Page - 80 - in The Austrian Business Cycle in the European Context

80 true order of the process is zero (which means the trend is station- ary), building first-order differences would mean an over-differenc- ing of the series, which shifts the spectral mass into the high- frequency area. If, however, the true process is integrated higher than order l (which is rather unlikely in the case of economic time series) it is shifted into the low-frequency direction. This first-order difference filter produces not only rather erratic trend behaviour, but - as can be seen in (3) - the cycle includes also an error term. In constructing growth rates this way, the miss- ing-end-of-the-sample problem is solved at the cost of a phase shift and the error variance is superimposed. The latter becomes apparent in the rather erratic movement of this series displayed in Figure 5. 8. 1.2 The HP filter The HP filter solves a lot of problems associated with de-trending reasonably well and is, for its easy application, probably the best practical choice. As it filters out deterministic as well as stochastic trends up to an integration of order four, no strong assumptions concerning the evolution of the trend have to be made implicitly. Due to its symmetry, no phase shift is introduced, and the end- point problem is implicitly solved by a mechanical trend extrapola- tion. High-frequency noise will remain, but is not superimposed as in the case of first-order differences. For all these reasons, the HP filter is still very popular in the literature and it is applied here, too. For our study, the smoothing parameter A is set at 1600, as it is most common in applications using quarterly data. According to Prescott ( 1986) and Baxter - King ( 1995) this corresponds to a filter that cancels out frequencies lower than 32 quarters, i.e. 8 years. 8. 1.3 The BK filter From a theoretical point of view, band-pass filters like the one pro- posed by Baxter - King ( 199 5) seem to be most suited for the kind