Question

In: Statistics and Probability

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

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)

Solutions

Expert Solution

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.


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