Toda-Yamamoto Granger Causality Test
Also known as: TY Causality Test, Modified Wald Granger Causality, MWALD Test, Toda-Yamamoto Nedensellik Testi
The Toda-Yamamoto (TY) causality test, introduced by Toda and Yamamoto (1995), provides a robust procedure for testing Granger non-causality in vector autoregressive (VAR) models when the variables may be integrated or cointegrated of arbitrary order. By intentionally over-fitting the VAR with extra lags equal to the maximum integration order, the method bypasses the need for pre-testing cointegration and preserves the standard asymptotic chi-squared distribution of the Wald statistic.
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When to use it
Use the Toda-Yamamoto test when you need to assess Granger causality among time series whose integration orders are uncertain or mixed, and you want to avoid the well-known pre-testing biases of the Engle-Granger or Johansen frameworks. It is particularly suitable for macroeconomic and financial panels where series may be I(0), I(1), or I(2). The key assumptions are correct specification of d_max and a sufficiently large sample to support the augmented lag structure. An important limitation is that test power can decline as d_max increases, especially in small samples. If cointegration is firmly established, an error-correction-based causality test may be preferred. Alternatives include the Dolado-Lutkepohl procedure and bootstrap-based Granger causality tests.
Strengths & limitations
- Avoids pre-testing for cointegration, eliminating the associated specification risk
- Applicable regardless of integration order (I(0), I(1), or I(2)) and cointegration status
- Preserves standard asymptotic chi-squared inference without simulation or bootstrapping
- Simple to implement in any standard VAR software by adding d_max extra lags
- Power decreases when d_max is large relative to sample size due to the over-fitted lag structure
- Requires reliable unit root pre-testing to set d_max correctly; misspecification invalidates the procedure
- Does not identify the direction or strength of causal effects, only their statistical presence
- Asymptotic results may not hold well in very small samples, where bootstrap critical values are advisable
Frequently asked
Does the Toda-Yamamoto test require the series to be cointegrated?
No. One of the primary advantages of the TY procedure is that it is valid regardless of whether the series are cointegrated or not. You do not need to pre-test for cointegration, which eliminates a major source of specification error common in alternative causality frameworks.
How do I choose d_max when the integration order is uncertain?
Use a battery of unit root tests (ADF, PP, KPSS) and select the maximum plausible integration order. If tests conflict, setting d_max = 1 is common in practice for most macroeconomic variables. Setting d_max conservatively high reduces power, so over-augmentation should be avoided when the sample is small.
Can I apply the Toda-Yamamoto test to more than two variables?
Yes. The procedure extends naturally to multivariate VAR systems with any number of variables. Each potential causal relationship is tested by placing MWALD restrictions on the relevant block of coefficients in the first p* lags, with the chi-squared degrees of freedom equal to p*.
Sources
- 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 2). Toda-Yamamoto Granger Causality Test. ScholarGate. https://scholargate.app/en/econometrics/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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- Granger CausalityEconometrics↔ compare
- VAR ModelEconometrics↔ compare