Page - 30 - in The Austrian Business Cycle in the European Context

(6) P(m)={~ ifjmjs~ itlml>~ 30 With the help of these frequency weights p(m) it is possible to de- rive the time series filter weights b{h) by applying the inverse Fourier transformation to the frequency response function The time series filter weights b(h) can be used for constructing the ideal low-pass filter in the time domain This ideal low-pass filter is symmetrical (as it goes from -oc to +oc} and is constructed as an infinite-order moving average. The weights in this moving average are b0 =[!!/tr, and bh = sin{hgj/h,r for h = 1, 2, ... 53• In practice, an approximation of this ideal filter will be enough, such that a shorter filter can be applied which solves the end point problem at the cost of a leakage. From this low-pass filter, a band-pass filter can be easily derived from two consecutive low-pass filters, one working at the lower boundary f!l and the other atm. The approximation to this ideal band-pass filter with the weighting scheme derived this way is called the Baxter-King filter or BK filter for short. Its weights are given by (9) b(L) = sin Liv -sinlm _1 _ 1f sin Liv -sinlm Ltr 2K + 1 l=-K Ltr 53 For details the reader is referred to Baxter -King ( 1995).