Vector autoregressions with unknown mixtures of I(0), I(1), and I(2) components

This paper develops a new estimation method for nonstationary vector autore gressions (VAR's) with unknown mixtures of I(0), 1(1), and 1(2) components. The method does not require prior knowledge on the exact number and locati on of unit roots in the system. It is, therefore, applicable for VAR's with any mixture of I(0), 1(1), and 1(2) variables, which may be cointegrated i n any form. The limit theory for the stationary component of our estimator is still normal, thereby preserving the usual VAR limit theory. Yet, the le ading term of the nonstationary component of the estimator has mixed normal limit distribution and does not involve unit root distribution. Our method is an extension of the FM-VAR procedure by Phillips (1995, Econometrica 63 , 1023-1078) and yields an estimator that is optimal in the sense of Philli ps (1991, Econometrica 59, 283-306). Moreover, we show for a certain class of linear restrictions that the Wald tests based on the estimator are asymp totically distributed as a weighted sum of independent chi-square: variates with weights between zero and one. For such restrictions, the limit distri bution of the standard Wald test is nonstandard and nuisance parameter depe ndent. This has a direct application for Granger-causality testing in nonst ationary VAR's.