There have been many attempts to develop forecasts based directly on a decomposition. The individual components are projected into the future and recombined to form a forecast of the underlying series.
Although this may appear a reasonable approach, in practice it rarely works well. The chief difficulty is in obtaining adequate forecasts of the components.
The trend-cycle is the most difficult component to forecast. It is sometimes proposed that it be modeled by a simple function such as a straight line or some other parametric trend model. But such models are rarely adequate. In the airline passenger example plotted in Figure 3-11, the trend-cycle does not follow any parametric trend model. But it does help identify “flat spots” and other features in the data which are not apparent from the time plot.
The other components are somewhat easier to forecast. The seasonal component for future years can be based on the seasonal
1STL code can be obtained from http://netlib.bell-labs.com/netlib/a/.
X-12-ARIMA code can be obtained from ftp://ftp.census.gov/pub/ts/x12a/.
3/7 Forecasting and decomposition 127
component from the last full period of data. But if the seasonal pattern is changing over time, this will be unlikely to be entirely adequate.
The irregular component may be forecast as zero (for additive decomposition) or one (for multiplicative decomposition). But this assumes that the irregular component is serially uncorrelated, which is often not the case. The decomposition of new one-family house sales (Figure 3-1) shows the irregular component with runs of positive or negative values. Clearly, if the irregular component at the end of the series is negative, it is more likely to be negative than zero for the first few forecasts, and so the forecasts will be too high.
One approach that has been found to work reasonably well is to forecast the seasonally adjusted data using Holt’s method (Chapter 4), then adjust the forecasts using the seasonal component from the end of the data. Makridakis et al. (1982) found that forecasts obtained in this manner performed quite well compared with several other methods.
However, we prefer to use decomposition as a tool for under-standing a time series rather than as a forecasting method in its own right. Time series decomposition provides graphical insight into the behavior of a time series. This can suggest possible causes of variation and help in identifying the structure of a series, thus leading to improved understanding of the problem and facilitating improved forecast accuracy. Decomposition is a useful tool in the forecaster’s toolbox, to be applied as a preliminary step before selecting and applying a forecasting method.
References and selected bibliography
Anderson, O. and U. Nochmals (1914) The elimination of spuri-ous correlation due to position in time or space, Biometrika, 10, 269–276.
Baxter, M.A. (1994) “A guide to seasonal adjustment of monthly data with X-11,” 3rd ed., Central Statistical Office, United King-dom.
Bell, W.R. and S.C. Hillmer (1984) Issues involved with the seasonal adjustment of economic time series, Journal of Business and Economic Statistics, 2, 291–320.
Brown, R.G. (1963) Smoothing, forecasting and prediction of dis-crete time series, Englewood Cliffs, N.J.: Prentice Hall.
Burman, J.P. (1979) Seasonal adjustment: A survey, TIMS Studies in Management Sciences, 12, 45–57.
Cleveland, R.B., W.S. Cleveland, J.E. McRae, and I. Ter-penning (1990) STL: A seasonal-trend decomposition procedure based on Loess (with discussion), J. Official Statistics, 6, 3–73.
Cleveland, W.S. (1983) Seasonal and calendar adjustment, in Handbook of statistics, vol. 3., 39–72, ed. D.R. Brillinger and P.R.
Krishnaiah. Elsevier Science Publishers B.V..
Cleveland, W.S. and S. Devlin (1988) Locally weighted regres-sion: an approach to regression analysis by local fitting, Journal of the American Statististical Association, 74, 596–610.
Cleveland, W.S., S. Devlin, and E. Grosse (1988) Regression by local fitting: methods, properties and computational algo-rithms, Journal of Econometrics, 37, 87–114.
Cleveland, W.S and I.J. Terpenning (1992) Graphical meth-ods for seasonal adjustment, Journal of the American Statistical Association, 77, 52–62.
Copeland, M.T. (1915) Statistical indices of business conditions, Quarterly Journal of Economics, 29, 522–562.
Dagum, E.B. (1982) Revisions of time varying seasonal filters, Journal of Forecasting, 1, 20–28.
Dagum, E.B. (1988) X-11-ARIMA/88 seasonal adjustment method:
foundations and users manual, Statistics Canada.
