top-down or bottom-up: aggregate versus disaggregate extrapolations

International Journal of Forecasting 8 (1992) 233-241

North-Holland 233

Top-down or bottom-up: Aggregate versus disaggregate extrapolations

Byron J. Dangerfield and John S. Morris

College of Business and Economics, University of Idaho, Moscow, ID 83843, USA

Abstract: Two approaches have been suggested for forecasting items in a product line. The top-down (TD) approach uses an aggregate forecast model to develop a summary forecast, which is then allocated to individual items on the basis of their historical relative frequency. The bottom-up (BU) approach employs an individual forecast model for each of the items in the family. The present study compares these two approaches by using over 15,000 aggregate series constructed by combining individual series from the M-competition database. The effects of correlation between individual items and the relative frequency of individual items in the family are examined. In most situations, BU forecasting of family items produces more accurate forecasts.

IKeywords: Family forecasts, Top-down forecasts, Aggregate forecasts, M-competition.

1. Introduction

Many organizations find it necessary to forecast individual items that make up a family or group classification. For example, a personal computer manufacturer may offer two models that make up their lap-top PC family. Two general approaches have been suggested for de- veloping forecasts for individual models or items. One approach might be referred to as a top- down (TD) strategy since a single forecast model is developed to forecast an aggregate -or family total which is then distributed to the individual items in the family based upon their historical proportion of the family total. The other approach might be labelled a bottom-up (BU) strategy since multiple forecast models based upon the individual item series are used to develop item forecasts.

Correspondence to: B.J. Dangerfield, College of Business

and Economics, University of Idaho, Moscow, ID 83843,

USA. Tel: (208) 885-6478; Fax: (208) 885-8939.

2. Previous research

A number of TD allocation methods have been suggested, all of which rely on using historical proportions for individual items. Brown (1962) suggested a vector smoothing method that updates the item proportions each period before the aggregate forecast is distributed. Initial item proportions are calculated as the ratio of the item’s cumulative demand to the family’s cumulative demand for some historical interval; the item proportions are then updated through an exponential smoothing formula based on the prior period’s item proportion and its most recent actual historical proportion of family demand. Item forecasts are developed by applying smoothed proportions to the aggregate or family forecast. Cohen (1966) developed a blending method for forecasting item demand; a forecast for the group is blended with the average demand for each item. Hausman and Sides (1973) used perhaps the simplest allocation scheme in their approach which consisted of apportioning the aggregate forecast based on the year-to-date percentage of total demand for each item.

0169-2070/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved

234 B. J. Dungerjield, J.S. Morris I Top-down or bottom-up

Support for the TD approach is usually based on the statistical fact that the variance of the aggregate demand is equal to the sum of the variances of independent item demands. For example. McLeavey and Narasimhan (1985), in their discussion of multi-item forecasting techniques, conclude that one ‘can generate a more accurate forecast for a group of items than for individual items in the group’ (p. 67). Fogarty and Hoffmann (1983) echo this sentiment in their discussion of this variance relationship: ‘This equation means that if we simply add to- gether the forecasts for the individual items, the variance will be quite large. Hence, it is usually better to forecast total demand directly than to sum component forecasts’ (p. 81). Of course the item demands may or may not be independent, and it is not clear that the TD allocation methods will improve forecast accuracy for individual items even if the independence assumption is correct.

Several authors have pointed out the weak- nesses of TD. Theil (1984) described conditions under which bias is introduced when aggregating from microeconomic to macroeconomic relations. Edwards and Orcutt (1969) and Orcutt, Watts and Edwards (1968) argued that information losses resulting from aggregated data could be substantial, and hence they generally sup- ported bottom-up approaches. Zellner (1969) ag- reed, noting that aggregating data involves an important loss of information. Aigner and Gold- feld (1974) addressed the impact of measure- ment error for independent variables and found no unequivocal superiority for the models constructed from aggregate data.

