RBFM-VAR: Residual-Based Fully Modified Vector Autoregression

A Python implementation of the Residual-Based Fully Modified Vector Autoregression (RBFM-VAR) methodology for nonstationary vector autoregressions with unknown mixtures of I(0), I(1), and I(2) components.

Overview

This package implements the methodology developed by Yoosoon Chang (2000): "Vector Autoregressions with Unknown Mixtures of I(0), I(1), and I(2) Components", published in Econometric Theory, Vol. 16, No. 6, pp. 905-926.

Key Features

• No Prior Knowledge Required: Estimates VAR models without requiring prior knowledge about the exact number and location of unit roots in the system
• Mixed Integration Orders: Handles any mixture of I(0), I(1), and I(2) variables that may be cointegrated in any form
• Optimal Estimation: Yields an estimator that is optimal in the sense of Phillips (1991)
• Mixed Normal Limit Theory: The nonstationary component has mixed normal limit distribution without unit root distributions
• Conservative Wald Tests: Provides asymptotically valid tests using conventional chi-square critical values
• Granger Causality Testing: Direct application for causality testing in nonstationary VARs

Installation

pip install rbfmvar

Or install from source:

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

Quick Start

import numpy as np
from rbfmvar import RBFMVAR

# Generate or load your data (T x n matrix)
# Example: bivariate system with 200 observations
T, n = 200, 2
np.random.seed(42)

# Simulate data (replace with your actual data)
data = np.cumsum(np.cumsum(np.random.randn(T, n), axis=0), axis=0)

# Fit RBFM-VAR model
model = RBFMVAR(lag_order=2, bandwidth='auto')
results = model.fit(data)

# Print summary
print(results.summary())

# Test for Granger causality
causality_test = results.granger_causality_test(
    caused_variable=0,
    causing_variables=[1]
)
print(causality_test)

Mathematical Background

The Model

The RBFM-VAR procedure estimates a p-th order VAR:

$$y_t = A_1 y_{t-1} + \cdots + A_p y_{t-p} + \varepsilon_t$$

which can be written in Error Correction Model (ECM) format:

$$\Delta^2 y_t = \Phi(L)\Delta^2 y_{t-1} + \Pi_1 \Delta y_{t-1} + \Pi_2 y_{t-1} + \varepsilon_t$$

RBFM-VAR Estimator

The estimator is defined as:

$$\hat{F}^+ = (Y'Z, Y^{+'} W + T\hat{A}^+)(X'X)^{-1}$$

where:

$$Y^{+'} = Y' - \hat{\Omega}{\varepsilon\hat{v}} \hat{\Omega}{\hat{v}\hat{v}}^{-1} \hat{V}'$$

$$\hat{A}^+ = \hat{\Omega}{\varepsilon\hat{v}} \hat{\Omega}{\hat{v}\hat{v}}^{-1} \hat{\Delta}_{\hat{v}\Delta w}$$

Key Results (Theorem 1)

(a) For the stationary component: $$\sqrt{T}(\hat{F}^+ - F)G^1 \xrightarrow{d} N(0, \Sigma_{\varepsilon\varepsilon} \otimes \Sigma_{x11}^{-1})$$

(b) For the nonstationary component: $$(\hat{F}^+ - F)G^b D_T \xrightarrow{d} \text{MN}(0, \Omega_{\varepsilon\varepsilon \cdot 2} \otimes (\int_0^1 \bar{B}_b \bar{B}_b')^{-1})$$

Modified Wald Test (Theorem 2)

The modified Wald statistic has a limit distribution that is a mixture of chi-square variates:

$$W_F^+ \xrightarrow{d} \chi^2_{q_1(q_\Phi + q_{A1})} + \sum_{i=1}^{q_1} d_i \chi^2_{q_{Ab}}(i)$$

This is bounded above by a $\chi^2_q$ distribution, allowing the use of conventional critical values.

API Reference

Main Classes

RBFMVAR

The main class for RBFM-VAR estimation.

RBFMVAR(
    lag_order: int = 1,
    bandwidth: Union[int, str] = 'auto',
    kernel: str = 'bartlett',
    trend: str = 'c'
)

Parameters:

• lag_order: Number of lags in the VAR model (p)
• bandwidth: Bandwidth for kernel estimation ('auto' or integer)
• kernel: Kernel function ('bartlett', 'parzen', 'qs')
• trend: Trend specification ('n' = none, 'c' = constant, 'ct' = constant + trend)

Methods:

• fit(data): Estimate the RBFM-VAR model
• predict(steps): Generate forecasts

RBFMVARResults

Results class containing estimation output.