References and selected bibliography 129 den Butter, F.A.G. and M.M.G. Fase (1991) Seasonal
adjust-ment as a practical problem, Amsterdam: North-Holland.
Findley, D.F. and B.C. Monsell (1989) Reg-ARIMA based pre-processing for seasonal adjustment, In Analysis of data in time, ed. A.C. Singe and P Whitridge, 117–123, Ottawa, Canada.
Findley, D.F., B.C. Monsell, H.B. Shulman, and M.G. Pugh (1990) Sliding spans diagnostics for seasonal and related ad-justments, Journal of the American Statistical Association, 85, 345–355.
Findley, D.F, B.C. Monsell, W.R. Bell, M.C. Otto, and B.-C. Chen (1997) New capabilities and methods of the X-12-ARIMA seasonal adjustment program, Journal of Business and Economic Statistics, to appear.
Hooker, R.H. (1901) The suspension of the Berlin produce ex-change and its effect upon corn prices, Journal of the Royal Statistical Society, 64, 574–603.
Kendall, M.G., A. Stuart, and K. Ord (1983) The advanced theory of statistics, Vol 3. London: Charles Griffin.
Lothian, J. and M. Morry (1978) A test of quality control statistics for the X-11-ARIMA seasonal adjustment program, Re-search paper, Seasonal adjustment and time series staff, Statistics Canada.
Macauley, F.R. (1930) The smoothing of time series, National Bureau of Economic Research.
Makridakis, S., A. Andersen, R. Carbone, R. Fildes, M. Hi-bon, R. Lewandowski, J. Newton, E. Parzen, and R. Win-kler (1982) The accuracy of extrapolation (time series) methods:
results of a forecasting competition, Journal of Forecasting, 1, 111–153.
Newbold, P. and T. Bos (1994) Introductory business and eco-nomic forecasting, 2nd ed., Cincinnati, Ohio: South-Western Publishing Co..
Poynting, J.H. (1884) A comparison of the fluctuations in the price of wheat and in the cotton and silk imports into Great Britain, Journal of the Royal Statistical Society, 47, 345–64.
“Rapport sur les indices des crises economiques et sur les mesures r´esultant de ces crises.” (1911). Government report, Ministry of Planning, Paris, France.
Shiskin, J. (1957) Electronic computers and business indicators, National Bureau of Economic Research, Occasional Paper 57.
(1961) Tests and revisions of bureau of the census methods of seasonal adjustments, Bureau of the Census, Techni-cal Paper No. 5.
Shiskin, J., A.H. Young, and J.C. Musgrave (1967) The X-11 variant of the Census II method seasonal adjustment program, Bureau of the Census, Technical Paper No. 15.
Spencer, J. (1904) On the graduation of the rates of sickness and mortality, Journal of the Institute of Actuaries, 38, 334.
Exercises 131
Exercises
3.1 The following values represent a cubic trend pattern mixed with some randomness. Apply a single 3-period moving average, a single 5-period moving average, a single 7-period moving average, a double 3 × 3 moving average, and a double 5 × 5 moving average. Which type of moving average seems most appropriate to you in identifying the cubic pattern of the data?
Period Shipments Period Shipments
1 42 9 180
2 69 10 204
3 100 11 228
4 115 12 247
5 132 13 291
6 141 14 337
7 154 15 391
8 171
3.2 Show that a 3 × 5 MA is equivalent to a 7-term weighted moving average with weights of 0.067, 0.133, 0.200, 0.200, 0.200, 0.133, and 0.067.
3.3 For quarterly data, an early step in seasonal adjustment often involves applying a moving average smoother of length 4 followed by a moving average of length 2.
(a) Explain the choice of the smoother lengths in about two sentences.
(b) Write the whole smoothing operation as a single weighted moving average by finding the appropriate weights.
3.4 Consider the quarterly electricity production for years 1–4:
Year 1 2 3 4
Q1 99 120 139 160 Q2 88 108 127 148 Q3 93 111 131 150 Q4 111 130 152 170
(a) Estimate the trend using a centered moving average.
(b) Using an classical additive decomposition, calculate the seasonal component.
