Chapter 15
Time Series Analysis and Forecasting
Learning Objectives
1.
Understand that the long-run success of an organization is often closely related to how well
management is able to predict future aspects of the operation.
2.
Know the various components of a time series.
3.
Be able to use smoothing techniques such as moving averages and exponential smoothing.
4.
Be able to use the least squares method to identify the trend component of a time series.
5.
Understand how the classical time series model can be used to explain the pattern or behavior of
the data in a time series and to develop a forecast for the time series.
6.
Be able to determine and model seasonal patterns for forecasting a time series.
7.
Know how curve-fitting can be used in forecasting.
8.
Know the definition of the following terms:
time series
forecast
trend component
cyclical component
seasonal component
irregular component
mean squared error
moving averages
weighted moving averages
smoothing constant
seasonal index
15 - 1
Solutions:
1.
The following table shows the calculations for parts (a), (b), and (c).
Time Series
Value
Forecast
Week
Absolute
Value of
Forecast
Error
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute Value
of Percentage
Error
1
18
2
13
18
-5
5
25
-38.46
38.46
3
16
13
3
3
9
18.75
18.75
4
11
16
-5
5
25
-45.45
45.45
5
17
11
6
6
36
35.29
35.29
6
14
17
-3
3
9
-21.43
21.43
Totals
22
104
-51.30
159.38
2.
a.
MAE = 22/5 = 4.4
b.
MSE = 104/5 = 20.8
c.
MAPE = 159.38/5 = 31.88
d.
Forecast for week 7 is 14
The following table shows the calculations for parts (a), (b), and (c).
Time Series
Value
Forecast
Week
Absolute
Value of
Forecast
Error
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute Value
of Percentage
Error
1
18
2
13
18.00
-5.00
5.00
25.00
-38.46
38.46
3
16
15.50
0.50
0.50
0.25
3.13
3.13
4
11
15.67
-4.67
4.67
21.81
-42.45
42.45
5
17
14.50
2.50
2.50
6.25
14.71
14.71
6
14
15.00
-1.00
1.00
1.00
-7.14
7.14
Totals
13.67
54.31
-70.21
105.86
3.
a.
MAE = 13.67/5 = 2.73
b.
MSE = 54.31/5 = 10.86
c.
MAPE = 105.89/5 = 21.18
d.
Forecast for week 7 is (18 + 13 + 16 + 11 + 17 + 14) / 6 = 14.83
The following table shows the measures of forecast error for both methods.
Exercise
15 - 2
Exercise
1
MAE
MSE
MAP
E
2
4.40
20.80
31.88
2.73
10.86
21.18
For each measure of forecast accuracy the average of all the historical data provided more accurate forecasts
than simply using the most recent value.
4.
a.
Month
Time Series
Value
Forecast
Squared
Forecast
Error
Forecast
Error
1
24
2
13
24
-11
121
3
20
13
7
49
4
12
20
-8
64
5
19
12
7
49
6
23
19
4
16
7
15
23
-8
64
Total
363
MSE = 363/6 = 60.5
Forecast for month 8 = 15
b.
Week
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
24
2
13
24.00
-11.00
121.00
3
20
18.50
1.50
2.25
4
12
19.00
-7.00
49.00
5
19
17.25
1.75
3.06
6
23
17.60
5.40
29.16
7
15
18.50
-3.50
12.25
Total
216.72
MSE = 216.72/6 = 36.12
Forecast for month 8 = (24 + 13 + 20 + 12 + 19 + 23 + 15) / 7 = 18
The average of all the previous values is better because MSE is smaller.
c.
5.
The average of all the previous values is better because MSE is smaller.
a.
15 - 3
The data appear to follow a horizontal pattern.
b.
Three-week moving average.
Time Series
Value
Forecast
Week
1
18
2
13
3
16
4
11
15.67
5
17
6
14
Forecast
Error
Squared
Forecast
Error
-4.67
21.78
13.33
3.67
13.44
14.67
-0.67
0.44
Total
35.67
MSE = 35.67/3 = 11.89
The forecast for week 7 = (11 + 17 + 14) / 3 = 14
c.
