Question

The accompanying data file contains 20 observations for t and y_t.

The data are plotted below.

t	1	2	3	4	5	6	7	8	9	10
yt	10.8	14.1	10.3	10.9	11.3	13.5	10.7	9.2	8.8	12
t	11	12	13	14	15	16	17	18	19	20
yt	9.8	11	15.1	12.5	12.9	12.3	9	14.9	10.1	11.9

b-1. Use the exponential smoothing method to make forecasts with α = 0.2. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

t	yt	yˆty^t
1	10.8	?
2	14.1	?
3	10.3	?
19	10.1	?
20	11.9	?
21

b-2. Compute the resulting MSE and MAD. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

c-1. Use the exponential smoothing method to make forecasts with α = 0.4. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

t	yt	yˆty^t
1	10.8	?
2	14.1	?
3	10.3	?
19	10.1	?
20	11.9	?
21

c-2. Compute the resulting MSE and MAD. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

d. Use the appropriate value of α to make a forecast for period 21. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

yˆty^t    

(Y-Hat sub t)

Answer 1

Here as we use exponential method to make forcasts with α = 0.2

y^_t+1 = α y_t + (1 - α ) y^_t

t	yt	Y^t
1	10.8	10.8
2	14.1	10.8
3	10.3	11.46
4	10.9	11.2280
5	11.3	11.1624
6	13.5	11.1899
7	10.7	11.6519
8	9.2	11.4615
9	8.8	11.0092
10	12.0	10.5674
11	9.8	10.8539
12	11.0	10.6431
13	15.1	10.7145
14	12.5	11.5916
15	12.9	11.7733
16	12.3	11.9986
17	9.0	12.0589
18	14.9	11.4471
19	10.1	12.1377
20	11.9	11.7302

for MSE and MAD

t	yt	Y^t	Abs. Error	Squared error
1	10.8	10.80	0.0000	0.0000
2	14.1	10.80	3.3000	10.8900
3	10.3	11.46	1.1600	1.3456
4	10.9	11.23	0.3280	0.1076
5	11.3	11.16	0.1376	0.0189
6	13.5	11.19	2.3101	5.3365
7	10.7	11.65	0.9519	0.9062
8	9.2	11.46	2.2615	5.1146
9	8.8	11.01	2.2092	4.8807
10	12.0	10.57	1.4326	2.0524
11	9.8	10.85	1.0539	1.1107
12	11.0	10.64	0.3569	0.1274
13	15.1	10.71	4.3855	19.2326
14	12.5	11.59	0.9084	0.8252
15	12.9	11.77	1.1267	1.2695
16	12.3	12.00	0.3014	0.0908
17	9.0	12.06	3.0589	9.3569
18	14.9	11.45	3.4529	11.9224
19	10.1	12.14	2.0377	4.1522
20	11.9	11.73	0.1698	0.0288
		Sum	30.9431	78.7689

MSE = 78.7689/19  =  4.15

MAD = 30.9431/19 = 1.63

(c-1) Here forcasts with α = 0.4

t	yt	Y^t	Abs. Error	Squared error
1	10.8	10.80	0.0000	0.0000
2	14.1	10.80	3.3000	10.8900
3	10.3	12.12	1.8200	3.3124
4	10.9	11.39	0.4920	0.2421
5	11.3	11.20	0.1048	0.0110
6	13.5	11.24	2.2629	5.1206
7	10.7	12.14	1.4423	2.0801
8	9.2	11.57	2.3654	5.5949
9	8.8	10.62	1.8192	3.3096
10	12.0	9.89	2.1085	4.4456
11	9.8	10.73	0.9349	0.8741
12	11.0	10.36	0.6390	0.4084
13	15.1	10.62	4.4834	20.1011
14	12.5	12.41	0.0901	0.0081
15	12.9	12.45	0.4540	0.2061
16	12.3	12.63	0.3276	0.1073
17	9.0	12.50	3.4965	12.2258
18	14.9	11.10	3.8021	14.4557
19	10.1	12.62	2.5188	6.3441
20	11.9	11.61	0.2887	0.0834
		Sum	32.7502	89.8206

for MSE and MAD

MAD = 32.7502/19 = 1.7237 (1.72)

MSE = 89.8206/19 = 4.7274 (4.73)

(d) Here appropriate value of alpha is 0.2

so here

y^ (21) = 0.2 * 11.9 + 0.8 * 11.7302 = 11.76