FreshX Toothpaste is a product that has stable demand with no discernible trends or seasonality. The following table shows the actual demand for FreshX over the last $9$ months, in thousands of tubes:

Month

$1$

$2$

$3$

$4$

$5$

$6$

$7$

$8$

$9$

$10$

$11$

Demand

$60$

$150$

$60$

$70$

$70$

$90$

$100$

$50$

$70$

$90$

$70$

FreshX would like to determine a method for forecasting toothpaste demand in future months.

Use months $8-11$ as a hold$-$out sample. That is$,$ pretend that data from months $8-11$ does not exist. Initialize your forecast in Month $2$ using the Month $1$ data. Then iteratively update your forecast using the next data point.

Use simple exponential smoothing $($alpha$=0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4)$ to forecast the demand for the hold out sample. PleasFor alpha $=0\mathrm{.}1,$ the forecasted demand for Month $8$ is what?

For alpha $=0\mathrm{.}2,$ the forecasted demand for Month $9$ is what?

For alpha $=0\mathrm{.}3,$ the forecasted demand for Month $10$ is what?

For alpha $=0\mathrm{.}4,$ the forecasted demand for Month $11$ is what?

Calculate the RMSE, MAD, and TS for each forecast. The RMSE is the root mean squared error. Just take the sqrt of MSE. Remember, we only care about the metrics for the holdout sample as that is where we are testing our forecast.

Which months do we include in our error metric calculations? $($Select all that apply$)$