Time Series, Forecasting, and Index Numbers

Chapter 12

time series, forecasting, and index numbers

12-1.Trend analysis is a quick method of determining in which general direction the data are moving through time. The method lacks, however, the theoretical justification of regression analysis because of the inherent autocorrelations and the intended use of the method in extrapolation beyond the estimation data set.

12-2.Problem 12-2:

(Using the template: “Trend Forecast.xls”)

Forecasting with Trend

Data
Period / t / Zt
jan / 1 / 544.2
feb / 2 / 548.5
mar / 3 / 551.8
apr / 4 / 550.5
may / 5 / 549.7
jun / 6 / 550.4
jul / 7 / 550.9
aug / 8 / 553
sep / 9 / 554.7
oct / 10 / 554.3
nov / 11 / 552.7
dec / 12 / 552.1
jan / 13 / 554.9
feb / 14 / 557.9
mar / 15 / 561.5
Forecast
t / Z-hat
16 / 558.759
17 / 559.545
18 / 560.33
19 / 561.116
20 / 561.902
21 / 562.688
22 / 563.473
23 / 564.259
24 / 565.045
25 / 565.83
26 / 566.616
27 / 567.402
Regression Statistics
r2 / 0.7561
MSE / 4.290168
Slope / 0.785714
Intercept / 546.1876

Forecast for April, 2004 = 558.8

12-3.The trend regression is:

b0 = 34.818b1 = 12.566 r 2 = .9858

(2005) = 198.182(2006) = 210.748

(Using the template: “Trend Forecast.xls”)

Forecasting with Trend

Data
Period / t / Zt
1993 / 1 / 53
1994 / 2 / 65
1995 / 3 / 74
1996 / 4 / 85
1997 / 5 / 92
1998 / 6 / 105
1999 / 7 / 120
2000 / 8 / 128
2001 / 9 / 144
2002 / 10 / 158
2003 / 11 / 179
2004 / 12 / 195
Forecast
t / Z-hat
13 / 198.182
14 / 210.748
15 / 223.315
16 / 235.881
17 / 248.448
18 / 261.014
19 / 273.58
20 / 286.147
21 / 298.713
22 / 311.28
23 / 323.846
24 / 336.413
Regression Statistics
r2 / 0.9858
MSE / 32.51189
Slope / 12.56643
Intercept / 34.81818

Forecast for 2005 = 198.182 and for 2006 = 210.748

12-4.Trend model through 2000

Forecast
t / Z-hat
25 / 1.63623
26 / 1.64622
27 / 1.6562
28 / 1.66619
29 / 1.67617
30 / 1.68616
31 / 1.69614
32 / 1.70613
33 / 1.71611
34 / 1.7261
35 / 1.73608
36 / 1.74607

Comparison to 2001: (not available when Solutions Manual written)

12-5.No, because of the seasonality.

12-6.No. Cyclicity is not well modeled by trend analysis.

12-7.The term, ‘seasonal variation’ is reserved for variation with a cycle of one year.

12-8.There will be too few degrees of freedom for error.

12-9.The weather, for one thing, changes from year to year. Thus sales of winter clothing, as an example, would have a variable seasonal component.

12-10.Beer sales at a local establishment, as an example: high during weekend nights, low at other times.

12-11.Using a computer:

Linear regression trend line:Zhat(t) = 372.876 + 0.8896 t

`t / (mon.) / data:
Z(t) / trend:
Zhat(t) / Centered
Moving
Average / C(t) =
CMA
Zhat(t) / Ratio
Moving
Average / Seasonal
Index
S / [Desea-
soned]
Z(t)/S%
1 / (Jun) / 375.00 / 373.77 / 99.52 / 376.83
2 / (Jul) / 370.00 / 374.66 / 98.87 / 374.22
3 / (Aug) / 374.00 / 375.54 / 99.25 / 376.82
4 / (Sep) / 378.00 / 376.43 / 99.74 / 378.97
5 / (Oct) / 376.00 / 377.32 / 99.78 / 376.82
6 / (Nov) / 380.00 / 378.21 / 100.48 / 378.20
7 / (Dec) / 384.00 / 379.10 / 378.62 / 0.999 / 101.42 / 102.33 / 375.26
8 / (Jan) / 380.00 / 379.99 / 379.37 / 0.998 / 100.16 / 100.95 / 376.43
9 / (Feb) / 378.00 / 380.88 / 380.29 / 0.998 / 99.40 / 99.84 / 378.62
10 / (Mar) / 380.00 / 381.77 / 381.12 / 0.998 / 99.70 / 99.39 / 382.31
11 / (Apr) / 382.00 / 382.66 / 382.08 / 0.998 / 99.98 / 100.09 / 381.64
12 / (May) / 383.00 / 383.55 / 383.17 / 0.999 / 99.96 / 99.76 / 383.92
13 / (Jun) / 382.00 / 384.44 / 384.46 / 1.000 / 99.36 / 99.52 / 383.86
14 / (Jul) / 381.00 / 385.33 / 386.00 / 1.002 / 98.70 / 98.87 / 385.35
15 / (Aug) / 385.00 / 386.22 / 387.37 / 1.003 / 99.39 / 99.25 / 387.91
16 / (Sep) / 387.00 / 387.11 / 388.37 / 1.003 / 99.65 / 99.74 / 387.99
17 / (Oct) / 390.00 / 388.00 / 389.25 / 1.003 / 100.19 / 99.78 / 390.85
18 / (Nov) / 392.00 / 388.89 / 390.12 / 1.003 / 100.48 / 100.48 / 390.14
19 / (Dec) / 403.00 / 389.78 / 390.96 / 1.003 / 103.08 / 102.33 / 393.82
20 / (Jan) / 398.00 / 390.67 / 391.83 / 1.003 / 101.57 / 100.95 / 394.26
21 / (Feb) / 393.00 / 391.56 / 392.54 / 1.003 / 100.12 / 99.84 / 393.65
22 / (Mar) / 389.00 / 392.45 / 393.21 / 1.002 / 98.93 / 99.39 / 391.37
23 / (Apr) / 394.00 / 393.34 / 393.79 / 1.001 / 100.05 / 100.09 / 393.63
24 / (May) / 392.00 / 394.23 / 394.33 / 1.000 / 99.41 / 99.76 / 392.94
25 / (Jun) / 393.00 / 395.12 / 394.92 / 0.999 / 99.51 / 99.52 / 394.91
26 / (Jul) / 391.00 / 396.01 / 395.42 / 0.999 / 98.88 / 98.87 / 395.46
27 / (Aug) / 392.00 / 396.89 / 396.12 / 0.998 / 98.96 / 99.25 / 394.96
28 / (Sep) / 396.00 / 397.78 / 397.25 / 0.999 / 99.69 / 99.74 / 397.02
29 / (Oct) / 395.00 / 398.67 / 398.12 / 0.999 / 99.22 / 99.78 / 395.86
30 / (Nov) / 400.00 / 399.56 / 398.75 / 0.998 / 100.31 / 100.48 / 398.11
31 / (Dec) / 409.00 / 400.45 / 102.33 / 399.69
32 / (Jan) / 404.00 / 401.34 / 100.95 / 400.21
33 / (Feb) / 404.00 / 402.23 / 99.84 / 404.67
34 / (Mar) / 405.00 / 403.12 / 99.39 / 407.47
35 / (Apr) / 399.00 / 404.01 / 100.09 / 398.63
36 / (May) / 402.00 / 404.90 / 99.76 / 402.97
------FORECAST------
37 / (Jun) / (Zhat = 405.79)(S = 99.52)/100 = 403.82

(Using the template: “Trend+Season Forecasting.xls”)