Smoothing constant = .2
15 - 4
Time Series
Value
Forecast
Week
Forecast
Error
Squared
Forecast
Error
1
18
2
13
18.00
-5.00
25.00
3
16
17.00
-1.00
1.00
4
11
16.80
-5.80
33.64
5
17
15.64
1.36
1.85
6
14
15.91
-1.91
3.66
Total
65.15
MSE = 65.15/5 = 13.03
The forecast for week 7 is .2(14) + (1 - .2)15.91 = 15.53
d.
The three-week moving average provides a better forecast since it has a smaller MSE.
e.
Alpha
Week
1
2
3
4
5
6
Time Series
Value
18
13
16
11
17
14
MSE =
0.367694922
Forecast
18
16.16
16.10
14.23
15.25
12.060999
15 - 5
Forecast
Error
Squared
Forecast
Error
-5.00
-0.16
-5.10
2.77
-1.25
Total
25.00
0.03
26.03
7.69
1.55
60.30
Chapter 15
6.
a.
The data appear to follow a horizontal pattern.
b.
Week
Three-week moving average.
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
24
2
13
3
20
4
12
19.00
-7.00
49.00
5
19
15.00
4.00
16.00
6
23
17.00
6.00
36.00
7
15
18.00
-3.00
9.00
Total
110.00
MSE = 110/4 = 27.5.
The forecast for week 8 = (19 + 23 + 15) / 3 = 19
15 - 6
Forecasting
c.
Smoothing constant = .2
Time Series
Value
Forecast
Week
Forecast
Error
Squared
Forecast
Error
1
24
2
13
24.00
-11.00
121.00
3
20
21.80
-1.80
3.24
4
12
21.44
-9.44
89.11
5
19
19.55
-0.55
0.30
6
23
19.44
3.56
12.66
7
15
20.15
-5.15
26.56
Total
252.87
MSE = 252.87/6 = 42.15
The forecast for week 8 is .2(15) + (1 - .2)20.15 = 19.12
d.
The three-week moving average provides a better forecast since it has a smaller MSE.
e.
Alpha
Month
1
2
3
4
5
6
7
Time Series
Value
24
13
20
12
19
23
15
0.351404848
Forecast
Forecast
Error
Squared
Forecast
Error
-11.00
-0.13
-8.09
1.75
5.14
-4.67
Total
121.00
0.02
65.40
3.08
26.40
21.79
237.69
24
20.13
20.09
17.25
17.86
19.67
MSE = 237.69/6 = 39.61428577
15 - 7
Chapter 15
7. a.
Week
1
2
3
4
5
6
7
8
9
10
11
12
b.
Time-Series
Value
17
21
19
23
18
16
20
18
22
20
15
22
4-Week
Moving
Average
Forecast
(Error) 2
20.00
20.25
19.00
19.25
18.00
19.00
20.00
18.75
Totals
4.00
18.06
1.00
1.56
16.00
1.00
25.00
10.56
77.18
5-Week
Moving
Average
Forecast
19.60
19.40
19.20
19.00
18.80
19.20
19.00
(Error) 2
12.96
0.36
1.44
9.00
1.44
17.64
9.00
51.84
MSE(4-Week) = 77.18 / 8 = 9.65
MSE(5-Week) = 51.84 / 7 = 7.41
c.
For the limited data provided, the 5-week moving average provides the smallest MSE.
8. a.
Week
1
2
3
4
5
6
7
8
9
10
11
12
Time-Series
Value
17
21
19
23
18
16
20
18
22
20
15
22
Weighted Moving
Average Forecast
Forecast
Error
19.33
21.33
19.83
17.83
18.33
18.33
20.33
20.33
17.83
3.67
-3.33
-3.83
2.17
-0.33
3.67
-0.33
-5.33
4.17
Total
b.
(Error)2
13.47
11.09
14.67
4.71
0.11
13.47
0.11
28.41
17.39
103.43
MSE = 103.43 / 9 = 11.49
Prefer the unweighted moving average here; it has a smaller MSE.
c.
