Question

Suppose these data show the number of gallons of gasoline sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks.

Week	Sales (1,000s of gallons)
1	17
2	21
3	19
4	23
5	18
6	16
7	21
8	19
9	23
10	21
11	16
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  for third most recent observation, compute a three-week weighted moving average for the time series. (Round your answers to two decimal places.)

Week	Time Series Value	Weighted Moving Average Forecast
1	17
2	21
3	19
4	23	?
5	18	?
6	16	?
7	21	?
8	19	?
9	23	?
10	21	?
11	16	?
12	23	?

(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? The MSE for the unweighted moving average is 10.91.

The weighted moving average is preferred because it has a smaller MSE compared to the unweighted moving average.

The unweighted moving average is preferred because it has a smaller MSE compared to the weighted moving average.    

The unweighted moving average is preferred because it has a larger MSE compared to the weighted moving average.

The weighted moving average is preferred because it has a larger MSE compared to the unweighted moving average.

(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? Why or why not?

Yes, because an unweighted moving average is just a weighted moving average where the weights are equal.

No, sometimes you need to let the weights sum to a number higher/lower than 1 in order to get a smaller MSE than the one for an unweighted moving average.    

No, an unweighted moving average always has a smaller MSE than a weighted moving average.

Yes, a weighted moving average always has a smaller MSE than an unweighted moving average.

Answer 1

(a) Lets consider week no. 9,10,11,12 to be the most recent observations , week no. 5,6,7,8  the second most recent observation and week no. 1,2,3,4 the third most recent observations. Let thier weight be 1/2 i.e. 0.5 , 1/3 i.e. 0.3333 and 1/6 i.e. 0.166667 respectively as given.

Weights   column shows weights alloted to the observations , sales*weights column gives product of sales and weights for each observation and formula for 3 week weighted moving average is

In this case, xi  are the sales for i th   time period/ observation, wi is the weight for ith   observation,n = 3 as we have to calculate 3 week MA.

week	Sales (1,000s of gallons)	weights	sales*weights	3 week weighted MA
1	17	0.166667	2.833333333
2	21	0.166667	3.5
3	19	0.166667	3.166666667	19
4	23	0.166667	3.833333333	21
5	18	0.333333	6	19.5
6	16	0.333333	5.333333333	18.2
7	21	0.333333	7	18.33
8	19	0.333333	6.333333333	18.67
9	23	0.5	11.5	21.29
10	21	0.5	10.5	21.25
11	16	0.5	8	20
12	23	0.5	11.5	20

(b) Fromula for MSE is

Where xi  is the actual sales value of ith  observation ,

  xi cap(^) is the moving average  value of ith  observation and m is the no. of observations (in this case 10)

The value for MSE for 3 week weighted moving average is 4.63144

Yes I prefer this weighted moving average to the unweighted moving average.

It is preferred because it has a smaller MSE compared to the unweighted moving average.

MSE is the measure of how close our predictions are to the actual value.Hence, smaller the MSE better is the prediction. Unweighted moving average is greater comparatively which means it has failed to reach the actual value.