Closed-form Bayesian inferences for the logit model via polynomial expansions

Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood extended with a population distribution of heterogeneity doesn’t yield closed-form inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closed-form Bayesian inferences for the logit model, which we obtain by approximating the logit likelihood via a polynomial expansion, and then positing a distribution of heterogeneity from a flexible family that is now conjugate and integrable. For problems where the response coefficients are independent, choosing the Gamma distribution leads to rapidly convergent closed-form expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closed-form expansions by positing a distribution of heterogeneity from a Multivariate Gamma distribution. The solution then comes from the moment generating function of the Multivariate Gamma distribution or in general from the multivariate heterogeneity distribution assumed. Closed-form Bayesian inferences, derivatives (useful for elasticity calculations), population distribution parameter estimates (useful for summarization) and starting values (useful for complicated algorithms) are hence directly available. Two simulation studies demonstrate the efficacy of our approach. JEL Classification C6 · C8 · M3     

Bell System scale economies estimated from a random coefficients model

This paper uses a variant of the Cobb-Douglas production function whose log-linear form looks like translog frontier. The parameters of the production function are estimated by a new method called the two stage ridge (TSR) estimator. It is applicable whenever multicollinearity is present both in the estimation of the means and variances of the random coefficients. The results seen to support the presence of scale economics in the Bell System. For our data it appears the TSR estimation is valid and provides additional insights. Thus it should be a valuable tool in similar models elsewhere.     

Bayesian estimation of discrete games of complete information

Estimation of discrete games of complete information, which have been applied to a variety of contexts such as market entry, technology adoption and peer effects, is challenging due to the presence of multiple equilibria. In this paper, we take a Bayesian MCMC approach to this problem, specifying a prior over multiple equilibrium selection mechanisms reflecting the analysts uncertainty over them. We develop a sampler, using the reversible jump algorithm to generate draws from the posterior distribution of parameters across these equilibrium selection rules. The algorithm is flexible in that it can be used both in situations where the equilibrium selection rule is identified and when it is not, and accommodates heterogeneity in equilibrium selection. We explore the methodology using both simulated data and two empirical applications, one in the context of joint consumption, using a dataset of casino visit decisions by married couples, and the second in the context of market entry by competing chains in the retail stationery market. We demonstrate the importance of accounting for multiple equilibrium selection rules in these applications and show that taking an empirical approach to the issue, such as the one we have demonstrated, can be useful.     

Bayesian estimation of multivariate-normal models when dimensions are absent

Multivariate economic and business data frequently suffer from a missing data phenomenon that has not been sufficiently explored in the literature: both the independent and dependent variables for one or more dimensions are absent for some of the observational units. For example, in choice based conjoint studies, not all brands are available for consideration on every choice task. In this case, the analyst lacks information on both the response and predictor variables because the underlying stimuli, the excluded brands, are absent. This situation differs from the usual missing data problem where some of the independent variables or dependent variables are missing at random or by a known mechanism, and the “holes” in the data-set can be imputed from the joint distribution of the data. When dimensions are absent, data imputation may not be a well-poised question, especially in designed experiments. One consequence of absent dimensions is that the standard Bayesian analysis of the multi-dimensional covariances structure becomes difficult because of the absent dimensions. This paper proposes a simple error augmentation scheme that simplifies the analysis and facilitates the estimation of the full covariance structure. An application to a choice-based conjoint experiment illustrates the methodology and demonstrates that naive approaches to circumvent absent dimensions lead to substantially distorted and misleading inferences.

A censored random coefficients model for the detection of zero willingness to pay

In this paper we address the problem of negative estimates of willingness to pay. We find that there exist a number of goods and services, especially in the fields of marketing and environmental valuation, for which only zero or positive WTP is meaningful. For the valuation of these goods an econometric model for the analysis of repeated dichotomous choice data is proposed. Our model restricts the domain of the estimates of WTP to strictly positive values, while also allowing for the detection of zero WTP. The model is tested on a simulated and a real data set.     

