State space trend-cycle decomposition of the ARIMA(1,1,1) process

It has been common practice to decompose an integrated time series into a random walk trend and a stationary cycle using the state space model. Application of state space trend-cycle decomposition, however, often results in a misleading interpretation of the model, especially when the observability of the state space model and the redundant relationships among the model parameters are not properly considered. In this study, it is shown that spurious trend-cycle decomposition, discussed by Nelson, results from an unobservable state space model, and the usual assumption of independent noise processes in the model results in parameter redundancy. Equivalence relationships for the ARIMA(1,1,1) process and the state space model consisting of a random walk trend and an AR(1) cycle, where the noise processes of the trend and of the cycle are generally correlated, are also derived.