Question

Problem 15-05 (Algorithmic)

Consider the following time series data.

Week	1	2	3	4	5	6
Value	16	13	18	11	15	14

1. Develop a three-week moving average for this time series. Compute MSE and a forecast for week 7. Round your answers to two decimal places.

   Week	Time Series Value	Forecast
   1	16
   2	13
   3	18
   4	11
   5	15
   6	14

   
   MSE: _______
   
   The forecast for week 7: ______
2. Use  = 0.2 to compute the exponential smoothing values for the time series. Compute MSE and a forecast for week 7. Round your answers to two decimal places.

   Week	Time Series Value	Forecast
   1	16
   2	13
   3	18
   4	11
   5	15
   6	14

   
   MSE: ___
   
   The forecast for week 7: _____
3. Use trial and error to find a value of the exponential smoothing coefficient that results in a smaller MSE than what you calculated for  = 0.2. Find a value of  for the smallest MSE. Round your answer to three decimal places.
   
   = ______

Answer 1

a).

Formula		(An-1 + An-2 + An-3)/3	A-F	(A-F)^2
Week (n)	Time Series Value (A)	Forecast (F)	Error (E)	Squared Forecast Error
1	16
2	13
3	18
4	11	15.67	(4.67)	21.78
5	15	14.00	1.00	1.00
6	14	14.67	(0.67)	0.44
7		13.33

Forecast for week 7 = 13.33

MSE = Sum of squared forecast error/number of errors =  (21.78 + 1.00 + 0.44)/3 = 7.74

b). For forecast at n = 2, F₂ is assumed to be the same as the actual value for n = 1, A₁. For the remaining forecasts, using alpha = 0.2, the forecast F_n is calculated as F_n-1 + alpha*Error_n-1

Formula		F2 = A1; Fn = Fn-1 + (a*En-1)	E = A-F	(A-F)^2
Week (n)	Time Series Value (A)	Forecast (F)	Forecast Error (E)	Sqaured Forecast Error
1	16
2	13	16.00	(3.00)	9.00
3	18	15.40	2.60	6.76
4	11	15.92	(4.92)	24.21
5	15	14.94	0.06	0.00
6	14	14.95	(0.95)	0.90
7		14.76

Forecast for week 7 = 14.76

MSE = Sum of squared forecast error/number of errors =  (9.00 + 6.76 + 21.21 + 0.00 + 0.90)/5 = 8.17

c). Using the table in part (b) and excel solver, the value of alpha at which MSE is minimized is 0.142 and MSE = 8.108

	alpha	0.142
Formula		F2 = A1; Fn = Fn-1 + (a*En-1)	E = A-F	(A-F)^2
Week (n)	Time Series Value (A)	Forecast (F)	Forecast Error (E)	Sqaured Forecast Error
1	16
2	13	16.00	(3.00)	9.00
3	18	15.57	2.43	5.88
4	11	15.92	(4.92)	24.19
5	15	15.22	(0.22)	0.05
6	14	15.19	(1.19)	1.42
7		15.02
			MSE	8.108