Why are Trend Cycle Decompositions of Alternative Models So Different?

When a certain procedure is applied to extract two component processes from a single observed process, it is necessary to impose a set of restrictions that defines two components. One popular restriction is the assumption that the shocks to the trend and cycle are orthogonal. Another is the assumption that the trend is a pure random walk process. The unobserved components (UC) model (Harvey, 1985) assumes both of the above, whereas the BN decomposition (Beveridge and Nelson, 1981) assumes only the latter. Quah (1992) investigates a broad class of decompositions by making the former assumption only. This paper provides a general framework in which alternative trend-cycle decompositions are regarded as special cases, and examines alternative decomposition schemes from the perspective of the frequency domain. We find that as long as the US GDP is concerned, the conventional UC model is inappropriate for the trend-cycle decomposition. We agree with Morley et al (2003) that the UC model is simply misspecified. However, this does not imply that the UC model that allows for the correlated shocks is a better model specification. The correlated UC model would lose many attractive features of the conventional UC model.