Question

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

Week	Sales (1000s of gallons)
1	17
2	21
3	19
4	24
5	18
6	15
7	21
8	19
9	22
10	19
11	15
12	23

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
					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 14.39 .
Prefer the unweighted moving average here; it has a - Select your answer -greatersmallerItem 42 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 43

Answer 1

Let Y(t) be the actual sales for period t.

the forecast for period t using the weighted moving average F(t) is calculated as

Weighted moving average is

Week	Sales (1000s of gallons)	Weighted Moving Average Forecast
1	17
2	21
3	19
4	24	1/2*19+1/3*21+1/6*17=19.3333
5	18	1/2*24+1/3*19+1/6*21=21.8333
6	15	1/2*18+1/3*24+1/6*19=20.1667
7	21	1/2*15+1/3*18+1/6*24=17.5
8	19	1/2*21+1/3*15+1/6*18=18.5
9	22	1/2*19+1/3*21+1/6*15=19
10	19	1/2*22+1/3*19+1/6*21=20.8333
11	15	1/2*19+1/3*22+1/6*19=20
12	23	1/2*15+1/3*19+1/6*22=17.5

Next we calculate the forecast error = actual sales - forecasted sales, and calculate the square of the error

If you are using a spreadsheet, prepare the following

get this

Week	Sales (1000s of gallons)	Weighted Moving Average Forecast	Forecast Error	(Error)2
1	17
2	21
3	19
4	24	19.33	4.67	21.78
5	18	21.83	-3.83	14.69
6	15	20.17	-5.17	26.69
7	21	17.50	3.50	12.25
8	19	18.50	0.50	0.25
9	22	19.00	3.00	9.00
10	19	20.83	-1.83	3.36
11	15	20.00	-5.00	25.00
12	23	17.50	5.50	30.25
			Total	143.28

b) MSE (mean square error) is

ans: MSE = 15.92

Do you prefer this weighted moving average to the unweighted moving average? Remember that the MSE for the unweighted moving average is 14.39 .

ans: Prefer the unweighted moving average here; it has a smaller 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?

If we have equal weights for all the observations in weighted moving average, then the forecasted value from the weighted moving average would be same as the unweighted moving average.

For example in the equation for weighted moving average above if the all the weights are the same and is 1/3 then

which is the 3-week unweighted moving average.

That is assigning equal weights would make the MSE as small for a weighted moving average as for an unweighted moving average

That means, we can 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

ans: Yes