What are the Differences in Trend Cycle Decompositions by Beveridge and Nelson and by Unobserved Component Models?

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 develops a convenient 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, although the conventional UC model is not necessarily a misspecification for describing the postwar U.S. GDP, choosing a univariate model among alternatives on the purely statistical grounds is difficult.