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	Test di Causalità Bayesiano di Toda-Yamamoto	Test di causalità di Granger	Test di causalità di Granger Toda-Yamamoto
Campo	Econometria	Econometria	Econometria
Famiglia	Regression model	Regression model	Hypothesis test
Anno di origine	1995 (base); Bayesian variant developed post-2000	1969	1995
Ideatore	Toda & Yamamoto (1995) for the frequentist base; Bayesian extension by subsequent applied econometricians	Clive W. J. Granger	Hiro Toda & Taku Yamamoto
Tipo	Causality test / VAR-based inference	Time-series predictive causality test	Modified Wald test on augmented VAR
Fonte seminale	Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1-2), 225-250.	Granger, C. W. J. (1969). Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica, 37(3), 424-438.	Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1–2), 225–250.
Alias	Bayesian TY causality, Bayesian modified Wald causality, Bayesian Granger non-causality in VAR, BTY causality	Granger causality test, Granger non-causality test, predictive causality test, Granger Nedensellik Testi	TY Causality Test, Modified Wald Granger Causality, MWALD Test, Toda-Yamamoto Nedensellik Testi
Correlati	3	5	3
Sintesi	The Bayesian Toda-Yamamoto causality procedure combines the Toda-Yamamoto VAR augmentation strategy — which sidesteps the need for pre-testing integration and cointegration — with Bayesian prior-posterior updating. It tests Granger non-causality between time series that may be integrated or cointegrated without requiring differencing or error-correction modeling, while incorporating prior information and producing full posterior distributions over the causal parameters.	The Granger causality test, introduced by Clive W. J. Granger in 1969, assesses whether the past values of one time series help predict another beyond what the latter's own past already explains. It defines causality in a strictly predictive sense rather than as a structural or physical cause.	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.