Forecasting market shares with disaggregate or pooled data: a comparison of attraction models

International Journal of Forecasting 10 (1994) 263-276

Forecasting market shares with disaggregate or pouled data: a comparison of attraction models

Youhua Chen* ,a , Vinay Kanetkar”, Doyle L. Weiss’ “Facutiy of ~a~ag~~e~~ University of Toronto, Toronto, CM., Canada “Faculty of Management University of Toronto, Toronto, ht., Canada

‘College of Business Administration University of Iowa Iowa City, IO. USA

Abstract

The objective of this paper is to compare market share forecasts using parameter estimates from three different ar~ngements of single source scanner data. The data arrangements are coupled with two different attraction models and a well known naive model. Each attraction model is specified with either autocorrelated errors or with a lagged attraction term on the right hand side. Forecasts are compared using data aggregated across stores and disaggregated by store. In the later case, parameters for each store are estimated and a composite (average) share forecast is formed or the data are pooled and a single set of estimated parameters provide the forecast.

The study finds that the full cross effects model with an autocorrelated error structure fits the data better than alternative models. However, the differential effects model with an autocorrelated error structure provides the best forecasts. With respect to the alternative data arrangements, the data aggregated across stores (i.e. chain level data) provides the best forecasts.

Keywords: Aggregate data; Disaggregate data: Empirical study; Predictive accuracy; Market share models; Pooiing data; Scanner data

Forecasting is an important activity that must be performed by every business. As a result, it is not surprising that researchers interested in branded packaged goods have conducted exten- sive research using market share response models to forecast brand shares. For the most part, this effort has paid limited attention to the

* Corresponding author. Fax: + 1 416-938-4312.

structural specifications and the refated issues of logical consistency and nested comparisons. It has also been somewhat casual in comparing models parameterised with data reflecting different collection periods and aggregations. It is not surprising to observe that many past efforts have produced some inconsistent results with respect to specification and forecasting performance. As a result, this paper has two general objectives; to controt for specification with the hope of explain- ing some of the past inconsistencies and to examine forecasting accuracy across different tevets and forms of data aggregation. We were

0169.2070/941$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDI 0169-2070(94)09003-N

264 R. Chen et al. I lnternational Journal of Forecasting 10 (1994) X-276

more successful with the second objective than with the first.

The inconsistencies mentioned above are evident in the literature. For example, Naert and Weverbergh (1981) found attraction share models to forecast consistently better than naive models or simple linear and multiplicative models. On the other hand, Brodie and de Kluyver (1984) found no significant difference between these and other specifications. Later, Brodie and de Kluyver (1987a) reported results in which naive market share models generally out-performed the other specifications. More recently Kumar and Heath (1990) provide results in which attraction models out-perform simple linear, multiplicative and naive models.

Bass (1987), Brodie and de Kluyver (1987a), Brodie and de Kluyver (1987b), and others have attempted to explain the inconsistent results mentioned above in terms of inherent random- ness, specification error, data interval aggregation, and the like. Kumar and Heath (1990) provide an effective summary of the important results from this literature and offer guidelines for additional efforts from their own results . In general, they conclude that:

(1) weekly data is preferred since it provides the variation necessary for model testing and conforms to the periodicity of in-store promotional activities;

(2) fully parameterised models (i.e. models including price and promotional variables) are

necessary for good forecasts; (3) GLS (SURE) estimation procedures are

generally preferred to OLS procedures; (4) unconstrained attraction (differential ef-

fects) models produce the most accurate results among the specifications tested.

The most complex specification examined by Kumar and Heath (1990) was the differential effects form of the attraction model. They pooled store-level data to provide their results.

This paper extends the results of Kumar and Heath (1990) and contributes to the associated literature in three important ways:

(1) First, it extends the structural specifications they examined to include the full cross effects model:

(2) Second, it examines two different error term specifications to account for the auto correlation often found in grocery products data.

(3) Third, it attempts to answer questions raised about the forecasting accuracy of contem- poraneous versus pooled data arrangements discussed by Wittink et al. (1992) and Blattberg and George (1991).

Specifically, with respect to data arrangement we answer the question, “When modeling and/ or forecasting, what is the appropriate aggregation of scanner data?” Is there one aggregation appropriate for modeling purposes (see for example Wittink et al. (1992)) and another for forecasting purposes?. We do not consider tem- poral aggregation. The collection period for our data is constant at 1 week.

Blattberg and George (1991) noted that models using pooled data provided plausible parameter estimates and more accurate forecasts than did the models using store by store (disaggregate) data. These researchers did not compare their models with naive models or with models which introduce lag structures directly. In this paper we provide such comparisons.

In the following section we describe the models that provide the basis for the response function specifications we compare and discuss alternative error structures imposed on these models. In Section 3 we describe our data, discuss estimation procedures, and provide a description of the statistics used to evaluate predictive accuracy. Section 4 presents and dis- cusses our results and Section 5 contains a summary and concludes our paper.