Forecasting with Trend and Seasonality

Data
t / Year / Month / Y / sn / Deseasonalized
1 / 2001 / 6 / Jun / 375 / 99.51539 / 376.826
2 / 2001 / 7 / Jul / 370 / 98.87112 / 374.225
3 / 2001 / 8 / Aug / 374 / 99.25034 / 376.825
4 / 2001 / 9 / Sep / 378 / 99.7436 / 378.972
5 / 2001 / 10 / Oct / 376 / 99.78185 / 376.822
6 / 2001 / 11 / Nov / 380 / 100.4756 / 378.201
7 / 2001 / 12 / Dec / 384 / 102.3298 / 375.257
8 / 2002 / 1 / Jan / 380 / 100.9482 / 376.431
9 / 2002 / 2 / Feb / 378 / 99.8351 / 378.624
10 / 2002 / 3 / Mar / 380 / 99.39496 / 382.313
11 / 2002 / 4 / Apr / 382 / 100.0938 / 381.642
12 / 2002 / 5 / May / 383 / 99.76035 / 383.92
13 / 2002 / 6 / Jun / 382 / 99.51539 / 383.86
14 / 2002 / 7 / Jul / 381 / 98.87112 / 385.35
15 / 2002 / 8 / Aug / 385 / 99.25034 / 387.908
16 / 2002 / 9 / Sep / 387 / 99.7436 / 387.995
17 / 2002 / 10 / Oct / 390 / 99.78185 / 390.853
18 / 2002 / 11 / Nov / 392 / 100.4756 / 390.145
19 / 2002 / 12 / Dec / 403 / 102.3298 / 393.825
20 / 2003 / 1 / Jan / 398 / 100.9482 / 394.262
21 / 2003 / 2 / Feb / 393 / 99.8351 / 393.649
22 / 2003 / 3 / Mar / 389 / 99.39496 / 391.368
23 / 2003 / 4 / Apr / 394 / 100.0938 / 393.631
24 / 2003 / 5 / May / 392 / 99.76035 / 392.942
25 / 2003 / 6 / Jun / 393 / 99.51539 / 394.914
26 / 2003 / 7 / Jul / 391 / 98.87112 / 395.464
27 / 2003 / 8 / Aug / 392 / 99.25034 / 394.961
28 / 2003 / 9 / Sep / 396 / 99.7436 / 397.018
29 / 2003 / 10 / Oct / 395 / 99.78185 / 395.864
30 / 2003 / 11 / Nov / 400 / 100.4756 / 398.107
31 / 2003 / 12 / Dec / 409 / 102.3298 / 399.688
32 / 2004 / 1 / Jan / 404 / 100.9482 / 400.205
33 / 2004 / 2 / Feb / 404 / 99.8351 / 404.667
34 / 2004 / 3 / Mar / 405 / 99.39496 / 407.465
35 / 2004 / 4 / Apr / 399 / 100.0938 / 398.626
36 / 2004 / 5 / May / 402 / 99.76035 / 402.966

Forecast for June, 2004 = 403.886

Forecasts
t / Year / Month / Y
37 / 2004 / 6 / Jun / 403.886
38 / 2004 / 7 / Jul / 402.146
39 / 2004 / 8 / Aug / 404.567
40 / 2004 / 9 / Sep / 407.46
41 / 2004 / 10 / Oct / 408.499
42 / 2004 / 11 / Nov / 412.229
43 / 2004 / 12 / Dec / 420.742
44 / 2005 / 1 / Jan / 415.954
45 / 2005 / 2 / Feb / 412.252
46 / 2005 / 3 / Mar / 411.314
47 / 2005 / 4 / Apr / 415.092
48 / 2005 / 5 / May / 414.592
Seasonal Indices
Month / Index
Jan / 100.95
Feb / 99.84
Mar / 99.39
Apr / 100.09
May / 99.76
Jun / 99.52
Jul / 98.87
Aug / 99.25
Sep / 99.74
Oct / 99.78
Nov / 100.48
Dec / 102.33

12-12.Using a computer:

