p.772 Table 18.2 n= 3
Moving Average
Sales (1000s (1000s of
of gallons) gallons)
Week Yt Ft | Yt - Ft |
1 17
2 21
3 19
4 23 19 4
5 18 21 3
6 16 20 4
7 20 19 1
8 18 18 0
9 22 18 4
10 20 20 0
11 15 20 5
12 22 19 3
F13 = 19
MAD = 2.6667 N= 9
a. Develop a 3-week moving average for this time series. What is the forecast for week 18?
b. Compute the MAD for the 3-week moving average.
HWK: p.778 #1 a-b For part b, compute MAD instead of MSE. (p.791 MAD)
p.775 Table 18.3 Alpha = 0.2
Sales Exponential Smmothing
(1000s (1000s of
of gallons) gallons)
Week Yt Ft | Yt - Ft |
1 17 17.00
2 21 17.00 4.00
3 19 17.80 1.20
4 23 18.04 4.96
5 18 19.03 1.03
6 16 18.83 2.83
7 20 18.26 1.74
8 18 18.61 0.61
9 22 18.49 3.51
10 20 19.19 0.81
11 15 19.35 4.35
12 22 18.48 3.52
F13 = 19.18
MAD = 2.5963 N= 11
HWK: p.778 #1 c-d For part d, compute MAD instead of MSE.
c. Use alpha = .2 to compute the exponential smoothing values for the time series. What is the forecast for week 13?
d. Compare the 3-week moving average forecast with the exponential smoothing forecast using alpha = .2. Which appears to provide the better forecast?
p.771 Table 18.1 Simple Linear Regression Model
Simple linear
Sales (1000s (1000s of
of gallons) gallons)
Week Yt t Ft | Yt - Ft |
1 17 1 19.12 2.12
2 21 2 19.14 1.86
3 19 3 19.16 0.16
4 23 4 19.19 3.81
5 18 5 19.21 1.21
6 16 6 19.24 3.24
7 20 7 19.26 0.74
8 18 8 19.29 1.29
9 22 9 19.31 2.69
10 20 10 19.34 0.66
11 15 11 19.36 4.36
12 22 12 19.38 2.62
F13 = 19.41
MAD = 2.0629 N= 12
SUMMARY OUTPUT
(1) Forecast sales for week 13.
Regression Statistics (2) Compute MAD.
Multiple R 0.034920211
R Square 0.001219421
Adjusted R Square -0.098658637
Standard Error 2.648855143
Observations 12
ANOVA
df SS MS F Significance F
Regression 1 0.085664336 0.085664336 0.012209 0.914203113
Residual 10 70.16433566 7.016433566
Total 11 70.25
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% U
Lower 95.0%pper 95.0%
Intercept 19.09090909 1.630257644 11.71036318 3.67E-07 15.45846807 22.72335 15.45847 22.72335
t 0.024475524 0.221508395 0.110494794 0.914203 -0.469076022 0.518027 -0.46908 0.518027
Supplementary: Quadratic Regression Model
Quadratic
Sales (1000s (1000s of
of gallons) gallons)
Week Yt t t-square Ft | Yt - Ft |
1 17 1 1 19.18 2.18
2 21 2 4 19.17 1.83
3 19 3 9 19.17 0.17
4 23 4 16 19.17 3.83
5 18 5 25 19.18 1.18
6 16 6 36 19.19 3.19
7 20 7 49 19.22 0.78
8 18 8 64 19.25 1.25
9 22 9 81 19.29 2.71
10 20 10 100 19.34 0.66
11 15 11 121 19.39 4.39
12 22 12 144 19.45 2.55
F13 = 19.52
MAD = 2.0604 N= 12
SUMMARY OUTPUT
(1) Forecast sales for week 13.
Regression Statistics (2) Compute MAD.