2. Multiplicative competitive interaction models

One of the frequently used functional forms for describing the effects of marketing mix variables on brand shares is the so-called multiplicative interaction model (see Nakanishi and Cooper (1982) and Nakanishi and Cooper (1974) for its early application and development). In its most elaborate form the cross effects brand attraction model can be written as

R. Chen et al. I International Journal of Forecasting 10 (1994) 263-276 265

ai, = exp(y, + pi,) fi fi X$y h=l j=l

(1)

where the ‘yi are brand specific constants, Xhjt is used to denote the value of mix variable h for brand j in period t, and the pi, are error terms. We present and discuss alternative specifications for the error terms below.

We also estimated multinomial logit variations of the attraction model where the product operators are replaced by summation operators and raised to the power e. We found this alternative specification had no impact on forecasting accuracy.

Note that there are H mix variables and n competing brands in the product market. For the fully extended model, attraction elasticities for brand i depend not only on the actions taken by its brand manager but also on the actions of all other brands. This interaction is implied by the set of coefficients, phij, in Eq. (1). The model is sometimes called the asymmetric response model (See Carpenter et al. 1988). In this paper, we will refer to it as the cross effects model.

In the marketing literature the a, are usually referred to as “brand attractions” and when they are divided by their sum a market share “attraction model” results (see Bell et al. 1975). That is

with the m, being attraction shares for brand i in period t. In practice it is customary for researchers to equate attraction shares to brand volume or choice shares. When brand volume share is used it is proper to think of the a,, functions as demand relationships and the phij values as demand elasticities. Market or attraction share elasticitiesaren,,ij = Phij - C~,,;m,f3hii wherem,is share for brand k.

Conceptually, for a set of parameters phij, all attraction models satisfy both the sum constraint (i.e. Cy=,mi = 1) and the range constraints (i.e. 0 =Z m, < 1 for all i). These conditions are associated with the issues of logical consistency discussed by

McGuire et. al (1968) and models which conform to them provide a level of “theoretical comfort” and correctness. Inthiswork,severalalternativemodels which maintain logical consistency, yet allow one to compare alternative specifications are evaluated. We think that this is an important feature of the present investigation.

In many product markets the flexibility provided by the cross effects model is needed because the effects of brand competition are not likely to be symmetric. That is, the cross mix attraction elasticities can be expected to be unequal between pairs of brands. Such inequality may arise for several reasons. It may be because of different positioning of the various brands, variations in product quality among the brands, differences in the effectiveness of brand promotions (including advertising), differences in distribution values, or differing consumer brand franchises and the like. Any of these conditions could cause weakly partitioned sub-markets (seg- ments) to arise naturally in a product class as an expression of unequal competition among the brands. Such inequality is effectively modeled by the asymmetric coefficients matrix of the cross effects model.

2.1. Error structure specifications

It is not unusual for researchers working with market share response models and grocery products data to find that the error components of the ai, exhibit some form of auto correlation. Several explanations have been provided to explain this finding; the most common being mis-specification due to missing variables. This is likely true for our data as well. Our purchase data is the result of a complex set of activities pursued by consumers, retailers and manufacturers. How- ever, our independent measures are incomplete as explanatory variables for these activities. For instance, we do not have measures for demo- graphic variables or advertising’s effect, to name two such variables. In addition, the parameter vector size and the complexity of the cross effects model’s functional form make it difficult to accommodate brand loyalty, carry over effects, partial adjustments phenomena, or other

266 R. Chen et al. I International Journal of Forecasting 10 (1994) 263-276

forms of inertia thought to induce auto correlation into the sales series.

Because of the numerous findings of auto correlation effects in the response model literature, we specify, test, and evaluate models with autoregressive and other error specifications to accommodate inadequacies in our specifications and data. For the first order autoregressive scheme the basic model is

&it = PPjr- I + Et, (3)

where p is an autoregressive parameter to be estimated from the data and assumed to lie within the interval minus one and plus one; E,, is a white noise error term. As an alternative to the first order scheme, one can argue that a some kind of an inertial phenomena is working with

respect to brand attraction (see for example, Weiss and Windal (1980)). In such a case, the structural model can be expressed as

h=l ,=I (4)

where Ai can be thought of as an indicator of habit or loyalty persistence and l it is a white noise term. In this context, A is expected to be positive and all of our estimation results support this expectation. As explained in the literature, (see Weiss and Windal (1980), and Leeflang et al. (1992)) the model specifications contained in Eqs. (1) and (4), and Eqs. (1) and (3) are nested and can be compared using likelihood ratio tests. That is, share models based on specifications (1) and (4) can be written in reduced forms which are also nested and whose parameters can be estimated using least squares procedures. How- ever, the reduced forms based on Eqs. (3) and (4) contain the same number of parameters and cannot be compared with likelihood ratio tests. This issue of model comparison is taken up in the next section.

3. A methodology for comparison of models and forecasts

In this section we discuss our data, establish the reduced forms and procedures used for

parameter estimation and discuss measures to be used to judge the forecasting accuracy of our results.

