>> jakub.ryšánek(‘homepage’)
  Curriculum Vitae (as of 12/2012)
Some econometrics:
  Hodrick-Prescott filter
(intro + online application)
for vector autoregressions
(supplement to my Master’s
thesis, in Czech only)

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Hodrick-Prescott Filter

To extract the business cycle information from the data, the HP filter processes a single time series. Thus, the technique itself can be labeled as univariate. Proper seasonal adjustment should be carried out prior to HP filtering.

Here I follow Kim's tutorial to formulate the HP filter as a function minimization problem. I also inherit Kim's notation (and ultimately entire mathematical expressions) - yt stands for the time series of interest, taut its trend component and ct its cyclical component:
where T is the sample size.

From a mathematical point of view, the HP filter is a discrete one.

Hodrick and Prescott (1997) suggest the following criterion to reveal the unobserved components, taut and ct, conditional on a choice of "smoothing parameter" λ:
An expert judgment for the choice of λ is necessary. In general, the close is λ to zero, the closer is filtered trend to the original series, since the second term in the criterion function does not bind. Likewise, if λ approaches infinity, the filtered trend becomes a straight line, since the second differences in trend get an infinite weight (delta2 denotes the second difference operator).

Hodrick and Prescott also provide default values for the smoothing parameter:

For a detailed description of the topic see:
  • Hodrick, R., Prescott, E. (1997): "Postwar U.S. Business Cycles: An Empirical Investigation", Journal of Money, Credit, and Banking, 29(1), pp. 1-16.
  • Beveridge, S., Nelson, C. R. (1981): "A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycle", Journal of Monetary Economics, No. 7, pp. 151-174.