Many forecasting models use parameters that are estimated using nonlinear optimization. A good example $is$ the Bass model introduced $in$ this chapter. Another example $is$ the

exponential smoothing forecasting model. The exponential smoothing model $is$ common $in$ practice. For instance, the basic exponential smoothing model for forecasting sales $is$

hat$(y{)}_{t+1}=y_t+(1-)hat(y{)}_t$

where

hat$(y{)}_{t+1}=$ forecast $of$ sales for period $t+1$

$y_t=$ actual value $of$ sales for period $t$

hat$(y{)}_t=$ forecast $of$ sales for period $t$

$=$ smoothing constant $0a1$

This model $is$ used recursively; the forecast for time period $t+1is$ based $on$ the forecast for period $t,$hat$(y{)}_{t^{\text{'}}}$ the observed value $of$ sales $in$ period $t,y_{t^{\text{'}}}$ and the smoothing parameter $.$ The

use $of$ this model $to$ forecast sales for $12$ months $is$ illustrated $in$ the table with the smoothing constant $=0\mathrm{.}3.(Note$: $We$ are displaying rounded intermediate values for practical

purposes. However, the calculations are made using the unrounded values.$)$

Exponential Smoothing Model for $=0\mathrm{.}3$ The forecast errors, $y_t-$hat$(y{)}_{t^{\text{'}}}$ are calculated $in$ the fourth column. The value $ofis$ often chosen $by$ minimizing the sum $of$ squared forecast errors. The last column $of$ the table shows the square $of$ the forecast error and the sum $of$ squared forecast errors. $In$ using exponential smoothing models, one tries $to$ choose the value $ofhat(y{)}_1=y_1=18$ hat$(y{)}_{t+1}is$ used

starting $in$ period $2.$ Make sure $to$ have a single cell corresponding $toin$ your spreadsheet model. After confirming the values $in$ the table below with $=0\mathrm{.}3,$ try different values $ofto$

see $if$ you can get a smaller sum $of$ squared forecast errors.

Use Excel Solver $to$ find the value $of$ a that minimizes the sum $of$ squared forecast errors. $(Round$ your answers $to$ three decimal places.$)$

$=$

$x,$ resulting $in$ a sum $of$ squared forecast errors equal $to$