Page - 28 - in The Austrian Business Cycle in the European Context

28 the trend component becomes linear. Typically a value of 1600 is chosen for applications using quarterly data48• The HP filter extracts a trend which can be stochastic but moves smoothly over time and is uncorrelated with the cyclical component49• A further property is its symmetry, so that no phase shift is introduced. In its ideal representation (its infinite sample version), it approximately places zero weight at the zero frequency (the trend) and close to unit weight at high frequency. This last property leads to a non- trended output but which carries high-frequency information like the noise component e, . Figure 2 depicts the gain function of this filter. It wipes out rather clear-cut all frequencies lower than n/16 (i.e. cycles longer than 32 quarters) and leaves all other frequen- cies unchanged (again, the loss of spectral mass at n/2 is due to the preceding seasonal adjustment process). In order to overcome the disadvantage of residual high frequen- cies, sometimes a kind of low-term moving average is applied af- ter the HP filter for smoothing the output. Artis - Marcellino - Proi- etti (2004) used a band-pass version of the HP filter. They com- bined two HP filters - which both are virtual approximations of high-pass filters - in order to get a smoother output, corresponding to the business cycle frequency band. The trend resulting from the minimisation process of (5) can also be represented as a linear symmetric filter. Like all symmetric filters, the HP filter is - in its ideal presentation - subjected to the endpoint problem, which means that there is a loss of several observations at the beginning and the end of the series. There exists a trade-off relation concerning symmetric filters, i.e. the more observations one is willing to lose, the more exactly the filtering procedure works. In the case of the practically applied HP filter, there is no such obvious end point problem as there is a kind of built-in me- 48 It can be shown that J = 1600 corresponds to a cut-off of frequencies lower than 32 quarters. See e.g. Prescott ( 1986) or Baxter -King ( 1995). 49 See Canova (1998).