Multiple R 0.038549428
R Square 0.001486058
Adjusted R Square -0.220405929
Standard Error 2.791765757
Observations 12
ANOVA
df SS MS F Significance F
Regression 2 0.104395604 0.052198 0.0066972 0.993330106
Residual 9 70.1456044 7.793956
Total 11 70.25
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Upper 95.0%
Lower 95.0%
Intercept 19.20454545 2.885370364 6.655834 9.311E-05 12.67737924 25.73171 12.67738 25.73171
t -0.024225774 1.020490186 -0.02374 0.9815785 -2.332736718 2.284285 -2.33274 2.284285
t-square 0.003746254 0.076417455 0.049024 0.9619712 -0.169122171 0.176615 -0.16912 0.176615
Multiplicative Time Series Model p.787 Table 18.7 HWK: p.795 #22
Question 1 Question 2 Question 3
(5)
Average of (4)
for same
(1) (3) (4) = (1)/(3) seasons (1)/(5) (2)
Seasonal
4-Quarter Centered Irregular Seasonal Deseasonalized
Moving Moving Value Index Sales
Year Quarter Sales ($1,000) Average Average (StIt) (St) (Yt/St = TtIt) t
1 1 4.8 0.932 5.149 1
2 4.1 0.838 4.894 2
3 6 5.350 5.475 1.096 1.093 5.488 3
5.600
4 6.5 5.738 1.133 1.143 5.685 4
5.875
2 1 5.8 5.975 0.971 0.932 6.222 5 5.350
6.075
2 5.2 6.300 6.188 0.840 0.838 6.207 6 5.600
3 6.8 6.350 6.325 1.075 1.093 6.219 7 5.875
4 7.4 6.450 6.400 1.156 1.143 6.472 8 6.075
3 1 6 6.625 6.538 0.918 0.932 6.436 9 6.300
2 5.6 6.725 6.675 0.839 0.838 6.684 10 6.350
3 7.5 6.800 6.763 1.109 1.093 6.860 11 6.450
4 7.8 6.875 6.838 1.141 1.143 6.822 12 6.625
4 1 6.3 7.000 6.938 0.908 0.932 6.758 13 6.725
2 5.9 7.150 7.075 0.834 0.838 7.043 14 6.800
3 8 1.093 7.317 15 6.875
4 8.4 1.143 7.347 16 7.000
7.150
Tt (Trend for deseasonalized sales)
SUMMARY OUTPUT
Regression Statistics Compare Figure 18.12 (p.790) with Figure 18.13 (p.792)
Multiple R 0.959578616
R Square 0.92079112 (1) Show the 4-quarter and centered moving average values.
Adjusted R Square0.915133342 (2) Compute seasonal index for the 4 quarters.
Standard Error 0.212671247 (3) Deseasonalize the sales and develop the trend equation for the deseasonalized data .
Observations 16 (4) The trend equation developed for the deseasonalized data was Tt = 5.0996 + 0.1471t.
Use trend adjusted for seasonal index to estimate sales for quarters 17, 18, 19, and 20.
ANOVA
df SS MS F Significance F
Regression 1 7.360933 7.360933 162.74786 4.24772E-09
Residual 14 0.633207 0.045229
Total 15 7.994139
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% U
Lower 95.0%pper 95.0%
Intercept 5.099610095 0.111526 45.72586 1.21E-16 4.860410953 5.338809237 4.860411 5.338809
X Variable 1 0.147138716 0.011534 12.75727 4.248E-09 0.12240133 0.171876102 0.122401 0.171876
QUESTION 4
Forecast sales for Quarters 17, 18, 19, and 20 (p.771)
Quarter 17: T17 = 7.600968 F17 = 7.601x0.932 = 7.086
Quarter 18: T18 = 7.748107 F18 = 7.748x0.838 = 6.491
Quarter 19: T19 = 7.895246 F19 = 7.895x1.093 = 8.632
Quarter 20: T20 = 8.042384 F20 = 8.042x1.143 = 9.195