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The Hodrick-Prescott filter or H-P filter is an algorithm for choosing smoothed values for a time series. The H-P filter chooses smooth values {st} for the series {xt} of T elements (t=1 to T) that solve the following minimization problem:
min { {(xt-st)2 … etc. }
Parameter l>0 is the penalty on variation, where variation is measured by the average squared second difference. A larger value of l makes the resulting {st} series smoother; less high-frequency noise. The commonly applied value of l is 1600.
For the study of business cycles one uses not the smoothed series, but the jagged series of residuals from it. H-P filtered data shows less fluctuation than first-differenced data, since the H-P filter pays less attention to high frequency movements. H-P filtered data also shows more serial correlation than first-differenced data. (Econterms).
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