Combining filter design with model-based filtering (with an application to business-cycle estimation)

Filters used to estimate unobserved components in time series are often designed on a priori grounds, so as to capture the frequencies associated with the component. A limitation of these filters is that they may yield spurious results. The danger can be avoided if the so-called ARIMA-model-based (AMB) procedure is used to derive the filter. However, parsimony of ARIMA models typically implies little resolution in terms of the detection of hidden components. It would be desirable to combine a higher resolution with consistency of the structure of the observed series. We show first that for a large class of a priori designed filters, an AMB interpretation is always possible. Using this result, proper convolution of AMB filters can produce richer decompositions of the series that incorporate a priori desired features of the components and fully respect the ARIMA model for the observed series (hence no additional parameter needs to be estimated). The procedure is discussed in detail in the context of business-cycle estimation by means of the Hodrick-Prescott filter applied to a seasonally adjusted series or a trend–cycle component.