That was my instinct as well. Thank you, trubador!
A brief update for fellow users interested in the HP filter:
I'm still investigating the proper procedures for detrending using either the HP or Corbae-Ouliaris FD filter with the ultimate goal of a side-by-side comparison of their empirical accuracy in forecasting applications. While the FD filter makes it easy to set a "window" for the cycle-lengths of interest, it was unclear how to apply the HP filter with such an equivalent window. After some more searching I found the OECD offers an alternative guideline for choosing the smoothing parameter (lambda), which allows users to approximate a window:
"In the OECD CLI methodology, the default settings allow to [sic] remove cyclical components that have a cycle length longer than 120 month and those that have a cycle length shorter than 12 months. They are equivalent of setting λ = 133107.94 and λ = 13.93 respectively.
Going from frequencies to λ parameter is achieved by substituting into the formula:
λ=[4(1-cos(ω0))2]-1.
Whereas ω0 is the frequency expressed in radians, and τ denotes the number of periods it takes to complete a full cycle. The two parameters are related through ω0=2π/τ. So the λ values above correspond to τ=120 months and τ=12 months."
See here:
http://www.oecd.org/fr/std/indicateursa ... aqs.htm#12
OECD literature (
http://stats.oecd.org/mei/default.asp?lang=e&subject=5) suggests first detrending and then smoothing (using the larger and then smaller smoothing parameter, respectively). After the first application of the HP filter (detrending, larger parameter), one is left with a cyclical and a trend component. The original series is detrended by dividing the original series by this trend component, thus implying a multiplicative approach. Multiplicative methods seem toto be the most popular and the Bank of Spain explains why in its TRAMO/SEATS literature, which has lots of useful information regarding additive versus multiplicate approaches: "Usually, the decomposition scheme is
multiplicative (either pure multiplicative or log-additive), because in most economic time series, the magnitudes
of the seasonal component appear to vary proportionally to the level of the series" (
http://epp.eurostat.ec.europa.eu/cache/ ... 006-EN.PDF). This detrended ratio-to-trend series is used in the second application of the HP filter (smoothing, smaller paramter) and one is left with a
smoothed, detrended ratio-to-trend series. This series fluctuates around 1, making the retrending stage easy: multiply the trend of the reference series by the standardized/averaged set of leading indicator components. Don't forget to rescale first - the amplitudes of these detrended ratio-to-trend leading indicator compnents must be adjusted to match the original reference series' ratio-to-trend.
I was unable to find an explanation of the
additive approach (typically after a log-transformation), which seems trickier in that the 'true' mean of the cyclical compenents is unclear. How do you add the cyclical component to the trend component if their means have been changed in the aggregation and data manipulation steps? Any comments and critiques are very welcome!
Regards,
Drew