Fourier NARDL Model

The Fourier NARDL extends the standard NARDL framework by incorporating Fourier trigonometric terms to capture smooth structural breaks without specifying break dates. Based on Zaghdoudi, Maktouf & Ochi (2023).

Overview

Traditional structural break tests require specifying break dates a priori or rely on multiple regime-switching models. The Fourier approach offers an elegant alternative by using trigonometric functions to approximate unknown structural changes smoothly.

Theoretical Background

The Fourier approach to structural breaks was pioneered by Becker, Enders & Lee (2006) and later applied to cointegration analysis. The key insight is that any smooth function can be approximated by a Fourier series:

In practice, a single frequency (k=1, 2, or 3) is typically sufficient to capture most structural variations, keeping the model parsimonious.

Fourier Approximation

For a sample of size T, we construct trigonometric terms at frequency \(k\):

The frequency \(k\) controls the periodicity of the breaks:

• \(k = 1\): One complete cycle over the sample (gradual, single break)
• \(k = 2\): Two complete cycles (two potential break periods)
• \(k = 3\): Three complete cycles (multiple gradual changes)

Model Specification

The Fourier NARDL model augments the standard NARDL with Fourier terms:

The Fourier terms (\(\gamma, \delta\)) capture time-varying intercepts (and potentially time-varying coefficients) that would otherwise bias the estimates if structural breaks are ignored.

Basic Usage

    

    Python

from nardl_fourier import FourierNARDL

# Fit Fourier NARDL model
model = FourierNARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp'],
    decomp_vars=['oil_price'],
    maxlag=4,
    max_freq=3,          # Maximum Fourier frequency to try
    ic='AIC',
    case=3
)

# View optimal frequency
print(f"Optimal Fourier frequency: k* = {model.best_freq}")

# Full summary
print(model.summary())
            

Additional Parameters

All other parameters are inherited from the standard NARDL class.

Optimal Frequency Selection

The optimal frequency \(k^*\) is selected automatically using the information criterion specified (AIC, BIC, or HQIC). The model estimates all frequencies from 1 to max_freq and selects the one that minimizes the criterion.

    

    Python

# Access optimal frequency and IC value
print(f"Optimal frequency: k* = {model.best_freq}")
print(f"Best {model.ic}: {model.best_ic:.4f}")

# The Fourier terms in the model
print("\\nFourier Terms Coefficients:")
for name in model.model.params.index:
    if 'sin' in name or 'cos' in name:
        print(f"  {name}: {model.model.params[name]:.6f}")
            

Fourier Terms Significance Test

To verify that the Fourier terms add explanatory power, we test the joint significance of the sine and cosine terms:

    

    Python

# Access Fourier significance test
if hasattr(model, 'fourier_test') and model.fourier_test:
    ft = model.fourier_test
    print("Fourier Terms Joint Significance Test:")
    print(f"  H₀: γ = δ = 0")
    print(f"  F-statistic: {ft['f_statistic']:.4f}")
    print(f"  p-value: {ft['p_value']:.4f}")
    print(f"  Significant: {ft['significant']}")
            

Complete Example

    

    Python

import pandas as pd
import numpy as np
from nardl_fourier import FourierNARDL, ResultsTable

# Simulate data with structural break
np.random.seed(42)
n = 200
t = np.arange(n)

# Create a smooth structural shift
shift = 2 * np.sin(2 * np.pi * t / n)

df = pd.DataFrame({
    'coal': shift + np.cumsum(np.random.randn(n) * 0.5),
    'oil_price': np.cumsum(np.random.randn(n)),
    'gdp': np.cumsum(np.random.randn(n))
})

# Fit Fourier NARDL
model = FourierNARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp'],
    decomp_vars=['oil_price'],
    maxlag=4,
    max_freq=3,
    ic='AIC'
)

# Summary
print(model.summary())

# Compare with standard NARDL
from nardl_fourier import NARDL

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

print("\\nModel Comparison:")
print(f"  Standard NARDL R²: {standard_model.model.rsquared:.4f}")
print(f"  Fourier NARDL R²:  {model.model.rsquared:.4f}")
print(f"  Improvement:       {(model.model.rsquared - standard_model.model.rsquared)*100:.2f}%")