Tag Archives: HP filter

Why is the Hodrick-Prescott filter often inappropriate?

The Hodrick-Prescott (HP) filter is the optimal estimator of the trend component in a smooth trend model with signal-to-noise ratio parameter fixed at 1/1600. It gives the detrended observations, X_{t},  for large samples and t not near the beginning or end of the series

 

\displaystyle X_{t}= \left[ \frac{(1-L)^{2}(1-L^{-1})^{2}}{\bar{q}_{\zeta}+ (1-L)^{2}(1-L^{-1})^{2}}\right] Y_{t}

 

where \bar{q}_{\zeta}= \sigma_{\zeta}^{2} / \sigma_{\epsilon}^{2} .

 

Bear in mind that if the  smooth trend model was believed to be the true model there would be no reason to apply the HP filter. The filtered data of a smooth trend model contain nothing more than white noise. The belief here is clearly different.

 

We can easily show that the gain from the detrending filter is given by:

G(\lambda) = \frac{4(1-cos \lambda)^{2}}{\bar{q}_{\zeta}+4(1-cos \lambda)^{2}} = \frac{16sin^{4}( \lambda /2)}{\bar{q}_{\zeta}+16sin^{4}( \lambda /2)}

Note that the smaller the {\bar{q}_{\zeta}} the more the filter concentrates on removing low frequencies.

 

Gain for HP filter

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