Exponential Smoothing Confidence Interval

Example using Real Statistics

Example 1: Use the Real Statistics’ Basic Forecasting data analysis tool to get the results from Example 2 of Simple Exponential Smoothing.

Press Ctrl-m and select the Basic Forecasting option from the Time S tab. Fill in the dialog box that appears as shown in Figure 4 of Simple Moving Average except that this time choose the Simple Exp Smoothing option and MAE optimization (and so Real Statistics will automatically select the alpha parameter that optimizes the MAE value). Note that for Example 1 of Simple Exponential Smoothing, we would specify Alpha to be .4 and set the optimization option to None.

After clicking on the OK button, the output shown in Figure 1 appears.

Figure 1 – Exponential Smoothing data analysis

Standard Error and Confidence Interval

Note that the standard error for the forecast at time t = n is given by the value

Thus, cell G23 contains the formula

=SQRT(SUMXMY2(E$9:E22,F$9:F22)/COUNT(F$9:F22))

You can also copy this formula into all the cells in range G10:G22. The 1-α confidence interval of the prediction is calculated by ŷ ± se · z_crit where z_crit = NORM.S.INV(1-α/2). E.g. the worksheet formula in cell H23 is =F23-G23*NORM.S.INV(1-I$3/2).

Note that 95% confidence intervals are displayed. You can change these to any other confidence interval by changing the value in cell I3 (e.g. insert .01 in I3 to obtain 99% confidence intervals).

Advanced Forecast

Figure 1 only provides a one-period-ahead forecast. If you require further forecast periods, they are all equal to the value shown in cell F23. The standard error would increase though. For k periods ahead the standard error would be

where se₁ is the one period ahead standard error. Thus for t = 17, the forecasted value is 82.02983 with a standard error of

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage

Reference

Nau, R. (2014) Forecasting with moving averages. Fuqua School of Business, Duke University
https://www.academia.edu/28074627/Notes_on_forecasting_with_moving_averages_Robert_Nau