps=2000, seed=1234)
res = spmg.fit(Y, X, var_names=["log(C/capita)", "log(GDP/capita)"])
res.summary()  # Prints rich table

# Access individual results
print(f"θ̂ = {res.theta_hat:.4f} ± {res.std_error:.4f}")
print(f"Bootstrap 95% CI: {res.ci_bootstrap}")
print(f"Unit coefficient (θ=1)? {res.is_unit_coefficient}")

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2. PMG — Pooled Mean Group

Single-equation version. Assumes long-run causality runs from x to y.

Paper: Pesaran, Shin & Smith (1999), JASA, 94, 621–634.

Model (conditional ARDL):

$$\Delta y_{it} = c_i - \phi_i(y_{i,t-1} - \theta x_{i,t-1}) + \text{short-run} + \varepsilon_{it}$$

where $\phi_i$ is scalar (single equation), and $\theta$ is pooled across units.

Syntax

from multicoint import PMG

estimator = PMG(
    p=2,                  # Lag order
    max_iter=500,         # Max iterations
    precision=1e-4,       # Convergence tolerance
    bootstrap_reps=2000,  # Bootstrap replications
    seed=1234,
)

result = estimator.fit(Y, X, bootstrap=True)

Key Differences from SPMG

Feature	PMG	SPMG
φ_i dimension	Scalar (1×1)	Vector (2×1)
Σ_i dimension	Scalar (σ²ᵢ)	Matrix (2×2)
Long-run causality	x→y only	x↔y (both directions)
Non-cointegrating	Robust	More robust
Normalization	Not invariant	Invariant

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3. Breitung — Two-Step Parametric

Parametric estimator that does not assume known causal direction.

Paper: Breitung (2005), Econometric Reviews, 24, 151–173.

Syntax

from multicoint import Breitung

estimator = Breitung(p=2)  # Only lag order needed
result = estimator.fit(Y, X)

Note: The Breitung estimator requires inverting α̂_i' Σ̂_i⁻¹ α̂_i per unit, which can be unstable when α̂_i → 0. SPMG is preferred for robustness.

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4. PDOLS — Panel Dynamic OLS

Adds leads and lags of Δx to absorb endogeneity.

Paper: Mark & Sul (2003), Oxford Bulletin of Economics and Statistics.

Syntax

from multicoint import PDOLS

# With 4 leads and lags (standard)
result4 = PDOLS(leads_lags=4).fit(Y, X)

# With 8 leads and lags (more conservative)
result8 = PDOLS(leads_lags=8).fit(Y, X)

Parameters

Parameter	Type	Default	Description
leads_lags	int	4	Number of leads AND lags of Δx to include

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5. MGMW — Müller-Watson Mean Group

Temporally aggregates data into q sub-periods, then runs pooled FE.

Paper: Müller & Watson (2018), Econometrica, 86, 775–804.

Syntax

from multicoint import MGMW

result = MGMW(q=5).fit(Y, X)  # q=5 sub-periods (matching MATLAB code)

Parameters

Parameter	Type	Default	Description
q	int	5	Number of temporal sub-periods

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6. MGDL — Mean Group Distributed Lag

Estimates impulse response functions of common observed shocks in panels with one or two cross-section dimensions.

Paper: Choi & Chudik (2024), "Mean Group Distributed Lag Estimation of IRFs in Large Panels", Fed Dallas GI WP 423.

Model:

$$x_{ijt} = a_{ij} + \sum_{\ell=0}^{h} b_{ij\ell} v_{t-\ell} + \phi'{hij} g{hijt} + e_{hijt}$$

where $v_t$ is the common observed shock, and $b_{ij\ell}$ are the IRF coefficients.

