VECTOR AUTOREGRESSIONS WITH UNKNOWN MIXTURES OF I(0), I(1), AND I(2) COMPONENTS
提出一种新的估计方法，适用于包含未知混合的I(0)、I(1)和I(2)分量的非平稳向量自回归模型，无需预先知道单位根的数量和位置，并给出了基于该估计量的Wald检验的渐近分布。
This paper develops a new estimation method for nonstationary vector autoregressions (VAR's) with unknown mixtures of I (0), I (1), and I (2) components. The method does not require prior knowledge on the exact number and location of unit roots in the system. It is, therefore, applicable for VAR's with any mixture of I (0), I (1), and I (2) variables, which may be cointegrated in any form. The limit theory for the stationary component of our estimator is still normal, thereby preserving the usual VAR limit theory. Yet, the leading 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 Phillips (1991, Econometrica 59, 283–306). Moreover, we show for a certain class of linear restrictions that the Wald tests based on the estimator are asymptotically distributed as a weighted sum of independent chi-square variates with weights between zero and one. For such restrictions, the limit distribution of the standard Wald test is nonstandard and nuisance parameter dependent. This has a direct application for Granger-causality testing in nonstationary VAR's.