Examples & Use Cases

Practical examples demonstrating how to use NARDL-Fourier for different research applications.

Example 1: Energy Economics

Analyzing asymmetric effects of oil prices on coal consumption following the methodology of Zaghdoudi et al. (2023).

    

    Python

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

# Load energy data
# Assuming annual data for a country
df = pd.read_excel('energy_data.xlsx')

# Variables:
# - coal: Coal consumption per capita (log)
# - oil_price: Real oil price (log)
# - gdp: GDP per capita (log)
# - gdp2: Squared GDP to capture EKC

print("Data preview:")
print(df.head())
print(f"\\nObservations: {len(df)}")

# Step 1: Fit Standard NARDL
print("\\n" + "="*60)
print("STANDARD NARDL MODEL")
print("="*60)

nardl = NARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp', 'gdp2'],
    decomp_vars=['oil_price'],
    maxlag=4,
    ic='AIC',
    case=3
)

print(nardl.summary())

# Step 2: Fit Fourier NARDL (account for structural breaks)
print("\\n" + "="*60)
print("FOURIER NARDL MODEL")
print("="*60)

fnardl = FourierNARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp', 'gdp2'],
    decomp_vars=['oil_price'],
    maxlag=4,
    max_freq=3,
    ic='AIC'
)

print(f"Optimal Fourier frequency: k* = {fnardl.best_freq}")
print(fnardl.summary())

# Step 3: Bootstrap test for robust inference
print("\\n" + "="*60)
print("BOOTSTRAP FOURIER NARDL")
print("="*60)

bnardl = BootstrapNARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp', 'gdp2'],
    decomp_vars=['oil_price'],
    maxlag=4,
    max_freq=3,
    n_bootstrap=1000,
    random_state=42
)

print(bnardl.summary())
print("\\nCointegration Decision:")
print(bnardl.cointegration_decision())
            

Interpretation

If the long-run multipliers show L⁺ ≠ L⁻ and the Wald test confirms asymmetry, coal consumption responds differently to oil price increases versus decreases. This has important policy implications for energy transitions.

Example 2: Exchange Rate Pass-Through

Examining asymmetric exchange rate pass-through to import prices.

    

    Python

import pandas as pd
from nardl_fourier import BootstrapNARDL, NARDLPlots

# Variables:
# - import_price: Import price index (log)
# - exchange_rate: Nominal exchange rate (log)
# - producer_price: Foreign producer prices (log)

df = pd.read_csv('erpt_data.csv')

# Model: Import prices ~ Exchange rate (asymmetric) + Foreign prices
model = BootstrapNARDL(
    data=df,
    depvar='import_price',
    exog_vars=['producer_price'],
    decomp_vars=['exchange_rate'],
    maxlag=4,
    max_freq=2,
    n_bootstrap=1000
)

# Results summary
print(model.summary())

# Extract key pass-through coefficients
lr = model.long_run['exchange_rate']

print("\\nExchange Rate Pass-Through (ERPT):")
print(f"  Depreciation (L⁺): {lr['positive']['coefficient']:.4f}")
print(f"  Appreciation (L⁻): {lr['negative']['coefficient']:.4f}")

# Interpretation
if abs(lr['positive']['coefficient']) > abs(lr['negative']['coefficient']):
    print("\\n→ Depreciations have LARGER effects on import prices")
    print("  (Firms pass through cost increases more readily)")
else:
    print("\\n→ Appreciations have LARGER effects on import prices")
    print("  (Consumers benefit more from currency appreciation)")

# Plot dynamic multipliers
plots = NARDLPlots(model)
fig = plots.dynamic_multipliers('exchange_rate', horizon=20)
fig.savefig('erpt_multipliers.png', dpi=300)
            

Example 3: Comparing All Three Models

A systematic comparison of NARDL, Fourier NARDL, and Bootstrap NARDL.

    

    Python

import pandas as pd
from nardl_fourier import NARDL, FourierNARDL, BootstrapNARDL

# Load data
df = pd.read_csv('data.csv')

# Common specification
spec = {
    'data': df,
    'depvar': 'y',
    'exog_vars': ['z'],
    'decomp_vars': ['x'],
    'maxlag': 4,
    'ic': 'AIC'
}

# Fit all three models
m1 = NARDL(**spec)
m2 = FourierNARDL(**spec, max_freq=3)
m3 = BootstrapNARDL(**spec, max_freq=3, n_bootstrap=1000)

# Comparison table
comparison = pd.DataFrame({
    'Metric': ['R²', 'Adj. R²', 'AIC', 'F-stat', 'Bounds Decision', 'ECT (ρ)'],
    'NARDL': [
        f"{m1.model.rsquared:.4f}",
        f"{m1.model.rsquared_adj:.4f}",
        f"{m1.best_ic:.2f}",
        f"{m1.bounds_test['f_statistic']:.4f}",
        m1.bounds_test['decision']['F_5%'],
        f"{m1.ect['coefficient']:.4f}"
    ],
    'Fourier NARDL': [
        f"{m2.model.rsquared:.4f}",
        f"{m2.model.rsquared_adj:.4f}",
        f"{m2.best_ic:.2f}",
        f"{m2.bounds_test['f_statistic']:.4f}",
        m2.bounds_test['decision']['F_5%'],
        f"{m2.ect['coefficient']:.4f}"
    ],
    'Bootstrap NARDL': [
        f"{m3.model.rsquared:.4f}",
        f"{m3.model.rsquared_adj:.4f}",
        f"{m3.best_ic:.2f}",
        f"{m3.bounds_test['f_statistic']:.4f}",
        m3.bootstrap_results['overall'],
        f"{m3.ect['coefficient']:.4f}"
    ]
})

print("Model Comparison:")
print(comparison.to_string(index=False))
            

Example 4: Generating Publication Tables

Export results in LaTeX format for academic papers.

    

    Python

from nardl_fourier import BootstrapNARDL, ResultsTable

# Fit model
model = BootstrapNARDL(...)

# Create table generator
table = ResultsTable(model)

# Individual tables as DataFrames
reg_df = table.regression_table()          # Full regression
lr_df = table.long_run_table()              # Long-run multipliers
sr_df = table.short_run_table()            # Short-run coefficients
diag_df = table.diagnostics_table()        # Diagnostic tests
bounds_df = table.bounds_test_table()      # Bounds test
wald_df = table.wald_test_table()          # Asymmetry tests

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

# Or get individual LaTeX tables
latex_lr = table.long_run_table(output='latex')
print(latex_lr)

# Bootstrap test summary
if model.bootstrap_test:
    boot_df = model.bootstrap_test.summary_table()
    print("\\nBootstrap Cointegration Tests:")
    print(boot_df.to_latex(index=False))
            

Sample LaTeX Output

    

    LaTeX

\begin{table}[htbp]
\centering
\caption{Long-Run Multipliers}
\label{tab:long-run_multipliers}
\begin{tabular}{lcccccc}
\toprule
Variable & Coeff. & Std. Error & t-stat & p-value & 95\% CI \\
\midrule
oil\_price$^{+}$ & 0.4521 & 0.0823 & 5.49 & 0.0000 & [0.290, 0.614] \\
oil\_price$^{-}$ & 0.2134 & 0.0756 & 2.82 & 0.0068 & [0.064, 0.363] \\
\bottomrule
\end{tabular}
\end{table}