9.
You could always find a weighted moving average at least as good as the unweighted one. Actually the
unweighted moving average is a special case of the weighted ones where the weights are equal.
The following tables show the calculations for = .1.
15 - 8
Forecasting
Time Series
Value
Forecast
17
21
17.00
19
17.40
23
17.56
18
18.10
16
18.09
20
17.88
18
18.10
22
18.09
20
18.48
15
18.63
22
18.27
Week
1
2
3
4
5
6
7
8
9
10
11
12
Forecast
Error
4.00
1.60
5.44
-0.10
-2.09
2.12
-0.10
3.91
1.52
-3.63
3.73
Totals
Absolute
Value of
Forecast
Error
4.00
1.60
5.44
0.10
2.09
2.12
0.10
3.91
1.52
3.63
3.73
28.24
Squared
Forecast
Error
16.00
2.56
29.59
0.01
4.37
4.49
0.01
15.29
2.31
13.18
13.91
101.72
Percentage
Error
Absolute Value
of Percentage
Error
19.05
8.42
23.65
-0.56
-13.06
10.60
-0.56
17.77
7.60
-24.20
16.95
65.67
19.05
8.42
23.65
0.56
13.06
10.60
0.56
17.77
7.60
24.20
16.95
142.42
The following tables show the calculations for = .2
Time Series
Value
Forecast
17
21
17.00
19
17.80
23
18.04
18
19.03
16
18.83
20
18.26
18
18.61
22
18.49
20
19.19
15
19.35
22
18.48
Week
1
2
3
4
5
6
7
8
9
10
11
12
a.
Forecast
Error
4.00
1.20
4.96
-1.03
-2.83
1.74
-0.61
3.51
0.81
-4.35
3.52
Totals
Absolute
Value of
Forecast
Error
4.00
1.20
4.96
1.03
2.83
1.74
0.61
3.51
0.81
4.35
3.52
28.56
Squared
Forecast
Error
16.00
1.44
24.60
1.06
8.01
3.03
0.37
12.32
0.66
18.92
12.39
98.80
Percentage
Error
Absolute Value
of Percentage
Error
19.05
6.32
21.57
-5.72
-17.69
8.70
-3.39
15.95
4.05
-29.00
16.00
35.84
MSE for = .1 = 101.72/11 = 9.25
MSE for = .2 = 98.80/11 = 8.98
= .2 provides more accurate forecasts based upon MSE
b.
MAE for = .1 = 28.24/11 = 2.57
MAE for = .2 = 28.56/11 = 2.60
= .1 provides more accurate forecasts based upon MAE; but, they are very close.
15 - 9
19.05
6.32
21.57
5.72
17.69
8.70
3.39
15.95
4.05
29.00
16.00
147.44
Chapter 15
c.
MAPE for = .1 = 142.42/11 = 12.95%
MAPE for = .2 = 147.44/11 = 13.40%
= .1 provides more accurate forecasts based upon MAPE.
10. a.
Yˆ13 = .2Y12 + .16Y11 + .64(.2Y10 + .8 Yˆ10 ) = .2Y12 + .16Y11 + .128Y10 + .512 Yˆ10
Yˆ13 = .2Y12 + .16Y11 + .128Y10 + .512(.2Y9 + .8 Yˆ9 ) = .2Y12 + .16Y11 + .128Y10 + .1024Y9 + .4096 Yˆ9
Yˆ13 = .2Y12 + .16Y11 + .128Y10 + .1024Y9 + .4096(.2Y8 + .8 Yˆ8 ) = .2Y12 + .16Y11 + .128Y10 + .1024Y9 + .
08192Y8 + .32768 Yˆ8
b.
The more recent data receives the greater weight or importance in determining the forecast. The moving
averages method weights the last n data values equally in determining the forecast.
11. a.
The first two time series values may be an indication that the time series has shifted to a new
higher level, as shown by the remainig 10 values. But, overall, the time series plot exhibits a
horizontal pattern.
15 - 10
Forecasting
b.
