Toda, Hiro Y.; Phillips, Peter C. B. Vector autoregressions and causality. (English) Zbl 0796.62104 Econometrica 61, No. 6, 1367-1393 (1993). The asymptotic properties of Granger-type causality tests in unrestricted and nonstationary vector autoregression models are discussed. A sufficiency condition involving a rank condition on the submatrix of the cointegrating matrix is developed under which the above causality tests are shown to be valid asymptotically as chi-square criteria.In the case when there is no cointegration, the Wald test statistic for causality is shown to have a nonstandard limiting distribution. For cointegrated systems the Wald tests for causality in error correction models are shown to be not always valid as chi-square criteria in an asymptotic sense. Reviewer: J.K.Sengupta (Santa Barbara) Cited in 44 Documents MSC: 62P20 Applications of statistics to economics 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) Keywords:asymptotic propertie; Granger-type causality tests; unrestricted and nonstationary vector autoregression models; sufficiency condition; rank condition; cointegrating matrix; chi-square criteria; Wald test statistic; nonstandard limiting distribution; error correction models PDFBibTeX XMLCite \textit{H. Y. Toda} and \textit{P. C. B. Phillips}, Econometrica 61, No. 6, 1367--1393 (1993; Zbl 0796.62104) Full Text: DOI