There has been only limited empirical testing of the two approaches, but the evidence so far supports the BU approach. Dunn, William and Spiney (1971) found that forecasts aggregated from lower-level modeling worked best in forecasting demand for telephones. Time series models were developed for each of nine local outlets. The models varied but included exponential smoothing and autoregressive integrated moving average (ARIMA) models. The authors mea- sured error by using mean absolute deviation (MAD), mean squared error (MSE), and a scaled-error criterion. Summing the forecasts from these local models proved more accurate than forecasting from aggregate data. Danger-

field and Morris (1988) examined the relative performance of TD and BU approaches in forecasting the returns of individual species of fish that make up the total run of anadromous fish in the Columbia River drainage. They developed exponential smoothing models for the total run, as well as for the individual species runs, and found that forecasts for the individual species were more accurate when separate exponential smoothing models for each species (bottom-up) were used. Similar results have been obtained using econometric methods to forecast earnings. Kinney (1971) found that disaggregating earnings data by market segments resulted in more accurate forecasts than when firm-level data were used. Collins (1976) compared segmented econometric models with aggregate models for a group of 96 firms. The segmented models using disaggregated data produced more accurate forecasts for both sales and profit forecasts.

Schwartzkopf, Tersine and Morris (1988) discussed the relative merits of TD and BU strategies and concluded that the relative performance of these two approaches depends upon three sources of error: estimation precision (variability of the estimate around the predicted value), bias (deviation of the mean of the estimate from the true value), and outlier influence (sensitivity to bad data). Using an analytical model of mean squared error for a two-item family, they showed that item correlation affect- ed the precision and bias error components in different ways. In TD forecasting, negative item correlation reduced variability in the aggregate series, but increased item model bias. They also found that item proportions in the aggregate series could have an affect on the relative performance of the two techniques. Similar item proportions increased the effects of model bias and outlier influence when the TD approach was used and therefore, favored BU forecasting. However, no specific guidance was given for selecting between the two approaches.

The purpose of this research was to examine the relative performance of the two different methods (TD or BU) using exponential smoothing models and simple two-item families. The two approaches were tested on over 15,000 aggregate time series, each consisting of two individual item series with different correlations and item proportions. In addition to observing

B.J. Dangerfield, J.S. Morris I Top-down or bottom-up 235

the overall performance of the two approaches, we hoped to observe differences due to the correlation and item proportion effects described by Schwartzkopf, Tersine and Morris (1988).

3. Methodology

The two forecasting approaches were compared using both carefully specified forecast models as well as models whose parameters were randomly specified. The results of the random model specification were used to examine the effect of model specification on the relative performance of the two methods. Next, the results using the carefully specified models were grouped into three classes of item correlation (high negative, low, high positive) and three classes of item one’s proportion of the aggregate series (low, medium, high). A detailed discussion of the experimental variables, along with any common factors and assumptions, is given below.

3.1. Time series

The M-competition data [Makridakis et al. (1982)] were selected to test the accuracy of the two approaches; these data consist of 1,001 time series that are classified as either micro or macro data. Each time series has two subsections: a specification subsection containing data points used to specify the model, and a holdout subsection to .test the accuracy of the model. In addition to the time series observations, the M- competition data include a set of seasonal indices for each series.

A subset of all 192 monthly series was chosen from the 1,001 series. Monthly series were selected since the literature favoring the TD methodology is largely found in the production/operations management field where short-term forecasts are the norm. The 192 series comprised the set of series from number 395 to 586. These 192 series were further reduced to 178 because we wished to consider only series that had at least 48 data points in the specification subset. A specification subsample of the most recent 48 observations was used to develop forecast models for the TD and BU approaches for the carefully specified models. Only the last four years of data

were used from each series so that all specification subsets had the same length. This 48-point restriction, while arbitrary, allowed equitable specification for all series as well as uniformity in the aggregation of separate series. The holdout or test sample consisted of 18 observations for the monthly series.

The series were then combined to create an aggregate family composed of two item series. A total of 15,753 aggregate time series was constructed using all possible unique combinations of pairs of the 178 series selected. Thus, each of the constructed aggregate or family series was composed of two of the 178 individual item series. These pairs of item series varied with respect to correlation (-0.96 < r < l.O), to seasonal and trend patterns, and to their relative proportion of the aggregate series. Exhibits 1 and 2 show the relative frequency distributions of families by item correlation and item proportion, both of which have been rounded to the nearest tenth.

3.2. Forecast models

To compare the performance of the two forecasting methodologies, exponential smoothing (ES) models were developed for both the TD and BU approaches. We chose to use Winters’ (1960) triple smoothing approach since it is commonly found in production/operations management texts [e.g. Tersine (1985), Vollmann, Berry and Whybark (1988)] and has performed well relative to other forecasting models in tests such as the M-competition. This model has three smoothing constants (a, b, and c) to smooth level, trend, and seasonal indices and is described in Makridakis, Wheelwright and McGee (1983). Two approaches were used to select smoothing constants.

