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 and is described in further detail in Chapter $15.$ For instance, the basic exponential smoothing model for forecasting sales is

where

This model is used recursively; the forecast for time period

$+1$ is based on the forecast for period

$,$

$,$ the observed value of sales in period

$,$

$,$ 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

$.$ The forecast errors,

$,$ are calculated in the fourth column. The value of

is often chosen by minimizing the sum of squared forecast errors, commonly referred to as 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.