Month
1
2
3
4
5
6
7
8
9
10
11
12
Yt
80
82
84
83
83
84
85
84
82
83
84
83
3-Month Moving
Averages
Forecast
82.00
83.00
83.33
83.33
84.00
84.33
83.67
83.00
83.00
α=2
Forecast
(Error) 2
1.00
0.00
0.45
2.79
0.00
5.43
0.45
1.00
0.00
11.12
(Error) 2
80.00
80.40
81.12
81.50
81.80
82.24
82.79
83.03
82.83
82.86
83.09
4.00
12.96
3.53
2.25
4.84
7.62
1.46
1.06
0.03
1.30
0.01
39.06
MSE(3-Month) = 11.12 / 9 = 1.24
MSE(α = .2) = 39.06 / 11 = 3.55
A 3-month moving average provides the most accurate forecast using MSE.
c.
We will use the 3-month moving average because we have found that it yields a superior MSE over our
time series. The 3-month moving average3-month moving average forecast = (83 + 84 + 83) / 3 = 83.3.
12. a.
The data appear to follow a horizontal pattern.
b.
Time-Series
3-Month Moving
15 - 11
4-Month Moving
Chapter 15
Month
1
2
3
4
5
6
7
8
9
10
11
12
Value
9.5
9.3
9.4
9.6
9.8
9.7
9.8
10.5
9.9
9.7
9.6
9.6
Average Forecast
(Error) 2
Average Forecast
(Error) 2
9.40
9.43
9.60
9.70
9.77
10.00
10.07
10.03
9.73
0.04
0.14
0.01
0.01
0.53
0.01
0.14
0.18
0.02
1.08
9.45
9.53
9.63
9.73
9.95
9.98
9.97
9.92
0.12
0.03
0.03
0.59
0.00
0.08
0.14
0.10
1.09
MSE(3-Month) = 1.08 / 9 = .12
MSE(4-Month) = 1.09 / 8 = .14
Use 3-Month moving averages.
c.
We will use the 3-month moving average because we have found that it yields a superior MSE over our
time series. The 3-month moving average3-month moving average forecast = (9.7 + 9.6 + 9.6) / 3 = 9.63.
13. a.
The data appear to follow a horizontal pattern.
15 - 12
Forecasting
b.
Month
1
2
3
4
5
6
7
8
9
10
11
12
Time-Series
Value
240
350
230
260
280
320
220
310
240
310
240
230
3-Month Moving
Average Forecast
273.33
280.00
256.67
286.67
273.33
283.33
256.67
286.67
263.33
(Error) 2
α = .2
Forecast
177.69
0.00
4010.69
4444.89
1344.69
1877.49
2844.09
2178.09
1110.89
17,988.52
240.00
262.00
255.60
256.48
261.18
272.95
262.36
271.89
265.51
274.41
267.53
(Error) 2
12100.00
1024.00
19.36
553.19
3459.79
2803.70
2269.57
1016.97
1979.36
1184.05
1408.50
27,818.49
MSE(3-Month) = 17,988.52 / 9 = 1998.72
MSE(α = .2) = 27,818.49 / 11 = 2528.95
Based on the above MSE values, the 3-month moving averages appears to be superior.
However, exponential smoothing was penalized by including month 2 which was difficult for
any method to forecast. Using only the errors for months 4 to 12, the MSE for exponential
smoothing is:
MSE(α = .2) = 14,694.49 / 9 = 1632.72
Thus, exponential smoothing was better considering months 4 to 12.
c.
Using exponential smoothing
Yˆ13 = α Y12 + (1 - α) Yˆ12 = .20(230) + .80(267.53) = 260
14. a.
The data appear to follow a horizontal pattern.
15 - 13
Chapter 15
b.
Smoothing constant = .3.
