Linganisha mbinu

	Mtihani wa Uhalali wa Granger wa Fourier Toda-Yamamoto	Kipimo cha Umuhimu cha Granger cha Toda-Yamamoto	Muundo wa Uhusiano wa Kiotomatiki wa Vecta (VAR)
Nyanja	Ekonometriki	Ekonometriki	Ekonometriki
Familia	Regression model	Hypothesis test	Regression model
Mwaka wa asili	2019	1995	2005
Mwanzilishi	Yilanci, Ozgur (building on Toda and Yamamoto 1995; Becker, Enders, and Hurn 2004)	Hiro Toda & Taku Yamamoto	Lütkepohl (textbook treatment); Sims (1980) macroeconometric tradition
Aina	Granger causality test	Modified Wald test on augmented VAR	Multivariate time-series model
Chanzo asilia	Yilanci, V., & Ozgur, O. (2019). Testing the Fourier Toda-Yamamoto causality test with an application to energy demand. Energy Economics, 84, 104498.	Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1–2), 225–250.	Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer.
Majina mbadala	FTY causality, Fourier TY causality, Toda-Yamamoto causality with Fourier approximation, FTY Granger causality	TY Causality Test, Modified Wald Granger Causality, MWALD Test, Toda-Yamamoto Nedensellik Testi	vector autoregression, VAR, VAR Modeli (Vektör Otoregresyon), vektör otoregresyon
Zinazohusiana	3	3	4
Muhtasari	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.	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.	Vector Autoregression is a multivariate time-series model that treats several interdependent series symmetrically, letting each variable depend on its own past values and the past values of all the others. It is the standard tool for capturing mutual causality and joint dynamics, developed in the modern multiple-time-series tradition treated by Lütkepohl (2005).