Holt’s Linear Trend Confidence Interval

Standard Error and Prediction Intervals

One calculates the standard error and prediction intervals for a Holt’s Linear Trend forecast as described in Exponential Smoothing Confidence Interval, except that now the k-step ahead standard error is

Data Analysis Tool

Example 1: Use the Basic Forecasting data analysis tool for Example 3 of Holt’s Linear Trend.

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 Holt’s Linear Trend option and MAE optimization (and so Real Statistics will automatically select the alpha parameter that optimizes the MAE value). 

After clicking on the OK button, the output shown in Figure 1 appears (although the chart, which is the same as in Figure 3 of Holt’s Linear Trend, is not displayed).

Figure 1 – Holt Linear Trend data analysis

Note that for Example 1 of Holt’s Linear Trend, we would proceed in the same way, except that we need to explicitly specify Alpha as .4, Beta as .7, and set the optimization option to None.

Key Formulas

The output contains the standard errors and 95% confidence intervals (based on the value in cell I3). The formula in cell H21 is

=$F$20+$G$20*(ROW(F21)-ROW($F$20))

and similarly for the other forecasted values. The formula used to calculate the standard error for the first forecasted value (shown in cell I23) is

=SQRT(SUMXMY2(E$7:E20,H$7:H20)/COUNT(H$7:H20))

The formula used to calculate the standard error of the fifth forecast value is

=I$21*SQRT((I24/I$21)^2-1+(E$28*(1+(ROW(H25)-ROW(H$21))*F$28))^2+1)

in cell I25. The corresponding upper confidence interval limit (cell K25) for this forecast value is calculated by the formula

=H25+NORM.S.INV(1-I$3/2)*I25

Optimization Approach

Note that the Real Statistics data analysis tool identifies values for alpha and beta that produce a lower value for MAE than that produced by Solver as shown in Example 2 of Holt’s Linear Trend. This is because the data analysis first calculates MAE for all values of alpha and beta between 0 and 1 in increments of .005. Once it finds the values of alpha and beta that produce the smallest MAE value, it then uses Solver based on these initial values to further optimize the value of MAE.

Reference

SAS/ETS Software (1999) Forecasting process details (chapter 30)
http://www.math.wpi.edu/saspdf/ets/chap30.pdf