Integrating the Dirichlet-multinomial and multinomial logit models of brand choice

This paper discusses the interpretative benefits that arise from merging the Dirichlet-multinomial (DM) model as a loyalty variable in the multinomial logit (MNL) model of brand choice. The estimated parameters of this hybrid model compare favorably to those of a pure DM model (with no marketing mix variables) as well as those of a standard MNL model with an exponentially smoothed loyalty measure. The hybrid model offers an index of consumer heterogeneity and estimates of each brand's preference share, adjusted for the effects of price and promotional activities. We illustrate the unique interpretations offered by these parameters across four different product categories, showing how changes in heterogeneity (across categories or model specifications) are closely related to changes in the overall impact of marketing mix variables.The author is grateful to Bruce Hardie, Barbara Kahn, Jim Lattin, Len Lodish, Bob Meyer, and Don Morrison for their comments and suggestions. He appreciates the support of Doug Honnold and Information Resources, Inc., for providing the data and computational facilities used in this paper.     

Flexible estimation of price response functions using retail scanner data

Kalyanam and Shively [1998. Estimating irregular pricing effects: a stochastic spline regression approach. Journal of Marketing Research 35 (1), 16–29] and van Heerde et al. [2001. Semiparametric analysis to estimate the deal effect curve. Journal of Marketing Research 38 (2), 197–215] have demonstrated the usefulness of nonparametric regression to estimate pricing effects flexibly. The empirical results of these two studies, however, also revealed that nonparametric regression may suffer from too much flexibility leading to nonmonotonic shapes for price effects. In this paper, we show how the problem of nonmonotonicity can be dealt with without losing the power of flexible estimation techniques. We propose a semiparametric approach based on Bayesian P-splines with monotonicity constraints imposed on own- and cross-price effects. In an empirical application, we illustrate that flexible estimation of own- and cross-price effects can improve the predictive validity of a sales response model substantially, even when price response curves were constrained to show a monotonic shape, as suggested by economic theory. We also discuss the consequences from an unconstrained estimation of price effects.     

Bayesian estimation of a simultaneous probit model using error augmentation: An application to multi-buying and churning behavior

Researchers in marketing are often interested in analyzing how an agent’s discrete choice decision affects a subsequent or concurrent discrete choice decision by the same or different agent. This analysis may necessitate the use of a simultaneous equations model with discrete and continuous endogenous variables as explanatory variables. In this paper, we offer an error augmentation approach to Hierarchical Bayesian estimation of a simultaneous bivariate probit model containing both discrete and continuous endogenous variables. We accomplish the error augmentation in our MCMC algorithm using a Metropolis-Hastings step that generates the error components of the latent variables in our model. Using simulated data, we demonstrate that our error augmentation algorithm recovers closely the true parameters of the simultaneous bivariate probit model. We then apply our algorithm to customer churn data from a wireless service provider. We formulate a simultaneous bivariate probit model to study the impact of a customer’s multiple product relationships with a firm (multi-buying) on the likelihood of churn by that customer. The empirical results show that the act of multi-buying significantly reduces churn even though the customers who are more predisposed to multi-buy have an inherently higher predisposition to churn.     

Uncovering the message from the mess of big data

User-generated content, such as online product reviews, is a valuable source of consumer insight. Such unstructured big data is generated in real-time, is easily accessed, and contains messages consumers want managers to hear. Analyzing such data has potential to revolutionize market research and competitive analysis, but how can the messages be extracted? How can the vast amount of data be condensed into insights to help steer businesses’ strategy? We describe a non-proprietary technique that can be applied by anyone with statistical training. Latent Dirichlet Allocation (LDA) can analyze huge amounts of text and describe the content as focusing on unseen attributes in a specific weighting. For example, a review of a graphic novel might be analyzed to focus 70% on the storyline and 30% on the graphics. Aggregating the content from numerous consumers allows us to understand what is, collectively, on consumers’ minds, and from this we can infer what consumers care about. We can even highlight which attributes are seen positively or negatively. The value of this technique extends well beyond the CMO's office as LDA can map the relative strategic positions of competitors where they matter most: in the minds of consumers.     