Linear regression trend line:Zhat(t) = 7.2043  0.0194 t

t / (mon.) / data:
Z(t) / trend:
Zhat(t) / Centered
Moving
Average / C(t) =
CMA
Zhat(t) / Ratio
Moving
Average / Seasonal
Index
S / [Desea-
soned]
Z(t)/S%
1 / (Jul) / 7.40 / 7.18 / 95.68 / 7.73
2 / (Aug) / 6.80 / 7.17 / 92.25 / 7.37
3 / (Sep) / 6.40 / 7.15 / 90.57 / 7.07
4 / (Oct) / 6.60 / 7.13 / 97.57 / 6.76
5 / (Nov) / 6.50 / 7.11 / 95.96 / 6.77
6 / (Dec) / 6.00 / 7.09 / 92.22 / 6.51
7 / (Jan) / 7.00 / 7.07 / 7.02 / 0.993 / 99.76 / 102.47 / 6.83
8 / (Feb) / 6.70 / 7.05 / 7.01 / 0.995 / 95.54 / 98.21 / 6.82
9 / (Mar) / 8.20 / 7.03 / 7.05 / 1.002 / 116.38 / 114.41 / 7.17
10 / (Apr) / 7.80 / 7.01 / 7.10 / 1.012 / 109.92 / 110.59 / 7.05
11 / (May) / 7.70 / 6.99 / 7.15 / 1.022 / 107.76 / 109.60 / 7.03
12 / (Jun) / 7.30 / 6.97 / 7.20 / 1.032 / 101.45 / 100.45 / 7.27
13 / (Jul) / 7.00 / 6.95 / 7.25 / 1.043 / 96.55 / 95.68 / 7.32
14 / (Aug) / 7.10 / 6.93 / 7.30 / 1.052 / 97.32 / 92.25 / 7.70
15 / (Sep) / 6.90 / 6.91 / 7.30 / 1.057 / 94.47 / 90.57 / 7.62
16 / (Oct) / 7.30 / 6.89 / 7.29 / 1.057 / 100.17 / 97.57 / 7.48
17 / (Nov) / 7.00 / 6.87 / 7.28 / 1.059 / 96.16 / 95.96 / 7.29
18 / (Dec) / 6.70 / 6.86 / 7.25 / 1.058 / 92.41 / 92.22 / 7.27
19 / (Jan) / 7.60 / 6.84 / 7.20 / 1.053 / 105.62 / 102.47 / 7.42
20 / (Feb) / 7.20 / 6.82 / 7.11 / 1.043 / 101.29 / 98.21 / 7.33
21 / (Mar) / 7.90 / 6.80 / 7.00 / 1.029 / 112.92 / 114.41 / 6.90
22 / (Apr) / 7.70 / 6.78 / 6.89 / 1.017 / 111.73 / 110.59 / 6.96
23 / (May) / 7.60 / 6.76 / 6.79 / 1.005 / 111.90 / 109.60 / 6.93
24 / (Jun) / 6.70 / 6.74 / 6.71 / 0.996 / 99.88 / 100.45 / 6.67
25 / (Jul) / 6.30 / 6.72 / 6.62 / 0.985 / 95.21 / 95.68 / 6.58
26 / (Aug) / 5.70 / 6.70 / 6.51 / 0.971 / 87.58 / 92.25 / 6.18
27 / (Sep) / 5.60 / 6.68 / 6.43 / 0.963 / 87.05 / 90.57 / 6.18
28 / (Oct) / 6.10 / 6.66 / 6.40 / 0.960 / 95.37 / 97.57 / 6.25
29 / (Nov) / 5.80 / 6.64 / 95.96 / 6.04
30 / (Dec) / 5.90 / 6.62 / 92.22 / 6.40
31 / (Jan) / 6.20 / 6.60 / 102.47 / 6.05
32 / (Feb) / 6.00 / 6.58 / 98.21 / 6.11
33 / (Mar) / 7.30 / 6.56 / 114.41 / 6.38
34 / (Apr) / 7.40 / 6.54 / 110.59 / 6.69
------FORECAST------
35 / (May) / (Zhat = 6.525)(S = 109.60)/100 = 7.15

Template Forecast is 7.045

12-13.(Using the template: “Trend Forecast.xls”)

Forecasting with Trend

Data
Period / t / Zt
mar / 1 / 860
apr / 2 / 860
may / 3 / 875
jun / 4 / 910
jul / 5 / 890
aug / 6 / 930
sep / 7 / 970
oct / 8 / 962
nov / 9 / 957
dec / 10 / 972
jan / 11 / 975
feb / 12 / 1000
mar / 13 / 985
apr / 14 / 1010

Forecast for May, 2004 = 1028.53

Forecast
t / Z-hat
15 / 1028.53
16 / 1040.37
17 / 1052.21
18 / 1064.05
19 / 1075.89
20 / 1087.74
21 / 1099.58
22 / 1111.42
23 / 1123.26
24 / 1135.1
25 / 1146.95
26 / 1158.79
Regression Statistics
r2 / 0.9122
MSE / 255.7634
Slope / 11.84176
Intercept / 850.9011

12-14.(Using the template: “Trend Forecast.xls”)

Forecasting with Trend

Data
Period / t / Zt
apr / 1 / 750
may / 2 / 840
jun / 3 / 895
jul / 4 / 890
aug / 5 / 900
sep / 6 / 940
oct / 7 / 955
nov / 8 / 962
dec / 9 / 1020
jan / 10 / 1055
feb / 11 / 1015
mar / 12 / 1050

Forecast for May, 2004 = 1094.92

Forecast
t / Z-hat
13 / 1094.92
14 / 1118.86
15 / 1142.8
16 / 1166.74
17 / 1190.67
18 / 1214.61
19 / 1238.55
20 / 1262.48
21 / 1286.42
22 / 1310.36
23 / 1334.29
24 / 1358.23
Regression Statistics
r2 / 0.9076
MSE / 834.21
Slope / 23.93706
Intercept / 783.7424

