Slutsky Exercise

Edward C. Prescott

CYCLES AS THE SUM OF RANDOM CAUSES

The purpose of this exercise is to show that time series that are sum of random causes display (apparent) cyclical behavior. This is based upon the work of Eugen Slutsky, "The Sum of Random Causes as the Source of Cyclical Processes".

Part 1: Flipping Coins

Flip a coin 160 times and record the sequence of heads and tails.

Click here to flip Canadian pennies. Recommendation: Flip 10 coins, reload page 16 times so it's easier to keep track. We emphasize here that every student should have a different series !

Part 2: Producing the series

Run program SUM. It produces the following 8 data series using the sequence of random events generated in Part 1.

Series 1: For observations t = 1,2, … , 160, enter 1 if that flip comes up head and -1 if that flip comes up tails. Call this data series {e_t}.

Series 2: Generate this series with

X_t = e_t+ 100 , t=1,2, … , 160

Series 3: Generate this series by

X_t+1 = r X_t + (1-r ) 100 + e_t , t=1, 2, … , 160

for r = 0.5 and for X₁= 100.

Series 4: The trend component for series 3, that is a smooth curve passing through plot of series 3 versus time.

Series 5: The deviation from trend of series 3. For each observation t, the value of series 5 is the value of series 3 minus the value of series 4.

Series 6: Generate this series by

X_t+1 = r X_t + (1-r ) 100 + e_t , t=1, 2, … , 160

for r = 0.95 and for X₁= 100.

Series 7: The trend component for series 6.

Series 8: The deviation component for series 6.

Part 3: Producing the plots.

Produce the plots A-E listed below using the time series that you generated in Part 1 and Part 2. Open your Excel file, select the columns that you wish to plot, click the chart wizard button, then click the selected area, and follow the instructions of the chart wizard.

Plot A: Series 2 versus time.

Plot B: Series 3 and 4 versus time.

Plot C: Series 5 versus time.

Plot D: Series 6 and 7 versus time.

Plot E: Series 8 versus time.

Part 4: Analyzing your results.

Read the section "Where are the Shocks?" in "Some Skeptical Observations on Real Business Cycle Theory" by L. Summers, MPLS FRB Quarterly Review of Fall 1986. The following paragraph is from "Response to a skeptic", by E. Prescott (same issue, p. 29):

"Another Summers question is: Where are the technology shocks? Apparently, he wants some identifiable shock to account for each of the half dozen postwar recessions. But our finding is not that infrequent large shocks produce fluctuations; it is, rather, that small shocks, every period do."

Write a short essay (maximum length 200 words) agreeing or disagreeing with Summers’ position. Support your position by using properties of one or more of the plots.

Instructions for carrying out the exercise.

1. Create a Slutsky folder

2. Click HERE to download sum.exe. Save this file in the Slutsky folder you just created.

3. Open NotePad or WordPad. Type in the first number (1 or -1), then press the enter key. Repeat until you have entered all 160 numbers. Save the file as sumin.txt in Slutsky folder.

4. Run program sum. This entails clicking on sum.exe in Slutsky folder. The output file is sumout.txt.

5. Open sumout.txt in Excel. It is a space delimited .txt file. Save it as sumout.xls

6. Click HERE for instructions to plot the time series.

What to turn in:

1. Five plots from Part 3.

2. The essay in Part 4.

11/20/02