In this box we 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.

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 we need to use table 1.1.6 from BEA.gov once again; starting in 1960:Q1 to the latest available year at a quarterly frequency. The smoothing parameter λ (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 (bea.gov), 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 [25 points] and calculate the standard deviation [25 points] of the cyclical components of GDP, consumption, and investment for each of the periods (plot the three variables in the same graph).
- Compare the four sample periods, summarize your findings [25 points] and explain in the context of the Great Moderation [25 points].

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.

https://ideas.repec.org/c/dge/qmrbcd/165.html

- 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 [the Excel function is =ln( )].
- 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).

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Example. Suppose you want to filter 5 data points in the first five cells of column A, say

A1:A5, at lambda = 1600.

Enter the following formula into cell B1: =HP(A1:A5,1600)

Then hit return or enter.

Then, starting with cell B1, select cells B1 through B5

Hit 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.

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5. 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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