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COINTEGRATION IN FUNCTIONAL AUTOREGRESSIVE PROCESSES
Econometric Theory ( IF 1.0 ) Pub Date : 2019-11-22 , DOI: 10.1017/s0266466619000306 Massimo Franchi , Paolo Paruolo
Econometric Theory ( IF 1.0 ) Pub Date : 2019-11-22 , DOI: 10.1017/s0266466619000306 Massimo Franchi , Paolo Paruolo
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:
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。
更新日期:2019-11-22
中文翻译:
功能自回归过程中的协整
本文定义的类