Fourier Toda-Yamamoto Granger Causality Test
Also known as: FTY causality, Fourier TY causality, Toda-Yamamoto causality with Fourier approximation, FTY Granger causality
The Fourier Toda-Yamamoto (FTY) causality test extends the classical Toda-Yamamoto procedure by embedding Fourier trigonometric terms in the augmented VAR to capture smooth, gradual structural breaks in the deterministic component. It retains the key advantage of the Toda-Yamamoto approach — Granger causality can be tested without pre-testing for integration or cointegration order — while dramatically improving size and power when breaks occur.
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When to use it
Use FTY causality when you have moderately long time series (T >= 50, preferably >= 80) and economic or financial variables that may be nonstationary and subject to gradual structural change — for example, oil prices, energy consumption, trade flows, or macroeconomic aggregates across different policy regimes. The test is preferable to the standard Toda-Yamamoto test whenever data span major structural events (financial crises, policy shifts, technological transitions) because misspecified deterministics inflate size distortion. Do not use FTY when the series exhibits sharp, abrupt breaks rather than smooth transitions — in that case a dummy-augmented or Hatemi-J procedure is more appropriate — and avoid it with very short series (T < 40) where Fourier terms consume too many degrees of freedom.
Strengths & limitations
- Handles smooth, gradual structural breaks without requiring the researcher to know their number or timing.
- Inherits the Toda-Yamamoto advantage: no pre-testing for integration or cointegration is needed before applying the MWALD test.
- The modified Wald statistic retains its asymptotic chi-squared distribution under mixed integration orders (I(0), I(1), I(2)).
- Low-dimensional Fourier approximation (usually k = 1 or 2) minimises the degrees-of-freedom cost relative to dummy-variable break approaches.
- Applicable to both bivariate and multivariate VAR systems.
- Requires a sufficiently long time series; with T < 50 the chi-squared approximation deteriorates and bootstrap critical values are advisable.
- Fourier terms can absorb smooth breaks but cannot adequately capture sharp, sudden structural breaks (use Hatemi-J or dummy-augmented tests for those).
- Adding d-max extra lags and Fourier regressors reduces degrees of freedom, which can reduce power in small samples.
- Optimal frequency selection by RSS minimisation can overfit if the upper bound for k is set too high.
- Like all VAR-based causality tests, results may be sensitive to lag length selection; different information criteria can yield different conclusions.
Frequently asked
How does FTY differ from the standard Toda-Yamamoto test?
The standard Toda-Yamamoto test augments the VAR with extra lags to handle nonstationarity but assumes a stable deterministic component. FTY additionally includes Fourier trigonometric terms that flexibly approximate smooth structural breaks in the intercept or trend, improving size and power when such breaks are present.
Do I need to test for unit roots or cointegration before applying FTY causality?
No. Like the original Toda-Yamamoto procedure, FTY only requires determining the maximum integration order d-max in the system (typically via unit root tests), not a full cointegration analysis. The MWALD statistic is valid regardless of whether the variables are I(0), I(1), or cointegrated.
How do I choose the Fourier frequency k?
Estimate the augmented VAR for each integer value of k from 1 to some upper bound (commonly 5) and choose the k that minimises the sum of squared residuals. A single Fourier frequency is usually sufficient; using more than two is rarely justified and costs degrees of freedom.
What sample size is needed for reliable results?
A minimum of around 50 observations is needed for the asymptotic chi-squared approximation to work reasonably well, and T >= 80 is preferred. For smaller samples, bootstrap critical values (e.g., 2000 replications) should replace the asymptotic chi-squared critical values.
Can FTY be used in a multivariate VAR with more than two variables?
Yes. The procedure extends naturally to a k-variable VAR: the Fourier terms and extra d-max lags are added to each equation, and the Wald test on each potential causal variable's first-p lags is conducted equation by equation.
Sources
- Yilanci, V., & Ozgur, O. (2019). Testing the Fourier Toda-Yamamoto causality test with an application to energy demand. Energy Economics, 84, 104498. link ↗
- Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1-2), 225-250. DOI: 10.1016/0304-4076(94)01616-8 ↗
How to cite this page
ScholarGate. (2026, June 3). Fourier Toda-Yamamoto Granger Causality Test. ScholarGate. https://scholargate.app/en/econometrics/fourier-toda-yamamoto-causality
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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