3.1. Data: automatic dishwashing detergent

Parameters for our models are estimated using automatic dishwashing detergent data collected and made available by A. C. Nielsen of Canada. One hundred 4-weeks of data are available for five stores belonging to the same supermarket chain in the province of Ontario, Canada. The data span the period September 1989 through August 1991. We have aggregated each individual stock keeping unit into weekly data for six brands. We divided the data into an estimation sample of 94 weeks and a hold-out sample of ten weeks. Because of the lagged and auto correlated models we ended up with 93 weeks of data for six brands sold at five stores.

Four marketing variables, price, deal amount, feature, and display are used in our analysis. Operational definitions are provided below.

Prices. Prices are dollars per kilogram and are based on the average unit price of all stock keeping units for a brand in that particular week. Each brand has, on average, 3.5 stock keeping units. We think that the definition of unit price used here best reAects the regular price of a brand.

Deal Amount. It is measured in dollars per kilogram and computed by subtracting the lowest unit price available in a store for a brand from the regular unit price. We recognize that our computation may not be the actual price pro- moted, but we think it is a reasonable proxy for it.

Feature and Display. We created two indices for measuring feature and display. Although it is common to find feature and display activities to be correlated, this is not a problem with this particular dataset. The highest correlation between feature and display is between brand 4 and brand 1 and is - 0.21. The indexed measure of feature (or display) reflects the particular brand being featured (or displayed) and also provides a measure relative to the other brands. For instance, if nr (or nJ denotes the number of brands featured (or displayed) among n compet-

R. Chen ec al. I International Journal of Forecasting 10 (1994) 263-276 261

ing brands in a given week, then the index for Subtracting log mk, given in Eq. (5) from log m,, feature for brand i (Xi,) is defined as given in Eq. (2), the simple form:

n - if brand i is featured and

$5 = nf

1 - z otherwise

Such an index was first proposed by Nakanishi et al. (1974) and used by Kumar and Heath (1990).

It must be noted that we have included variables that are commonly available in the scanner databases. It is acknowledged that mass advertising and store shelf and inventory variables are not included. It is possible that these variables influence store brand sales and hence shares. We have assumed that the impact of missing variables will be reflected in the error structure of the models. A number of such alternative specifications and their reduced forms are presented below.

We review below the so-called log ratio reduced form used with least squares procedures to provide share estimates for the various specifications described above. Cooper and Nakanishi (1988) provide an alternative but similar transformation. For a comparison of the two trans- formations and their implications for estimation purposes, see Houston et al. (1991).

We show that the log ratio reduced form provides share estimates that are both sum and range constrained (see above) and are therefore logically consistent. In addition, the transformation nests both the so-called naive models with forms of the attraction models found in the forecasting literature. The nested structure is important since it provides a convenient frame- work with which to compare the various models specified in the paper.

3.2. The log ratio transformation

A system of reduced form equations for the cross effects share model becomes immediately available by noting that

log mk, = log akt - log@ aJ

log mrr - log mk, = log a,, - log ak, (6)

is obtained. The above transformation is implicit in the work of Theil (1969) and was used by McGuire et al. (1977) as a means for dealing with some of the issues of logical consistency associated with market share models. For con- venience we will refer to this reduced form as the log ratio transformation.

Finally, note that unique estimates of the structural coefficients (p,,) of Eq. (2) cannot be recovered from the estimated coefficients of the log-ratio reduced form. There are Hn structural

P ,,,, values and only H(n - 1) reduced form coefficients. (See Kanetkar et al. (1993) for a discussion of this identification problem.) How- ever, share forecasts can be made from estimates of the reduced form coefficients by using the sum constraint imposed by the definition of market shares.

3.3. Reduced form error specifications

If the model contains an auto regressive error structure as given in Eq. (3), we may write the reduced form as

log mi, - log mn, = POog m,,- 1 - 1% m,,- 1 >

+ Cl- PKY, - xl>

+ 2 i (Phij - Phnj)(lOg xhjl - P log xhjt- 1) h=l j=l

+ ‘it - ‘,I (7)

where II denotes the base brand (in our usage the last brand). There are (n - 1) such equations in the reduced form. Based on the results of Kumar and Heath (1990), we have made the simplifying assumption of a homogeneous auto correlation effect; i.e. a single p. Kumar and Heath (1990) report auto correlation coefficients for three brands of 0.304, 0.273, 0.369, and 0.142, 0.132, and 0.138.

If, however, the auto regression results from habit persistence, brand loyal or other inertial processes as described in equation (4), then the log-ratio transformation for the model may be written as

268 R. Chen et al. J International Journal of Forecasting 10 (1994) 263-276

Further, if we let A, = A, for all i = 1, 2, . . . , n, the model reduces to:

log rni, - log %, = (n - r,,)

+ ,$, ,g, (Phij - Phnj)loS x!zjt

+ h(lOg”ir-I -l”gm,,_,) + ‘jr -S,r (8)

There are several features of these two reduced forms which are important to this discussion. First, if p is invariant across brands, Dur- bin’s two stage method or Cochrane-Orcutt’s iterative least squares procedure applied to Eq. (7) will provide unbiased and efficient estimates of the reduced form parameters (see Kennedy (1992, p.123)). Second, several alternative models reported in the literature are special cases of these two reduced forms. For instance, if &ij = & when i = j; and phi, = 0 when i #j we have the differential effects models used by Kumar and Heath (1990). Third, note that when the structural modei contains auto regressive errors, the models used by Kumar and Heath (1990) and Brodie and de Kiuyver (1987a) are mis-specified. Note also that, if A = 0 and p = 0, we obtain a model that contains no time depen- dency and forecasts depend only on current marketing variables. Finally, when Phij = 0 for all i, j = 1,. . . , n and A f 0, we have a naive model that contains lagged market share as the only explanatory variable. As can be seen from the above discussion, two structural models and their resulting reduced forms cover a wide variety of the models reported in the marketing and forecasting literature.