Month t
1
2
3
4
5
6
7
8
9
10
11
12
Forecast Error
Squared Error
Yt - Yˆt
(Yt - Yˆt )2
Forecast Yˆt
Time-Series Value
Yt
105
135
120
105
90
120
145
140
100
80
100
110
105.00
114.00
115.80
112.56
105.79
110.05
120.54
126.38
118.46
106.92
104.85
30.00
6.00
-10.80
-22.56
14.21
34.95
19.46
-26.38
-38.46
-6.92
5.15
Total
900.00
36.00
116.64
508.95
201.92
1221.50
378.69
695.90
1479.17
47.89
26.52
5613.18
MSE = 5613.18 / 11 = 510.29
Forecast for month 13: Yˆ13 = .3(110) + .7(104.85) = 106.4
c.
Alpha
Month
1
2
3
4
5
6
7
8
9
10
11
12
0.032564518
Time Series
Value
105
135
120
105
90
120
145
140
100
80
100
110
Forecast
105
105.98
106.43
106.39
105.85
106.31
107.57
108.63
108.35
107.43
107.18
MSE = 5056.62 / 11 = 459.6929489
15 - 14
Forecast
Error
Squared
Forecast
Error
30.00
14.02
-1.43
-16.39
14.15
38.69
32.43
-8.63
-28.35
-7.43
2.82
Total
900.00
196.65
2.06
268.53
200.13
1496.61
1051.46
74.47
803.65
55.14
7.93
5056.62
Forecasting
15. a.
You might think the time series plot shown above exhibits some trend. But, this is simply
due to the fact that the smallest value on the vertical axis is 7.1, as shown by the following
version of the plot.
In other words, the time series plot shows an underlying horizontal pattern.
15 - 15
Chapter 15
b.
Alpha
Futures
Index
7.35
7.4
7.55
7.56
7.6
7.52
7.52
7.7
7.62
7.55
Week
1
2
3
4
5
6
7
8
9
10
0.910230734
Forecast
Forecast
Error
Squared
Forecast
Error
0.05
0.15
0.02
0.04
-0.08
-0.01
0.18
-0.06
-0.08
Total
0.00
0.02
0.00
0.00
0.01
0.00
0.03
0.00
0.01
0.08
7.35
7.40
7.54
7.56
7.60
7.53
7.52
7.68
7.63
MSE = .08/9 = .008507355
16. a.
The number of homes is generally increasing over time, so we see a positive trend over the
years the Super Bowl has been played. The trend appears to be linear with some random
variation around the positive trend from year to year.
b.
Because this time series plot indicates a possible linear trend in the data, so forecasting methods
discussed in this chapter are appropriate to develop forecasts for this time series.
15 - 16
Forecasting
c.
The following values are needed to compute the slope and intercept:
 t 1128
t
2
Y
35720
t
 tY
1808715
t
48566536
Computation of slope:
b1 
 tY    t  Y  / n  48566536   1128   1808715  / 47 596.3663
35720   1128  / 47
t   t / n
t
t
2
2
2
Computation of intercept:
b0 Y  b1 t (38483.30/47) – (596.366)(1128/47) = 24170.506
Equation for linear trend: yˆ t  24170.506  596.366t
The annual increase in households viewing the Super Bowl is approximately 596,366.
17. a.
The time series plot shows a linear trend.
b.
Minimizing SSE is the same as minimizing MSE:
b0
4.70
b1
2.10
Squared
Year
Sales
1
6.00
Forecast
6.80
15 - 17
Forecast
Forecast
Error
Error
-0.80
0.64
Chapter 15
2
11.00
8.90
2.10
4.41
3
9.00
11.00
-2.00
4.00
4
14.00
13.10
0.90
0.81
5
15.00
15.20
-0.20
0.04
6
17.30
Total
9.9
MSE = 9.9/5 = 1.982.475
c.
Yˆ6 4.7  2.1(6) 17.3
18. a.
b.
Alpha
Period
1st-2012
2nd-2012
3rd-2012
4th-2012
1st-2013
2nd-2013
3rd-2013
4th-2013
1st-2014
2nd-2014
Stock %
29.8
31
29.9
30.1
32.2
31.5
32
31.9
30
MSE =
1.222838367
c.
