In this paper, we define and characterize cointegrated solutions of continuous-time linear state-space models driven by Lévy processes. Cointegrated solutions of such models are shown to be representable as a sum of a Lévy process and a stationary solution of a linear state-space model, analogous to the Granger representation for cointegrated VAR models. Moreover, we prove that the class of cointegrated multivariate Lévy-driven autoregressive moving-average (MCARMA) processes, the continuous-time analogues of the classical vector ARMA processes, is equivalent to the class of cointegrated solutions of continuous-time linear state-space models. Necessary conditions for MCARMA processes to be cointegrated are given as well extending the results of Comte [Discrete and continuous time cointegration, J. Econometrics 88 (1999), pp. 207–226] for MCAR processes. The conditions depend only on the autoregressive polynomial if we have a minimal model. Finally, we investigate cointegrated continuous-time linear state-space models observed on a discrete time-grid and calculate their linear innovations. Based on the representation of the linear innovations, we derive an error correction form. The error correction form uses an infinite linear filter in contrast to the finite linear filter for VAR models. Some sufficient identifiable criteria are also given.

在本文中，我们定义并表征了由Lévy过程驱动的连续时间线性状态空间模型的协整解。此类模型的协整解显示为可表示为Lévy过程和线性状态空间模型的平稳解的总和，类似于协整VAR模型的Granger表示。此外，我们证明了经典矢量ARMA过程的连续时间类似物的协变量多元Lévy驱动的自回归移动平均（MCARMA）过程的类等效于连续时间线性状态空间的协积分解的类楷模。给出了将MCARMA过程进行协整的必要条件，并扩展了Comte [离散和连续时间协整的结果，J。Econometrics 88（1999），第207-226页]。如果我们有最小模型，则条件仅取决于自回归多项式。最后，我们研究了在离散时间网格上观察到的协整连续时间线性状态空间模型，并计算了它们的线性创新。基于线性创新的表示，我们得出了误差校正形式。与VAR模型的有限线性滤波器相比，纠错形式使用无限线性滤波器。还给出了一些足够的可识别标准。