3.3. Best jit model specification

Models were formulated for each of the aggregate series for the TD method as well as for each of the item series used in the BU approach using the 4%point specification subset. Smooth- ing constraints were selected after a grid search over both the family series (TD) and each of the individual item series (BU). The initial trend and level for this search process were estimated by

236 B.J. Dangerfield, J.S. Morris I Top-down or bottom-up

0.15

, 0.1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Correlation Coefficient

Exhibit 1. Frequencies of correlation coefficient.

0.05

J 01 0 0.2 0.4 0.6 0.8 1

Proportion of Item 1

Exhibit 2. Frequencies of item 1 proportion.

fitting a regression line to each of the aggregate and individual series. The seasonal indices sup- plied in the M-competition data were used as initial indices for item series in the constant search procedure. Initial aggregate series indices were calculated as proportionally weighted averages of these item indices. The combination of constants that minimized mean absolute percen-

tage error (MAPE) was selected for use in the respective models. Makridakis, Wheelwright and McGee (1983, pp. 45-47) suggest that a relative error criterion such as MAPE is more appropri- ate than mean squared error (MSE) in model specification due to the potential problem of ‘overfitting which is equivalent to including ran- domness as part of the generating process’.

B. J. Dange$eld, J.S. Morris i Top-down or bottom-up 237

The best fit smoothing constants were then used to generate an initial forecast for the 18 period holdout sample. This was accomplished by recursive application of Winter’s model to the 48-period specification subset using the best fit smoothing constants. The smoothed level, trend and seasonal indices from the specification subset were then used to generate an initial forecast for the test sample.

3.4. Randomly selected smoothing constants

We also investigated the effect of random model specification to test the robustness of the two approaches to poorly specified models. Ran- dom values for each of the three smoothing constants were selected for both the aggregate series used in TD and each of the time series used in BU. Initial trend, level, and seasonal indices for the holdout sample were generated in the same manner as the best fit models.

3.5. Item forecasts

Item forecasts using BU were generated by applying the individually specified ES models to the holdout sample. Item forecasts for the TD approach were developed by multiplying the aggregate forecast by an allocation parameter, pi, which was equal to item i’s fractional share of the aggregate demand during the specification period. The specific item forecast model is given below:

f,,,=Pi*F, 1

where A,, is the forecast for item i in period t, F, is the forecast for the aggregate series in period t, and p, is the fraction that item i represented in the aggregate series during the specification period. This allocation procedure is consistent with those discussed in the literature and ob- served in practice. While other TD allocation methods exist, all involve the calculation of an allocation parameter based upon an item’s share of the aggregate demand over some period of time. Neither the proportionality factors nor smoothing constants were updated during the actual forecasting competition with the holdout sample.

3.6. Performance measures

We assumed that the objective of the firm was to forecast as accurately as possible each of the two individual item demands. Armstrong (1985) points out that there is no universally accepted measure of accuracy. However, Armstrong and Collopy (1992) used M-competition series to test a number of error measures for reliability, con- struct validity, outlier protection, and sensitivity and concluded that MSE should not be used for generalizing about the level of accuracy of alter- native forecasting methods because of its low reliability. Therefore, our analysis used MAPE for comparing the TD and BU methods. We do, however, report the results using MSE in Ap- pendix A and discuss them below in Section 5 since MSE is a commonly used measure.

Comparison of the results of the two forecasting approaches was complicated by the fact that the aggregate forecasts generated only one result (i.e. only one MAPE), while the disaggregate forecasts generated two. To allow direct comparison of the methods, we constructed a summary error measure consisting of the average of the two item MAPEs.

Next, the natural log of the ratios of error measures for the two approaches was computed, i.e. ln( top-down MAPE/ bottom-up MAPE). If this log relative error is positive, BU is more accurate than TD; if it is negative, TD is more accurate. The natural log was used to eliminate the potential basis in interpretation that can result when summary statistics are computed using simple ratios. Because of the limited range of simple ratios (in this case they cannot be less than zero), the frequency distribution of ratios is positively skewed [Alexander and Francis (1986, p. 145)]. For example, consider the results from the hypothetical test of the two approaches given in Exhibit 3. The overall average error ratio for the two series tested is 1.25 ((2 + 0.5)/2) for the simple ratio, indicating inferior overall performance for TD even though there is really no difference in performance between TD and BU. BU is twice as good as TD in series A, while the reverse is true for series B. When the natural log of the ratios is used, the overall average of this statistic is zero. Thus, the average log relative error measure provides an unbiased summary

238

Exhibit 3

B.J. Dangerfield, J.S. Morris I Top-down or bonom-up

Series

A

B

TD MAPE BU MAPE TD MAPEiBU MAPE In(TD MAPEiBU MAPE)

2 1 2.0 0.69

1 2 0.5 -0.69

statistic which can be used to evaluate performance.