0.467307293
Forecast
Forecast
Error
Squared
Forecast
Error
29.8
30.36
30.15
30.12
31.09
31.28
31.62
31.75
30.93
1.20
-0.46
-0.05
2.08
0.41
0.72
0.28
-1.75
Total
1.44
0.21
0.00
4.31
0.16
0.51
0.08
3.06
9.78
Forecast for second quarter 2014 = 30.93
19. a.
15 - 18
Forecasting
The time series plot shows a linear trend.
b.
b0
119.71
b1
-4.9286
Squared
Observed
Period
Value
Forecast
Error
Error
1
120.00
114.79
5.21
27.19
2
110.00
109.86
0.14
0.02
3
100.00
104.93
-4.93
24.29
4
96.00
100.00
-4.00
16.00
5
94.00
95.07
-1.07
1.15
6
92.00
90.14
1.86
3.45
7
88.00
85.21
2.79
7.76
8
c.
Forecast
Forecast
80.29
Yˆ8 119.714  4.9286(8) 80.29
20. a.
15 - 19
Total
79.857143
Chapter 15
The time series plot exhibits a curvilinear trend.
b.
b0
b1
c.
4.7167
1.4567
Ŷ10 =4.7167 + 1.4567(10) = 19.28
21. a.
This time series plot indicates a possible negative linear trend in the data.
b.
The following values are needed to compute the slope and intercept:
15 - 20
Forecasting
 t 66
t
2
506
Y
t
 tY
228.7
t
1335.2
Computation of slope:
b1 
 tY    t  Y  / n 1335.2   66   228.7  / 11 0.3364
506   66  / 11
t    t / n
t
t
2
2
2
Computation of intercept:
b0 Y  b1 t (228.7/11) – (-0.3364)(66/11) = 22.8909
Equation for linear trend: Yˆt  22.89096  0.3364t
The annual decrease in the percent of adults who smoke is approximately 0.3364%.
c.
The forecast of the percent of adults who smoke for 2020 is
Yˆ20  22.89096  0.3364  20  16.0818
The regression model from part (b) does suggest that the OSH is not on target to meet this goal.
The forecast of the percent of adults who smoke falls below 12% in 2033:
Yˆ33  22.89096  0.3364  33  11.702
15 - 21
Chapter 15
22. a.
The time series plot shows an upward linear trend
b.
b0
19.9928
b1
1.7738
Forecast
Year
1
2
3
4
5
6
7
Cost/Unit($)
Forecast
20.00
21.77
24.50
23.54
28.20
25.31
27.50
27.09
26.60
28.86
30.00
30.64
31.00
32.41
8
36.00
9
34.18
35.96
MSE =
Squared
Forecast
Error
-1.77
0.96
2.89
0.41
-2.26
-0.64
-1.41
Error
3.12
0.92
8.33
0.17
5.12
0.40
1.99
1.82
3.30
Total
23.34619
2.9183
c. The average cost/unit has been increasing by approximately $1.77 per year.
d. T9 19.9928  1.7738(9) 35.96
23.
a.
15 - 22
Forecasting
This time series plot indicates a possible positive linear trend in the data.
b. The following values are needed to compute the slope and intercept:
 t 120
t
2
1240
Y
t
723.8
 tY
t
5990.7
Computation of slope:
b1 
 tY    t  Y  / n  5990.7   120   723.8  / 15 0.7154
1240   120  / 15
t   t / n
t
t
2
2
2
Computation of intercept:
b0 Y  b1 t  (723.8/15) – (0.7154)(120/15) = 42.5305
Equation for linear trend: Yˆt  42.5305  0.7154t
The annual increase in the percent of adults who report that they exercise for 30 or more
minutes at least three times per week is approximately 0.7154%.
c. The forecast of the percent of adults who will report that they exercise for 30 or more minutes
at least three times per week smoke next year (year 16 of the study) is
Yˆ16  42.5305  0.7154  16  53.9762 or 53.98%.
d. The linear trend we observed in the time series plot from part (a) appears to be stable, so the
trend equation from part (b) can be used to forecast the percentage of adults three years from
now (year 18 of the study) who will report that they exercise for 30 or more minutes at least
three times per week. The forecast of the percent of adults who will report that they exercise
15 - 23
Chapter 15
for 30 or more minutes at least three times per week smoke three years from now (year 18 of
the study) is Yˆ18  42.5305  0.7154  18  55.4069 or 55.41%
24. a.
