Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of Houston Houston, Texas USA Damped Trend Forecasting: You know it makes sense!

A trend is a trend is a trend, But the question is, will it bend? Will it alter its course Through some unforeseen force And come to a premature end? Sir Alec Cairncross, in Economic Forecasting, 1969

Linear Trend Smoothing (Holt)

PastPresentFuture

PastPresentFuture

PastPresentFuture

PastPresentFuture Exponential Smoothing

PastPresentFuture Exponential Smoothing

PastPresentFuture Exponential Smoothing Damped Trend Forecasting

Strong Linear Trend in Data   usual Linear Trend forecast Erratic/Weak Linear Trend   Trend levels off to constant No Linear Trend   Simple Exponential Smoothing

Demonstrated ( ) on a large database of time series that using the method on all non-seasonal series gave more accurate forecasts at longer horizons, but lost little, if any accuracy, even at short ones. Damping trend may seem – perhaps sensibly conservative – but arbitrary. However, works extremely well in practice…. …. two academic reviewer comments from large empirical studies… “… it is difficult to beat the damped trend when a single forecasting method is applied to a collection of time series.” (2001) Damped Trend can “reasonably claim to be a benchmark forecasting method for all others to beat.” (2008)

Reason for Empirical Success? Pragmatic View Projecting a Linear Trend indefinitely into the future is simply far too optimistic (pessimistic) in practice. Damped Trend is more conservative for longer-term, more reasonable, and so more successful, but ……

…….. leaves unanswered the question: How can we model what is happening in the observed time series that makes Damped Trend Forecasting a successful approach?

Modelling View: Amongst models used in forecasting, can we find one  which has intuitive appeal and  for which Trend –Damping yields an optimal approach?

SSOE State Space Models Linear Trend model:

SSOE State Space Models Linear Trend model …. Reduced Form is an ARIMA(0,2,2)

Damped Linear Trend model: Reduced Form: ARIMA(1,1,2)

Strong Linear Trend in Data   usual Linear Trend forecast Erratic/Weak Linear Trend   Trend levels off to constant No Linear Trend   Simple Exponential Smoothing

Our Approach: use as a measure of the persistence of the linear trend, i.e. how long any particular linear trend persists, before changing slope …… Have RUNS of a specific slope with each run ending as the slope revision equation RESTARTS anew.

where are i.i.d. Binary r.v.s with New slope revision equation form

A Random Coefficient State Space Model for Linear Trend

Reduced version is a Random Coefficient ARIMA(1,1,2)

with probability :

Has the same correlation structure as the standard ARIMA(1,1,2) …and hence same MMSE forecasts … and so Damped Trend Smoothing offers an optimal approach

Optimal for a wider class of models than originally realized, including ones allowing gradient to change not only smoothly but also suddenly. Argue that this is more likely in practice than smooth change, and so Damped Trend Smoothing should be a first approach. (rather than just a reasonable approximation) Another – but clearly related – possibility is that the approach can yield forecasts which are optimal for so many different processes that every possibility is covered. To explore both ideas, used the method on the M3 Competition database of 3003 time series, and noted which implied models were identified.

ParameterValues Method Identified InitialValues LocalGlobal LevelTrendDamping %-ages 1 Damped Trend Linear Trend SES with Damped Drift SES with Drift SES RW with Damped Drift RW with Drift RW - Random Walk Modified Expo Trend Straight Line Simple Average

ParameterValues Method Identified InitialValues LocalGlobal LevelTrendDamping %-ages 1 Damped Trend Linear Trend SES with Damped Drift SES with Drift SES RW with Damped Drift RW with Drift RW - Random Walk Modified Expo Trend Straight Line Simple Average Series requiring Damping: 84% 70%

ParameterValues Method Identified InitialValues LocalGlobal LevelTrendDamping %-ages 1 Damped Trend Linear Trend SES with Damped Drift SES with Drift SES RW with Damped Drift RW with Drift RW - Random Walk Modified Expo Trend Straight Line Simple Average Series with some kind of Drift or Smoothed Trend term 98.9% 99.4%

ParameterValues Method Identified InitialValues LocalGlobal LevelTrendDamping %-ages 1 Damped Trend Linear Trend SES with Damped Drift SES with Drift SES RW with Damped Drift RW with Drift RW - Random Walk Modified Expo Trend Straight Line Simple Average

1. SES with Drift: 2. SES with Damped Drift: 3. Random Walk with Drift & Damped Drift:as 1 & 2 above with 4. Modified Exponential Trend:

1. SES with Drift: 2. SES with Damped Drift: 3. Random Walk with Drift & Damped Drift:as 1 & 2 above with Both correspond to random gradient coefficient models in which the drift term or slope satisfies.. As before, but with no error. Thus, slope is subject to changes of constant values at random times 4. Modified Exponential Trend:

Additive Seasonality (period: n)

with probability :

State Space Models: Non-constant variance models

Random Coefficient version:

with probability : where