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 ‘no-purchase’ 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.