# 1 Why the damped trend works Everette S. Gardner, Jr. Eddie McKenzie.

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1 Why the damped trend works Everette S. Gardner, Jr. Eddie McKenzie

2 Empirical performance of the damped trend  “The damped trend can reasonably claim to be a benchmark forecasting method for all others to beat.” (Fildes et al., JORS, 2008)  “The damped trend is a well established forecasting method that should improve accuracy in practical applications.” (Armstrong, IJF, 2006)

3 Why the damped trend works  Practice Optimal parameters are often found at the boundaries of the [0, 1] interval. Thus fitting the damped trend is a means of automatic method selection from numerous special cases.  Theory The damped trend and each special case has an underlying random coefficient state space (RCSS) model that adapts to changes in trend.

4 The damped trend method

5 Special case when ø = 1

6 Special cases when β = 0

7 Fit periods for M3 Annual series # YB067

8 Special cases when β = 0, continued

9 Special cases when α = β = 0

10 Fitting the damped trend to the M3 series  Multiplicative seasonal adjustment  Initial values for level and trend  Local: Regression on first 5 observations  Global: Regression on all fit data  Optimization (Minimum SSE)  Parameters only  Parameters and initial values (no significant difference from parameters only)

11 M3 mean symmetric APE (Horizons 1-18) Makridakis & Hibon (2000) with backcasted initial values 13.6% Gardner & McKenzie (2010) with local initial values 13.5 with global initial values 13.8

12 MethodLocalGlobal Damped trend 43.0% 27.8% Holt 10.0 1.8 SES w/ damped drift 24.8 23.5 SES w/ drift2.4 11.6 SES0.80.6 RW w/ damped drift7.89.6 RW w/ drift2.58.4 RW0.0 Modified exp. trend8.38.7 Linear trend0.17.9 Simple average0.30.0 Methods identified in the M3 time series Initial values

13 Initial values Components identified in the M3 time series ComponentLocalGlobal Damped trend 51.3% 36.5% Damped drift32.6 33.6 Trend10.1 9.7 Drift 4.9 20.0 Constant level 1.2 0.6

14 MethodAnn.Qtr.Mon. Damped trend 25.9% 47.1% 47.5% Holt 17.4 14.2 3.6 SES w/ damped drift 17.7 16.7 33.6 SES w/ drift3.63.7 1.1 SES0.20.4 1.5 RW w/ damped drift 18.39.0 1.9 RW w/ drift7.82.1 0.4 RW0.0 Modified exp. trend9.16.0 10.1 Linear trend0.20.1 Simple average0.00.80.2 Methods identified by type of data (Local initial values)

15 Rationale for the damped trend  Brown’s (1963) original thinking:  Parameters are constant only within local segments of the time series  Parameters often change from one segment to the next  Change may be sudden or smooth  Such behavior can be captured by a random coefficient state space (RCSS) model  There is an underlying RCSS model for the damped trend and each of its special cases

16 SSOE state space models for the damped trend  {A t } are i.i.d. binary random variates  White noise innovation processes ε and are different  Parameters h and h* are related but usually different

17 Runs of linear trends in the RCSS model  With a strong linear trend, {A t } will consist of long runs of 1s with occasional 0s.  With a weak linear trend, {A t } will consist of long runs of 0s with occasional 1s.  In between, we get a mixture of models on shorter time scales, i.e. damping.

18 Advantages of the RCSS model  Allows both smooth and sudden changes in trend.  is a measure of the persistence of the linear trend. The mean run length is thus and  RCSS prediction intervals are much wider than those of constant coefficient models.

19 Conclusions  Fitting the damped trend is actually a means of automatic method selection.  There is an underlying RCSS model for the damped trend and each of its special cases.  SES with damped drift was frequently identified in the M3 series and should receive some consideration in empirical research.

20 References Paper and presentation available at: www.bauer.uh.edu/gardner

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