Attributes:

• coefficients: Estimated coefficient matrices
• residuals: Model residuals
• sigma_epsilon: Estimated error covariance matrix
• wald_statistic: Modified Wald test statistic

Methods:

• summary(): Print formatted summary
• granger_causality_test(): Test for Granger causality
• wald_test(R, r): General linear hypothesis test

Utility Functions

from rbfmvar import (
    kernel_covariance,      # Long-run covariance estimation
    bartlett_kernel,        # Bartlett kernel function
    parzen_kernel,          # Parzen kernel function
    qs_kernel,              # Quadratic Spectral kernel
    optimal_bandwidth       # Automatic bandwidth selection
)

Examples

Example 1: Basic Estimation

import numpy as np
import pandas as pd
from rbfmvar import RBFMVAR

# Load data
data = pd.read_csv('your_data.csv').values

# Estimate model
model = RBFMVAR(lag_order=2)
results = model.fit(data)

# Get coefficient estimates
print("OLS-VAR Coefficients:")
print(results.ols_coefficients)

print("\nRBFM-VAR Coefficients:")
print(results.coefficients)

Example 2: Granger Causality Testing

from rbfmvar import RBFMVAR

# Estimate model
model = RBFMVAR(lag_order=2)
results = model.fit(data)

# Test if variable 1 Granger-causes variable 0
causality = results.granger_causality_test(
    caused_variable=0,
    causing_variables=[1]
)

print(f"Modified Wald Statistic: {causality['statistic']:.4f}")
print(f"P-value (conservative): {causality['p_value']:.4f}")
print(f"Degrees of Freedom: {causality['df']}")

Example 3: Monte Carlo Simulation

from rbfmvar import monte_carlo_simulation

# Run simulation study
results = monte_carlo_simulation(
    dgp='case_a',      # Data generating process
    T=150,             # Sample size
    n_reps=1000,       # Number of replications
    seed=42
)

# Display results
print(results.summary())

Data Generating Processes

The package includes three DGPs from the paper:

• Case A (ρ₁=1, ρ₂=0): Both y₁ and y₂ are I(2) with no cointegration
• Case B (ρ₁=0.5, ρ₂=0): y₁ is I(1), y₂ is I(2), no Granger causality
• Case C (ρ₁=-0.3, ρ₂=-0.15): y₁ is I(1), y₂ is I(2), y₂ Granger-causes y₁

Citation

If you use this package in your research, please cite:

@article{chang2000var,
  title={Vector Autoregressions with Unknown Mixtures of {I}(0), {I}(1), and {I}(2) Components},
  author={Chang, Yoosoon},
  journal={Econometric Theory},
  volume={16},
  number={6},
  pages={905--926},
  year={2000},
  publisher={Cambridge University Press}
}

@software{roudane2024rbfmvar,
  title={RBFM-VAR: Python Implementation of Residual-Based Fully Modified VAR},
  author={Roudane, Merwan},
  year={2024},
  url={https://github.com/merwanroudane/rbfmvar}
}

References

• Chang, Y. (2000). Vector Autoregressions with Unknown Mixtures of I(0), I(1), and I(2) Components. Econometric Theory, 16(6), 905-926.
• Chang, Y. & Phillips, P.C.B. (1995). Time series regression with mixtures of integrated processes. Econometric Theory, 11, 1033-1094.
• Phillips, P.C.B. (1991). Optimal inference in cointegrated systems. Econometrica, 59, 283-306.
• Phillips, P.C.B. (1995). Fully modified least squares and vector autoregression. Econometrica, 63, 1023-1078.
• Johansen, S. (1995). A statistical analysis of cointegration for I(2) variables. Econometric Theory, 11, 25-59.
• Toda, H. & Phillips, P.C.B. (1993). Vector autoregressions and causality. Econometrica, 61, 1367-1393.

License

MIT License - see LICENSE for details.

Author

Dr. Merwan Roudane

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

Changelog

Version 2.0.0 (2025)

• Complete implementation of RBFM-VAR methodology
• Long-run covariance estimation with multiple kernel options
• Modified Wald tests for hypothesis testing
• Granger causality testing
• Monte Carlo simulation tools
• Publication-ready output formatting