Consumer demand analysis attracts considerable attention. It remains an open question, however, whether estimating demand with aggregate data is reliable when disaggregate store-level data is given. Demand models may produce biased results when applied to data aggregated across stores with different pricing strategies. In this study, the graphical model is used to investigate the following question: Do we find the same structure when we fit causal models on sub-groupings of stores, as we find when we fit models on aggregate data from all stores?causal analysis, aggregation bias, Demand and Price Analysis, C01,

Bessler, David A.

Lai, Pei-Chun

English

Research Papers in Economics

On Aggregation Bias in Structural Demand Models  
 
 
 
 
 
 
Pei-Chun Lai and David A. Bessler 
 
Department of Agricultural Economics, Texas A&M University, College 
Station, TX 77843 
 
 
 
 
 
 
 
 
 
 
 
Poster prepared for presentation at the Agricultural & Applied Economics Association 2010 
AAEA,CAES, & WAEA Joint Annual Meeting, Denver, Colorado, July 25-27, 2010 
 
 
 
 
 
 
 
 
 
Copyright 2010 by Pei-Chun Lai and David A. Bessler. All rights reserved. Readers may make 
verbatim copies of this document for non-commercial purposes by any means, provided that this 
copyright notice appears on all such copies. 
 
 Introduction
Consumer demand analysis attracts considerable attention. 
It remains an open question, however, whether estimating 
demand with aggregate data is reliable when disaggregate 
store-level data is given. Demand models may produce 
biased results when applied to data aggregated across 
stores with different pricing strategies. In this study, the 
graphical model is used to investigate the following 
question: Do we find the same structure when we fit causal 
models on sub-groupings of stores, as we find when we fit models 
on aggregate data from all stores?
Graphical methods for the discovery of causal 
connection in structural equation models (SEM) provide 
interesting tools to justify causal claims between variables. 
Nevertheless, an observed relation among variables might 
reflect the influence of a hidden common cause, thus 
making the correlation spurious. Fast Causal Inference 
(FCI) algorithm is developed to explore the causal 
structural when latent confounders exist.
We apply constraint based FCI algorithm on the 
Dominick’s scanner data and zip code information for the 
chain stores. The data set contains weekly sales 
information (03/ 02/ 95-03/ 06/ 96) of Coke 6 package with 
12 fl oz about 74 supermarket chain stores in Chicago area. 
The sales information includes supermarket’s retail price (    
), manufacturer’s wholesale price (     ), weekly sold 
quantity (Q), and store-specific median family income (I). 
Materials and methods
We do not impose an a priori causal flow among the four 
demand related variables studied here.  The usual 
structure of demand has the following causal graph: 
Acknowledgments
We gratefully acknowledge the provision of panel scanner data by the 
University of Chicago, James M. Kilts Center, Graduate School of 
Business.
If there is a association between the corresponding 
error terms (i.e.            ), for SEM with correlated errors, the 
possible influence of latent (unobserved) confounders can 
be taken into account by implementing the FCI algorithm.
Since we attempt to detect the existence of aggregation 
bias,  we classify the whole data into aggregate and 
disaggregate groups Figure 1 illustrates the processing 
flow of our analysis.
Conclusions
Demand estimates based on aggregate data is possibly 
biased when stores are heterogeneous. In this study, we use 
FCI algorithm to test if an aggregation bias exists when 
aggregating data across stores with different geographical 
population distribution. 
The question we ask is: does aggregation across stores 
give us the same result as disaggregate analysis? The 
answer is no! The aggregate result is not precisely 
consistent with disaggregate result, but they are similar to 
each. Our result suggests that when aggregating data, 
some association between variables may spuriously exist. 
However, how to obtain a properly modified aggregate 
demand framework to avoid this problem is not answered 
in this poster. 
Unlike traditionally statistical method, we detect the 
causal patterns between variables to examine the existence 
of aggregation bias. Causal discovery techniques usually 
assume that all causes are observed and known a priori. 
This is the so-called causal sufficiency assumption. 
However, this presumption is not always true. FCI 
algorithm is helpful to check the possible unobserved 
latent confounders between variables when there is causal 
insufficiency. 
Finally, as several previous studies in marketing 
indicate, our results show retail price and consumers’ 
family income may have effects on purchase behavior. We 
found this result without imposing the causal structure a 
priori. 
On aggregation bias in structural demand models
Pei-Chun Lai and David A Bessler
Department of Agricultural Economics, Texas A&M University, College Station, Texas 77843
Figure 1. Flow of model processing
For further information
Please contact PCLai@ag.tamu.edu. 
Results
The disaggregate-level data is defined by using 1990 U.S. 
Census information. The stores that fall into group one are 
those that face a consumer base whose median family 
income is less than $35,597 (first quartile). Stores that reside 
in zip codes characterized by median household incomes 
greater than $48,705 define our second disaggregate group 
(third quartile). We ignore stores where median family 
incomes are between the first and third quartiles. Figures 2 
and 3 display the PAGs of aggregate-level and 
disaggregate-level data. Our findings show that:
• For the variables     ,     , and I, they have a direct 
effect on sold quantity, or their relation with Q is 
due to a common cause, or a combination of both.
• In the aggregate and low median household 
income graphs, either manufacturer may have 
more pricing power over supermarket retailer, or 
supermarket retailer has more pricing power over 
manufacturer, or there is a latent common cause 
of      and    , or there is a combination of these.
• For stores that face median family income 
greater than $48,705, there is no relation between      
and     .
• We find agreement in 3 edges and directions but 
we miss one edge.
Table 1. Statistics of store-specifically median household income. The first 
quartile and third quartile are used to make the disaggregate groups.
Figure 2. PAG of aggregate-level 
data (p=0.01).
Be sure to separate figures from other figures by generous use 
of white space. When figures are too cramped, viewers get 
confused about which figures to read first and which legend 
goes with which figure.
Temme, D. 2006. Constraint-Based Inference Algorithms for Structural 
Models with Latent Confounders-Empirical Application and 
Simulations. Computational Statistics 21:151-182.
Tenn, S. 2006. Avoiding Aggregation Bias in Demand Estimation: a 
Multivariate Promotional Disaggregation Approach. Journal of 
Machine Learning Research 9:1437-1474.
Literature cited
Kwon, D.H. 2007. Causality and Aggregation Economics: the Use of High 
Dimensional Panel Data in Micro-Econometrics and Macro-
Econometrics, Ph. D. Dissertation, Department of Agricultural 
Economics, Texas A&M University.
Pearl, J. 2009. Causality: Models, Reasoning, and Inference. Cambridge 
University Press, 2nd edition.
4
3
2 1
1 2 1 Q


