The following table reports the percentage of stocks in a portfolio for the nine previous quarter

Quarter Stock $\%$

$1$st$$Year $129\mathrm{.}8$

$2$nd$$Year $131\mathrm{.}0$

$3$rd$$Year $129\mathrm{.}9$

$4$th$$Year $130\mathrm{.}1$

$1$st$$Year $232\mathrm{.}2$

$2$nd$$Year $231\mathrm{.}5$

$3$rd$$Year $232\mathrm{.}0$

$4$th$$Year $231\mathrm{.}9$

$1$st$$Year $330\mathrm{.}0$

Use trial and error to find a value of the exponential smoothing coefficient alpha that results in a relatively small MSE. Consider smoothing constants of alpha $=0\mathrm{.}3,0\mathrm{.}6,$ and $0\mathrm{.}8.($Round your answers to two decimal places.$)$

Period Time Series

Value $($in $\%)$ alpha $=0\mathrm{.}3$

Forecasts $($in $\%)$ alpha$=0\mathrm{.}6$

Forecasts $($in $\%)$ alpha$=0\mathrm{.}8$

Forecasts $($in $\%)$

$129\mathrm{.}8$

$231\mathrm{.}0$

$329\mathrm{.}9$

$430\mathrm{.}1$

$532\mathrm{.}2$

$631\mathrm{.}5$

$732\mathrm{.}0$

$831\mathrm{.}9$

$930\mathrm{.}0$

What value of the smoothing constant provides the most accurate forecasts?

A smoothing constant of alpha $=0\mathrm{.}6$ provides the best forecast because it has a smaller MSE than alpha$=0\mathrm{.}3$ or $0\mathrm{.}8.$

A smoothing constant of alpha $=0\mathrm{.}3$ provides the best forecast because its MSE is the median of the three MSE values obtained using smoothing constants of alpha $=0\mathrm{.}3,0\mathrm{.}6,$ and $0\mathrm{.}8.$

A smoothing constant of alpha $=0\mathrm{.}8$ provides the best forecast because it has a larger MSE than alpha$=0\mathrm{.}3$ or $0\mathrm{.}6.$

All three values of the smoothing constant provide the same MSE, so none provide a more accurate forecast than the others.

$($c$)$

Using the exponential smoothing model you developed in part $($b$),$ what is the forecast of the percentage of stocks in a typical portfolio for the second quarter of Year $3?$