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 $ex-$

ponential smoothing forecasting model. The exponential smoothing model $is$ common $in$

practice and $is$ described $in$ further detail $in$ Chapter $15.$ For instance, the basic exponential

smoothing model for forecasting sales $is$

$F_{t+1}=Y_t+(1-)F_t$

where

$F_{t+1}=$ forecast $of$ sales for period $t+1$

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

$F_t=$ forecast $of$ sales for period $t$

$=$ smoothing constant $01$

This model $is$ used recursively; the forecast for time period $t+1is$ based $on$ the

forecast for period $t,F_t,$ the observed value $of$ sales $in$ period $t,Y_t,$ and the smoothing

parameter $.$ The use $of$ this model $to$ forecast sales for $12$ months $is$ illustrated $in$ Table $8\mathrm{.}9$

with the smoothing constant $=0\mathrm{.}3.$ The forecast errors, $Y_t-F_t,$ are calculated $in$ the

fourth column. The value $ofis$ often chosen $by$ minimizing the sum $of$ squared fore$-$

cast errors, commonly referred $toas$ the mean squared error $(MSE).$ The last column $of$

Table $8\mathrm{.}9$ shows the square $of$ the forecast error and the sum $of$ squared forecast errors.

TABLE $8\mathrm{.}9$ EXPONENTIAL SMOOTHING MODEL FOR $=0\mathrm{.}3$

Week Observed Value

$(t)$

Forecast

Forecast Error

$(Y_t-F_t)$

Squared Forecast Error

$\begin{array}{cc}(Y_t-F_t{)}^2& 0\mathrm{.}00\\ 16\mathrm{.}00& 0\mathrm{.}64\\ 20\mathrm{.}79& 3\mathrm{.}27\\ 10\mathrm{.}66& 2\mathrm{.}94\\ 0\mathrm{.}64& 11\mathrm{.}83\\ 0\mathrm{.}17& 22\mathrm{.}23\\ 13\mathrm{.}69& \text{S}\text{U}\text{M}\end{array}=102\mathrm{.}86$