Page - 34 - in The Austrian Business Cycle in the European Context

34 end-point problem at the cost of risking forecasting errors of possi- bly the same size56_ 3.2.4.6 The Christiano-Fitzgerald filter A somewhat different band-pass filter has been proposed by Christiano - Fitzgerald (2003). It also represents an optimal ap- proximation of an ideal band-pass filter, by imposing a somewhat different criterion of optimality. This criterion minimises the sum of the squared approximation errors which are weighted by their spectral density fx ( m} of the data being filtered57. (10) min J" IB(e-i"')-BP•f(e-i"')l2 fx(w)dw ·p,f . -,r Bi ,;=-f, ... ,p Furthermore, the filter length is allowed to vary over the time series and is not restricted to being symmetrical. Moving towards the start and the end of the time series the filter becomes more and more asymmetric, which allows circumventing the end-point prob- lem. Christiano - Fitzgerald (2003) have shown that the length of the BK filter window not only determines the degree of approximation to the ideal band-pass filter, but influences especially its capability of filtering out long-term movements. The filter proposed by them can also be adjusted to filter a specific frequency band, but - in contrast to the BK filter - its length is a result of the optimisation process and cannot be altered by the applier. Therefore, it is not possible to define explicitly the business cycle with regard to a cer- tain frequency band. 56 In order to extend the HP filter, Kaiser - Morovo/1 ( 1999) proposed on IMA (2.2) time series model for forecasting. 57 The weights are thus generated by a trigonometric function.