Previous research has shown that seasonal factors provide one of the most important ways to improve forecast accuracy. For example, in forecasts over an 18-month horizon for 68 monthly economic series from the M-Competition, Makridakis et al. (1984, Table 14) found that seasonal adjustments reduced the MAPE from 23.0 to 17.7 percent, an error reduction of 23%. On the other hand, research has also shown that seasonal factors sometimes increase forecast errors (e.g., Nelson, 1972). So, when forecasting with a data series measured in intervals that represent part of a year, should one use seasonal factors or not? Statistical tests have been devised to answer this question, and they have been quite useful. However, some people might say that the question is not fair. Why does it have to be either/or? Shouldn^Rt the question be ^Sto what extent should seasonal factors be used for a given series?^Tseasonal factors, forecast, accuracy

J. S. Armstrong

English

Research Papers in Economics

  1
Damped Seasonality Factors: Introduction 
 
Forthcoming, International Journal of Forecasting 
 
J. Scott Armstrong 
The Wharton School 
University of Pennsylvania, Philadelphia, PA 
 
armstrong@wharton.upenn.edu 
 
 
Previous research has shown that seasonal factors provide one of the most important 
ways to improve forecast accuracy. For example, in forecasts over an 18-month horizon 
for 68 monthly economic series from the M-Competition, Makridakis et al. (1984, Table 
14) found that seasonal adjustments reduced the MAPE from 23.0 to 17.7 percent, an 
error reduction of 23%. On the other hand, research has also shown that seasonal factors 
sometimes increase forecast errors (e.g., Nelson, 1972). 
 
So, when forecasting with a data series measured in intervals that represent part of a year, 
should one use seasonal factors or not? Statistical tests have been devised to answer this 
question, and they have been quite useful. However, some people might say that the 
question is not fair. Why does it have to be either/or? Shouldn’t the question be “to what 
extent should seasonal factors be used for a given series?”  
 
Damping as a Basic Strategy for Forecasting 
 
One solution to “to what extent” relies on damping. Basically, damping says that the 
forecaster is more conservative as uncertainty increases. In 1978, in summarizing 
research by others, I concluded that trends should be damped (Armstrong 1978, p.153). 
As nearly as I can tell, only two people took action: Gardner and Mackenzie (1985) 
provided convincing evidence that damping improved accuracy. Just as important, they 
provided an operational procedure. Their effort led to one of the more important advances 
in extrapolation.  
 
Again drawing on the research of others, I concluded in Armstrong (1978, 148-150) that 
seasonal factors should be damped. Here also only two people listened. Miller and 
Williams (2003, 2004) have obtained convincing evidence that damped seasonal factors 
improve forecast accuracy. They have also developed an operational procedure for doing 
this.  
 
Is the Miller-Williams’ procedure optimal? Along with the panelists (including Miller 
and Williams), I am skeptical that it is. However, I expect that their statistical procedures 
for damping seasonality will prove to be nearly optimal. Their two papers provide 
evidence from simulations and from analyses of monthly series. For the real data, 
damping improved accuracy in about 60% of the series, with about a 4% error reduction. 
   2
 
Process for the Special Section on Damped Seasonal Factors 
 
I reviewed early versions of the current Miller-Williams paper in 2001, then in 2002. 
Following that, I sent their version to five reviewers widely recognized as leaders in the 
use of seasonal factors. All who were asked agreed to participate. Three others joined as 
coauthors on the commentaries.  In doing their reviews, all checked the Miller-Williams 
procedures and conducted additional analyses. 
 
Miller and Williams made revisions to address reviewers’ concerns. Authors of two 
commentaries conducted additional analyses. A revised paper and written commentaries 
were prepared for presentation at the International Symposium on Forecasting in Merida, 
Mexico, June 2003. The session drew much interest. It also generated a heated discussion 
by some of those in the audience.  
 
Miller and Williams conducted extensive analyses to test possible threats to validity that  
were raised at the symposium. This revised version was then sent to the commentators so 
they could revise their commenty. Additional suggestions were made by the 
commentators and by another independent reviewer, and Miller and Williams again 
revised their paper in light of the comments. In all, the process extended over several 
years and involved ten reviewers. While there have been many additional analyses that 
have led to substantial improvements in the paper, the original findings have held up well 
over time. 
 