 
  


 
  
I
P
P P
I P
w
w r
r w P
r P
r P
w P
Q
I
4 3   
Coke 6/12 fl oz weekly 
sales data.
Handle the data into aggregate 
and disaggregate parts.
Aggregate
Each store is treated as 
homogeneous.
Disaggregate
The whole supermarket 
stores are categorized into 
low-consumers’ income 
and high-consumer’s 
income groups.
The output of the FCI algorithm is a partial ancestral 
graph (PAG) and the edges in a PAG can be interpreted as 
follows:
: a is a cause of b.
: there is a latent common cause of a and b so that a 
does not cause b and b does not cause a.
: a is a cause of b, or there is a latent common cause 
of a and b, or both.
a ◦—◦b : either a is a cause of b or b is a cause of a, or there 
is a latent common cause of a and b, or there is a 
combination of these.
Make series of statistical tests on 
partial correlation and conditional 
independence relationships 
among related variables.
Compare the ultimate causal 
structures.
b a 
b a 
b a    
Mean  Median 
Value  First Quartile  Third 
Quartile 
42486.7  42065  35597  48705 
 
(a) (b)
Figure 3. PAGs of disaggregate-level data (p=0.01). The two disaggregate-
level groups are defined along the lines of median family income: (a) PAG of 
group one (b) PAG of group two.  
w P r P
w P r P
w P
r P

On aggregation bias in structural demand models

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