With the gasoline time series data from Table 17.1, show the exponential smoothing forecasts using α = .1.
- a. Applying the MSE measure of forecast accuracy, would you prefer a smoothing constant of α = . 1 or α = .2 for the gasoline sales time series?
- b. Are the results the same if you apply MAE as the measure of accuracy?
- c. What are the results if MAPE is used?
a.
Explain the smoothing constant of 0.1 or 0.2 that is preferable for the gasoline sales using the mean squared error.
Answer to Problem 9E
The smoothing constant 0.2 provides the more accurate forecast value when compared to the smoothing constant 0.1.
Explanation of Solution
Calculation:
The given data represents the gasoline sales.
The formula for mean squared error (MSE) is as follows:
The formula for exponential smoothing is as follows:
Here
Calculate the absolute value of forecast error for 0.1 smoothing constant as follows:
Week |
Time series Value | Forecast |
Forecast error | Absolute forecast error | Squared forecast error | Absolute percentage error |
1 | 17 | |||||
2 | 21 | 17 | 4 | 4 | 16 | 19.05 |
3 | 19 | 1.6 | 1.6 | 2.56 | 8.42 | |
4 | 23 | 5.44 | 5.44 | 29.59 | 23.65 | |
5 | 18 | –0.1 | 0.1 | 0.01 | 0.56 | |
6 | 16 | –2.09 | 2.09 | 4.37 | 13.06 | |
7 | 20 | 2.12 | 2.12 | 4.49 | 10.60 | |
8 | 18 | –0.10 | 0.10 | 0.01 | 0.56 | |
9 | 22 | 3.91 | 3.91 | 15.29 | 17.77 | |
10 | 20 | 1.52 | 1.52 | 2.31 | 7.60 | |
11 | 15 | –3.63 | 3.63 | 13.18 | 24.20 | |
12 | 22 | 3.73 | 3.73 | 13.91 | 16.95 | |
Total | 28.24 | 101.72 | 142.42 |
Here, the forecast value is the previous week time series value.
Forecast error = Time series value – Forecast value.
The value of MSE is as follows:
Thus, the value of mean squared error is 9.25.
Calculate the absolute value of forecast error for 0.2 smoothing constant as follows:
Week |
Time series Value | Forecast |
Forecast error | Absolute forecast error | Squared forecast error | Absolute percentage error |
1 | 17 | |||||
2 | 21 | 17 | 4.00 | 4.00 | 16.00 | 19.05 |
3 | 19 | 1.20 | 1.20 | 1.44 | 6.32 | |
4 | 23 | 4.96 | 4.96 | 24.60 | 21.57 | |
5 | 18 | –1.03 | 1.03 | 1.07 | 5.73 | |
6 | 16 | –2.83 | 2.83 | 7.98 | 17.66 | |
7 | 20 | 1.74 | 1.74 | 3.03 | 8.70 | |
8 | 18 | –0.61 | 0.61 | 0.37 | 3.38 | |
9 | 22 | 3.51 | 3.51 | 12.34 | 15.97 | |
10 | 20 | 0.81 | 0.81 | 0.66 | 4.05 | |
11 | 15 | –4.35 | 4.35 | 18.94 | 29.01 | |
12 | 22 | 3.52 | 3.52 | 12.38 | 15.99 | |
Total | 28.56 | 98.80 | 147.43 |
The value of MSE is as follows:
Thus, the value of mean squared error is 8.98.
By comparing the MSE values of smoothing constants 0.1 and 0.2, the MSE values of smoothing constant 0.2 has less error than the constant 0.1.
Thus, the smoothing constant 0.2 provides the more accurate forecast value when compared to the smoothing constant 0.1.
b.
Explain the smoothing constant of 0.1 or 0.2 that is preferable for the gasoline sales using the MAE.
Answer to Problem 9E
The smoothing constant 0.1 provides the more accurate forecast value when compared to the smoothing constant 0.2.
Explanation of Solution
Calculation:
The formula for mean absolute error (MAE) is as follows:
For smoothing constant 0.1, the value of MAE is as follows:
Thus, the value of mean absolute error is 2.57.
For smoothing constant 0.2, the value of MAE is as follows:
Thus, the value of mean absolute error is 2.60.
By comparing the MAE values of smoothing constants 0.1 and 0.2, the MAE values of smoothing constant 0.1 has less error than the constant 0.2.
Thus, the smoothing constant 0.1 provides the more accurate forecast value when compared to the smoothing constant 0.2. However, in these two cases, the MAE values are closer.
c.
Explain the smoothing constant of 0.1 or 0.2 that is preferable for the gasoline sales y using the MAPE.
Answer to Problem 9E
The smoothing constant 0.1 provides the more accurate forecast value when compared to the smoothing constant 0.2.
Explanation of Solution
Calculation:
The formula for mean absolute percentage error (MAPE) is as follows:
For smoothing constant 0.1, the value of MAPE is as follows:
Thus, the value of mean absolute percentage error is 12.95.
For smoothing constant 0.2, the value of MAE is as follows:
Thus, the value of mean absolute error is 13.40.
By comparing the MAPE values of smoothing constants 0.1 and 0.2, the MAPE value of smoothing constant 0.1 is less than the constant 0.2.
Thus, the smoothing constant 0.1 provides the more accurate forecast value when compared to the smoothing constant 0.2.
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Chapter 17 Solutions
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