Bootstrap Fourier NARDL

The Bootstrap Fourier NARDL combines the Fourier-NARDL framework with bootstrap-based cointegration tests following Bertelli, Vacca & Zoia (2022). This approach eliminates the inconclusive zone problem inherent in PSS bounds tests.

Overview

The PSS bounds test has a well-known limitation: when the F-statistic falls between the I(0) and I(1) critical value bounds, the test is inconclusive. The bootstrap approach resolves this by generating data-specific critical values through resampling.

Motivation

The traditional PSS bounds test creates three possible outcomes:

• F > I(1): Cointegration (clear result)
• F < I(0): No cointegration (clear result)
• I(0) < F < I(1): Inconclusive (problematic!)

The inconclusive zone can be frustrating for researchers. Moreover, tabulated critical values may not be accurate for small sample sizes or non-standard error distributions.

Theoretical Framework

Bertelli, Vacca & Zoia (2022) propose a comprehensive testing strategy with three tests:

Three Bootstrap Tests

1. Overall F-test (F_ov)

Tests whether all level coefficients are jointly zero. This is the primary cointegration test.

2. t-test for Error Correction

Tests whether the error correction coefficient (\(\rho\)) is zero. If \(\rho = 0\), the dependent variable does not adjust to disequilibrium.

3. Independent Variables F-test (F_ind)

Tests whether the independent variables have zero long-run effect. If we reject \(H_0\) for F_ov but fail to reject for F_ind, we have a "degenerate" relationship.

Bootstrap Algorithm

The bootstrap procedure follows these steps:

1. Estimate the unrestricted model and obtain residuals \(\hat{\varepsilon}_t\)
2. For each bootstrap replication \(b = 1, \ldots, B\):
   • Resample residuals with replacement
   • Recenter resampled residuals (Davidson & MacKinnon, 2005)
   • Generate bootstrap \(y^*\) values
   • Re-estimate the model and compute test statistics
3. Compute bootstrap critical values from the empirical distribution
4. Compare original statistics to bootstrap critical values

Basic Usage

    

    Python

from nardl_fourier import BootstrapNARDL

# Fit Bootstrap Fourier NARDL
model = BootstrapNARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp'],
    decomp_vars=['oil_price'],
    maxlag=4,
    max_freq=3,
    n_bootstrap=1000,      # Number of bootstrap replications
    random_state=42       # For reproducibility
)

# View summary with bootstrap tests
print(model.summary())

# Get cointegration decision
print(model.cointegration_decision())
            

Bootstrap Parameters

Interpreting Results

The bootstrap test follows a decision flowchart:

Possible Outcomes

    

    Python

# Access detailed bootstrap results
results = model.bootstrap_results

print("Bootstrap Cointegration Tests:")
print("=" * 50)

for test in ['F_ov', 't', 'F_ind']:
    stat = results['statistics'][test]
    cv = results['critical_values'][test]['5%']
    pval = results['p_values'][test]
    decision = '✓ Reject H₀' if results['conclusions'][test] else '✗ Fail to reject'
    
    print(f"\\n{test}:")
    print(f"  Statistic:      {stat:.4f}")
    print(f"  Bootstrap CV 5%: {cv:.4f}")
    print(f"  p-value:        {pval:.4f}")
    print(f"  Decision:       {decision}")

print("\\n" + "=" * 50)
print(f"Overall: {results['overall']}")
print(f"{results['interpretation']}")
            

Visualization

Visualize the bootstrap distributions to see how your test statistics compare:

    

    Python

# Plot bootstrap distributions
fig = model.plot_bootstrap_distributions(figsize=(15, 5))

# The plot shows:
# - Histogram of bootstrap statistics
# - Red line: observed statistic
# - Green dashed line: 5% critical value

fig.savefig('bootstrap_distributions.png', dpi=300, bbox_inches='tight')
            

Complete Example

    

    Python

import pandas as pd
import numpy as np
from nardl_fourier import BootstrapNARDL

# Load or create data
np.random.seed(42)
n = 150
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 Bootstrap Fourier NARDL
model = BootstrapNARDL(
    data=df,
    depvar='coal',
    exog_vars=['gdp'],
    decomp_vars=['oil_price'],
    maxlag=4,
    max_freq=2,
    n_bootstrap=1000,
    random_state=42
)

# Full summary output
print(model.summary())

# Bootstrap summary table
if model.bootstrap_test:
    table = model.bootstrap_test.summary_table()
    print("\\nBootstrap Tests Summary:")
    print(table.to_string(index=False))

# Final decision
print("\\n" + "=" * 50)
print(model.cointegration_decision())

# Compare with standard bounds test
print("\\nComparison with PSS Bounds Test:")
bt = model.bounds_test
print(f"  PSS F-statistic: {bt['f_statistic']:.4f}")
print(f"  PSS Decision: {bt['decision']['F_5%']}")
print(f"  Bootstrap Decision: {model.bootstrap_results['overall']}")
            

References