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

In this box you will work with the HP-filter. This is a procedure to decompose economic variables into a trend and a cycle. The HP-filter allows for a flexible time-varying trend, in contrast to imposing a constant trend over long periods of time —say, decades. Since long-term trends change (eventually), this provides a better measure of the cycles i.e. deviations from trend. (1200-1500 words)The purpose of the box is to calculate the co-movement and volatility of consumption and investment relative to GDP, all measured in real terms. For this exercise you will use table 1.1.6 from www.bea.gov (Links to an external site.)Links to an external site. once again; starting in 1960:Q1 to the latest available year at a quarterly frequency. The smoothing parameter ? (lambda) for quarterly data is 1,600.The box has to describe the nature of the filter and report the correlation and volatility of the cyclical components of GDP, consumption, and investment over time. The focus will be on the Great Moderation which is known as the period under which the U.S. economy experienced a break in the volatility of most of its macroeconomic variables. The literature has identified it as beginning in the mid-1980s, and ending in 2008 with the Great Recession. Part of the exercise is to identify the Great Moderation in the data and see whether the 2008 broke that trend; that is, has the U.S. economy become as volatile as before 1984, or has the moderation persisted?To create the box follow these steps:Go to NIPA Table 1.1.6 in the Bureau of Economic Analysis website (www.bea.gov (Links to an external site.)Links to an external site.), and download the quarterly data for GDP, consumption and investment for the period 1960:Q1 to latest available.Use the HP-filter to calculate the trend of the three variables for the whole period, and then for the periods 1960-83; 1984-2007; and 2008 to latest available. Use a value of 1,600 for lambda.Detrend the original series and plot and calculate the standard deviation of the cyclical components of GDP, consumption and investment.Compare the four sample periods and summarize your findings.The end.+++ Detailed instructions for point 2 above (follow each step verbatim) +++Download the excel add-in for the HP-filter.http://ideas.repec.org/c/dge/qmrbcd/165.html (Links to an external site.)Links to an external site.Make sure that the HPFilter.xla file is in the same directory as where you are going to save your excel file. Double click on HPFilter.xla to launch it in Excel and make sure to enable macros. This should enable the HP-Filter function. You will always have to first start the macro every time you use the workfile (in that order).Transpose the variables into columns and then take the natural log of each of the variables.HP-filter the natural log of each variable, the function in Excel is: HP(timeseries to be filtered, value for lambda). This is an array formula (columns).—————————————————————————————————————————————Example. Suppose you want to filter 5 data points in the first five cells of column A, sayA1:A5, at lambda = 1600.Enter the following formula into cell B1: =HP(A1:A5,1600)Then hit return.Then, starting with cell B1, select cells B1 through B5Hit F2.Then hit the Ctrl, Shift, and Enter buttons simultaneously. This will HP-Filter the entire set of cells from A1-A5 and place the filtered series into B1-B5.—————————————————————————————————————————————-For the case of GDP, the variable on column A should be the ln (GDP) and column B represents the trend, ln (GDP*). The next step is to compute the cycle ln (GDP) – ln (GDP*) in column C. Now you can plot it and calculate its standard deviation. Do the same for consumption and investment.

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ECON 562
Macroeconomic Analysis & Public Policy
Module 6: Business Cycles
Copyright 2017 Montclair State University
Trends vs
ECON 562
Macroeconomic Analysis & Public Policy
Module 6a: Business Cycles
Business Cycles
The economy grows over time, but
goes through phases of expansions
and contractions, these are the
business cycles. The parts of the
cycle are:
(2 quarters
or more of
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
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?
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
Module 6c: Business Cycle Statistics
Business Cycle Statistics
The cyclical component of
US series arising from the
HP-Filter reveal these
business cycle facts:
• Deviations of GDP from trend are
• 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
Business Cycle Statistics
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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