Time series models for trend-cycle decomposition with applications to business cycle and new product sales forecasts

Abstract

In this study, we have developed two decomposition methods of business and economic time series into the long run trend and the short run cycle: (i) a trend-cycle decomposition based on a state space model and (ii) a growth-cycle decomposition based on a diffusion model, where the long run trend can be characterized by either a stochastic trend or a deterministic growth curve. In addition, we have applied the developed model (i) and (ii) to the US real GNP to understand the business cycle and to the annual sales of room air conditioners in Korea to incorporate the short run effects of sales environment into the diffusion model, respectively. It has been a 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 misleads the interpretation of the model, especially when the basic properties of the state space model, such as observability, controllability and model equivalency, are not properly considered. In this study, it is shown that the spurious trend-cycle decomposition results from the unobservable state space model, and the usual assumption of independent noises in the model results in the parameter redundancy. The equivalent relationships between the ARIMA process and the state space model of a random walk trend and an AR cycle, where the noises of the trend and the cycle are generally correlated, are also derived. We have found that though there exist infinitely many numbers of SS model equivalent to an ARIMA model, all the SS models provide the same forecast obtained from the ARIMA model when the steady state values of Kalman gains are used in the SS model. It has been considered that the real aspect of economy can be regarded as more important than its monetary aspect to the business cycle, if the variance of the trend is greater than that of the cycle, and vice versa. According to the application result to the...