Fourier Engle-Granger Cointegration Test
Also known as: Fourier EG cointegration, Enders-Jones cointegration test, smooth structural break cointegration, FEGC test
The Fourier Engle-Granger cointegration test extends the classic two-step Engle-Granger procedure by embedding low-frequency trigonometric (Fourier) terms in the cointegrating regression. This accommodates an unknown number of smooth structural breaks in the deterministic components without specifying their dates, producing a more powerful test when long-run relationships shift gradually over time.
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When to use it
Use the Fourier Engle-Granger test when you have two or more time series that are each integrated of order one, I(1), and you suspect a long-run equilibrium relationship that may have shifted gradually over the sample — for instance due to structural reforms, regime changes, or slow globalisation effects. It is especially valuable when you cannot pinpoint break dates or when the number of breaks is unknown. Prefer it over the standard Engle-Granger test whenever diagnostic plots or Bai-Perron tests hint at parameter instability, but avoid it when series are I(0) or I(2), when the sample is very short (below roughly 80 observations), or when the relationship is inherently nonlinear in a way that Fourier sine-cosine terms cannot represent well — in that case, threshold or smooth-transition cointegration models may be more appropriate.
Strengths & limitations
- Accommodates an unknown number of smooth structural breaks without requiring break-date specification.
- More powerful than standard Engle-Granger when the long-run relationship drifts gradually over time.
- Parsimonious: a single Fourier frequency pair (two extra regressors) often suffices to capture substantial parameter variation.
- Extends naturally to systems with more than two variables by applying the same Fourier-augmented first stage.
- Avoids the pre-testing bias introduced by step-dummy structural break methods that depend on correctly identifying break locations.
- Requires longer time series (at least 80–100 observations) to estimate Fourier terms reliably and achieve adequate test power.
- Fourier terms can only approximate smooth, low-frequency breaks; sudden, sharp structural breaks may require dummy-based methods such as Gregory-Hansen or Hatemi-J.
- Nonstandard critical values must be used; applying standard ADF critical values to the residuals leads to incorrect inference.
- The two-variable (bivariate) framework cannot directly handle cointegrating relationships with more than one cointegrating vector — use Johansen-based Fourier extensions in that case.
Frequently asked
How does the Fourier Engle-Granger test differ from the standard Engle-Granger test?
The standard Engle-Granger test regresses one I(1) variable on another with only a constant and trend, then tests residuals for stationarity. The Fourier version adds sine and cosine terms at the optimal frequency to that regression, allowing the intercept and slope to shift smoothly. If structural breaks are present, the standard test has low power and may fail to detect genuine cointegration that the Fourier test finds.
How do I select the optimal Fourier frequency?
Estimate the Fourier-augmented cointegrating regression for each candidate integer frequency k (commonly k = 1, 2, 3) and choose the k that minimises the residual sum of squares. This data-driven selection avoids arbitrary choices but means the final critical values are those tabulated for the frequency-selection procedure, not for a fixed known frequency.
Can this test be applied to more than two variables?
Yes, in the bivariate sense: add additional I(1) regressors to the Fourier cointegrating regression. However, when multiple cointegrating vectors may exist, a Johansen-type framework with Fourier augmentation is preferable because the two-step approach only identifies a single cointegrating relationship.
What should I do after finding cointegration with the Fourier EG test?
Estimate a Fourier-augmented error-correction model (ECM) in which the lagged residual from the cointegrating regression enters as the error-correction term. This captures how the variables adjust toward equilibrium in the short run, separating long-run cointegration from short-run dynamics.
What if the Fourier terms are jointly insignificant?
If an F-test on the Fourier sine and cosine terms fails to reject their joint insignificance, the data provide no evidence of smooth structural change; you may then fall back on the standard Engle-Granger test, which is more powerful in the absence of breaks.
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 ↗
- Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55(2), 251–276. DOI: 10.2307/1913236 ↗
How to cite this page
ScholarGate. (2026, June 3). Fourier Engle-Granger Cointegration Test. ScholarGate. https://scholargate.app/en/econometrics/fourier-engle-granger-cointegration
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