A Zero Phase Shift Band Pass Filter
Many economic time series can be decomposed into transitory phenomena, cyclical movements and permanent shifts. Band pass ﬁlters such as those proposed by Bax-ter and King (BK) and Christiano and Fitzgerald (CF) can be used to decompose such series without necessitating strong model assumptions on the time series dynamics.
However, both these ﬁlters have some known drawbacks. The BK ﬁlter can remove trends at the zero frequency effectively, but does not adequately suppress other un-desired low frequency ﬂuctuations. Furthermore, improving the approximation of the frequency response of ideal ﬁlters requires sacriﬁcing more observations. The CF ﬁl-ter improves the frequency response approximation, but has deteriorating performance near the end points of the data, and induces phase shift distortions. In this paper we present an approximate band pass ﬁlter which overcomes the draw-backs of the previous ﬁlters. The ﬁlter works by recursively ﬁtting multiple trigonometric functions in the time domain to observed series, and ﬁltering the remaining residual directly in the frequency domain. By only retaining the frequencies in the desired pass band, a band pass ﬁlter with desired properties is deﬁned. In particular, the ﬁlter does not induce phase shifts and does not require down weighting or removal of any data. We provide two empirical applications of the ﬁlter. In the ﬁrst application we ﬁlter Eu-ropean macro-economic output series with business cycle pass bands and make a comparative analysis with the ﬁlter output of the BK and CF ﬁlters. In the second ap-plication, we perform a bi-orthogonal decomposition using a combination of band pass ﬁltering and principal components analysis to construct a business cycle indicator.