In this paper we derive the Beveridge–Nelson (BN) decomposition and the state space representation for various multivariate (co)integrated time series subject to Markov switching in regime. Then we provide explicit expressions for the BN trend and cyclical components in terms of the matrices involved in the state space representation of the considered process. Our matrix expressions in closed form improve computational performance since they are readily programmable and greatly reduce the computational cost. Then we develop impulse-response function analysis and represent the BN trend component as a random walk. An empirical application on the world economy illustrates the feasibility of the proposed approach.

在本文中，我们推导了受马尔可夫状态切换影响的各种多元（共）积分时间序列的贝弗里奇-尼尔森（BN）分解和状态空间表示。然后，我们根据所考虑过程的状态空间表示中涉及的矩阵，提供 BN 趋势和循环分量的显式表达式。我们的封闭形式的矩阵表达式提高了计算性能，因为它们易于编程并大大降低了计算成本。然后我们进行脉冲响应函数分析并将 BN 趋势分量表示为随机游走。对世界经济的实证应用说明了该方法的可行性。