(c) Explain how you handled the end points.
3.5 The data in Table 3-11 represent the monthly sales of product A for a plastics manufacturer for years 1 through 5.
1 2 3 4 5
Jan 742 741 896 951 1030
Feb 697 700 793 861 1032
Mar 776 774 885 938 1126
Apr 898 932 1055 1109 1285 May 1030 1099 1204 1274 1468 Jun 1107 1223 1326 1422 1637 Jul 1165 1290 1303 1486 1611 Aug 1216 1349 1436 1555 1608 Sep 1208 1341 1473 1604 1528 Oct 1131 1296 1453 1600 1420 Nov 971 1066 1170 1403 1119 Dec 783 901 1023 1209 1013
Table 3-11: Monthly sales of product A for a plastics manufacturer (in 1,000s).
(a) Plot the time series of sales of product A. Can you identify seasonal fluctuations and/or a trend?
(b) Use a classical multiplicative decomposition to calculate the trend-cycle and monthly seasonal indices.
(c) Do the results support the graphical interpretation from part (a)?
Exercises 133
3.6 The following are the seasonal indices for Exercise 3.5 calcu-lated by the classical multiplicative decomposition method.
Seasonal Seasonal
Indices Indices
Jan 76.96 Jul 116.76
Feb 71.27 Aug 122.94
Mar 77.91 Sep 123.55
Apr 91.34 Oct 119.28
May 104.83 Nov 99.53
Jun 116.09 Dec 83.59
Assuming the trend in the data is Tt= 894.11 + 8.85t, where t = 1 is January of year 1 and t = 60 is December of year 5, prepare forecasts for the 12 months of year 6.
3.7 The sales data in Table 3-12 are for quarterly exports of a French company.
Sales Sales
(thousands (thousands
Year Quarter Period of francs) Year Quarter Period of francs)
1 1 1 362 4 1 13 544
2 2 385 2 14 582
3 3 432 3 15 681
4 4 341 4 16 557
2 1 5 382 5 1 17 628
2 6 409 2 18 707
3 7 498 3 19 773
4 8 387 4 20 592
3 1 9 473 6 1 21 627
2 10 513 2 22 725
3 11 582 3 23 854
4 12 474 4 24 661
Table 3-12: Quarterly exports of a French company.
(a) Make a time plot of the data. Note the pronounced seasonality.
(b) Use a classical multiplicative decomposition to estimate the seasonal indices and the trend.
(c) Comment on these results and their implications for forecasting.
3.8 Figure 3-13 shows the result of applying STL to the number of persons in the civilian labor force in Australia each month from February 1978 to August 1995.
(a) Say which quantities are plotted in each graph.
(b) Write about 3–5 sentences describing the results of the seasonal adjustment. Pay particular attention to the scales of the graphs in making your interpretation.
(c) Is the recession of 1991/1992 visible in the estimated components?
3.9 A company’s six-monthly sales figures for a five-year period are given below (in millions of dollars).
1992 1993 1994 1995 1996 Jan–June 1.09 1.10 1.08 1.04 1.03 July–Dec 1.07 1.06 1.03 1.01 0.96
(a) Obtain a trend estimate using a centered 2 MA smoother and compute the de-trended figures assuming an additive decomposition.
(b) Assuming the seasonality is not changing over time, cal-culate seasonally adjusted figures for 1996.
(c) Suppose several more years of data were available. Ex-plain in about two sentences how the classical decom-position method could be modified to allow for seasonal effects changing over time.
Exercises 135
650075008500 650075008500
-100050100 -200-1000100
1980 1985 1990 1995
Australian civilian labor force (’000 persons)
-100-50050100
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Figure 3-13: STL decomposition of the number of persons in the civilian labor force in Australia each month from February 1978 to August 1995.
