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

2. 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. 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. 4 The damped trend method

5. 5 Special case when ø = 1

6. 6 Special cases when β = 0

7. 7 Fit periods for M3 Annual series # YB067

8. 8 Special cases when β = 0, continued

9. 9 Special cases when α = β = 0

10. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 20 References Paper and presentation available at: www.bauer.uh.edu/gardner