The Hodrick-Prescott Filter
The Hodrick and Prescott (HP) (1997) filter has two justifications: one theoretical and one statistical. The theoretical part of the HP filter is connected with real business cycle (RBC) literature, for instance, in the RBC world the trend of a time series is not intrinsic to the data but it is a representation of the preferences of the researcher and depends on the economic question being investigated. The popularity of the HP filter among applied macroeconomists results from its flexibility, which can accommodate these needs since the implied trend line resembles what an analyst would draw by hand through the plot of the data. The selection mechanism that economic theory imposes on the data via the HP filter can be justified using the statistical literature on curve fitting.
The conceptual framework presented by Hodrick-Prescott can be summarized as follows:
Where T is the sample size. A given series yt is the sum of a growth component gt and a cyclical component ct. Actually, there is also a seasonal component but, as the data is seasonally adjusted, this component has already been removed by those preparing the data series.
In this framework the HP filter optimally extracts a trend (gt) that is stochastic but moves smoothly over ...