Holt’s Linear Trend

The data in Figure 3 of Simple Exponential Smoothing (as well as previous figures on that webpage) shows a distinct upwards trend. The Moving Average and Simple Exponential Smoothing methods don’t adequately model this, but Holt’s Linear Trend Method (aka Double Exponential Smoothing) does. This is accomplished by adding a second single exponential smoothing model to capture the trend (either upwards or downwards). The model takes the following form for all i > 1

where 0 < α ≤ 1 and 0 ≤ β ≤ 1.

An alternative form of these equations is

where

Note that if β = 0, then the Holt model is equivalent to the Single Exponential Smoothing model.

Example 1: Redo Example 1 of Simple Exponential Smoothing using Holt’s Linear Trend Method where α = .4 and β = .7.

The result is shown in Figure 1. Here the cell C4 contains the formula =B4, cell D4 contains the value 0, cell C5 contains the formula =B$21*B5+(1-B$21)*(C4+D4), cell D5 contains the formula =C$21*(C5-C4)+(1-C$21)*D4 and cell E5 contains the formula =C4+D4.

Figure 1 – Holt’s Linear Trend

Example 2: Find the best fit Holt’s approximation to the data in Example 1, using the MAE measure of accuracy.

Using Solver, we minimize the value of MAE (cell E21 of Figure 2) by changing the value in range B21:C21 subject to the constraint that B21 <= 1.0 and C21 <= 1.0.

The result shown in Figure 2 is that α = .271817 and β = .598161, with MAE = 6.74.

Figure 2 – Holt’s Method Optimized

Observation: For any value of i, the forecast at time i+h is given by the formula

Example 3: Forecast the next 5 values in the times series from Example 2.

The result is shown in Figure 3.

Figure 3 – Forecast using Holt’s Method

The y and predicted y values shown in Figure 3 for i = 1 to 15 are the same as shown in Figure 2. The next five values are shown in range S19:S23. E.g. to calculate that ŷ₁₆ = 95.16312, we use the formula =C$18+(Q19-Q$18)*D$18. We get the other four values by highlighting the range S19:S23 and pressing Ctrl-D.

The graph on the right side of Figure 3 shows that the forecasted values after i = 15 are linear.