The time series plot shows a horizontal pattern. But, there is a seasonal pattern in the data.
For instance, in each year the lowest value occurs in quarter 2 and the highest value occurs in
quarter 4.
b.
The fitted equation is:
Value = 77.0 - 10.0 Qtr1 - 30.0 Qtr2 - 20.0 Qtr3
c.
The quarterly forecasts for next year are as follows:
Quarter 1 forecast = 77.0 - 10.0(1) - 30.0(0) - 20.0(0) = 67
Quarter 2 forecast = 77.0 - 10.0(0) - 30.0(1) - 20.0(0) = 47
Quarter 3 forecast = 77.0 - 10.0(0) - 30.0(0) - 20.0(1) = 57
Quarter 4 forecast = 77.0 - 10.0(0) - 30.0(0) - 20.0(0) = 77
15 - 24
Forecasting
25. a.
The time series plot shows a linear trend and a seasonal pattern in the data.
b.
The fitted regression model is:
Value = 3.42 + 0.219 Qtr1 - 2.19 Qtr2 - 1.59 Qtr3 + 0.406 t
c.
The quarterly forecasts for next year (t = 13, 14, 15, and 16) are as follows:
Quarter 1 forecast = 3.42 + 0.219(1) - 2.19(0) - 1.59(0) + 0.406(13) = 8.92
Quarter 2 forecast = 3.42 + 0.219(0) - 2.19(1) - 1.59(0) + 0.406(14) = 6.92
Quarter 3 forecast = 3.42 + 0.219(0) - 2.19(0) - 1.59(1) + 0.406(15) = 7.92
Quarter 4 forecast = 3.42 + 0.219(0) - 2.19(0) - 1.59(0) + 0.406(16) = 9.92
26. a.
There appears to be a seasonal pattern in the data and perhaps a moderate upward linear trend.
15 - 25
Chapter 15
b.
The fitted regression model is:
Value = 2492 - 712 Qtr1 - 1512 Qtr2 + 327 Qtr3
c.
The quarterly forecasts for next year are as follows:
Quarter 1 forecast = 2492 – 712(1) – 1512(0) + 327(0) = 1780
Quarter 2 forecast = 2492 – 712(0) – 1512(1) + 327(0) = 980
Quarter 3 forecast = 2492 – 712(0) – 1512(0) + 327(1) = 2819
Quarter 4 forecast = 2492 – 712(0) – 1512(0) + 327(0) = 2492
d.
The fitted regression model is:
Value = 2307 - 642 Qtr1 - 1465 Qtr2 + 350 Qtr3 + 23.1 t
The quarterly forecasts for next year are as follows:
Quarter 1 forecast = 2307 – 642(1) – 1465(0) + 350(0) + 23.1(17) = 2058
Quarter 2 forecast = 2307 – 642(0) – 1465(1) + 350(0) + 23.1(18) = 1258
Quarter 3 forecast = 2307 – 642(0) – 1465(0) + 350(1) + 23.1(19) = 3096
Quarter 4 forecast = 2307 – 642(0) – 1465(0) + 350(0) + 23.1(20) = 2769
27. a.
The time series plot indicates a seasonal pattern in the data and perhaps a slight upward linear
trend.
b.
The fitted regression model is:
Level = 21.7 + 7.67 Hour1 + 11.7 Hour2 + 16.7 Hour3 + 34.3 Hour4 + 42.3 Hour5 + 45.0
Hour6 + 28.3 Hour7 + 18.3 Hour8 + 13.3 Hour9 + 3.33 Hour10 + 1.67 Hour11
c. The hourly forecasts for the next day can be obtained very easily using the estimated
regression equation. For instance, setting Hour1 = 1 and the rest of the dummy variables equal
to 0 provides the forecast for the first hour; setting Hour2 = 1 and the rest of the dummy
variables equal to 0 provides the forecast for the second hour; and so on.
