# How does using HP filter allows us to estimate long-run values for variables such as output and unemployement?

I'm reading the paper "Okun's Law: Fit at 50?" written by Laurence Ball, Daniel Leigh and Prakash Loungani.

In it, in order to estimate Okun's Law in its level form, they use the HP filter to find the long-run values of the variable. I quote:

In this case, the tricky problem is to measure the natural rate $$U_t^*$$ and potential output $$Y_t^*$$. In most of our analysis, we use the most obvious method: we smooth the output and unemployment series with the Hodrick Prescott (HP) filter.

Could anyone explain to me how that works? From what I could gather, the HP filter is used to find the cyclical and trend components of a time series. How does one go from that to estimating long-run values?

Thanks!

## 1 Answer

I've found the answer to my question. By using the HP filter, we are able to extract the long-term component of a time-series, namely it's trend component. Using that, we can then take the difference between the series and its trend component at every observation. This leaves us with a cyclical component; if we are interested in estimating gaps between observed values and potential values or natural rates, this is what we are looking for.

Of course, this is the english explanation of what happens; for a mathematical/statistical approach, I recommend reading the original paper.