Syntax

from multicoint import MGDL

estimator = MGDL(
    h=4,                  # IRF horizon (quarters)
    augmented_var=True,    # Use augmented variance (eq 9-10) for robust inference
    bonferroni=True,       # Bonferroni correction for family-wise coverage
    seasonal=True,         # Include seasonal dummies
)

result = estimator.fit(
    X_panel,              # np.ndarray, shape (M, N, T) — panel of outcomes
    shock,                # np.ndarray, shape (T,) — common shock series
    product_names=[…],    # Optional: M product names
    location_names=[…],   # Optional: N location names
)

Result Attributes (MGDLResult)

result.product_irfs      # (M, h+1) — Mean group IRF per product
result.location_effects  # (N, h+1) — Location effects ĉ_j
result.cum_multipliers   # (M,) — Cumulative multiplier δ̂_{i,h}
result.cum_ci_lower      # (M,) — Lower bound of family-wise 95% CI
result.cum_ci_upper      # (M,) — Upper bound of family-wise 95% CI
result.significant       # (M,) bool — Whether each product is significant
result.M, result.N, result.T, result.h  # Dimensions

Example: Oil Price Pass-Through

import multicoint as mc

# X_panel: (43 products, 41 cities, 104 quarters)
# oil_shock: (104,) first-differenced log WTI crude oil prices

mgdl = mc.MGDL(h=4, augmented_var=True, bonferroni=True)
res = mgdl.fit(X_panel, oil_shock, product_names=product_list)
mgdl.summary()  # Rich table of cumulative multipliers

# Plot significant IRFs
mc.plot_impulse_response(res)

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7. PME — Pooled Minimum Eigenvalue

Estimates multiple long-run relations in panels where n >> T. No prior method exists for this setting.

Paper: Chudik, Pesaran & Smith (2025), "Analysis of Multiple Long-Run Relations in Panel Data Models", arXiv:2506.02135v3.

Key innovations:

• Works when n >> T (e.g., n=1000 firms, T=20 years)
• Estimates r₀ ≥ 1 long-run relations simultaneously
• No need to model short-run dynamics (semi-parametric)
• No need to specify causal ordering
• Robust to interactive time effects (latent factors)

Algorithm

1. Split T observations into q ≥ 2 non-overlapping sub-samples
2. Compute pooled covariance matrix Q_{w̄w̄} from sub-sample means
3. Eigendecompose Q_{w̄w̄}: the r₀ smallest eigenvalues → long-run relations
4. Threshold eigenvalues at T^{-δ} to estimate r₀
5. Identify coefficients via exact restrictions

Syntax

from multicoint import PME

estimator = PME(
    q=2,          # Number of sub-samples (minimum 2, default 2)
    delta=0.25,   # Thresholding exponent for r₀ estimation
    r0=None,      # If known, fix r₀. Otherwise estimated automatically.
)

result = estimator.fit(
    W,            # np.ndarray, shape (n, T, m) — panel of m variables
    var_names=["exports", "imports", "GDP"],  # Optional
)

Result Attributes (PMEResult)

result.r_hat         # int — Estimated number of long-run relations
result.eigenvalues   # (m,) — Ordered eigenvalues of Q_{w̄w̄}
result.B_hat         # (m, r_hat) — Eigenvectors (long-run directions)
result.Theta_hat     # (m-r_hat, r_hat) — Identified long-run coefficients
result.std_errors    # Standard errors of vec(Θ̂)
result.t_ratios      # t-ratios for H₀: θ_{jk} = 0
result.p_values      # Two-sided p-values
result.ci_95         # List of (lower, upper) 95% CIs
result.Q_ww          # (m, m) — Pooled covariance matrix
result.n, result.T, result.m, result.q  # Dimensions

Example: Multiple Cointegrating Relations

import multicoint as mc

# W: (200 countries, 50 years, 3 variables: exports, imports, GDP)
pme = mc.PME(q=2, delta=0.25)
res = pme.fit(W, var_names=["Exports", "Imports", "GDP"])
pme.summary()  # Prints eigenvalue analysis + coefficient table

print(f"Number of long-run relations: r̂₀ = {res.r_hat}")
print(f"Eigenvalues: {res.eigenvalues}")
print(f"Long-run coefficients:\n{res.Theta_hat}")

