The exact linkage between the Beveridge–Nelson decomposition and other permanent-transitory decompositions

The econometric literature has recently focused attention on the relationship between the Beveridge–Nelson decomposition and unobserved components processes when decomposing time series into permanent and transitory shocks. This paper shows the existence of an algebraic linkage between reduced and structural forms parameters of some unobserved components processes. Results allow measuring how close standard unobserved components processes and unrestricted ARIMA models are regardless of the number of structural/reduced form parameters. Results are provided when the reduced forms are ARIMA(2,1,2) and ARIMA(0,2,2). For the latter, the exact relation between the Hodrick–Prescott filter and the IMA(2,2) reduced form is also shown.