Hodrick-Prescott Filter

In 1997, Robert J. Hodrick and Edward C. Prescott proposed a new procedure to decompose a time series into its trend and cyclical components. Rather than understanding mathematically, I think you should first see how it works.

I H-P filtered Japanese output growth rate (see the figure below). You can see three lines, actual data (blue), cyclical component of data (green), and "trend" of data (red). The point: we see ups and downs of growth rate (that's the business cycle!) if H-P filter is not used. When H-P filter is used, we can see the "trend" of output growth rate. So, this filtering is useful when one wants to see the data in the long run (of course, H-P filter is not complete device. When using, you have to be careful). We see that Japan's trend growth rate in the 21st century has been, roughly saying, between 0% and 1%...very weak growth.

Below I applied H-P filtering to Japanese inflation rate:

to U.S. output growth rate:

and to U.S. inflation rate:

Most macroeconomic variables are not stationary. Applying H-P filtering to unemployment rate (or anything else) turns out to be meaningful, I think.