# Scree plot of eigenvalues with threshold
mc.plot_eigenvalues(res)

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🔬 Diagnostics

Cross-Section Dependence Test

from multicoint import cd_test

# residuals: (T, n) array of panel residuals
result = cd_test(residuals)
print(f"CD statistic: {result.cd_stat:.3f}")
print(f"p-value: {result.p_value:.4f}")
print(f"Average pair-wise correlation: {result.avg_rho:.4f}")

Panel Unit Root Tests

from multicoint import panel_adf, panel_kpss

# ADF test (H₀: unit root)
adf_results = panel_adf(data, const=True)  # data: (T, n)
for r in adf_results:
    print(f"Unit {r['unit']}: ADF={r['adf_stat']:.3f}, reject={r['reject_5pct']}")

# KPSS test (H₀: stationarity)
kpss_results = panel_kpss(data)
for r in kpss_results:
    print(f"Unit {r['unit']}: KPSS={r['kpss_stat']:.3f}, reject={r['reject_5pct']}")

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📊 Visualization

Forest Plot of Confidence Intervals

import multicoint as mc

results = [spmg_result, pmg_result, breitung_result, pdols_result, mgmw_result]
fig, ax = mc.plot_confidence_intervals(
    results,
    theta0=1.0,           # Reference line (unit coefficient)
    figsize=(10, 6),
    save="forest_plot.png"
)

Bar Chart of Long-Run Coefficients

# Compare across multiple ratios
ratio_results = {
    "C/GDP": [spmg_cg, pmg_cg, breit_cg],
    "I/GDP": [spmg_ig, pmg_ig, breit_ig],
    "Debt/GDP": [spmg_dg, pmg_dg, breit_dg],
}
fig, ax = mc.plot_long_run_coefficients(ratio_results, save="ratios.png")

PME Eigenvalue Scree Plot

fig, ax = mc.plot_eigenvalues(pme_result, save="eigenvalues.png")

MGDL Impulse Response Functions

fig, axes = mc.plot_impulse_response(
    mgdl_result,
    product_idx=[0, 5, 25],  # Specific products to plot
    save="irfs.png"
)

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📋 Comparison Tables

Single Estimator Summary

result.summary()  # Rich-formatted terminal output

Multi-Estimator Comparison

import multicoint as mc

text, df = mc.comparison_table(
    [spmg_result, pmg_result, breitung_result, pdols4_result, mgmw_result],
    title="Great Ratios: Consumption / GDP",
    print_it=True,
)

# Export to various formats
df.to_csv("comparison.csv", index=False)
df.to_excel("comparison.xlsx", index=False)
df.to_latex("comparison.tex", index=False)

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🧪 Simulated Datasets

Built-in DGP simulators matching the Monte Carlo designs from all three papers:

1. Bivariate Panel with Single Long-Run Relation

from multicoint.datasets import simulate_great_ratios

Y, X = simulate_great_ratios(
    n=30,              # Cross-section units
    T=100,             # Time periods
    theta=1.0,         # True long-run coefficient
    phi_range=(0.1, 0.3),  # Speed of adjustment range
    pi_noncoint=0.2,   # Fraction of non-cointegrating units
    seed=42,
)
# Y, X: np.ndarray of shape (T, n)

2. Multivariate Panel with Multiple Long-Run Relations

from multicoint.datasets import simulate_multiple_lr

W, B0_true = simulate_multiple_lr(
    n=200,     # Cross-section units
    T=50,      # Time periods
    m=3,       # Number of variables
    r0=2,      # Number of long-run relations
    seed=42,
)
# W: (n, T, m), B0_true: (m, r0) true coefficient matrix

3. Panel for MGDL IRF Estimation

from multicoint.datasets import simulate_irf_panel

X_panel, shock, true_irf = simulate_irf_panel(
    M=43,      # Products
    N=41,      # Cities
    T=104,     # Quarters
    h=4,       # IRF horizon
    seed=42,
)
# X_panel: (M, N, T), shock: (T,), true_irf: (M, h+1)