 
Implementation 
 
Major software providers such as Forecast Pro and SAS have gone on record as being 
willing to provide new features upon requests from clients. If you are using seasonal 
factors to make forecasts, you should ask them and other software developers to 
implement the Miller-Williams Seasonality procedures.  
 
However, you do not need to wait for the software providers. Miller and Williams have 
provided full disclosure of their procedures. In addition, they provide spreadsheet 
software for their procedures as part of the public domain. (See under software at 
forecastingprinciples.com.) You could direct your current software provider to this 
procedure. 
 
 
Beyond Miller-Williams 
 
The major potential for gain beyond the Miller-Williams procedures will come from 
using additional information. Two sources are promising: domain knowledge and 
analogous series. 
   3
In a pilot study involving a small number of time series, Fred Collopy and I (1998) found 
that damping based on the amount of data and on domain experts’ expectations about 
trend reduced forecast error by more than six percent (see these results at the Researchers 
page of forecastingprinciples.com). Miller and Williams (2003) showed that if our 
adjustment were based only on the number of years of data (fewer years implying more 
uncertainty and thus more damping) there was no improvement over the traditional 
approach. In effect it likely led to too much damping. Thus, the overall gain we found 
was apparently due to domain knowledge. 
 
Another approach to damping is to draw the seasonal factors towards the estimates of the 
seasonal factors for one or more analogous series, For example, you could average 
seasonal factors for a brand with those for that brand’s product class. This is expected to 
be especially useful when the analogous series involve considerably more data. Bunn and 
Vassilopoulos (1999) provide procedures for combining seasonal factors across 
analogous data, along with supporting evidence for such a procedure. 
 
 
Conclusion 
 
Historically, seasonal factors have been concerned with estimates of historical data, not 
with forecasting.  The Miller-Williams procedures enable organizations to adjust seasonal 
factors so as to make more accurate forecasts. This has an immediate payoff to 
forecasters – they can reduce forecast errors by about four percent. In addition, their 
papers have paved the way for improvements in the estimation of seasonal factors for 
forecasting, such as through the use of domain knowledge and analogous series.  
 
 
References 
Armstrong, J. S. (1978, 1985, 2
nd ed.). Long-Range Forecasting: From Crystal Ball to 
Computer. New York: John Wiley. 1985 edition available in full-text at 
hops.wharton.upenn.edu/forecast. 
 
Armstrong, J. S. & F. Collopy (1998), Speculations about seasonal factors. In full text on 
Researchers’ page at forecastingprinciples.com. 
 
Bunn, D.W. & A. I. Vassilopoulos, (1999). Comparison of seasonal estimation methods 
in multi-item short-term forecasting. International Journal of Forecasting, 15, 431-443. 
 
Gardner, E. S., Jr. & E. McKenzie (1985). Forecasting trends in time-series. Management 
Science, 31, 1237-1246. 
Makridakis, S., A. Andersen, R. Carbone, R. Fildes, M. Hibon, R. Lewandowski, J. 
Newton, E. Parzen & R. Winkler (1984). The Forecasting Accuracy of Major Time-series 
Methods. Chichester: John Wiley.   4
Miller, D. M. & D. Williams (2004). Shrinkage estimators for damping X-12-ARIMA 
seasonals. International Journal of Forecasting, forthcoming. 
Miller, D. M. & D. Williams (2003). Shrinkage estimators of time series seasonal factors 
and their effect on forecasting accuracy. International Journal of Forecasting, 19, 669-
684. 
Nelson, C.R. (1972). The prediction performance of the FRB-MIT-Penn model of the 
U.S. economy. American Economic Review, 5, 902-917. 
 
 
 

2 nd ed.). Long-Range Forecasting: From Crystal Ball to Computer.

Comparison of seasonal estimation methods in multi-item short-term forecasting.

Forecasting trends in time-series.

Shrinkage estimators of time series seasonal factors and their effect on forecasting accuracy.

Speculations about seasonal factors.

The Forecasting Accuracy of Major Time-series Methods.

The prediction performance of the FRB-MIT-Penn model of the U.S.

Damped Seasonality Factors: Introduction

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Damped Seasonality Factors: Introduction

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