3.4. Models, estimation procedures and data urrangements

Models We estimate seven alternative structural models based on the reduced forms given in Eqs. (7) and (8). Referring to these equations the models are:

(1) The naive model; formed when Phjj = 0 for all i, j and h and p = 0.

(2) The current effects model; formed when p = 0 and h = 0. There are two alternative structural specifications for this model.

(a) The differential effects model; formed

when Phi, = &j for i = j and phij = 0 for i #j (b) The cross effects model. (3) The brand loyal model; formed when p =

0. There are also two alternative structural specifications for this model.

(a) The differential effects model; formed when phii = &,, for i = j and Pijh = 0 for i # j

(b) The cross effects model. (4) The auto correlated error model; formed

when h = 0, There are also two alternative structural specifications as described above for the current effects and brand loyal models.

We use data for six brands and four mix variables; price, deal, feature and display. As a result, the proliferation of parameters caused the full cross effects models to be too large for our estimation routines. Cross effects were main- tained only for price and deal amount. Feature and display were treated as differential effects variables in the cross effects models. We estimated cross effects model for these variables. The likelihood ratio test statistics indicated that the differential effects is preferred over the cross effects model for these two variables.

3.5, Estimation procedures

There is mixed evidence as to whether maximum likelihood procedures, generalized least squares procedures (GLS) or ordinary least squares (OLS) procedures are better at providing estimates and forecasts when multiple equations are involved. Fair (1973) reported that maximum likelihood procedures worked well. Kumar and Heath (1990) reported that GLS was not always preferred to OLS. Moreover, Brodie and de Kluyver (1984), Chambers (1990) and Ghosh et al. (1984) provided support in favor of ordinary least squares. In our case, several of the models involve across equation constraints on parame-


ters and Zellner’s SUR estimation procedures are preferred. However, because of the number of parameters involved in our models we had little choice except to estimate parameters equation by equation using OLS.

As can be seen from the tables provided in Section 4, when disaggregate data are used (Table 3) the number of parameters to be estimated for the naive models is 26, while the cross effects models require estimates for 386 parameters. Even for pooled data (Table 3), 98 parameters are required for the cross effects models. As a result, we are unable to provide GLS estimates for many of the differential effects and cross effects models. We used equation by equation OLS coupled with Cochrane-Orcutt procedures for autocorrelation. The alternative was to aggregate the data and form a reduced set of brands. For instance, we could have arranged the data to reflect two major brands and an all other brand as is sometimes done..

A related issue concerns the parameters p and h. A test of the hypothesis that pi St p for all i was rejected. For with disaggregate data x2 = 18 for 20 degrees of freedom. Similar values were obtained for the other model-data combinations. However, because of using OLS we cannot constrain the estimates for p or A to be equal across equations. Consequently, we estimate p and X for each equation and choose the value p or h that minimizes the sum of squared differences between the observed and estimated shares for the estimation sample.

Data arrangements Parameters for all of the models are estimated

with three arrangements of the data as described below.

(I) Disaggregate: separate models are estimated for each of the five stores. Consequently, information across stores is not strictly accounted for and parameters reflecting the impact of marketing mix variables are different across stores. For instance, the effect of across chain promotions or manufacturer sponsored promotions are not accounted for in the model.

(2) Pooled: marketing effects are estimated as fixed across stores and store differences are

accounted for with brand constants varying across stores. (3) Aggregate: data are aggregated across stores and parameters are estimated. In this instance, sales are first aggregated across stores and then brand shares are calculated.

We assess forecasting effectiveness by calculat- ing three measures of predictive accuracy: root mean squared errors (RMSE), the average absolute error (AAE), and the median relative absolute error (MdRAE), a measure proposed by Armstrong and Collopy (1992). The median relative absolute error is a function of the relative absolute error (RAE) which along with the other measures is defined below. That is

m rt T I!2

RMSE = ,z ,; Tgir - AiJ

(mnT)

,,I n T

AAE = ,z 2 g';ir - AjJ)

mnT and

il: i: I$, - A,;,1

RAE, = ‘==!I ‘;I

where Fji, and Aii, are forecasted and actual values of shares at store j for brand i in week T, y.;, is the random walk forecast of share at store jfor brand i in week T, T is the number of weeks in the hold out sample or forecast horizon, II is the number of competing brands in the product market, and m is number of stores. It should be noted that for aggregate data, m = 1; for the other data forms m = 5.