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📚 Full API Reference

Estimator Classes

Class	Paper	Variables	Long-run relations	Causality	n vs T
SPMG	[1]	2	1	Both (x↔y)	T ≫ n
PMG	[1]	2	1	One-way (x→y)	T ≫ n
Breitung	[1]	2	1	Both	T ≫ n
PDOLS	[1]	2	1	One-way (x→y)	T ≫ n
MGMW	[1]	2	1	Both	T ≫ n
MGDL	[2]	M×N	IRFs	Shock→outcome	N,M,T large
PME	[3]	m ≥ 2	r₀ ≥ 1	Any	n ≫ T

Common Parameters

Parameter	Type	Default	Used by
p	int	2	SPMG, PMG, Breitung
max_iter	int	500	SPMG, PMG
precision	float	1e-4	SPMG, PMG
bootstrap_reps	int	2000	SPMG, PMG
seed	int	1234	SPMG, PMG
leads_lags	int	4	PDOLS
q	int	5/2	MGMW / PME
h	int	4	MGDL
delta	float	0.25	PME
r0	int	None	PME (auto-estimated if None)

Data Format

Estimator	Input Shape	Description
SPMG/PMG/etc.	Y(T,n), X(T,n)	Bivariate panel: T periods, n units
MGDL	X(M,N,T), v(T,)	Two cross-sections + shock series
PME	W(n,T,m)	m-variate panel: n units, T periods

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📐 Mathematical Background

Convergence Rates

Estimator	Rate	Conditions
SPMG	$T\sqrt{(1-\pi)n}$	$T \gg n$
PMG	$T\sqrt{n}$	$T \gg n$, strict exogeneity
MGDL	$\sqrt{N}$ and $\sqrt{M}$	$N/T \to \kappa_1 \geq 0$
PME	$\sqrt{nT}$	$T \approx n^d$, $d > 1/2$

Bootstrap Methods

Method	Used by	Description
Wild (Rademacher)	SPMG, PMG	$\kappa_t \sim \pm 1$ with prob 1/2 each
Conditional on X	PMG	Holds X fixed, resamples y equation only
Unconditional	SPMG	Resamples both y and x equations jointly

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📖 References

1. Chudik, A., Pesaran, M.H. & Smith, R.P. (2023). "Revisiting the Great Ratios Hypothesis." Federal Reserve Bank of Dallas, Globalization Institute Working Paper No. 415. [SPMG, PMG, Breitung, PDOLS, MGMW]

2. Choi, C.-Y. & Chudik, A. (2024). "Mean Group Distributed Lag Estimation of Impulse Response Functions in Large Panels." Federal Reserve Bank of Dallas, GI Working Paper No. 423. [MGDL]

3. Chudik, A., Pesaran, M.H. & Smith, R.P. (2025). "Analysis of Multiple Long-Run Relations in Panel Data Models." arXiv:2506.02135v3. [PME]

Supporting References

• Pesaran, M.H., Shin, Y. & Smith, R.P. (1999). "Pooled Mean Group Estimation." JASA, 94, 621–634.
• Breitung, J. (2005). "A Parametric Approach to the Estimation of Cointegration Vectors in Panel Data." Econometric Reviews, 24, 151–173.
• Mark, N.C. & Sul, D. (2003). "Cointegration Vector Estimation by Panel DOLS." Oxford Bulletin of Economics and Statistics.
• Müller, U.K. & Watson, M.W. (2018). "Long-Run Covariability." Econometrica, 86, 775–804.
• Pesaran, M.H. (2004). "General Diagnostic Tests for Cross Section Dependence in Panels." CESifo Working Paper 1229.

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👨‍🔬 Author

Dr. Merwan Roudane

• 📧 Email: merwanroudane920@gmail.com
• 🐙 GitHub: github.com/merwanroudane

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📄 License

MIT License. See LICENSE for details.

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