# HP-Filter_Decompose economic variables into a trend and a cycle.

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ECON 562
Macroeconomic Analysis & Public Policy
Introduction
Cycles
Trends vs
Cycles
Detrending
Variables
Cycle
Statistics
ECON 562
Macroeconomic Analysis & Public Policy
The economy grows over time, but
goes through phases of expansions
and contractions, these are the
business cycles. The parts of the
cycle are:
Expansion
Trough
Peak
Recession
(2 quarters
or more of
negative
growth)
Trends vs Cycles
Why study business cycles (separately from growth)?
 Macro variables have very different long-run trends than cyclical patterns.
ECON 562
Macroeconomic Analysis & Public Policy
Module 6b: Detrending Variables
Detrending Variables
To start we need to separate the “trend” from the “cycle” in the major macro
variables.
One “trend-cycle decomposition” of real GDP is to plot a straight line through
the series. Here,
 Straight line: trend
 Deviations from straight line: cycle
A straight trend-line in real GDP means a constant growth rate ? in GDP
every period,
*
*
?

?
?
?-?
?? ?*? – ?? ?*?-? 
= ?.
*
??-?
Detrending Variables
With the trend at hand, we can then calculate the cycle defined as the
deviation from the trend,
????? = ?? ?? – ?? ?*?
?? – ?*?

?*?
The cycle is equal to the percent deviation of real GDP (?? ) from its trend
(?*? ).
Detrending Variables
But wouldnt a straight-line trend be to constraining? (think over long periods
of time)
Could we have a trend rate of growth of real GDP that changes (slows down
or accelerate) over time?
Yes!
The HP-Filter
To allow trend rate of growth to change over time we follow the procedure
proposed by Hodrick and Prescott (HP-Filter).
Detrending Variables
Compute ?? ?*? in each period to minimize the following:
*
*
s??=? ?? ?? – ?? ?*? ? + ? s?-?
?=? ??? ??+? – ??? ??
where ??? ?*? = ?? ?*? – ?? ?*?-? .
?
,
The parameter ? is a smoothing parameter that depends on the frequency of the
data (quarterly is set at 1,600, for annual at 100).
 ? = ? implies that changes to trend are unimportant (2nd term drops out) and the
solution becomes ?? ?? = ?? ?*?
 ? ? 8 implies straight line trend all the weight goes to the 2nd term so that
??? ?*?+? = ??? ?*? = ?.
Detrending Variables
Out of the two methods the HP-Filter meets better the criteria for the properties of a
cyclical variable:
1. The average value of the variable is close to zero in any sub-sample of
the data.
2. The variable crosses the zero line at a relatively frequent pace.
Cycles based on the HP-Filter trend have both properties, the reason: the trend growth
of GDP changes over time!
ECON 562
Macroeconomic Analysis & Public Policy
The cyclical component of
US series arising from the
HP-Filter reveal these
 Deviations of GDP from trend are
persistent
 Consumption, investment, GDP, hours
worked “pro-cyclical” and persistent
 Consumption is less volatile than GDP
 Investment is more volatile than GDP
 Hours worked are about as volatile as
GDP
Summing up, the key differences between US trends (growth) & cycles:
 Trend properties:
 Trend consumption, investment, GDP increase at same rate.
 Hours worked (per-capita) is trendless.
 Cycle properties:





Deviations of GDP from trend are persistent
Consumption, investment, GDP, hours worked “pro-cyclical”
Consumption is less volatile than GDP
Investment is more volatile than GDP
Hours worked are as volatile as GDP

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