#### Transcript Determining the smoothness parameter in the HP filter

```Nikhil Vellodi, Bank of Papua New Guinea
PFTAC Workshop, Apia, Samoa,
15th-23rd November, 2011,
“Improving Analytical Tools for Better Understanding”
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What is the HP filter and what is it used for?
What is lambda and why does it matter?
How did Hodrick and Prescott calculate lambda?
Does it hold for other countries?
The Ravn and Marcet approach.
◦ Intuition
◦ Their findings
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The Eviews Program
Conclusion
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First proposed in Hodrick and Prescott (1997), "Postwar U.S.
Business Cycles: An Empirical Investigation," Journal of Money, Credit,
and Banking.
Statistical technique used to separate cyclical component from raw
data series.
◦ Cyclical may not mean periodic. In this case, simply a residual from a certain
fitted line through the data.
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Suppose we split our output series into trend and cyclical
components:
Then, for a given lambda, the HP filter determines the series by
minimizing:
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Minimizing the first term penalized the cyclical term.
◦ i.e makes it as small as possible.
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Minimizing the second term penalizes variations in the
growth rate of the trend term.
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The Lambda parameter determines the smoothness of the
trend component:
◦ The larger the value of lambda, the higher the penalty in the second
term.
 The smoother the trend component.
 The more of the data the cyclical component accounts for.
 Output gap will be larger magnitude, as well as displaying longer “cycles”.
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“Our prior view is that a 5 percent cyclical component is
moderately large, as is a one-eight of 1 percent change in
the growth rate in a quarter.This led us to select…
lambda = 1600… With this value, the implied trend path
for the logarithm of real GNP is close to the one that
students of the business cycle and growth would draw
through a time plot of the series.”
They used basic economic rationale and experience.
 What they considered reasonable in terms of the size of
the cyclical component, as well as the variability of the
trend component.
 Based on quarterly US data, 1950-1979.
No reason to believe this will be the same for SPIs today!
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A. Marcet and M. Ravn, “The HP-Filter in Cross-Country
Comparisons”, CEPR Discussion Papers, 2003.
◦ Applying the same lambda across countries and time assumes the same
level of volatility of the trend component for all countries.
◦ Preserves only the relative variance of the trend and cycle
components, not the volatility.
◦ If a country is believed in reality to have longer cycles, applying the
same lambda will force the trend component to be too volatile, i.e.
absorb too much of the data away from the cyclical component.
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Propose re-interpreting HP filter:
◦ Rather than keep the relative variance of the trend and cycle
components fixed, we now keep the relative volatility fixed as well.
◦ This is more in keeping with the nature of the business cycle.
 Allows for varying levels of persistence in the cycle component, as
determined by the structure of the actual output series for the country in
question.
◦ Modification carried out via two “adjustment rules”
 AR1: Keeps relative volatility of growth in trend and cycle constant.
 AR2: Keeps absolute volatility of growth in trend constant.
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Key results…
◦ Spanish data implies lambda = 8000, approx.
 More in keeping with historical picture.
◦ Most other OECD (except Italy, Japan), 1600 is reasonable value.
◦ Interpolation versus actual data:
 Find that if quarterly output series is based on interpolating annual, AR1
brings lambda up, whilst AR2 lowers it.
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