Fourier Granger Causality Test
Fourier Approximation Granger Causality Test · Also known as: Fourier Granger causality test, Enders-Jones Granger causality, smooth structural break Granger test, spectral Granger causality
The Fourier Granger causality test extends the classic Granger causality framework by embedding low-frequency Fourier terms in the VAR equation, allowing the causal relationship to shift gradually over time without requiring the researcher to pre-specify the number or location of structural breaks.
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When to use it
Use Fourier Granger causality when you suspect the causal relationship between two time series has shifted gradually over time — for example, due to globalisation, policy changes, or technological transitions — and you cannot pinpoint exact break dates. It is appropriate for moderately long stationary (or stationarised) time series, typically at least 80–100 observations. Do not use it as a substitute for the Toda-Yamamoto procedure when variables are clearly I(1) and not cointegrated; in that case, first-differencing or Toda-Yamamoto with Fourier augmentation is more appropriate. Also avoid it when very sharp, abrupt breaks are expected — a Bai-Perron or Zivot-Andrews framework may be more suitable.
Strengths & limitations
- Accounts for smooth, gradual structural breaks without requiring the researcher to specify break dates or their number.
- More powerful than standard Granger tests when underlying structural change is present, reducing the risk of false non-rejection.
- Computationally straightforward — implemented via standard OLS augmented with a small number of trigonometric regressors.
- The Fourier frequency selection step is data-driven and transparent.
- Applicable across diverse fields including finance, energy economics, and macroeconomics where regime shifts are common.
- Relies on the assumption that structural change is smooth and gradual; abrupt breaks are not well captured by low-frequency Fourier terms.
- Requires stationarity (or pre-tested stationarity) of the series; applying the test directly to I(1) variables without appropriate modification inflates Type I error.
- Performance deteriorates in short samples (fewer than 80 observations) because the trigonometric terms consume degrees of freedom.
- The test is sensitive to lag-length selection; an inappropriate lag order can distort size and power.
- Does not identify the direction of the structural shift or the time at which it occurs.
Frequently asked
How is Fourier Granger causality different from standard Granger causality?
Standard Granger causality assumes constant parameters throughout the sample. Fourier Granger causality adds trigonometric terms to the regression to absorb smooth structural change before testing for predictive causality, making it robust to gradual shifts in the causal relationship.
What Fourier frequency should I select?
Select the integer frequency k (from 1 to a maximum of around 5) that minimises the residual sum of squares or an information criterion such as AIC in the augmented equation. A single low frequency (k = 1 or 2) is often sufficient and preferred to avoid overfitting.
Can I use this test when my series are I(1)?
Not directly in the standard form. The test requires stationary series. If your variables are I(1), you should either first-difference them and check for cointegration, or use a Toda-Yamamoto-style approach with Fourier augmentation that adds extra lags of the levels to account for integration without differencing.
How many observations do I need?
The test performs well with at least 80–100 observations. Shorter samples reduce power and risk overfitting the Fourier terms relative to the causal signal of interest.
Is this the same as spectral Granger causality?
No. Spectral Granger causality decomposes causal influence by frequency band using spectral methods. Fourier Granger causality uses Fourier terms purely to model smooth structural change in the time domain before conducting a standard causality F-test.
Sources
- Enders, W., & Jones, P. (2016). Grain prices, oil prices, and multiple smooth breaks in a VAR. Studies in Nonlinear Dynamics and Econometrics, 20(4), 399–419. DOI: 10.1515/snde-2014-0101 ↗
- Nazlioglu, S., Gormus, N. A., & Soytas, U. (2016). Oil prices and real estate investment trusts (REITs): Gradual-shift causality and volatility transmission analysis. Energy Economics, 60, 168–175. DOI: 10.1016/j.eneco.2016.09.009 ↗
How to cite this page
ScholarGate. (2026, June 3). Fourier Approximation Granger Causality Test. ScholarGate. https://scholargate.app/en/econometrics/fourier-granger-causality
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