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	21
3	19
4	24
5	18
6	16
7	19
8	17
9	23
10	20
11	15
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 13.69.
Prefer the unweighted moving average here; it has a (greater/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?
(Yes/No)

Answer 1

Answer a)

Now, the following table shows the calculations of the corresponding error metrics:

Total = ΣError² = 23.3611+14.6944+17.3611+1+0.6944+30.25+0.1111+30.25+16 = 133.7221

Answer b)

MSE = ΣError²/n = 133.7221/9

MSE = 14.86

In this case, MSE for weighted moving average (14.86) is greater than MSE for unweighted moving average (13.69). Therefore, we prefer unweighted moving average.

Prefer the unweighted moving average here; it has a smaller MSE.

Answer c) Yes

Explanation

You could always find a weighted moving average at least as good as the unweighted one; actually the unweighted moving average is a special case of the weighted ones where the weights are equal.