METHODS
4/1 The forecasting scenario . . . 138 4/2 Averaging methods . . . 141 4/2/1 The mean . . . 141 4/2/2 Moving averages . . . 142 4/3 Exponential smoothing methods . . . 147 4/3/1 Single exponential smoothing . . . 147 4/3/2 Single exponential smoothing: an adaptive
ap-proach . . . 155 4/3/3 Holt’s linear method . . . 158 4/3/4 Holt-Winters’ trend and seasonality method . . . 161 4/3/5 Exponential smoothing: Pegels’ classification . . 169 4/4 A comparison of methods . . . 171 4/5 General aspects of smoothing methods . . . 174 4/5/1 Initialization . . . 174 4/5/2 Optimization . . . 176 4/5/3 Prediction intervals . . . 177 References and selected bibliography . . . 179 Exercises . . . 181
136 Chapter 4. Exponential Smoothing Methods
In Chapter 2 the mean was discussed as an estimator that mini-mizes the mean squared error (MSE). If the mean is used as a forecast-ing tool, then, as with all forecastforecast-ing methods, optimal use requires a knowledge of the conditions that determine its appropriateness.
For the mean, the condition is that the data must be stationary, stationary
meaning that the process generating the data is in equilibrium around a constant value (the underlying mean) and that the variance around the mean remains constant over time.
Thus, if a time series is generated by a constant process subject to random error (or noise), then the mean is a useful statistic and can be used as a forecast for the next period(s). However, if the time series involves a trend (in an upward or downward direction), or a seasonal effect (strong sales of heating oil in winter months, for example), or both a trend and a seasonal effect, then the simple average is no longer able to capture the data pattern. In this chapter we consider a variety of smoothing methods that seek to improve upon the mean as the forecast for the next period(s).
Before discussing any particular methods, we introduce in Section 4/1 a general forecasting scenario and strategy for evaluating forecast-ing methods. This is used throughout the chapter when appraisforecast-ing and comparing methods.
The classification of the forecasting methods discussed in this chapter is done in Table 4-1 where two distinct groupings are evident.
The group called “averaging methods” conform to the conventional averaging methods
understanding of what an average is—namely, equally weighted ob-servations. Two examples from this class of methods are examined in Section 4/2.
The second group of methods applies an unequal set of weights to past data, and because the weights typically decay in an exponential manner from the most recent to the most distant data point, the methods are known as exponential smoothing methods. This is some-exponential
smoothing methods thing of a misnomer since the methods are not smoothing the data in the sense of estimating a trend-cycle; they are taking a weighted average of past observations using weights that decay smoothly.
All methods in this second group require that certain parameters be defined, and these parameter values lie between 0 and 1. (These parameters will determine the unequal weights to be applied to past data.) The simplest exponential smoothing method is the single
Averaging methods Simple average (4/2/1) Moving averages (4/2/2)
Exponential smoothing methods Single exponential smoothing
one parameter (4/3/1) adaptive parameter (4/3/2) Holt’s linear method (4/3/3)
(suitable for trends) Holt-Winters’ method (4/3/4)
(suitable for trends and seasonality) Pegels’ classification (4/3/5)
Table 4-1: A classification of smoothing methods.
exponential smoothing (SES) method, for which just one parameter needs to be estimated. Another possibility is to allow the value of the parameter in SES to change over time in response to changes in the data pattern. This is known as adaptive SES and one variety to be discussed is known as Adaptive Response Rate Single Exponential Smoothing (ARRSES). Holt’s method makes use of two different parameters and allows forecasting for series with trend. Holt-Winters’
method involves three smoothing parameters to smooth the data, the trend, and the seasonal index. The exponential smoothing methods are discussed in Section 4/3.
Other exponential smoothing methods are possible and are dis-cussed in Section 4/3/5. These methods are based on Pegels’ (1969)
classification of trend and seasonality patterns depending on whether Pegels’ classification they are additive or multiplicative. Patterns based on this
classifi-cation are shown in Figure 4-1. For the practitioner of forecasting, the usage of an appropriate model is of vital importance. Clearly, an inappropriate forecasting model, even when optimized, will be inferior to a more appropriate model.
In Section 4/4 we apply all of the methods discussed in this chapter to a data set with both trend and seasonality, and compare the results. Finally, in Section 4/5 we look at some general issues in the practical implementation and use of exponential smoothing methods.
138 Chapter 4. Exponential Smoothing Methods
Old Figure 3-4b about here
Figure 4-1: Patterns based on Pegels’ (1969) classification.