# 6.7 Exercises

1. Show that a $3\times 5$ MA is equivalent to a 7-term weighted moving average with weights of 0.067, 0.133, 0.200, 0.200, 0.200, 0.133, and 0.067.
2. The data below represent the monthly sales (in thousands) of product A for a plastics manufacturer for years 1 through 5 (data set plastics).
1 2 3 4 5
Jan 742 741 896 951 1030
Feb 697 700 793 861 1032
Mar 776 774 885 938 1126
Apr 898 932 1055 1109 1285
May 1030 1099 1204 1274 1468
Jun 1107 1223 1326 1422 1637
Jul 1165 1290 1303 1486 1611
Aug 1216 1349 1436 1555 1608
Sep 1208 1341 1473 1604 1528
Oct 1131 1296 1453 1600 1420
Nov 971 1066 1170 1403 1119
Dec 783 901 1023 1209 1013
1. Plot the time series of sales of product A. Can you identify seasonal fluctuations and/or a trend?
2. Use a classical multiplicative decomposition to calculate the trend-cycle and seasonal indices.
3. Do the results support the graphical interpretation from part (a)?
4. Compute and plot the seasonally adjusted data.
5. Change one observation to be an outlier (e.g., add 500 to one observation), and recompute the seasonally adjusted data. What is the effect of the outlier?
6. Does it make any difference if the outlier is near the end rather than in the middle of the time series?
7. Use a random walk with drift to produce forecasts of the seasonally adjusted data.
8. Reseasonalize the results to give forecasts on the original scale.
3. Figure 6.13 shows the result of decomposing the number of persons in the civilian labor force in Australia each month from February 1978 to August 1995.
Figure 6.13: Decomposition of the number of persons in the civilian labor force in Australia each month from February 1978 to August 1995.
1. Write about 3–5 sentences describing the results of the seasonal adjustment. Pay particular attention to the scales of the graphs in making your interpretation.
2. Is the recession of 1991/1992 visible in the estimated components?