qardl 1.0.2

Quantile Autoregressive Distributed Lag (QARDL) Models - Cho, Kim & Shin (2015)

QARDL: Quantile Autoregressive Distributed Lag Models

Complete Python implementation of Quantile Autoregressive Distributed Lag (QARDL) models 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.

What's New in v1.0.1

CRITICAL BUG FIX:

• Fixed ordering issue in _prepare_data() method
• X_effective and y_effective now correctly set BEFORE building matrices
• This bug caused "AttributeError: X_effective doesn't exist" errors

All theoretical corrections from the original paper are maintained and working properly.

Features

✅ Correct Implementation - All formulas match Cho, Kim & Shin (2015) exactly ✅ Proper Standard Errors - H_t projection with correct asymptotic variance ✅ Correct Wald Tests - n² scaling for long-run, n scaling for short-run ✅ Complete ECM - Full Error Correction Model representation ✅ Lag Selection - OLS-based BIC/AIC at mean (icmean.m method) ✅ Rolling Estimation - Time-varying parameter analysis ✅ Multi-Quantile - Cross-quantile inference ✅ Publication Ready - Output formatted for academic journals

Installation

pip install qardl

For development:

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

Quick Start

import numpy as np
from qardl import QARDLCorrected

# Generate data
np.random.seed(42)
n = 200
y = np.cumsum(np.random.normal(0, 1, n))
X = np.cumsum(np.random.normal(0, 1, (n, 2)), axis=0)

# Estimate QARDL(2,1) at median
model = QARDLCorrected(y, X, p=2, q=1, tau=0.5)
results = model.fit()
results.summary()

Core Functionality

1. QARDL Estimation

from qardl import QARDLCorrected

# Single quantile
model = QARDLCorrected(y, X, p=2, q=1, tau=0.5)
results = model.fit()

# Multiple quantiles
model_multi = QARDLCorrected(y, X, p=2, q=1, tau=[0.25, 0.50, 0.75])
results_dict = model_multi.fit()

2. Hypothesis Testing

from qardl import WaldTestsCorrected

# Initialize tests
wald = WaldTestsCorrected(results)

# Long-run parameters (uses n² scaling)
lr_test = wald.test_long_run_parameters()
print(f"LR Wald: {lr_test['statistic']:.4f}, p-value: {lr_test['pvalue']:.4f}")

# Short-run parameters (uses n scaling)
sr_test = wald.test_short_run_parameters()
print(f"SR Wald: {sr_test['statistic']:.4f}, p-value: {sr_test['pvalue']:.4f}")

# Symmetry test
sym_test = wald.test_symmetry()

3. ECM Representation

from qardl import QARDLtoECM, ECMWaldTests

# Convert to ECM
ecm = QARDLtoECM(results)
ecm.print_summary()

# ECM-specific tests
ecm_wald = ECMWaldTests(ecm)
speed_test = ecm_wald.test_speed_of_adjustment()

4. Lag Selection

from qardl import select_qardl_orders

# Automatic lag selection
best_orders = select_qardl_orders(
    y, X,
    max_p=4,
    max_q=4,
    tau=0.5,
    criterion='bic'
)
print(f"Optimal lags: p={best_orders['p']}, q={best_orders['q']}")

5. Rolling Estimation

from qardl import RollingQARDL

# Rolling window estimation
rolling = RollingQARDL(y, X, p=2, q=1, tau=0.5, window=100)
rolling_results = rolling.estimate()

# Plot results
rolling.plot_coefficients(['beta_y_lag1', 'delta_x1'])

Complete Example

See examples/ folder for detailed examples:

• example_01_basic_qardl.py - Basic estimation
• example_02_wald_tests.py - Hypothesis testing
• example_03_ecm.py - ECM representation
• example_04_lag_selection.py - Lag selection
• example_05_multiple_quantiles.py - Multi-quantile analysis
• example_06_rolling.py - Rolling estimation

Mathematical Background

QARDL(p,q) Model

The model estimates conditional quantiles:

Q_τ(y_t | X_t) = β_0(τ) + β_1(τ)y_{t-1} + Σ γ_i(τ)Δy_{t-i} + 
                  Σ_j [δ_j(τ)x_{jt} + Σ_i φ_{ji}(τ)Δx_{jt-i}]

Key Corrections

1. Standard Errors (Theorem 1):

   • Uses H_t projection: H_t = K_t - E[K_tW_t']E[W_tW_t']^{-1}W_t
   • Asymptotic variance: Var(β̂) = τ(1-τ) * (M^{-1} Σ M^{-1}) / n²

2. Long-Run Wald Tests (Corollary 1):

   • Statistic: W = n²(Rβ-r)'[R(Σ⊗M^{-1})R']^{-1}(Rβ-r)
   • Uses n² scaling (not n)

3. Short-Run Wald Tests (Corollaries 2-3):

   • Statistic: W = n(Rφ-r)'[RΠR']^{-1}(Rφ-r)
   • Uses n scaling

4. M Matrix (Theorem 2):

   • Formula: M = n^{-2}X'[I - W(W'W)^{-1}W']X
   • Critical for long-run inference

API Reference

Classes

• QARDLCorrected - Main estimation class
• QARDLResultsCorrected - Results container
• WaldTestsCorrected - Hypothesis testing
• QARDLtoECM - ECM representation
• RollingQARDL - Rolling window estimation

Functions

• select_qardl_orders() - Automatic lag selection
• select_orders_sequential() - Sequential testing
• compare_orders() - Compare multiple specifications
• quantile_regression() - Basic quantile regression
• bandwidth_hall_sheather() - Bandwidth selection

Requirements

• Python >= 3.7
• NumPy >= 1.20.0
• SciPy >= 1.7.0
• pandas >= 1.3.0
• statsmodels >= 0.13.0
• matplotlib >= 3.4.0
• seaborn >= 0.11.0

Citation

If you use this package in your research, 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}
}

And for the software:

@software{qardl2024,
  title={QARDL: Quantile Autoregressive Distributed Lag Models},
  author={Roudane, Merwan},
  year={2024},
  url={https://github.com/merwanroudane/qardl}
}

License

MIT License - see LICENSE file

Author

Dr. Merwan Roudane

• Email: [e-mail]
• GitHub: https://github.com/merwanroudane

Acknowledgments

This implementation is based on the original MATLAB and GAUSS codes accompanying Cho, Kim & Shin (2015). All theoretical results follow their paper exactly.

Support

• Issues: https://github.com/merwanroudane/qardl/issues
• Documentation: https://github.com/merwanroudane/qardl
• Email: [e-mail]

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Important: This package implements ALL corrections from the original paper. Previous versions may have had bugs - always use v1.0.1 or later.