Numerical Integration in Random Coefficient Models of Demand

The causal relation between a product's purchased quantity and observable characteristics of the product itself or its substitutes lies at the heart of many economic questions. Demand models relate these two types of variables and allow, for example, to answer industrial organization related questions, which analyze changes in the environment on market outcomes to derive policy implications. In a business context, the optimization of dynamic pricing strategies by predicting consumers' reactions to price changes gives another interesting and highly relevant application. Due to their fundamental importance, the evolution of demand models has a long history and is still in progress. The model of Berry, Levinsohn and Pakes (1995, BLP hereafter) provides the most recent and powerful framework to estimate the demand of differentiated products with aggregate data. Their model treats products as a bundle of characteristics and explicitly considers individual consumer reactions to changes of these characteristics without actually observing individual decisions. Moreover, the usual problem of co-movements between observed and unobserved variables is addressed in a standard instrumental variable approach. Identifying the effect of a product's price is a well-known example, where valid instruments allow to rule out the impact of numerous unobservable quality components. By combining these properties with an efficient way of calculating a large amount of parameters, BLP created a tool to make realistic economic predictions. In the presence of consumer heterogeneity, this results in consumers, who rather switch to another product with similar characteristics, if they are confronted with an unfavorable change in the environment (e.g. a price increase). The economic use of considering consumers' heterogeneity comes at the cost of a more complex and computationally burdensome model. When economists face problems with heterogeneity or other forms of uncertainty, the approximation of integrals without analytic solutions often becomes necessary. In the BLP model, a fundamental step of the estimation algorithm consists of numerous aggregations of individual purchase probabilities for a given product. The aggregation is realized by solving a market share without an analytical solution. This thesis deals with different challenges related to the numerical integration problem. More precisely, harmful effects of an inaccurate numerical integration in the integral above (chapter 2), adaptive approaches to mitigate this inaccuracies (chapter 3) and the problem of having only limited information about observed heterogeneity (chapter 4) are addressed in three different chapters. Each of them is explained in greater detail in the following. Chapter 2 entitled ``Simulation Error Causes Weak Identification in Random Coefficient Logit Models Using Aggregate Data'' (co-authored with Florian Heiss, Andre Romahn and Constantin Weiser) investigates the propagation of integration error in the GMM objective function. It is shown that inaccurate integration impacts the shape of the GMM objective severely and produces wrong point estimates. This has immediate consequences for economic predictions: implied average own price elasticities, for example, based on different stochastic integration rules range from -44 to -2.3 with the lowest integration accuracy in BLPs' car data. With the most accurate integration rule, this range collapses to -11 to -6.8. Accurate integration does not only contribute to parameter reliability but fixes many other problems reported in the literature (Knittel and Metaxoglou, 2014). Reducing the integration error in BLPs' model is further examined in chapter 3 (``Implications of Adaptive Integration Rules for the Performance of Random Coefficient Models of Demand''). Contrary to the strategy of the former chapter, integrals are not accurately approximated by a large amount of function evaluations but fewer and more cleverly placed ones. The idea is to evaluate the integral at an initial ``standard'' set of points and then, based on information about the function's shape, perform an adjustment of the initial set. Adaptive integration rules are of particular use, if unfortunate parameter combinations cause very small regions on the integrand's support with a high contribution to the integral. This is demonstrated with simulated data and Nevo's cereal data (Nevo, 2000), where the gain of adaptive approaches in integration and parameter precision is impressive. Especially in the BLP model, adaptive approaches cause significant additional computational cost. This issue is addressed by proposing a new integration rule combining the advantages of standard and adaptive rules. Chapter 4 entitled ``Copulas for Demand Estimation Models with Partly Observed Heterogeneity'' deals with the problem of limited information about the joint density of observed heterogeneity. BLP incorporate observed consumer heterogeneity by demographic variables and the integration problem processes them by integrating over their joint distribution. The approximation of the integral requires a precise knowledge of this distribution, which is often not available. This problem frequently arises with official statistics like census data that have to protect individuals' privacy. To deal with some forms of incomplete information, copula functions are proposed to model the demographics' joint distribution. The power of this approach is illustrated with an application from the banking literature by estimating the demand for deposits in commercial banks. Exclusive information about marginal distributions of demographics, age and income, on a fine geographic level is combined with a dependence structure from a broader geographic level. In this example, copulas enable the researcher to work with incomplete data and predict bank products with similar deposit rates to be good substitutes. To summarize, this thesis discusses different challenges that can be encountered when making use of the great flexibility in BLPs' demand model and contributes by proposing possible solutions. In applied work, the numerical accuracy of the integral approximations constitutes an important trade-off between computational cost and accurate estimates. The results of chapter 2 indicate that this trade-off should be decided in favor of accuracy, because the integral approximation is a major source of error and easily eliminates the estimator's reliability. Chapter 3 discusses adaptive integration methods to deal with difficult integrands, where an acceptable error level cannot be obtained by standard methods. This is of particular use, if estimation results from very accurate standard integration rules are still impacted by significant integration noise. An accurate approximation is not possible without knowing the density in the integration problem, which gives the starting point of chapter 4. It is motivated by a highly practical problem: the joint distribution for public and official data is often not available on fine geographical levels. In this case, copula functions allow a combination of information from different sources, which enables the researcher to proceed with the demand estimation.