Endogeneity and Heterogeneity in a Probit Demand Model: Estimation Using Aggregate Data

Two issues that have become increasingly important while estimating the parameters of aggregate demand functions to study firm behavior are the endogeneity of marketing activities (typically, price) and heterogeneity across consumers in the market under consideration. Ignoring these issues in the estimation of the demand function parameters can lead to biased and inconsistent estimates for the effects of marketing activities. Endogeneity and heterogeneity have achieved prominence in large measure because of the increasing popularity of logit models to characterize demand functions using aggregate data. The logit model accounts for purchase incidence and brand choice by including a ‘‘nopurchase’’ alternative in the consumer’s choice set. This allows for category sales to change as a function of the marketing activities of brands in the category. There are three issues with using the logit model with the no-purchase option to characterize demand when studying competitive interactions among firms. (1) The marketing literature dealing with brand choice behavior at the consumer level has found that the IIA restriction is not appropriate, as each brand in the choice set is more similar to some brands than it is to others. (2) Studies have found that the purchase incidence decision is distinct from the brand choice decision. Hence, it may not be appropriate to model the no-purchase decision as just another alternative in the choice set with the IIA restriction holding across all brands and the no-purchase option. (3) Even if the distinction between the purchase incidence and brand choice decisions is accounted for via, for example, a nested logit specification, accounting for the purchase incidence decision with aggregate data requires assumptions for computing the share of the no-purchase alternative which is otherwise unobserved. In this paper, we propose a probit model as an alternative to the logit model to specify the aggregate demand functions of firms competing in oligopoly markets. The probit model avoids the IIA property that affects the logit model at the individual consumer level. Furthermore, the probit model can naturally account for the distinction between the purchase incidence and brand choice decisions due to the general covariance structure assumed for the utilities of the alternatives. We demonstrate how the parameters of the proposed model can be estimated using aggregate time series data from a product market. In the estimation, we account for the endogeneity of marketing variables as well as for heterogeneity across consumers. Our results indicate that both endogeneity as well as heterogeneity need to be accounted for even after allowing for a non-IIA specification at the individual consumer level. Specific to our data, we also find that ignoring endogeneity has a bigger impact on the estimated price elasticities than ignoring the effects of heterogeneity. A comparison of the elasticities obtained from the probit model with those from the corresponding logit specification indicates that the range of elasticities obtained from the probit model across brands is larger than that obtained from the logit. The results have implications for issues such as firm-level pricing. In addition to specifying a probit model and providing comparisons with the logit model, the paper also addresses the third issue raised above. We propose a simple alternative to the purchase incidence/brand choice specification by decomposing the demand for a brand into a category demand equation and a conditional brand choice share equation. We provide a comparison of results from this specification to those from the specification that includes the no-purchase alternative and find that estimated elasticities are sensitive to the specification used. We also estimate the demand function parameters using a traditional specification such as the double-logarithmic model. Here, we find that the estimated elasticities could be signed in such a manner as to be not useful for firm-level pricing decisions. One of the key limitations of the proposed model is that while it accounts for the purchase incidence and brand choice decisions of households, it does not account for differences across consumers in their purchase quantities. The model and analysis are best suited for product categories in which consumers typically make single-unit purchases. Another limitation is more practical in nature. While recent advances have been made in computing probit probabilities, it could nevertheless be a challenge to do so when the number of alternatives is large. (Heterogeneity; Endogeneity; Probit Model; Logit Model) ENDOGENEITY AND HETEROGENEITY IN A PROBIT DEMAND MODEL: ESTIMATION USING AGGREGATE DATA MARKETING SCIENCE/Vol. 20, No. 4, Fall 2001 443 Introduction The recent literature in marketing and in economics has seen an explosion of studies dealing with the analysis of firm-level behavior. The principal motivation behind these studies is to measure market power of firms and to understand interfirm competitive behavior. For example, Kadiyali (1996) studies the competitive interactions between Kodak and Fuji in the U.S. market to investigate whether or not the rivalry between these two firms is as intense as indicated by the popular press. Sudhir (2001) looks at the competitive interactions among firms within various segments of the automobile industry to determine whether these interactions vary significantly across product segments. Nevo (2001) investigates the extent of pricing rivalry in the ready-to-eat breakfast cereal market to determine whether observed prices reflect market power associated with product differentiation or collusion by firms in the industry. The fundamental building block for the analysis of firm behavior is the demand function for each of the players in the marketplace. The demand functions relate the sales of the brands to their prices, promotions, and other marketing variables. Two issues that have become increasingly important while estimating the parameters of such aggregate demand functions to study firm behavior are the endogeneity of marketing activities (typically, price) and the heterogeneity across consumers in the market under consideration. The endogeneity problem arises when there are variables for which data are not available (such as shelf space allocation, shelf location, store coupons, etc.) that could influence a brand’s sales in a given week and if these variables are correlated with the included marketing variables such as price (lowering the price of brand in a given week may be accompanied by giving it more shelf facings). These other marketing activities are part of the error term in the estimation and the correlation between the price variable and the error term results in the endogeneity problem. Not accounting for this correlation will give incorrect estimates for the effects of the included marketing variables. The issue with heterogeneity is the same as it is with household data. If the observed data at the store or market level are the aggregation of consumers with different brand preferences and sensitivities to marketing instruments, then ignoring this heterogeneity in the estimation of the demand function parameters can lead to biased and inconsistent estimates for the marketing activities. The issues of endogeneity and heterogeneity have achieved prominence in large measure because of the increasing popularity of discrete-choice models to specify the demand functions to study firm behavior. The aforementioned studies by Sudhir and Nevo, along with others by Berry et al. (1995), have used discrete-choice-based demand functions. The main advantages of discrete-choice models are: (i) They are derived from utility maximizing behavior of consumers in the marketplace. (ii) They require estimation of fewer numbers of parameters, as compared to linear (and log–log and semilog) demand functions (instead of estimating 100 price parameters in a market with 10 brands, usually a single price parameter is estimated). (iii) They seldom result in parameter estimates with incorrect signs for own and cross effects, as is the case with linear demand systems and their variants. The most widely used specification of discretechoice demand function in such studies thus far has been the logit model. To allow the total demand for the category to vary over time, the model treats the no-purchase option or the ‘‘outside good’’ as an additional alternative available in the choice set. The specification embodies all the advantages of discretechoice models noted above. Additionally, it is also easy to estimate. All these advantages appear to justify the model’s widespread use in the literature. Hence, researchers have used the logit demand model and have accounted for the issues of endogeneity and heterogeneity while estimating the parameters of these models with aggregate store (Besanko et al. 1998), chain, or market (Sudhir 2001) data. Accounting for heterogeneity in the logit model also alleviates the problem of restrictive cross-elasticities that are obtained from this model because of the ‘‘independence of irrelevant alternatives’’ (IIA) property at the individual consumer level (see the discussion in Nevo 2001).