Refer again to the gasoline sales time series data in Table $6\mathrm{.}1.$

Using a weight of $1/2$ for the most recent observation, $1/3$ for the second most recent, and $1/6$ for third most recent, compute a three$-$week weighted moving average for the time series.

SHOW ANSWER

Compute the MSE for the weighted moving average in part $($a$).$ Do you prefer this weighted moving average to the unweighted moving average? Remember that the MSE for the unweighted moving average is $10\mathrm{.}22.$

SHOW ANSWER

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 smaller for a weighted moving average than for an unweighted moving average? Why or why not?