Armstrong and Collopy (1992) rank the RAE observations and define the median relative absolute error measure (MdRAE) as the median value of the series; that is, observation n + l/2 if y1 is odd, or the mean value of observations n/2 and (n/2) + 1 if n is even. The MdRAE is chosen for two reasons. First, as Armstrong and Collopy point out, when only smali sets of series are

270 R. Chen et al, I International Journal of Forecasting 10 (1994) 263-276

available (we have only six brands), the MdRAE is appropriate in terms of reliability, construct validity, and outlier protection which are pri- mary criteria for comparing forecasting methods. Second, the RMSE has been criticized by au- thors for its unreliability (Armstrong and Col- lopy) when outlier or extreme observations are present. The RMSE and AAE are included because of their simplicity and popularity.

4. Results

In Table 1 we provide a summary of estimated elasticities using three different data arrangements and for six different structural models. Table 2 summarizes estimated own market share

Table 1

Number of estimated market share elasticities with the

expected signs”

Model Price

Estimated using pooled data Current Diff. 24” effects cross 33 Brand Diff. 18 loyal Cross 33

Auto- Diff. 30 correlated Cross 32

Deal Feature Display

36 36 36

29 36 24

36 36 36

29 36 24

30 36 36

30 36 36

Estimated using disaggregate data Current Diff. 90 174

effects cross 133 129 Brand Diff. 108 174 loyal Cross 136 132 Auto- Diff. 180 138 correlated cross 133 129

132 156

138 138

126 162

120 162

108 138

126 162

Estimated using aggregate data Current Diff. 18 30 24 36 effects Cross 19 25 24 36 Brand Diff. I8 30 18 36 loyal Cross 21 24 24 36 Auto- Diff. 30 30 24 30 correlated Cross 27 24 30 30

” It is expected that own price, cross deal, feature and display elasticities will be negative and cross price, own deal. feature

and display elasticities will be positive. ‘There are 36 elasticities estimated for models based on

pooled and aggregated data and 180 elasticities per mix

variable estimated for models based on d&aggregated data

for each of six models.

Table 2

Own market share elasticities for a randomly chosen brand

by data arrangement and structural model”

Model Price Deal Feature Display

Estimated using pooled data

Current Diff. -4.010 2.138 0.154 0.260 effects Cross -4.846 2.116 0.171 0.313 Brand Diff. -3.548 1.752 0.128 0.319 loyal Crass -4.846 2.116 0.171 0.313 Auto- Diff. -4.024 2.138 0.154 0.260

correlated Cross -4.725 2.097 0.127 0.267

Estimated using disaggregate data Current Diff. -2.152’ 2.096 0.235 0.274

effects cross - 1.407 2.742 0.138 0.255 Brand Diff. -2.319 2.047 0.185 0.265 loyal cross - 1.312 2.908 0.137 0.256 Auto- Diff. -3.961 2.784 0.178 0.234 correlated Cross -4.486 3.092 0.139 0.243

Estimated using aggregate data Current Diff. - 1.222 1.539 0.001 0.312

effects Cross -0.201 2.385 0.053 0.238 Brand Diff. -0.286 1.249 -0.032 0.310 loyal Cross -0.429 1.954 0.013 0.264

Auto- Diff. - 1.087 2.458 -0.210 0.539 correlated Cross -0.883 2.149 -0.054 0.236

d Estimated elasticities are obtained by using estimates of

structural parameters as well as own and competing brand

market shares (see text). Consequently, it is difficult to

estimate the standard error associated with these elasticities.

’ Elasticities are obtained separately for each store and the

mean of five stores is reported.

elasticities, evaluated at the mean values of the marketing mix variables for one of the six brands. Elasticities from three alternative data arrangements are used. Log-likelihood values and xZ statistics for model comparisons are reported in Tables 3 through 5. Tables 6 through 11 report the measures of forecasting accuracy (RMSE, AAE and MdRAE) discussed in Sec- tion 3. We provide comments and observations concerning the results reported in these tables below.

4.1. Face validity and model testing

Face validity We establish face validity for our specifications

by examining signs and magnitudes of estimated elasticities for the four marketing mix variables.

R. Chen et al. i International Journal of Forecasting 10 (1994) 263-276 271

Table 1 provides a summary of the number of proper signs for estimated own and cross elasticities for each of six models with three different data arrangements. We reject the constant effects model. That is, we test whether price parameters are identical across brands and reject the null model.

We find that when pooled data are used, 90% of the estimated elasticities (for all of our models) have the expected signs. It is expected that own price, cross deal, feature and display elasticities will have negative signs and cross price, own deal, feature, and display elasticities will have positive signs.

For pooled data, the best model appears to be the auto correlated cross effects model. It has 134 out of 144 or 93.1% of the estimated elasticities with the expected signs. The differential effects model (both the current effects and brand loyal forms) are also very close with 91.7% signs estimated correctly.

When disaggregate data is used, we estimated 36 elasticities per store per marketing mix variable or 720 elasticities per model. Although we estimated six and 30 parameters per mix variable for the differential and cross effects models respectively, the number of estimated elasticities per mix variable is identical for both models. We find that the differential effects model, with the brand loyal specification included, performed very well with 79.2% of estimated elasticities having correct signs. The cross effects model with an auto correlated error structure, however, is a close second with 78.3% of the estimated elasticities having the expected signs.

