Analogously to low pass filters it is possible to

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Analogously to low-pass filters, it is possible to construct high-pass filters. Figure 6.5 compares the transfer function of an ideal high-pass filter with two filters truncated at q = 8 and q = 32, respectively. Obviously, the transfer function with the higher q approximates the ideal filter better. In the neighborhood of the critical frequency, in our case π/ 16, however, the approximation remains inaccurate. This is known as the Gibbs phenomenon. 6.5.2 The Hodrick-Prescott Filter The Hodrick-Prescott filter (HP-Filter) has gained great popularity in the macroeconomic literature, particularly in the context of the real business cy- cles theory. This high-pass filter is designed to eliminate the trend and cycles of high periodicity and to emphasize movements at business cycles frequen- cies (see Hodrick and Prescott, 1980; King and Rebelo, 1993; Brandner and Neusser, 1992). One way to introduce the HP-filter is to examine the problem of decom- posing a time series { X t } additively into a growth component { G t } and a cyclical component { C t } : X t = G t + C t . This decomposition is, without further information, not unique. Following the suggestion of Whittaker (1923), the growth component should be ap- proximated by a smooth curve. Based on this recommendation Hodrick and Prescott suggest to solve the following restricted least-squares problem given
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136 CHAPTER 6. SPECTRAL ANALYSIS AND LINEAR FILTERS a sample { X t } t =1 ,...,T : T X t =1 ( X t - G t ) 2 + λ T - 1 X t =2 [( G t +1 - G t ) - ( G t - G t - 1 )] 2 -→ min { G t } . The above objective function has two terms. The first one measures the fit of { G t } to the data. The closer { G t } is to { X t } the smaller this term becomes. In the limit when G t = X t for all t , the term is minimized and equal to zero. The second term measures the smoothness of the growth component by looking at the discrete analogue to the second derivative. This term is minimized if the changes of the growth component from one period to the next are constant. This, however, implies that G t is a linear function. Thus the above objective function represents a trade-off between fitting the data and smoothness of the approximating function. This trade-off is governed by the meta-parameter λ which must be fixed a priori. The value of λ depends on the critical frequency and on the periodicity of the data (see Uhlig and Ravn, 2002, for the latter). Following the proposal by Hodrick and Prescott (1980) the following values for λ are common in the literature: λ = 6 . 25 , yearly observations; 1600 , quarterly observations; 14400 , monthly observations. It can be shown that these choices for λ practically eliminate waves of pe- riodicity longer than eight years. The cyclical or business cycle component is therefore composed of waves with periodicity less than eight years. Thus, the choice of λ implicitly defines the business cycle. Figure 6.5 compares the transfer function of the HP-filter to the ideal high-pass filter and two approximate high-pass filters. 6 As an example, figure 6.6 displays the HP-filtered US logged GDP to-
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