Question

Consider the following gasoline sales time series data. Click on the datafile logo to reference the data.

Week	Sales (1000s of gallons)
1	16
2	22
3	18
4	23
5	18
6	16
7	20
8	17
9	22
10	20
11	16
12	22

a. Using a weight of 1/2 for the most recent observation, 1/3 for the second most recent observation, and 1/6 third the most recent observation, compute a three-week weighted moving average for the time series (to 2 decimals). Enter negative values as negative numbers.

Week	Time-Series Value	Weighted Moving Average Forecast	Forecast Error			(Error)2
1
2
3
4
5
6
7
8
9
10
11
12
					Total

b. Compute the MSE for the weighted moving average in part (a).
MSE =

Do you prefer this weighted moving average to the unweighted moving average? Remember that the MSE for the unweighted moving average is 9.19 .
Prefer the unweighted moving average here; it has a - Select your answer -(greatersmallerItem) MSE.

c. Suppose you are allowed to choose any weights as long as they sum to 1. Could you always find a set of weights that would make the MSE at least as small for a weighted moving average than for an unweighted moving average?
- Select your answer -(YesNoItem)

Answer 1

a)Using a weight of 1/2 for the most recent observation, 1/3 for the second most recent observation, and 1/6 third the most recent observation, we have calculate weighted moving average.

Weighted moving average of cell 4 is computedby = (18*1/2) + (22*1/3) + (16*1/6) = 19 and so on.

Forecast error of cell 4 is computed by, 23-19 = 4

Error square is nothing but ( Forecast error)2

Week	Sales	Weighted Moving Average Fore cast	Forecast error	(error)^2
1	16
2	22
3	18
4	23	19	4	16
5	18	21.17	-3.17	10.05
6	16	19.67	-3.67	13.47
7	20	17.83	2.17	4.71
8	17	18.33	-1.33	1.77
9	22	17.83	4.17	17.38
10	20	20	0	0
11	16	20.17	-4.17	17.38
12	22	18.33	3.67	13.47
				94.23

b)  The MSE for the weighted moving average is

  MSE = 94.23/9 = 10.47

   The MSE for the unweighted moving average is 9.19 .

   We prefer the unweighted moving average here as it has a smaller MSE.

c) Suppose we are allowed to choose any weights as long as they sum to 1.

  No, we couldn't find a set of weights that would make the MSE at least as small for a weighted moving average than for an unweighted moving average. Because, we put equal weights to the unweighted moving average. So, the error eventually reduced.

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