Finally, when the data are aggregated, the auto correlated error structure coupled with the differential and cross effects models provided the best results with 83.3% and 81.3% of the estimated elasticities (respectively) having correct signs.

To summarize, in comparing signs of estimated parameters, we find that the relative frequencies of correct signs are highest when pooled data is used. Overall, however, the disaggregate and aggregated data arrangements provide comparable results. With respect to the models, best results are observed when the auto

correlated error structure is coupled with either the differential or cross effects model. These models provide estimates with the most correct signs and are, on that basis, preferred to the other models.

Since it is not feasible to present the three hundred or so estimated coefficients and associated elasticities for the various models, we re- strict our attention to one brand, chosen randomly, and report all of its own elasticities under the same set of situations reported above. Table 2 contains this information and provides additional evidence of face validity for our specifications and the various data arrangements used. Because we have not modeled cross effects for feature and display variables, estimated elasticities for these two variables appear similar for both pooled and disaggregate data. For aggregate data, estimated display elasticities are generally small and often have incorrect signs. In addition, estimated feature elasticities show high variability across the various models; a sign that these parameters may be unstable.

Estimated price and deal elasticities, vary not only across the various data arrangements but also across the structural specifications as well. When data are pooled, the cross effects models, generally produce stable estimates for both price and deal amount elasticities. The differential effects model, however, understated price sen- sitivity in comparison with the cross effects model. There is no such clear pattern for the deal variable.

When the data arrangment is disaggregate, price elasticities lacked the clear pattern evident for pooled data. In addition, estimated deal elasticities are generally understated by the differential effect model when compared to the cross effects. When aggregated data are used, price and deal elasticities are substantially small in absolute terms when compared to those of other data arrangements. We take this to be a sign that while the aggregated data may maintain the micro relation’s directionally, the estimated elasticities may be substantially biased.

In summary, we note that models using pooled data provides better face validity than models estimated with either disaggregate or aggregate

272 R. Chen et al. I International Journal of Forecasting 111 (1904) 26%276

data. That is, the estimates from pooled data are more consistent in expected sign, and reasonable in magnitude. Although we do not report the results here, the pooled data arrangment produced parameters that are mostly significant. As a result, we believe that our models based on pooled data pass the test for face validity. If one has to choose between the disaggregate and aggregate data, then it appears there might be tradeoffs between sign and magnitude of the estimates. For the various structural models there was not dramatic differences in face validity and all of the models appear to be accept- able by this measure.

Model testing Tables 3, 4, and 5 presents likelihood ratio statistics, estimated chi square values, and critical values for selected likelihood ratio tests. For the models we use, any pair, except those pairing lagged and auto correlated models, can be nested. However, critical values necessary to test all possible pairs are not provided. Even so, there is little uncertainty that the auto correlated cross effects model is the preferred model when likelihood values are used as criteria.

As can be seen from the series of likelihood ratio tests in Tables 3, 4, and 5, the naive model is rejected for all cases in which it can serve as the null hypothesis. In addition, the differential effects model is rejected in all cases when it can be compared with the cross effects models. Moreover, it is clear that the lagged effects models are superior to their current effects counterparts. And as expected, the current effects mode1 is rejected when compared to the auto correlated cross effects model; in any pairing. For Table 3, comparing the current cross effects model and the auto cortelated cross effects model, the degrees of freedom are 5, the x2 statistic is 2136 and the critical value is 11 at p

G 0.05. These results are robust in that they are

consistent across all three data arrangements and we are left with the choice between the non- nested pairings of the lagged effects models and the auto correlated models.

Choosing between the non-nested pairings of

Table 3

Comparison hetwecn alternative model fits and number of

estimated parameters models estimated by store--disaggre-

gate approach

Estimated

model

Naive”

Number of Log- XI XL parameters likelihood Estimate critical

statistics value

30 -9494

Current Effects

Differential” 145

Full crossL 38.5

-9125 73Sd 141

_- 8883 484’ 277

Lagged Effects

Differential 150 -9101 4x’? II

Full cross 390 -- 8849 504” 277

Autocorrelated

Differential 1 so -8879 492” 11

Full cross 390 -7811 2136’ 277

“There five brand constants per store and one parameter to

capture the lagged effect

hThcre are 29 parameters per store (four marketing mix

variables and each mix variable has six parameters and five

brand constants).

’ Note that price and deal amount variables have 30 parame-

ters each, feature and display have six parameters each and

five brand constants. Consequently, there arc 77 parameters

per store.

’ Compared to the naive model. ’ Compared to the differential effects model.

’ Compared to the current differential effects model.

’ Compared to the lagged differential effects model.

” Compared to the current differential effects model.

’ Compared to the autocorrelated differential effects model.

the lagged effects models and the auto correlated models is difficult. While some theory and some ad hoc procedures have been developed for single equation models, we are aware of no such development for systems of equations. Still, based on the changes in likelihood values it appears that the auto correlated cross effects model fits the data best. We find this somewhat surprising and suggest some reasons for it in Section 5 below.