4. Results and discussion

In most situations, BU forecasting of family items produced more accurate forecasts. Exhibit 4, part (a), compares the two methods using both best fit models and randomly specified models. BU was preferable in 74% of the time series tested when best fit ES models were used and

(a) Overall results

73% of the time series when models were randomly specified. In addition, the average log relative error was positive in both the best fit and randomly specified models indicating better overall MAPE performance for BU forecasting. The average log relative error of 0.29 in the best fit models and 0.30 in the randomly specified models translates into a 34-35% higher MAPE, on average, for TD models. These results indicate that the fit of the forecast model does not have a substantial effect on the relative performance of TD and BU forecasting.

Model

Specification

Best fit

smoothing constant

Randomly selected

smoothing constant

Percent when

top-down better

(In(TDIBU) > 0)

26

27

Percent when

no difference

(In(TDIBU) = 0)

0

0

Percent when

bottom-up better

(ln(TDIBU) CO)

74

73

Average error

(Mean In(TD/BU))

0.29

0.30

(h) Results by item 1 proportion

Item 1

proportion

Low (O<p, CO.35)

Medium (0.35 < < 0.65) pI High (0.65 <pI < 1.0)

Percent when

top-down better

(In(TDIBU) > 0)

31

35

19

Percent when

no difference

(In(TD/BU) = 0)

0

0

0

Percent when bottom-up better

(In(TDIBU) C 0)

69

65

81

Average error

(Mean In(TDIBU))

0.26

0.21

0.36

(c) Results by item correlation

Item

correlation

High negative

correlation (-l.O< r< -0.25)

Low correlation

(-0.25 < r < 0.25) High positive

correlation CO.25 < I < 1 .O)

Percent when Percent when

top-down better no difference

(ln(TD/BU) > 0) (In(TDIBU) = 0)

18 0

27 0

30 0

Percent when

bottom-up better

(In(TDIBU) < 0)

82

73

70

Average error

(Mean In(TD/BU))

0.50

0.26

0.24

Exhibit 4. TD vs. BU performance using log relative MAPE


Effects of item proportion and item correlation in best jit models

In Exhibit 4, part (b), the results from the best fit models were separated into three categories based on the relative proportion (low, medium, and high) of item one (p,) in the group demand. BU forecasting was consistently superior to TD regardless of the relative distribution of demands for the two items that made up a family group. However, we did not observe as strong a symmetry as expected in the low and high categories of item 1 proportion. One might expect that these categories should produce not only the same choice of forecasting method, but also should do so about the same percentage of the time, all other things being equal. Moreover, these results are not consistent with those predicted by Schwartzkopf, Tersine and Morris (1988). Their model of TD forecast error predicted minimum total error when one item domi- nates the family series; however, they used MSE as their accuracy measure.

The influence of item series correlation is considered in Exhibit 4, part (c). The time series were separated into three categories of item dependence: high positive correlation, low correlation, and high negative correlation. BU forecasting performed better for all three categories

of correlation resulting in lower MAPE in 70- 82% of the series. Schwartzkopf, Tersine and Morris (1988) describe two opposing effects with respect to negative item correlation: model difference error which favors BU forecasting and reduced variability in the aggregate series which favors TD forecasting. The results indicate that the effect of model differences outweighs the improved stability of the aggregate series since BU forecasting is preferred 82% of the time for these families. The results for positively correlated item series are as expected; BU forecasting should produce better results owing to the increased variability in the aggregate series.

The interaction between item distribution ( pl) and item series correlation (r) is illustrated in Exhibit 5. This exhibit plots the average log relative MAPE errors for groups of time series with the same rounded item series correlation for three different levels of p,. The three MAPE plots indicate superior BU performance across all values of r for each of the three ranges of pl, although the log relative ratios are particularly high for item series that have a strong inverse relationship. The results for extreme correlation values (i.e. 1.0 or -1.0) should be viewed with caution, however, owing to the low frequencies in these categories.

