7.3 Exponential trend method

A variation from Holt’s linear trend method is achieved by allowing the level and the slope to be multiplied rather than added:

\begin{align*} \pred{y}{t+h}{t} &= \ell_{t} b_{t}^h\\ \ell_{t} &= \alpha y_{t} + (1 - \alpha)(\ell_{t-1} b_{t-1})\\ b_{t} &= \beta^*\frac{\ell_{t}}{ \ell_{t-1}} + (1 -\beta^*)b_{t-1} \end{align*}

where $b_t$ now represents an estimated growth rate (in relative terms rather than absolute) which is multiplied rather than added to the estimated level. The trend in the forecast function is now exponential rather than linear, so that the forecasts project a constant growth rate rather than a constant slope. The error correction form is

\begin{align*} \ell_{t} &= \ell_{t-1} b_{t-1}+\alpha e_{t}\\ b_{t} &= b_{t-1}+ \alpha \beta^*\frac{e_{t}}{\ell_{t-1}} \end{align*} where $e_t=y_{t} -(\ell_{t-1}b_{t-1})=y_{t}-\pred{y}{t}{t-1}$.

Example 7.2    Air Passengers (continued)

In Table 7.3 we also demonstrate the application of the exponential trend method. As for Holt’s linear method we set, $\alpha=0.8$, $\beta^*=0.2$ and $\ell_0 = y_1$, however $b_0 = y_2 / y_1$ as suggested in Table 7.9.

Notice the difference between the trend and the growth rate in the two methods in the columns labelled $b_t$. In Holt’s linear method, $b_t$ is added to the corresponding level term in the calculations; in the exponential trend method, $b_t$ is multiplied with the corresponding level term in the calculations.