12-15.(Using the template: “Trend+Season Forecasting.xls”)

Forecasting with Trend and Seasonality (quarterly)

t / Year / Q / Y / Deseasonalized
1 / 2002 / 1 / 3.4 / 3.869621
2 / 2002 / 2 / 4.5 / 4.150717
3 / 2002 / 3 / 4 / 4.258289
4 / 2002 / 4 / 5 / 4.554288
5 / 2003 / 1 / 4.2 / 4.78012
6 / 2003 / 2 / 5.4 / 4.98086
7 / 2003 / 3 / 4.9 / 5.216404
8 / 2003 / 4 / 5.7 / 5.191888
9 / 2004 / 1 / 4.6 / 5.23537
Forecasts
t / Year / Q / Y
10 / 2004 / 2 / 6.20676
11 / 2004 / 3 / 5.56327
12 / 2004 / 4 / 6.71894
Seasonal Indices
Q / Index
1 / 87.86
2 / 108.42
3 / 93.93
4 / 109.79
400

Forecast for Q2, 2004 = 6.20676

12-16.Using a computer:

w = 0.6Zhat(1) = Z(1) = 142

Zhat( 2):0.6(142.00) + 0.4(142.00) = 142.00

Zhat( 3):0.6(137.00) + 0.4(142.00) = 139.00

Zhat( 4):0.6(143.00) + 0.4(139.00) = 141.40

Zhat( 5):0.6(142.00) + 0.4(141.40) = 141.76

Zhat( 6):0.6(149.00) + 0.4(141.76) = 146.10

Zhat( 7):0.6(143.00) + 0.4(146.10) = 144.24

Zhat( 8):0.6(151.00) + 0.4(144.24) = 148.30

Zhat( 9):0.6(150.00) + 0.4(148.30) = 149.32

Zhat(10):0.6(151.00) + 0.4(149.32) = 150.33

Zhat(11):0.6(146.00) + 0.4(150.33) = 147.73

Zhat(12):0.6(144.00) + 0.4(147.73) = 145.49

Zhat(13):0.6(145.00) + 0.4(145.49) = 145.20

------FORECAST------

Zhat(14):0.6(147.00) + 0.4(145.20) = 146.28

By experimenting, we find that lower values of w in this case produce values that agree more closely with the raw data at the end of the series.

(Using the template: “Exponential Smoothing.xls”)

Exponential Smoothing

w / 0.6
t / Zt / Forecast
1 / 142 / 142
2 / 137 / 142
3 / 143 / 139
4 / 142 / 141.4
5 / 149 / 141.76
6 / 143 / 146.104
7 / 151 / 144.242
8 / 150 / 148.297
9 / 151 / 149.319
10 / 146 / 150.327
11 / 144 / 147.731
12 / 145 / 145.492
13 / 147 / 145.197
14 / 146.279

Forecast for July, 2004 = 146.279

12-17.Using a computer:

