## Hodrick-Prescott Filter

The Hodrick-Prescott (HP) filter is a specialized filter for trend and business cycle estimation (no seasonal component). Suppose a time series yt can be additively decomposed into a trend and business cycle component. Denote the trend component gt and the cycle component ct. Then ${y}_{t}={g}_{t}+{c}_{t}.$

The HP filter finds a trend estimate, ${\stackrel{^}{g}}_{t}$, by solving a penalized optimization problem. The smoothness of the trend estimate depends on the choice of penalty parameter. The cycle component, which is often of interest to business cycle analysts, is estimated as ${\stackrel{^}{c}}_{t}={y}_{t}-{\stackrel{^}{g}}_{t}$.

`hpfilter` returns the estimated trend and cycle components of a time series.