1 Exponential smoothing in the telecommunications data Everette S. Gardner, Jr.

2 Exponential smoothing in the telecommunications data  Empirical research in exponential smoothing  Summary of Fildes et al. (IJF, 1998)  Data analysis  Re-examination of the smoothing methods  Conclusions  Chatfield’s thoughtful approach to exponential smoothing

3 Empirical research in exponential smoothing:  Total of 65 coherent empirical studies  Excluding M-competitions and related papers  Excluding studies based on simulated data  Some form of exponential smoothing performed well in all but 7 studies  All of these exceptions should be re-examined  The most surprising exception is Fildes et al.’s (IJF, 1998) study of telecommunications data

4 Fildes et al. (IJF, 1998)  Study of 261 monthly, nonseasonal series  71 observations each  Forecasts through 18 steps ahead from origins 23, 31, 38, 45, and 53  Steady trends with negative slopes  Numerous outliers  Methods tested  Robust trend  Exponential smoothing  Holt’s additive trend  Damped additive trend

5 The robust trend method  Underlying model  ARIMA (0, 1, 0) with drift  Drift term  Estimated by median of the differenced data  Subject to complex adjustments

6 Fildes et al. (IJF, 1998) continued  Robust trend was the most accurate method  Holt’s additive trend was more accurate than the damped trend  Contrary to theory and all other empirical studies in the literature  Armstrong (IJF, 2006) recommends replication

7

8 Re-examination: Methods tested  Holt’s additive trend  Damped additive trend  Theta method (Assimakopoulos & Nikolopoulos, IJF, 2000)  SES with drift (Hyndman & Billah, IJF, 2003)

9 SES with drift SES with drift is equivalent to the Theta method when drift equals ½ the slope of a classical linear trend. However, Hyndman and Billah recommend optimization of the drift term. Fixed drift

10 Re-examination: Fitting the methods  Two sets of data were fitted through each forecast origin  Original data  Trimmed data – observations prior to an early trend reversal were discarded  Initial values for smoothing methods  Intercept and slope of a classical linear trend  Fit criteria  MSE  MAD (to cope with outliers)

11 Fitting continued  Parameter choice  Usual [0,1] interval  Full range of invertibility  SES with drift  Initial level and drift were optimized simultaneously with smoothing parameter  Theta method  Initial level only was optimized simultaneously with smoothing parameter  Drift fixed at half the slope of the fit data

12 Effects of model-fitting on the damped trend FitParametersDataCriterionMAPE 1ApproximateOriginalMSE9.7 2OptimalOriginalMSE7.8 3OptimalTrimmedMSE7.2 4OptimalTrimmedMAD6.8 Note: MAPE is the average of all forecast origins and horizons.

13 Effects of model-fitting on the Holt method FitParametersDataCriterionMAPE 1ApproximateOriginalMSE 8.1* 2OptimalOriginalMSE8.1 3OptimalTrimmedMSE7.9 4OptimalTrimmedMAD7.4 * We were unable to replicate Fildes et al.’s Holt results.

14 Effects of model-fitting on SES with drift FitParametersDataCriterionMAPE 1OptimalOriginalMSE7.4 2OptimalTrimmedMSE7.1 3OptimalTrimmedMAD6.2

15 Why did SES with drift perform so well?  Trends in most series are so consistent that there is no need to change initial estimates obtained by least-squares regression  Smoothing parameter was fitted at 1.0 almost half the time  This produces an ARIMA (0,1,0) with drift, the underlying model for the robust trend

16 Revised empirical comparisons MethodMAPE Robust trend6.2 SES with drift6.2 Damped additive trend6.8 Holt’s additive trend7.4 Theta method7.6 All methods except robust trend fitted to trimmed data to minimize the MAD.

17 Conclusions  Contrary to Fildes et al., the damped trend is in fact more accurate than the Holt method.  SES with drift:  Simplest method tested  Drift term should be optimized  More accurate than the Theta method  About the same accuracy as the robust trend

18 Recommendations  Trim irrelevant early data  Use a MAD fit to cope with outliers  Optimize smoothing parameters  Follow Chatfield’s (AS, 1978) “thoughtful” approach to exponential smoothing

19  Re-examination of Newbold and Granger (JRSS, 1974), who found the Box-Jenkins procedure was far more accurate than exponential smoothing  Findings  Newbold and Granger’s empirical comparisons were biased  It was easy to improve the performance of exponential smoothing Chatfield (AS,1978)

20 Chatfield’s thoughtful approach to exponential smoothing 1.Plot the series and look for trend, seasonality, outliers, and changes in structure 2.Adjust or transform the data if necessary 3.Choose an appropriate form of trend and seasonality 4.Fit the method and produce forecasts 5.Examine the errors and verify the adequacy of the method