w = 0.3Zhat(1) = Z(1) = 57w = 0.8

Zhat( 2):0.3(57.00) + 0.7(57.00) = 57.000.8(57.00) + 0.2(57.00) = 57.00

Zhat( 3):0.3(58.00) + 0.7(57.00) = 57.300.8(58.00) + 0.2(57.00) = 57.80

Zhat( 4):0.3(60.00) + 0.7(57.30) = 58.110.8(60.00) + 0.2(57.80) = 59.56

Zhat( 5):0.3(54.00) + 0.7(58.11) = 56.880.8(54.00) + 0.2(59.56) = 55.11

Zhat( 6):0.3(56.00) + 0.7(56.88) = 56.610.8(56.00) + 0.2(55.11) = 55.82

Zhat( 7):0.3(53.00) + 0.7(56.61) = 55.530.8(63.00) + 0.2(55.82) = 53.56

Zhat( 8):0.3(55.00) + 0.7(55.53) = 55.370.8(55.00) + 0.2(53.56) = 54.71

Zhat( 9):0.3(59.00) + 0.7(55.37) = 56.460.8(59.00) + 0.2(54.71) = 58.14

Zhat(10):0.3(62.00) + 0.7(56.46) = 58.120.8(62.00) + 0.2(58.14) = 61.23

Zhat(11):0.3(57.00) + 0.7(58.12) = 57.790.8(57.00) + 0.2(61.23) = 57.85

Zhat(12):0.3(50.00) + 0.7(57.79) = 55.450.8(50.00) + 0.2(57.85) = 51.57

Zhat(13):0.3(48.00) + 0.7(55.45) = 53.210.8(48.00) + 0.2(51.57) = 48.71

Zhat(14):0.3(52.00) + 0.7(53.21) = 52.850.8(52.00) + 0.2(48.71) = 51.34

Zhat(15):0.3(55.00) + 0.7(52.85) = 53.500.8(55.00) + 0.2(51.34) = 54.27

Zhat(16):0.3(58.00) + 0.7(53.50) = 54.850.8(58.00) + 0.2(54.27) = 57.25

Zhat(17):0.3(61.00) + 0.7(54.85) = 56.690.8(61.00) + 0.2(57.25) = 60.25

The w = .8 forecasts follow the raw data much more closely. This makes sense because the raw data jump back and forth fairly abruptly, so we need a high w for the forecasts to respond to those oscillations sooner.

12-18.Using a computer:

w = 0.7Zhat(1) = Z(1) = 195

Zhat( 2):0.7(195.00) + 0.3(195.00) = 195.00

Zhat( 3):0.7(193.00) + 0.3(195.00) = 193.60

Zhat( 4):0.7(190.00) + 0.3(193.60) = 191.08

Zhat( 5):0.7(185.00) + 0.3(191.08) = 186.82

Zhat( 6):0.7(180.00) + 0.3(186.82) = 182.05

Zhat( 7):0.7(190.00) + 0.3(182.05) = 187.61

Zhat( 8):0.7(185.00) + 0.3(187.61) = 185.78

Zhat( 9):0.7(186.00) + 0.3(185.78) = 185.94

Zhat(10):0.7(184.00) + 0.3(185.94) = 184.58

Zhat(11):0.7(185.00) + 0.3(184.58) = 184.87

Zhat(12):0.7(198.00) + 0.3(184.87) = 194.06

Zhat(13):0.7(199.00) + 0.3(194.06) = 197.52

Zhat(14):0.7(200.00) + 0.3(197.52) = 199.26

Zhat(15):0.7(201.00) + 0.3(199.26) = 200.48

Zhat(16):0.7(199.00) + 0.3(200.48) = 199.44

Zhat(17):0.7(187.00) + 0.3(199.44) = 190.73

Zhat(18):0.7(186.00) + 0.3(190.73) = 187.42

Zhat(19):0.7(191.00) + 0.3(187.42) = 189.93

Zhat(20):0.7(195.00) + 0.3(189.93) = 193.48

Zhat(21):0.7(200.00) + 0.3(193.48) = 198.04

Zhat(22):0.7(200.00) + 0.3(198.04) = 199.41

Zhat(23):0.7(190.00) + 0.3(199.41) = 192.82

Zhat(24):0.7(186.00) + 0.3(192.82) = 188.05

Zhat(25):0.7(196.00) + 0.3(188.05) = 193.61

Zhat(26):0.7(198.00) + 0.3(193.61) = 196.68

Zhat(27):0.7(200.00) + 0.3(196.68) = 199.01

------FORECAST------

Zhat(28):0.7(200.00) + ).3(199.01) = 199.70

Exponential Smoothing
MAE / MAPE / MSE
w / 0.7 / 4.8241 / 2.52% / 34.8155
t / Zt / Forecast / |Error| / %Error / Error2
1 / 195 / 195
2 / 193 / 195
3 / 190 / 193.6 / 3.6 / 1.89% / 12.96
4 / 185 / 191.08 / 6.08 / 3.29% / 36.9664
5 / 180 / 186.824 / 6.824 / 3.79% / 46.567
6 / 190 / 182.047 / 7.9528 / 4.19% / 63.247
7 / 185 / 187.614 / 2.61416 / 1.41% / 6.83383
8 / 186 / 185.784 / 0.21575 / 0.12% / 0.04655
9 / 184 / 185.935 / 1.93527 / 1.05% / 3.74529
10 / 185 / 184.581 / 0.41942 / 0.23% / 0.17591
11 / 198 / 184.874 / 13.1258 / 6.63% / 172.287
12 / 199 / 194.062 / 4.93775 / 2.48% / 24.3814
13 / 200 / 197.519 / 2.48132 / 1.24% / 6.15697
14 / 201 / 199.256 / 1.7444 / 0.87% / 3.04292
15 / 199 / 200.477 / 1.47668 / 0.74% / 2.18059
16 / 187 / 199.443 / 12.443 / 6.65% / 154.828
17 / 186 / 190.733 / 4.7329 / 2.54% / 22.4004
18 / 191 / 187.42 / 3.58013 / 1.87% / 12.8173
19 / 195 / 189.926 / 5.07404 / 2.60% / 25.7459
20 / 200 / 193.478 / 6.52221 / 3.26% / 42.5392
21 / 200 / 198.043 / 1.95666 / 0.98% / 3.82853
22 / 190 / 199.413 / 9.413 / 4.95% / 88.6046
23 / 186 / 192.824 / 6.8239 / 3.67% / 46.5656
24 / 196 / 188.047 / 7.95283 / 4.06% / 63.2475
25 / 198 / 193.614 / 4.38585 / 2.22% / 19.2357
26 / 200 / 196.684 / 3.31575 / 1.66% / 10.9942
27 / 200 / 199.005 / 0.99473 / 0.50% / 0.98948
28 / 199.702

