Page - 29 - in The Austrian Business Cycle in the European Context

29 chanical forecast feature which allows de-trending even at the endpoints50. The HP filter calculates the trend component and identifies the cyclical component as the difference between the original series and the trend component51 • The end-point problem is therefore concentrated on changes in the trend component. This makes the HP filter very attractive for practical purposes, but the end-point problem is solved at the cost of the accuracy of the filter. Especially at the end points, the filter works imprecisely in that it shows a stronger leakage. This means that some frequencies be- longing to the trend can pass, whereas fluctuations of cyclical na- ture (especially those of low frequency order) are filtered out. Cogley - Nason ( 1995) and Canova ( 1998) pointed out that de- spite the HP filter's ability to de-trend difference stationary time se- ries, it distorts the frequency spectrum so that business cycle ex- traction could be problematic. 3.2.4.5 The Baxter-King filter Band-pass filters are understood as frequency filters, which give - in their ideal representation - zero weight to the frequency band to be filtered out and unit weight to the rest52. Baxter -King (1995) proposed such a filter for business cycle filtering, which is an opti- mal linear approximation to an ideal band-pass filter. They con- structed a band-pass filter by starting from a low-pass filter, which allows all frequencies w (below or equal a certain threshold .Q?) to pass. This requires that all frequencies get a unit weight p if they are above this threshold and zero otherwise: 50 See Baxter -King ( 1995). 51 See Kranendonk - Bonenl<omp - Verbruggen (2004). 52 In a wider sense, high-pass filters, which filter out all frequencies below a certain frequency threshold (e.g. the trend) as well as low-poss filters, which let pass only frequencies below a certain threshold, can be regarded as band-pass filters. But what is meant here are only the filters which capture the band between two thresholds not belonging to extreme ends.