4.2. Predictive accuracy

Tables 6 through 8 provide measures for one week ahead forecasts while Tables 9 through 11 provide measures for 10 week ahead forecasts.


Table 4

Comparison between alternative model fits and number of

estimated parameters models estimated by store-pooling

approach”

Estimated Number of Log- X2 X2 model parameters likelihood Estimate critical

statistics value

Naive” 26 -9613 -

Current Effects

Differential 49 -9227 772 35

Full cross 97 -9167 120 67

Lagged Effects


Full cross 98 -9001 264 65

Autocorrelated


Full cross 98 -8989 86 65

a See footnotes to Table 3.

In all cases, the models have been used to forecast brands shares for the aggregate five stores. For instance, for the disaggregate store- by-store models, brand shares for individual stores are forecast and then compared to actual store by store sales. The results are then aggregated for the five stores. No weighting is applied to account for store volume.

As is noted in the literature, good parameter

Table 5

Comparison between alternative model fits and number of

estimated parameters models estimated by aggregating across stores=

Estimated Number of Log- X2 X2 model parameters likelihood Estimate critical

statistics value

Naive 6 -1254 -

Current Effects

Differential 29 - 1222 64 35 Full cross 77 -1165 114 65

Lagged Effects Differential 30 -1182 72 4 Full cross 78 -1144 84 65

Autocorrelated

Differential 30 -1180 76 4 Full cross 78 -1138 96 65

a See footnotes to Table 3.

Table 6

Short term forecasting (one week ahead) performance of

various models estimated by StoreAisaggregate approach

Specification RMSE AAE MdRAE

Lagged market Naive 111.75 80.76 61.25 share model model

Current effects Differential 92.52 67.10 57.58 model effects

Full cross 81.06 58.46 49.30

effects

Lagged market Differential 91.14 68.86 59.17

shared model effects

Full cross 74.93 52.55 43.98

Effects

Autocorrelated Differential 68.05 48.42 41.44 Error effect

Full cross 75.52 52.19 42.67

effects

RMSE and AAE x lo3 and MdRAE x 10’.

estimates and better fit between data and model are necessary but not sufficient conditions with which to obtain accurate forecasts. The measures of predictive accuracy in Tables 6 through 11 are consistent with this observation. For example, in Table 6 the best predictive accuracy occurs when the model is specified to be the differential

Table 7

Short term forecasting (one week ahead) performance of

various models estimated by store-pooling approach


Lagged market Naive 124.52 89.34 62.46

share model model

Current effects Differential 70.94 53.01 43.48

model effects

Full cross 66.81 46.28 37.30

effects

Lagged market Differential 63.33 44.62 37.55

shared model effects

Full cross 64.87 45.69 38.59

Effects

Autocorrelated Differential 60.49 43.25 36.56

Error effect

Full cross 64.60 45.32 37.17

effects

RMSE and AAE x 10’ and MdRAE x 102.

214 R. Chen et al. I International Journal of Forecasting 10 (1994) 263-276

Table 8 Short term forecasting (one week ahead) performance of

various models estimated by aggregating across stores



Current effects Differential 69.53 S2.63 46.05 model effects

Full cross 62.61 49.42 42.58 effects

Lagged market Differential 49.32 36.03 30.55 shared model effects

Full cross 4x.41 34.67 34.81 Effects


Full cross 45.19 32.34 29.25 effects

RMSE and AAE X IO’ and MdRAE x 10’.

effects model with an auto correlated error structure. However, we report above that the cross effects mode1 with an auto correlated error structure is the preferred specification when likelihood ratios are the criteria (see Table 3). It appears that the parameters added by the cross effects mode1 improve fit but add little to forecasting accuracy, a disappointing and difficult to explain finding.

Note also that the models with auto correlated

Table 9

Medium term forecasting (lO-week ahead) performance of

various models estimated by storedisaggregate approach




Full cross 76.60 54.24 45.59

Effects

Autocorrelated Differential 69.32 50.21 41.22 Autocorrelated Differential 47.39 32.94 25.31 Error effect Error effect

Full cross 80.46 66.18 47.35 Full cross 47.46 32.71 30.05 effects effects

RMSE and AAE X 10’ and MdRAE x IO’. RMSE and AAE x IO’ and MdRAE x 10’.

Table 10

Medium term forecasting (lo-week ahead) performance of

various models estimated by store-pooling approach


Lagged market Naive 99.18 71.53 63.36

share model model


Full cross 66.64 47.OY 39.84

Effects


Full cross 66.05 46.40 37.74

effects

RMSE and AAE X 10’ and MdRAE x 10’.

error structures improve predictive accuracy by as much as 30%, for the disaggregate data, to more than 50% when pooled or aggregate data are used and the comparison is with the lagged models. Although alternative measures of accuracy are not completely consistent across the models and data arrangement pairings, overall we find that the differential effects mode1 with auto correlated errors offers the best forecasts. This is true for both one week and ten week forecasting horizons.