12-19.Using a computer:

w = 0.6Zhat(1) = Z(1) = 16.4

Zhat( 2):0.6(16.40) + 0.4(16.40) = 16.40

Zhat( 3):0.6(17.10) + 0.4(16.40) = 16.82

Zhat( 4):0.6(16.90) + 0.4(16.82) = 16.87

Zhat( 5):0.6(17.30) + 0.4(16.87) = 17.13

Zhat( 6):0.6(17.50) + 0.4(17.13) = 17.35

Zhat( 7):0.6(17.20) + 0.4(17.35) = 17.26

Zhat( 8):0.6(17.30) + 0.4(17.26) = 17.28

Zhat( 9):0.6(17.10) + 0.4(17.28) = 17.17

Zhat(10):0.6(16.90) + 0.4(17.17) = 17.01

Zhat(11):0.6(17.00) + 0.4(17.01) = 17.00

Zhat(12):0.6(17.10) + 0.4(17.00) = 17.06

------FORECAST------

Zhat(13):0.6(17.20) + 0.4(17.06) = 17.14

(Using the template: “Exponential Smoothing.xls”)

Exponential Smoothing

w / 0.6
t / Zt / Forecast
1 / 16.4 / 16.4
2 / 17.1 / 16.4
3 / 16.9 / 16.82
4 / 17.3 / 16.868
5 / 17.5 / 17.1272
6 / 17.2 / 17.3509
7 / 17.3 / 17.2604
8 / 17.1 / 17.2841
9 / 16.9 / 17.1737
10 / 17 / 17.0095
11 / 17.1 / 17.0038
12 / 17.2 / 17.0615
13 / 17.1446

Forecast for May, 2004 = 17.1446

12-20.Answers will vary.

12-21.Equation (12-11):

The same equation for (shifting all subscripts back by 1):

= wZt1 + w(1w)Z t2 + w(1-w) 2Z t3 + w(1-w) 3Z t4 + …

Now multiplying this second equation throughout by (1w) gives:

(1w) = w(1w)Zt-1 + w(1w) 2Z t-2 + w(1w) 3Z t3 = w(1w) 4Z t-4 + …

Now note that all the terms on the right side of the equation above are identical to all the terms in Equation (12-11) on the top, after the term wZ t. Hence we can substitute in Equation (12-11) the left hand side of our last equation, (1w) for all the terms past the first. This gives us:

which is Equation (12-12).

12-22.Equation (12-13) is: Multiplying out we get:

,

which is Equation (12-12).

12-23.Simply divide each CPI by ; thus:

yearold CPInew CPI

195072.124.9

195177.826.9

195279.527.5

195380.127.7

...

...

...

12-24.168.77 in July 2000 and 173.48 in June 2001.

12-25.A simple price index reflects changes in a single price variable of time, relative to a single base time.

12-26.Index numbers are used as deflators for comparing values and prices over time in a way that prevents a given inflationary factor from affecting comparisons. They are also used to provide an aggregate measure of changes over time in several related variables.

