NARDL Model

The Nonlinear Autoregressive Distributed Lag (NARDL) model extends the linear ARDL framework to capture asymmetric effects in cointegrating relationships. Based on the seminal work of Shin, Yu & Greenwood-Nimmo (2014).

Overview

Traditional cointegration analysis assumes symmetric adjustments to equilibrium. However, many economic relationships exhibit asymmetry—variables may respond differently to positive versus negative changes. The NARDL model addresses this by decomposing regressors into partial sums of positive and negative changes.

Theoretical Framework

The NARDL approach rests on the concept of asymmetric cointegration. Consider a long-run relationship between \(y_t\) and \(x_t\):

where \(x_t^+\) and \(x_t^-\) are partial sum processes of positive and negative changes in \(x_t\):

Model Specification

The NARDL(p,q) error correction model is specified as:

Where:

• \(\rho\) is the error correction coefficient (expected to be negative)
• \(\theta^+, \theta^-\) capture the level (long-run) effects
• \(\pi_j^+, \pi_j^-\) capture the short-run dynamic effects
• Long-run multipliers: \(L^+ = -\theta^+ / \rho\) and \(L^- = -\theta^- / \rho\)

Basic Usage

    

    Python

from nardl_fourier import NARDL
import pandas as pd

# Load your data
df = pd.read_csv('data.csv', index_col=0, parse_dates=True)

# Fit NARDL model
model = NARDL(
    data=df,
    depvar='coal',              # Dependent variable
    exog_vars=['gdp', 'gdp2'],   # Control variables (linear)
    decomp_vars=['oil_price'],  # Variables to decompose (asymmetric)
    maxlag=4,                   # Maximum lag order
    ic='AIC',                   # Information criterion
    case=3                      # Case for bounds test
)

# View summary
print(model.summary())
            

Parameters Explained

Partial Sum Decomposition

The core of NARDL is the partial sum decomposition. For any variable \(x\), we compute:

    

    Python

from nardl_fourier.utils import partial_sum_decomposition

# Decompose a variable
x_positive, x_negative = partial_sum_decomposition(df['oil_price'])

# x_positive: cumulative sum of positive changes
# x_negative: cumulative sum of negative changes
print(f"Final positive sum: {x_positive[-1]:.2f}")
print(f"Final negative sum: {x_negative[-1]:.2f}")
            

Long-Run Multipliers

Long-run multipliers measure the cumulative impact of a unit change in \(x\) on \(y\):

    

    Python

# Access long-run multipliers
lr = model.long_run['oil_price']

print("Long-Run Multipliers:")
print(f"  L⁺ (positive): {lr['positive']['coefficient']:.4f}")
print(f"     Std.Error:  {lr['positive']['std_error']:.4f}")
print(f"     p-value:    {lr['positive']['p_value']:.4f}")
print(f"  L⁻ (negative): {lr['negative']['coefficient']:.4f}")
print(f"     Std.Error:  {lr['negative']['std_error']:.4f}")
print(f"     p-value:    {lr['negative']['p_value']:.4f}")
            

Short-Run Dynamics

Short-run coefficients capture the immediate dynamic adjustments:

    

    Python

# Error Correction Term
ect = model.ect
print(f"ECT (ρ): {ect['coefficient']:.4f}")
print(f"Half-life: {ect['half_life']:.2f} periods")

# Short-run coefficients table
from nardl_fourier import ResultsTable
table = ResultsTable(model)
sr_df = table.short_run_table()
print(sr_df)
            

Bounds Test for Cointegration

The PSS bounds test examines whether a long-run relationship exists:

    

    Python

# Access bounds test results
bt = model.bounds_test

print("PSS Bounds Test:")
print(f"  F-statistic: {bt['f_statistic']:.4f}")
print(f"  Critical Values (5%):")
print(f"    I(0): {bt['critical_values']['F']['5%']['I(0)']:.4f}")
print(f"    I(1): {bt['critical_values']['F']['5%']['I(1)']:.4f}")
print(f"  Decision: {bt['decision']['F_5%']}")
            

Interpretation

• F > I(1) bound: Reject H₀ → Evidence of cointegration
• F < I(0) bound: Fail to reject H₀ → No cointegration
• I(0) < F < I(1): Inconclusive (consider Bootstrap NARDL)

Testing for Asymmetry

Wald tests formally examine whether positive and negative effects are statistically different:

    

    Python

# Access Wald test results
wald = model.wald['oil_price']

# Short-run asymmetry
print("Short-Run Asymmetry Test:")
print(f"  H₀: π⁺ = π⁻ at all lags")
print(f"  F-statistic: {wald['short_run']['f_statistic']:.4f}")
print(f"  p-value: {wald['short_run']['p_value']:.4f}")
print(f"  Asymmetric: {wald['short_run']['asymmetric']}")

# Long-run asymmetry
print("\\nLong-Run Asymmetry Test:")
print(f"  H₀: L⁺ = L⁻")
print(f"  F-statistic: {wald['long_run']['f_statistic']:.4f}")
print(f"  p-value: {wald['long_run']['p_value']:.4f}")
print(f"  Asymmetric: {wald['long_run']['asymmetric']}")
            

Complete Example

    

    Python

import pandas as pd
import numpy as np
from nardl_fourier import NARDL, ResultsTable, NARDLPlots

# Create sample data
np.random.seed(42)
n = 200
df = pd.DataFrame({
    'coal': np.cumsum(np.random.randn(n)),
    'oil_price': np.cumsum(np.random.randn(n)),
    'gdp': np.cumsum(np.random.randn(n))
})

# Fit NARDL model
model = NARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp'],
    decomp_vars=['oil_price'],
    maxlag=4,
    ic='AIC'
)

# Print full summary
print(model.summary())

# Generate publication tables
table = ResultsTable(model)
print("\\nLong-Run Multipliers:")
print(table.long_run_table())

print("\\nDiagnostics:")
print(table.diagnostics_table())

# Export to LaTeX
table.to_latex('nardl_results.tex')

# Create plots
plots = NARDLPlots(model)
plots.dynamic_multipliers('oil_price', horizon=20)
plots.cusum()