Question

Consider the following gasoline sales time series. If needed, round your answers to two-decimal digits. Week Sales (1,000s of gallons) 1 18 2 22 3 15 4 24 5 18 6 15 7 21 8 19 9 21 10 20 11 16 12 22 (a) Show the exponential smoothing forecasts using α = 0.1, and α = 0.2. Exponential Smoothing Week α = 0.1 α = 0.2 13 (b) Applying the MSE measure of forecast accuracy, would you prefer a smoothing constant of α = 0.1 smoothing constant provides the more accurate forecast, with an overall MSE of (c) Are the results the same if you apply MAE as the measure of accuracy? An a= 0.1 smoothing constant provides the more accurate forecast, with an overall MAE of (d) What are the results if MAPE is used? An a=0.1 smoothing constant provides the more accurate forecast, with an overall MAPE of =

Answer 1

Let shows the actual value for year t and shows the forecasted value of year t. So exponential smoothing forecast formula is

(a)

Following table shows the forecasted values for a =0.1 and a =0.2:

Week	Sales (1,000s of gallons), Yt	Ft, alpha=0.1	Ft, alpha=0.2
1	18	18	18
2	22	18	18
3	15	18.4	18.8
4	24	18.06	18.04
5	18	18.65	19.23
6	15	18.59	18.98
7	21	18.23	18.18
8	19	18.51	18.74
9	21	18.56	18.79
10	20	18.8	19.23
11	16	18.92	19.38
12	22	18.63	18.7
13		18.97	224.07

(b)

Following table shows the calculations for MSE:

Week	Sales (1,000s of gallons), Yt	Ft, alpha=0.1	Ft, alpha=0.2	e^2=(Yt-Ft)^2, alpha=0.1	e^2=(Yt-Ft)^2, alpha=0.2
1	18
2	22	18	18	16	16
3	15	18.4	18.8	11.56	14.44
4	24	18.06	18.04	35.2836	35.5216
5	18	18.65	19.23	0.4225	1.5129
6	15	18.59	18.98	12.8881	15.8404
7	21	18.23	18.18	7.6729	7.9524
8	19	18.51	18.74	0.2401	0.0676
9	21	18.56	18.79	5.9536	4.8841
10	20	18.8	19.23	1.44	0.5929
11	16	18.92	19.38	8.5264	11.4244
12	22	18.63	18.7	11.3569	10.89
Total				111.3441	119.1263

So MSE for a =0.1:

So MSE for a =0.2:

​ Applying the MSE measure of forecast accuracy, you should prefer a smoothing constant of α = 0.1 smoothing constant provides the more accurate forecast, with an overall MSE of 10.12.

(c)

Week	Sales (1,000s of gallons), Yt	Ft, alpha=0.1	Ft, alpha=0.2	|e|=|Yt-Ft|, alpha=0.1	|e|=|Yt-Ft|, alpha=0.2
1	18
2	22	18	18	4	4
3	15	18.4	18.8	3.4	3.8
4	24	18.06	18.04	5.94	5.96
5	18	18.65	19.23	0.65	1.23
6	15	18.59	18.98	3.59	3.98
7	21	18.23	18.18	2.77	2.82
8	19	18.51	18.74	0.49	0.26
9	21	18.56	18.79	2.44	2.21
10	20	18.8	19.23	1.2	0.77
11	16	18.92	19.38	2.92	3.38
12	22	18.63	18.7	3.37	3.3
Total				30.77	31.71

So MAD for a =0.1:

So MSE for a =0.2:

​ Applying the MAD measure of forecast accuracy, you should prefer a smoothing constant of α = 0.1 smoothing constant provides the more accurate forecast, with an overall MAD of 2.80.

(d)

Week	Sales (1,000s of gallons), Yt	Ft, alpha=0.1	Ft, alpha=0.2	|Yt-Ft|/Yt, alpha=0.1	|Yt-Ft|/Yt, alpha=0.2
1	18
2	22	18	18	0.181818182	0.181818182
3	15	18.4	18.8	0.226666667	0.253333333
4	24	18.06	18.04	0.2475	0.248333333
5	18	18.65	19.23	0.036111111	0.068333333
6	15	18.59	18.98	0.239333333	0.265333333
7	21	18.23	18.18	0.131904762	0.134285714
8	19	18.51	18.74	0.025789474	0.013684211
9	21	18.56	18.79	0.116190476	0.105238095
10	20	18.8	19.23	0.06	0.0385
11	16	18.92	19.38	0.1825	0.21125
12	22	18.63	18.7	0.153181818	0.15
Total				1.600995823	1.670109535

So MAD for a =0.1:

So MSE for a =0.2:

​ Applying the MPAE measure of forecast accuracy, you should prefer a smoothing constant of α = 0.1 smoothing constant provides the more accurate forecast, with an overall MPAE of 14.55%.