12-27.a.1988

b.Just divide each index number by =

c.It fell, from 145% of the 1988 output down to 133% of that output.

d.Big increase in the mid ‘80’s, then a sharp drop in 1986, tumbling for three more years, then slowly climbing back up until 1995, then a drop-off.

a)

Price Index
BaseYear / 1988 / 100 / Base
Year / Price / Index
1984 / 175 / 175
1985 / 190 / 190
1986 / 132 / 132
1987 / 96 / 96
1988 / 100 / 100
1989 / 78 / 78
1990 / 131 / 131
1991 / 135 / 135
1992 / 154 / 154
1993 / 163 / 163
1994 / 178 / 178
1995 / 170 / 170
1996 / 145 / 145
1997 / 133 / 133

c)

Price Index
BaseYear / 1993 / 163 / Base
Year / Price / Index
1984 / 175 / 107.36
1985 / 190 / 116.56
1986 / 132 / 80.982
1987 / 96 / 58.896
1988 / 100 / 61.35
1989 / 78 / 47.853
1990 / 131 / 80.368
1991 / 135 / 82.822
1992 / 154 / 94.479
1993 / 163 / 100
1994 / 178 / 109.2
1995 / 170 / 104.29
1996 / 145 / 88.957
1997 / 133 / 81.595

12-28.Divide each data point by

Jun. ’03: 98.6Jul. ’03: 95.14 …

12-29.Since a yearly cycle has 12 months and there are only 18 data points, a seasonal/cyclical decomposition isn’t feasible. Simple linear regression, with the successive months numbered 1,2,..., gives SALES = 4.23987  .03870MONTH, thus for July 1995 (month #19), the forecast is 3.5046.

(Using the template: “Trend Forecast.xls”)

Forecasting with Trend

Data
Period / t / Zt
jan / 1 / 4.4
feb / 2 / 4.2
mar / 3 / 3.8
apr / 4 / 4.1
may / 5 / 4.1
jun / 6 / 4
jul / 7 / 4
aug / 8 / 3.9
sep / 9 / 3.9
oct / 10 / 3.8
nov / 11 / 3.7
dec / 12 / 3.7
jan / 13 / 3.8
feb / 14 / 3.9
mar / 15 / 3.8
apr / 16 / 3.7
may / 17 / 3.5
jun / 18 / 3.4
Forecast
t / Z-hat
19 / 3.50458
20 / 3.46588
21 / 3.42718

The forecast of sales for July, 2004 is 3.5 million units.

Regression Statistics
r2 / 0.7285
MSE / 0.016906
Slope / -0.0387
Intercept / 4.239869

12-30.Trend analysis is a quick, if sometimes inaccurate, method that can give good results. The additive and multiplicative TSCI models are sometimes useful, although they lack a firm theoretical framework. Exponential smoothing methods are good models. The ones described in this book do not handle seasonality, but extensions are possible. This author believes that Box-Jenkins ARIMA models are the way to go. One limitation of these models is the need for large data sets.

12-31.Exponential smoothing models smooth the data of sharp variations and produce forecasts that follow a type of “average” movement in the data. The greater the weighting factor w, the closer the exponential smoothing series follows the data and forecasts tend to follow the variations in the data more closely.

12-32.The trend regression is:

(Using the template: “Trend Forecast.xls”)

Forecasting with Trend

Data
Period / t / Zt
1999 / 1 / 340
2000 / 2 / 370
2001 / 3 / 350
2002 / 4 / 660
2003 / 5 / 1620
Forecast
t / Z-hat
6 / 1523
7 / 1808

Forecast for 2004 = 1523

Regression Statistics
r2 / 0.6747
MSE / 130543.3
Slope / 285
Intercept / -187

12-33.An exponential regression from Minitab gives:

Yt = 4.47535(1.14077)t

y(1998) = 62.343

12-34. a)raised the seasonal index to 99.38 for April from 99.29

We would expect to see the April index change by a significant amount. The reason it did not is due to the calculations involving moving average.

b)raised the seasonal index to 122.27 for April from 99.29

c)raised the seasonal index to 100.16 for December from 100.09 We would expect the December index to change by a significant amount. It did not due to the calculations for moving average.

d)very high or low values for data points at the beginning or end of a series have little impact on the seasonal index due to their limited influence in the moving average computations.

12-35. (Using the template: “Trend Forecast.xls”)

Forecasting with Trend

Data
Period / t / Zt
1998 / 1 / 6.3
1999 / 2 / 6.6
2000 / 3 / 7.3
2001 / 4 / 7.4
2002 / 5 / 7.8
2003 / 6 / 6.9
2004 / 7 / 7.8
Forecast
t / Z-hat
8 / 7.95714
9 / 8.15714
10 / 8.35714

Forecast for 2005 = 7.957

Regression Statistics
r2 / 0.5552
MSE / 0.179429
Slope / 0.2
Intercept / 6.357143

12-36.Answers will vary.

1