Chapter 17.3, Problem 9E

With the gasoline time series data from Table 17.1, show the exponential smoothing forecasts using α = .1.

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?
2. b. Are the results the same if you apply MAE as the measure of accuracy?
3. c. What are the results if MAPE is used?

To determine

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:

$\begin{array}{c}MSE=\text{Average of the sum of squared forecast errors}\\ \text{=}\frac{\text{Total of the sum of squared forecast errors}}{\text{Number of weeks/years/months}}\end{array}$

The formula for exponential smoothing is as follows:

$F_{t+1}=\alpha Y_t+\left(1-\alpha \right)F_t$

Here

 $\begin{array}{c}{F}_{t+1}=\text{forecast of the times series of period }t+1\\ Y_t=\text{Actual value of the time series in period }t\\ \alpha =\text{Smoothing constant }\left(0\le \alpha \le 1\right)\end{array}$

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	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 21\right)+\\ \left(0.9\times 17\right)\end{array}\right)\\ =17.4\end{array}$	1.6	1.6	2.56	8.42
4	23	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 19\right)+\\ \left(0.9\times 17.4\right)\end{array}\right)\\ =17.56\end{array}$	5.44	5.44	29.59	23.65
5	18	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 23\right)+\\ \left(0.9\times 17.56\right)\end{array}\right)\\ =18.1\end{array}$	–0.1	0.1	0.01	0.56
6	16	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 18\right)+\\ \left(0.9\times 18.1\right)\end{array}\right)\\ =18.09\end{array}$	–2.09	2.09	4.37	13.06
7	20	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 16\right)+\\ \left(0.9\times 18.09\right)\end{array}\right)\\ =17.88\end{array}$	2.12	2.12	4.49	10.60
8	18	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 20\right)+\\ \left(0.9\times 17.88\right)\end{array}\right)\\ =18.10\end{array}$	–0.10	0.10	0.01	0.56
9	22	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 18\right)+\\ \left(0.9\times 18.1\right)\end{array}\right)\\ =18.09\end{array}$	3.91	3.91	15.29	17.77
10	20	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 22\right)+\\ \left(0.9\times 18.09\right)\end{array}\right)\\ =18.48\end{array}$	1.52	1.52	2.31	7.60
11	15	$\begin{array}{l}\left(\begin{array}{l}\left(0.1\times 20\right)+\\ \left(0.9\times 18.48\right)\end{array}\right)\\ =18.63\end{array}$	–3.63	3.63	13.18	24.20
12	22	$\begin{array}{l}\left(0.1\times 15\right)+\left(0.9\times 18.63\right)\\ =18.27\end{array}$	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.

$\text{Percentage error}=\frac{\text{Sum of forecast errors}}{\text{Number of forecast errors}}\times 100$

The value of MSE is as follows:

$\begin{array}{c}MSE=\frac{\text{101}\text{.72}}{11}\\ =9.25\end{array}$

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	$\begin{array}{l}\left(0.2\times 21\right)+\left(0.8\times 17\right)\\ =17.8\end{array}$	1.20	1.20	1.44	6.32
4	23	$\begin{array}{l}\left(0.2\times 19\right)+\left(0.8\times 17.4\right)\\ =18.04\end{array}$	4.96	4.96	24.60	21.57
5	18	$\begin{array}{l}\left(0.2\times 23\right)+\left(0.8\times 18.04\right)\\ =19.03\end{array}$	–1.03	1.03	1.07	5.73
6	16	$\begin{array}{l}\left(0.2\times 18\right)+\left(0.8\times 19.03\right)\\ =18.83\end{array}$	–2.83	2.83	7.98	17.66
7	20	$\begin{array}{l}\left(0.2\times 16\right)+\left(0.8\times 18.83\right)\\ =18.26\end{array}$	1.74	1.74	3.03	8.70
8	18	$\begin{array}{l}\left(0.2\times 20\right)+\left(0.8\times 18.26\right)\\ =18.61\end{array}$	–0.61	0.61	0.37	3.38
9	22	$\begin{array}{l}\left(0.2\times 18\right)+\left(0.8\times 18.61\right)\\ =18.49\end{array}$	3.51	3.51	12.34	15.97
10	20	$\begin{array}{l}\left(0.2\times 22\right)+\left(0.8\times 18.49\right)\\ =19.19\end{array}$	0.81	0.81	0.66	4.05
11	15	$\begin{array}{l}\left(0.2\times 20\right)+\left(0.8\times 19.19\right)\\ =19.35\end{array}$	–4.35	4.35	18.94	29.01
12	22	$\begin{array}{l}\left(0.2\times 15\right)+\left(0.8\times 19.35\right)\\ =18.48\end{array}$	3.52	3.52	12.38	15.99
Total				28.56	98.80	147.43

The value of MSE is as follows:

$\begin{array}{c}MSE=\frac{\text{98}\text{.80}}{11}\\ =8.98\end{array}$

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.

To determine

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:

$\begin{array}{c}MAE=\text{Average of the absolute value of forecast errors}\\ \text{=}\frac{\text{Total of the absolute value of forecast errors}}{\text{Number of weeks/years/months}}\end{array}$

For smoothing constant 0.1, the value of MAE is as follows:

$\begin{array}{c}MAE=\frac{\text{28}\text{.24}}{11}\\ =2.57\end{array}$

Thus, the value of mean absolute error is 2.57.

For smoothing constant 0.2, the value of MAE is as follows:

$\begin{array}{c}MAE=\frac{\text{28}\text{.56}}{11}\\ =2.60\end{array}$

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.

To determine

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:

$\begin{array}{c}MAPE=\text{Average of the absolute value of percentage forecast errors}\\ \text{=}\frac{\text{Total of the absolute value of percentage forecast errors}}{\text{Number of weeks/years/months}}\end{array}$

For smoothing constant 0.1, the value of MAPE is as follows:

$\begin{array}{c}MAPE=\frac{\text{142}\text{.42}}{11}\\ =12.95\end{array}$

Thus, the value of mean absolute percentage error is 12.95.

For smoothing constant 0.2, the value of MAE is as follows:

$\begin{array}{c}MAE=\frac{\text{147}\text{.43}}{11}\\ =13.40\end{array}$

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.