# Understanding the HP Filter

The HP Filter has two objectives, with the importance of each objective denoted by the user given value of lambda:

Objective 1: minimize the $\tau_t$ in the term in the square brackets such that we minimize the changes in the estimated growth rate over time.

Objective 2: We want to bring the $\tau_t$ to be as close as possible to $y_t$ to minimize the first sum in the equation.

What I am failing to understand is why these acts of minimization will help us discover what the cyclical component of GDP is? Is there a fundamental gap in my knowledge?

• I don't think the purpose of the filter is to discover a cyclical component. – Michael Greinecker Apr 5 '18 at 11:47
• It's to separate the trend and cyclical components, no? – Joseph Apr 13 '18 at 6:49

## 1 Answer

The function $\tau$ is the 'line of best fit', the growth trend. The 'residual' $y-\tau$ is the output gap. So, $\tau$ detrends $y$ over time, to give the detrended business cycle $y-\tau$. $\quad y-\tau$ fluctuates around 0—it is positive when the output gap is positive and the economy is in a boom, and negative when the economy is in recession.

The first term makes the line of best fit follow the points in a more curvy way than just a straight line; the second term makes the line of best fit straighter (the higher $\lambda$ is).

• ahorn explained it beautifully and link below goes into more detail. uni-goettingen.de/de/document/download/...en.pdf/Filtering.pdf – mark leeds Jun 29 '18 at 15:16
• Joseph: be aware that there is quite a bit of controversy amongst econometricians regarding this filter. there's one group that says that its a particularly flawed ( induces spurious cycles ) filter and shouldn't be used. then there's another group that says its no more problematic than any other filter and the criticisms are flawed. I have not read deeply enough to even have an opinion but you should be aware of this if you use it. google for "hamilton hodrick prescott" or "pollock hodrick prescott" for more info. – mark leeds Jun 29 '18 at 15:18
• one link is here but be aware that there are a whole bunch of top econometricians who disagree. econweb.ucsd.edu/~jhamilto/hp.pdf. note that this paper is pretty recent so he may be responding to the critics of the critics. I don't know. – mark leeds Jun 29 '18 at 15:21
• not sure why but link to introduction doesn't work. but one can google for "introduction to hodrick prescott filter" and it's the second one in the result. – mark leeds Jun 29 '18 at 15:27