The functional form of the aggregate import demand equation

The theory of international trade gives little guidance on the appropriate functional form to use when specifying and estimating an import demand equation. The choice between the standard linear and log-linear specifications is thus essentially an empirical one. The purpose of this paper is to decide on empirical grounds the form of the aggregate import demand equation for the United States, Canada and Japan. A test that allows the data to dictate the appropriate functional form was performed on standard import equations and it was discovered that for these countries a log-linear specification was better than a linear one.     

Legal reform and aggregate small and micro business bankruptcy rates: evidence from the 1997 Belgian bankruptcy code

Many Continental European countries recently reformed their bankruptcy legislations to stimulate reorganization and firm survival. We show that the Belgian 1997 bankruptcy code reform, which implemented several international best practice recommendations, significantly reduced aggregate small and micro business bankruptcy rates. However, using distributed lag models to control for the relationship between bankruptcy rates and macroeconomic variables such as real GDP growth, consumer confidence, inflation, etc., we find that the new code’s impact is not the same for all types of companies. Specifically, while the beneficial effect of the reform is largely similar between small firms (i.e. stock corporations) and micro firms (i.e. partnerships), it is only significant in certain industries (manufacturing and trade). Overall, our results indicate that especially the measures taken to limit domino bankruptcy effects are likely to have had a substantial impact. Our findings have several policy implications for the evaluation and modification of the bankruptcy system.

A Bayesian Approach for Analyzing the Services of Banking Institutions

This study illustrates an adaptive Bayesian framework for examining consumers' perceptions of the services of banking institutions in a target market. The approach uses multiple-measures data to estimate the significant determinants of consumers' preferences for competing banks in the market. This is accomplished essentially by incorporating prior structural information into analyses, information that is used to obtain a stabilized version of the observed predictor-criterion covariance matrix. Bayesian structural regression estimates are shown to provide a banking institution with reliable information for use in positioning itself in its financial marketplace. The paper also provides a number of implications for the use of the methodology practical marketing response applications.     

Is the performance of the dollar a random process?

It is still frequently argued that a random process would be the best way of predicting the DM/dollar exchange rate. In the following empirical analysis, however, Dr. Rolf Schneider comes to the conclusion that the behaviour of the dollar can be explained largely in terms of basic economic variables.     

离散型随机变量数学期望的求法探究

大庆油田有限责任公司采油工程研究院是我国石油系统中较早从事采油工艺研究、开发、应用的综合性科研机构,多年以来,该单位始终坚持"高质量研究、高速度推广、高水平服务"的发展战略,以科学发展观为先导,名牌战略为依托,针时油田生产中的疑难问题,展开攻关,企业得以快速发展.     

离散型随机变量数学期望的求法探究

离散型随机变量的数学期望的解法是一个重要内容.离散型随机变量在计算中和生活实际中都有着广泛的应用.离散型随机变量的数学期望的解法多种多样、灵活多变,具有一定的规律性和技巧性.本文对离散型随机变量数学期望的求法技巧进行了深入探讨.     

浅谈关系数据模型的矢量数据空间叠加

本文对已有空间数据模型的特点进行了评述,指出数据存储方式决定了空间叠加方式的选取,然后基于当前流行的关系数据模型,对矢量数据的空间叠加进行了详细的探讨,提出了将拓扑复合数据结构引入关系数据模型的空间叠加设想.     

Adaptive grids for the estimation of dynamic models

Quantitative Marketing and Economics - This paper develops a method to flexibly adapt interpolation grids of value function approximations in the estimation of dynamic models using either NFXP...     

A probability model of aggregate meat consumption in the U.S.A.*

This study illustrates the use of Markov analysis to investigate the nature of competition among demand-related products when only aggregate consumption data are available. Applied to meat consumption in the U.S.A. before and after 1976, the year in which per capita beef consumption began to decline, the results indicate that beef has not lost dramatically to chicken. This is contrary to conventional wisdom where consumers are thought to substitute chicken for beef because of the perceived high fat and cholesterol content of beef, and because of the relatively cheaper price of chicken.