Forecast for hour 1 = 21.667 + 7.667(1) + 11.667(0) + 16.667(0) + 34.333 (0) + 42.333(0) +
45.000(0) + 28.333(0) + 18.333(0) + 13.333(0) + 3.333(0) + 1.667(0) = 29.33
15 - 26
Forecasting
Forecast for hour 2 = 21.667 + 7.667(0) + 11.667(1) + 16.667(0) + 34.333 (0) + 42.333(0) +
45.000(0) + 28.333(0) + 18.333(0) + 13.333(0) + 3.333(0) + 1.667(0) = 33.33
The forecasts for the remaining hours can be obtained similarly. But, since there is no trend
the data the hourly forecasts can also be computed by simply taking the average of the three
time series values for each hour.
Hour
July 15
July 16
July 17
Average
1
25
28
35
29.33
2
28
30
42
33.33
3
35
35
45
38.33
4
50
48
70
56.00
5
60
60
72
64.00
6
60
65
75
66.67
7
40
50
60
50.00
8
35
40
45
40.00
9
30
35
40
35.00
10
25
25
25
25.00
11
25
20
25
23.33
12
20
20
25
21.67
In other words, the forecast for hour 1 is the average of the three observations for hour 1 on
July 15, 16, and 17, or 29.33; the forecast for hour 2 is the average of the three observations
for hour 1 on July 15, 16, and 17, or 33.33; and so on. Note that the forecast for the last hour
is 21.67, the value of b0 in the estimated regression equation.
d.
The fitted regression model is:
Level = 11.2 + 12.5 Hour1 + 16.0 Hour2 + 20.6 Hour3 + 37.8 Hour4 + 45.4 Hour5 + 47.6
Hour6 + 30.5 Hour7 + 20.1 Hour8 + 14.6 Hour9 + 4.21 Hour10 + 2.10 Hour11 + 0.437 t
Hour 1 on July 18 corresponds to Hour1 = 1 and t = 37.
Forecast for hour 1 on July 18 = 11.167 + 12.479(1) + .4375(37) = 39.834
Hour 2 on July 18 corresponds to Hour2 = 1 and t = 38.
Forecast for hour 2 on July 18 = 11.167 + 16.042(1) + .4375(38) = 43.834
The forecasts for the other hours are computed in a similar manner. The following table
shows the forecasts for the 12 hours on July 18.
Hour 1
39.834
Hour 2
43.834
Hour 3
48.834
Hour 4
66.500
Hour 5
74.501
15 - 27
Chapter 15
Hour 6
77.167
Hour 7
60.501
Hour 8
50.500
Hour 9
45.501
Hour 10
35.500
Hour 11
33.834
Hour 12
32.167
28. a.
The time series plot shows both a linear trend and seasonal effects.
b.
The fitted regression model is:
Revenue = 70.0 + 10.0 Qtr1 + 105 Qtr2 + 245 Qtr3
Quarter 1 forecast = 70.0 + 10.0(1) + 105(0) + 245(0) = 80
Quarter 2 forecast = 70.0 + 10.0(0) + 105(1) + 245(0) = 175
Quarter 3 forecast = 70.0 + 10.0(0) + 105(0) + 245(1) = 315
Quarter 4 forecast = 70.0 + 10.0(0) + 105(0) + 245(0) = 70
c.
The fitted regression model is:
Revenue = - 70.1 + 45.0 Qtr1 + 128 Qtr2 + 257 Qtr3 + 11.7 1eriod
Quarter 1 forecast = -70.1 + 45.0(1) + 128(0) + 257(0) + 11.7(21) = 221
Quarter 2 forecast = -70.1 + 45.0(0) + 128(1) + 257(0) + 11.7(22) = 315
Quarter 3 forecast = -70.1 + 45.0(0) + 128(0) + 257(1) + 11.7(23) = 456
Quarter 4 forecast = -70.1 + 45.0(0) + 128(0) + 257(0) + 11.7(24) = 211
15 - 28