Absolute Best Cereal $($ABC$)$ is a small$-$size cereal$-$production company in a mysterious countryside. Through some efforts aided by AI$,$ its annual cereal production data $(C_t,$ in tons$)$ in the pre$-$pandemic period from $2010$ to $)=(110$ was obtained as below. You are asked to answer the following questions with clear steps $($including necessary formulas, calculations, explanations or discussions$)$

$\backslash$table$[[$Year$,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019],[t,1,2,3,4,5,6,7,8,9,10],[C_t,107\mathrm{.}02,114\mathrm{.}86,116\mathrm{.}07,84\mathrm{.}04,117\mathrm{.}70,102\mathrm{.}05,86\mathrm{.}85,147\mathrm{.}87,162\mathrm{.}09,182\mathrm{.}94]]$

Q$6.(5$ marks$)$ Forecast ABC's cereal production $(hat(C{)}_t)$ for years $2011-2020$ using the simple

exponential smoothing $($SES$)$ method with smoothing parameter $=0\mathrm{.}85[$assuming that the

forecast for the starting year of $)=(1$ is $111\mathrm{.}111].$

Hint: SES forecast hat$(C{)}_{t+1}=C_t+(1-)hat(C{)}_t,t=1,2,$dotsdots

Q$7.(15$ marks$)$ A simple and useful way to compare the performances of different forecasting

methods is by way of Mean Squared Error $($MSE$).$ Define forecasting$-$error $(e_t)=$ actual $-$

forecast $=C_t-$hat$(C{)}_t,$ and

MSE$=\frac{{\displaystyle \underset{t=1}{\overset{h}{}}}(C_{s+t}-\text{h}\text{a}\text{t}(C{)}_{s+t}{)}^2}{h}=\frac{{\displaystyle \underset{t=1}{\overset{h}{}}}(e_{s+t}{)}^2}{h}$

where $h$ is the number of forecasts made, $s$ is a given time value or point, and the forecasts to be

evaluated are for times $s+1$ to $s+h.$ Calculate the MSEs for your forecasts in the above Q$5$ and

Q$6,$ and compare the forecasting accuracies of the SMA and SES methods based on their MSEs

$[$you should give the two methods' forecasting errors for $2019].$ Note that $s=3$ and $h=7($for

times $4-10$ or years $2013-2019)$ in both cases. Based on the equation of MSE, would a more

accurate method$/$model give a larger or smaller MSE? Which one of the two methods is more

accurate $[$and thus will be used for real forecasting$]$ as a result of your calculations, and what

will be your forecasting for ABC's cereal production in $2021,$ and why?

Q$8.(6$ marks$)$ Rewrite the equation of SES forecast in terms of forecasting error $(e_t=C_t-$hat$(C{)}_t)$

defined in the above Q$6.$ An unexpected decline in ABC's cereal production occurred in

$2020-2022$ due to the pandemic in the three years, and it did not rebound until $2023.$ Given this

development, do you think the SES model with a larger or smaller $$ will make the resulting

forecasts more capable in capturing the rebound in $2023$ and afterwards? Briefly explain.

Q$9.(20$ marks$)$ Based on your answers on the stability $($or steadiness$)$ of ABC's cereal production

time series $\{C_t\}$ in Q$1$ and Q$2,$ do you think the SES model with a larger or smaller $$ will make

the in$-$sample forecasts $($for $2013-2019)$ more accurate? Re$-$do the in$-$sample forecasting based on

the SES model with $=0\mathrm{.}5$ and $=0\mathrm{.}2,$ and calculate the resulting MSEs $[$for $2013-2019$

forecasts$][$you should give the forecasts and forecasting errors for $2019].$ By comparing the

MSEs of the SES forecasts with $=0\mathrm{.}85,0\mathrm{.}5$ and $=0\mathrm{.}2,$ what conclusion can you make?