The forecasts generated by Holt’s linear method display a constant trend (increasing or decreasing) indefinitely into the future. Empirical evidence indicates that these methods tend to over-forecast, especially for longer forecast horizons. Motivated by this observation, Gardner & McKenzie (1985) introduced a parameter that “dampens” the trend to a flat line some time in the future. Methods that include a damped trend have proven to be very successful, and are arguably the most popular individual methods when forecasts are required automatically for many series.
In conjunction with the smoothing parameters alpha and beta (with values between 0 and 1 as in Holt’s method), this method also includes a damping parameter 0 < phi (ϕ) < 1:
Forecast equation: ŷ = l + (phi + phi^2 + ... + phi^h) * b
Level equation: l = alpha * y + (1 - alpha) * (l + phi * b)
Trend equation: b = beta * (l - l) + (1 - beta) * phi * b
If phi=1, the method is identical to Holt’s linear method. For values between 0 and 1, phi dampens the trend so that it approaches a constant some time in the future. In fact, the forecasts converge to lT + phi * bT/(1 - phi) as h → ∞ for any value 0< phi <1. This means that short-run forecasts are trended while long-run forecasts are constant. In practice, phi is rarely less than 0.8 as the damping has a very strong effect for smaller values. Values of phi close to 1 will mean that a damped model is not able to be distinguished from a non-damped model. For these reasons, we usually restrict phi to a minimum of 0.8 and a maximum of 0.98.
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