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[hodrick prescott filter]

The Hodrick Prescott filter (HP filter), introduced by Hodrick and Prescott (1980), is a flexible detrending method that

is widely used in empirical macro research. Let's suppose that the original series  is composed of a trend component  and a cyclical component

The HP-Filter isolates the cycle component by following minimization problem.

The first term is a measure of the fitness of the time series while the second term is a measure of the smoothness. There is a conflict between "goodness of fit" and "smoothness". To keep track of this problem there is a "trade-off"-parameter .Note that  is 0, the trend component becomes equivalent to the original series while  diverges to infinity, the trend component approaches a linear trend.

As you can see the HP filter acts to remove a trend from the data by solving a least square problem. In matrix notation we get

with ,  and

It can be shown that the solution of the minimization problem is be given by  where  is the identity matrix with dimension T.

The height of the value  depends on the frequency of the data. In the literature the following values are suggested.


yearly data


quarterly data


monthly data



The solution of the HP-filter must satisfies

The computation could be done by a native Gauss algorithm. Unfortunately this method is not very efficient especially if you want detrend a lot of data points. Note, that the computational complexity of Gaussian elimination is . A precise look at the matrix  shows that this matrix has got a pentadiagonal structure. If we use this property, we can accelerate the calculations strongly.

In the HP-filter Add-In I used an algorithm which is described in Späth, Helmuth "Numerik: Eine Einführung für Mathematiker und Informatiker". Vieweg-Verlag Braunschweig/Wiesbaden (1994)

[related links]

All links will be open in a new window

wikipedia A description of the Hodrick Prescott filter at wikipedia. (HTML)

Hodrick Prescott reference, by Hyeongwoo Kim. A brief introduction. (PDF)

Hodrick Prescott Filter, by Yossi Yakhin. A brief introduction. (PDF)

Links to other sites from these pages are for information only and Kurt Annen accepts no responsibility or liability for access to, or the material on, any site which is linked from or to this site.


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The hodrick prescott filter for matlab was written by Kurt Annen. This program is freeware. But I would highly appreciate if you could give me credit for my work by providing me with information about possible open positions as an economist. My focus as an economist is on econometrics and dynamic macroeconomics. If you like the program, please send me an email.

[hodrick prescott (matlab/octave)]
© 2005 Kurt Annen