QARDL - Quantile Autoregressive Distributed Lag Models

Version 1.0.3 - GAUSS/MATLAB Compatible Implementation

Exact Python implementation of the QARDL methodology from:

Cho, J.S., Kim, T.-H., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300.

Features

• ✅ GAUSS Compatible: Design matrix construction matches original GAUSS qardl.src exactly
• ✅ Correct Long-run Formula: β = θ₀ / (1 - Σφ) using only current X coefficient
• ✅ Proper Wald Tests: Exact scaling factors (n-1)² for long-run, (n-1) for short-run
• ✅ ECM Representation: MATLAB qardlecm.m compatible Error Correction Model
• ✅ Rolling Estimation: rollingQardl with Wald tests at each window
• ✅ Simulation: qardlAR2Sim for Monte Carlo experiments
• ✅ Plotting: plotQARDL equivalent visualization
• ✅ BIC Lag Selection: Automatic optimal (p, q) selection

Installation

pip install qardl

Or from source:

pip install -e .

Quick Start

import numpy as np
from qardl import qardl, QARDL, pq_order

# Prepare data: [y | X1 | X2 | ...]
data = np.column_stack([y, X])

# Select optimal lag orders
p_opt, q_opt = pq_order(data, p_max=7, q_max=7)
print(f"Optimal lags: p={p_opt}, q={q_opt}")

# Estimate QARDL model
tau = np.array([0.10, 0.25, 0.50, 0.75, 0.90])
qaOut = qardl(data, p_opt, q_opt, tau)

# Results
print("Long-run β:", qaOut.bigbt)
print("Short-run φ:", qaOut.phi)
print("Short-run γ:", qaOut.gamma)
print(qaOut.summary())

Wald Tests

from qardl import wtestlrb, wtestsrp, wtestsrg

# Define restrictions: R * β = r
# Example: Test β₁(τ₁) = β₁(τ₂)
R = np.zeros((1, len(qaOut.bigbt)))
R[0, 0] = 1   # β₁(τ₁)
R[0, k] = -1  # β₁(τ₂)
r = np.zeros(1)

# Long-run parameter test (uses (n-1)² scaling)
wt, pv = wtestlrb(qaOut.bigbt, qaOut.bigbt_cov, R, r, data)
print(f"Wald statistic: {wt:.4f}, p-value: {pv:.4f}")

# Short-run tests (use (n-1) scaling)
wt_phi, pv_phi = wtestsrp(qaOut.phi, qaOut.phi_cov, R_phi, r_phi, data)
wt_gam, pv_gam = wtestsrg(qaOut.gamma, qaOut.gamma_cov, R_gam, r_gam, data)

Rolling QARDL Estimation

from qardl import rolling_qardl, create_wald_restrictions

# Create Wald test restrictions
tau = np.array([0.25, 0.50, 0.75])
wctl = create_wald_restrictions(k=2, p_max=7, num_tau=len(tau))

# Run rolling estimation
rqaOut = rolling_qardl(data, p_max=7, q_max=7, tau=tau, wctl=wctl)

# Results
print(f"Window size: {rqaOut.window_size}")
print(f"Beta array shape: {rqaOut.bigbt.shape}")  # (k x num_est x num_tau)

# Get specific estimates
beta1_05 = rqaOut.get_beta(var_idx=0, tau_idx=1)  # β₁(τ=0.5)

# Wald test results
print("Wald beta:", rqaOut.rWaldOut.wald_beta)

ECM Representation

from qardl import qardl_ecm, QARDLECM

# Direct ECM estimation
ecmOut = qardl_ecm(data, p, q, tau)

# Results
print("Adjustment speed (ζ):", ecmOut.zeta)
print("Long-run β:", ecmOut.beta)
print("Half-life:", ecmOut.get_half_life(0))  # For first quantile

Simulation

from qardl import qardl_ar2_sim, DGPParams, simulate_wald_tests

# Generate QARDL data (matches GAUSS qardlAR2Sim)
y, X = qardl_ar2_sim(n=500, alpha=1.0, phi=0.25, rho=0.5, 
                      theta0=2.0, theta1=3.0, seed=42)

# With custom parameters
params = DGPParams(alpha=1.0, phi=0.25, theta0=2.0, theta1=3.0)
print(f"Long-run β = {params.beta:.4f}")

# Monte Carlo simulation for Wald test size
results = simulate_wald_tests(n=500, n_iter=1000, p=1, q=2, 
                               tau=np.array([0.2, 0.4, 0.6, 0.8]))

Plotting

from qardl import plot_qardl, plot_beta, plot_rolling

# Plot all parameters
fig = plot_qardl(qaOut)
plt.show()

# Plot with confidence intervals
fig = plot_beta(qaOut, with_ci=True, alpha=0.05)
plt.show()

# Plot rolling results
fig = plot_rolling(rqaOut, var_idx=0, tau_idx=1, param_type='beta')
plt.show()

Design Matrix Structure

The design matrix follows GAUSS ordering exactly:

ONEX = [1 | eei | xxi | yyi]

Where:

• 1: Intercept
• eei: Lagged first differences of X (Δx_{t}, Δx_{t-1}, ..., Δx_{t-q+1})
• xxi: Current levels of X (x_t)
• yyi: Lagged dependent variable (y_{t-1}, ..., y_{t-p})

Complete Function Reference

Core Estimation

Wald Tests

Rolling Estimation

ECM

Simulation

Plotting

Output Structure (matches GAUSS)

Where k = number of X variables, s = number of quantiles, p = AR order.

Citation

If you use this package, please cite:

@article{cho2015quantile,
  title={Quantile cointegration in the autoregressive distributed-lag modeling framework},
  author={Cho, Jin Seo and Kim, Tae-hwan and Shin, Yongcheol},
  journal={Journal of Econometrics},
  volume={188},
  number={1},
  pages={281--300},
  year={2015},
  publisher={Elsevier}
}

Author

Dr. Merwan Roudane
Email: merwanroudane920@gmail.com

License

MIT License