This article defines the class of ${\cal H}$-valued autoregressive (AR) processes with a unit root of finite type, where ${\cal H}$ is a possibly infinite-dimensional separable Hilbert space, and derives a generalization of the Granger–Johansen Representation Theorem valid for any integration order $d = 1,2, \ldots$. An existence theorem shows that the solution of an AR process with a unit root of finite type is necessarily integrated of some finite integer order d, displays a common trends representation with a finite number of common stochastic trends, and it possesses an infinite-dimensional cointegrating space when ${\rm{dim}}{\cal H} = \infty$. A characterization theorem clarifies the connections between the structure of the AR operators and (i) the order of integration, (ii) the structure of the attractor space and the cointegrating space, (iii) the expression of the cointegrating relations, and (iv) the triangular representation of the process. Except for the fact that the dimension of the cointegrating space is infinite when ${\rm{dim}}{\cal H} = \infty$, the representation of AR processes with a unit root of finite type coincides with the one of finite-dimensional VARs, which can be obtained setting ${\cal H} = ^p $ in the present results.

中文翻译：

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功能自回归过程中的协整

本文定义的类${\cal H}$具有有限类型单位根的值自回归 (AR) 过程，其中${\cal H}$是一个可能无限维的可分 Hilbert 空间，并推导出对任何积分阶有效的 Granger-Johansen 表示定理的推广$d = 1,2, \ldots$. 存在定理表明，具有有限类型单位根的 AR 过程的解必然是某个有限整数阶的积分d, 显示具有有限数量的常见随机趋势的共同趋势表示，并且当它具有无限维协整空间时${\rm{dim}}{\cal H} = \infty$. 一个刻画定理阐明了 AR 算子的结构和 (一世) 积分顺序, (ii）吸引子空间和协整空间的结构，（三) 协整关系的表达式，和 (iv) 过程的三角形表示。除了协整空间的维数是无限的${\rm{dim}}{\cal H} = \infty$，具有有限类型单位根的 AR 过程的表示与有限维 VAR 的表示一致，可以设置为${\cal H} = ^p $在目前的结果中。