2.5

. ‘\,

-1 -0.6 -0.2 0.2 0.6 1 r, rounded

- o<p<.35 _ .35<p<.65 _ .65<p<l

Exhibit 5. Average log relative MAPE, rounded r’s,

240 B.J. Dangerfield, J.S. Morris I Top-down or bottom-up

5. Limitations

The above results apply to families of two items. In addition, this research provides guidance on the more general question of whether an individual item series should be forecasted separ- ately from a family series. The two-item series, in this situation, would consist of an individual item series and another series that consists of the sum of the remaining family of items. By applying this logic recursively, one can continue to decompose the remaining aggregate series until all item forecast decisions have been made.

Other performance measures could and, in fdCt, did produce different results. Results using MSE as the accuracy criterion are reported in

Appendix A. Using MSE, BU forecasting produced better forecasts in only 34-49% of the more than 15,000 cases tested. However, given the unreliability of this measure reported by Armstrong and Collopy (1992), these results must be viewed with caution. The results using MSE are, however, consistent with the effects expected by Schwartzkopf, Tersine and Morris (1988). Items with similar proportions produced better BU forecasts.

In addition, care should be taken in interpret- ing the categorical results for item proportion and correlation. While care was taken to ensure that the item proportion categories had compar-

able densities, the distribution of series within categories was not consistent. Item correlation categories were formed on the basis of the statistical meaning of correlation and as a result did not contain comparable densities of the time series tested.

Winters’ exponential smoothing model was the only model tested. While it has performed well in previous studies, the external validity of these results could be extended by testing other time series models as well as casual forecasting models, Other methods for apportioning the family forecast to individual items should also be examined.

Finally, this research does not address other important factors such as cost and data accuracy.

6. Summary and conclusions

Bottom-up forecasting resulted in more accurate forecasts for nearly three out of four series tested regardless of model fit; the result was more pronounced when items were highly correlated and/or when one item dominated the aggregate series. We found no combination of item correlation and/or proportion where top-down forecasting produced a lower total MAPE than forecasts developed using individual ES models.

Appendix A. TD vs. BU performance using log relative MSE

(a) Overall results

Model

specification

Percent when top-down better

(In(TD/BU) > 0)

Percent when

no difference

(In(TD/BU) = 0)

Percent when bottom-up better

(ln(TD/BU) < 0)

Average error (Mean In(TD/BU))

Best fit smoothing constant

Randomly selected smoothing constant

63 3 34 0.02

51 0 49 0.03

(b) Results by item 1 proportion

Item 1

proportion

Low (Oip, CO.35)

Medium (0.35 < < 0.65) p, High (0.65 < p, < 1.0)

Percent when Percent when Percent when

top-down better no difference bottom-up better

(In(TD/BU) > 0) (ln(TD/BU) = 0) (In(TD/BU) < 0)

65 3 32

46 0 54 67 3 30

Average error

(Mean ln(TD/BU))

-0.02

0.19 -0.01


(c) Results by item correlation

Item

correlation

High negative

correlation

(-l.O< r< -0.25)

Low correlation

(-0.25 < r < 0.25)

High positive

correlation

(0.25 < r < 1 .O)

Percent when top-down better

(In(TDIBU) >O)

59

65

62

Percent when

no difference

(In(TD/BU) = 0)

2

3

3

Percent when

bottom-up better

(ln(TD/BU) < 0)

39

32

35

Average error

(Mean In(TDIBU))

0.13

-0.01

0.01

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Biographies: Byron J. DANGERFIELD is Associate Pro- fessor of Information Systems in the College of Business and Economics at the University of Idaho. He also serves as the Dean of the College of Business and Economics. He is a member of the Association for Computing Machinery and the American Decision Sciences Institute and has published in the Production and Inventory Management Journal and the International Journal of Operations and Production Manage- ment. He has presented papers at many conferences, including the Information Systems Education Conference, the De- cision Sciences Institute, and the Mountain Plains Manage- ment Association.

John S. MORRIS is Associate Professor of Production/ Operations Management in the College of Business and Economics at the University of Idaho. He is a member of APICS, TIMS. and the Production/Operations Management Society and has published articles in the International Journal of Production Research, Journal of Cost Analysis, Production and Inventory Management Journal, Journal of Manufactur- ing and Operations Management, Journal of Marketing Edu- cation, International Journal of Operations and Production Management, and Management Science. He has also presented papers at a number of national and local professional meetings including APICS, TIMSIORSA, the Decision Sci- ences Institute, and the Winter Simulation Conference.

top-down or bottom-up: aggregate versus disaggregate extrapolations

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