1. Exponential Smoothing All statistical fitting methods involve some type of smoothing e.g., in regression, a trend line is fitter to the data points Consider the following forecasting scheme f t = wy t-1 + (1-w)f t-1 or, if you update by one period f t+1 = wy t + (1-w)f t Another way to write the process is f t+1 = f t + w(y t – f t ) The forecast is the current forecast plus a fraction of the current forecast error.

2. Exponential Smoothing Weighting Past Values Exponential smoothing places declining weights on past values of a series. Consider f t+1 = wy t + (1-w)f t = wy t + (1-w)[wy t-1 + (1-w)f t-1 ] = w[y t + (1-w)y t-1 ] + (1-w) 2 (f t-1 ) = w[y t + (1-w)y t-1 ] + (1-w) 2 [wy t-2 + (1-w)f t-2 ] = w[y t + (1-w)y t-1 +(1-w) 2 y t-2 ] + (1-w) 3 f t-3 Exponential smoothing places geometrically declining weights on past valued of {y t }

3. Exponential Smoothing Properties of the weights Note that w must be between 0 and 1 A large value of w places a high weight on the current realization of the series The sum of the weights on past values of {y t } is unity. The spreadsheet shows a value of w = 0.8

4. Exponential Smoothing Holt-Winters Other smoothing methods involve smoothing a trend lines fit to a data series f t+1 = wy t + (1-w)(f t +T t ) T t+1 = b(f t+1 – f t ) +(1-b)T t The one-step ahead forecast is H t+1 = F t+1 + T t+1 The m=step ahead forecast is H t+m = F t+1 + mT t+1 Winter’s method includes seasonals In forecast x you select alpha for the level beta for the seasonal gamma for the trend

5. Exponential Smoothing Hints The moving average and exponential smoothing methods will work well with series that do not have a trend or structural breaks. Exponential smoothing will be preferable if you want to heavily weight near-term events. Use the Holt-Winters Method for data that has trend and seasonals. Select values alpha, beta and gamma near 1 if you want forecasts to change strongly with new information. Forecast-x will also select the smoothing coefficients for you. The Mean Square Forecast Error (or RMSE) is often used a a criterion as to the best smoothing constant. The adaptive-response model is limited; it will not be discussed in class