## 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