It is also noteworthy that the differential effects mode1 with auto correlated errors pro-

Table 11

Medium term forecasting (IO-week ahead) performance of

various models estimated by aggregating across stores



Lagged market Differential so.15 37.64 35.81 shared model effects

Full cross 54.15 37.64 35.81

Effects

R. Chen et al. I International Journal of Forecasting IO (1994) 263-276 27s

duced substantially different forecasts than those produced by the other two versions of the same model. On the other hand, the three versions of the cross effects model produced essentially similar forecasts.

Comparing alternative data arrangements provides additional insights. As might be expected, the pooled data produce better forecasts than the disaggregate data. However, it is somewhat surprising that forecasts derived from aggregate data are consistently preferred to those from pooled or disaggregate data.

In summary for our data, the auto correlated differential effects model used with aggregate data provides the best forecasts. This result is robust in that it is independent of the measures of predictive accuracy used for comparisons, arrangements of data used to fit the models, and forecasting horizon.

5. Summary and discussion

This study is designed to compare various market share response (attraction) models across three different data arrangements. Base comparisons are made with a common naive model.

Parameters for the differential and cross effects models with auto correlated and lagged structures are estimated using aggregate, disaggregate and pooled (weekly) data. A. C. Nielsen, Inc. provided scanner data for dishwashing detergent for the study. The study compares the models on the basis of likelihood values (structural integrity) and forecasting accuracy (managerial interest).

Structural comparisons indicate the auto correlated cross effects model is preferred to the other models. This result is stable across the three data arrangements used. However, it is our judgment that better or more reasonable parameter estimates are associated with the pooled data.

The superiority of the cross effects models does not hold for measures of forecasting accuracy. The more complex cross effects models provide no better forecasts than the differential effects models. However, as with the structural comparisons, the auto correlated error specifica-

tion is superior to the lagged model. These results for auto correlation are independent of the particular measure of forecasting accuracy used and are somewhat surprising. It might be the case that auto correlation is evident because of missing variables. One obvious candidate for a missing variable is television advertising which is not included in our data. Also missing are details concerning promotional execution. For instance, the size and location of a display in a store is not part of our data. Finally, store inventory (stock outs) may be important and it is also missing from our data.

With respect to alternative data arrangements, we find that pooled data provide better estimates of parameters although data aggregated across stores (chain level) provide the best forecasts. From our results it appears that there is not an appropriate data level for both forecasting and structural testing purposes.

We end with a comment about forecasting and the real world. Because we are using data from the past, we are able to use the actual values of marketing mix variables to provide forecasts. A chain or store manager does not have such luxury but instead must forecast own and competitors’ actions before the models can be used to provide forecasts. In other words, our response function forecasts are contingent on estimates of competitive marketing mix decisions. Such decisions are difficult to estimate and managers can be expected to use the response functions as ‘what if’ mechanisms to optimize mix decisions. Consequently, this will make judgments of forecast accuracy difficult in the real world and suggests a need for a methodology to forecast mix decisions of competitors.

Acknowledgments

We thank A.C. Nielsen of Canada for making the data used in this paper available. Vinay Kanetkar acknowledges the support received from the member companies of the Canadian Centre for Marketing Information Technologies (C2MIT), and the financial assistance of the Social Sciences and Humanities Research Coun-

cil, Canada (Grant #al&92-1292) and Humanities and Social Sciences Committee of the University of Toronto.

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Biographies: Youhua CHEN is doctoral student in Opera-

tions Management. at the Faculty of M~in~~~ern~nt, the

University of Toronto. He t-eccived B.E. in ~~~clianical

Engineering from the Qinghua University. China and M.A.

Economics from the University of Watcrlno. Canada. His

research interests arc in econometric methods in business

applications. and the interface between manufacturing and

distribution activies.

Vinay KANETKAR is Assistant Professor of Marketing at

the LJniversity of Toronto. He holds Ph.D. and M.Sc.

degrees from the University of British Columbia. His rc-

starch and teaching interests conccntratc on marketing

rcscarch and models. particularly in their applications to a

variety of marketing nl~~n~~gcrnent problems. He is research

associate with Canadian Centre for Marketing Inf~~rrn~ltion

Techn~~i~~~ies at the University of Toronto and this capacity

he has been developing variety of forecasting models for

retailer and manufacturing organizations. His most recent

work has appeared in Jourd of Marketing Rexwch, Or-

gar~imtionul Uehol~ior cd Humc~n LlcGion Proce.w~~s. Jaw-

rml of Ir~drr,stritrl Econornic~ and Markding .kie~m~.

Doyle L. WEISS is the John F. Murray Professor of Markct-

ing at the University of Iowa. Previous to Iovva he has held

appointments at the University of British Columbia, Purdue

University, University of Pittsburgh and Indiana University. Professor Weiss holds a Ph.D. in Industrial Administration

from Carnegie Mellon University. His writing has appeared in marketing and behavioral science journals such as Re-

~1~1~~~~~11 Science. ~0llrFld of &~sitie.w. ~~~1~~~~~1~ of Marketing,

Journd of Markdng Revecrrih, and ,~urk~t~ng Science.

Forecasting market shares with disaggregate or pooled data: a comparison of attraction models

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