Capturing Flexible Heterogeneous Utility Curves: A Bayesian Spline Approach
提出一种贝叶斯样条方法，在个体层面估计灵活形状的效用函数，用于选择模型。该方法能处理价格反应单调性等约束，在联合分析和面板数据中预测改进6%-20%以上，揭示个体与市场层面价格曲线的差异。
Empirical evidence suggests that decision makers often weight successive additional units of a valued attribute or monetary endowment unequally, so that their utility functions are intrinsically nonlinear or irregularly shaped. Although the analyst may impose various functional specifications exogenously, this approach is ad hoc, tedious, and reliant on various metrics to decide which specification is “best.” In this paper, we develop a method that yields individual-level, flexibly shaped utility functions for use in choice models. This flexibility at the individual level is accomplished through splines of the truncated power basis type in a general additive regression framework for latent utility. Because the number and location of spline knots are unknown, we use the birth-death process of Denison et al. (1998) and Green’s (1995) reversible jump method. We further show how exogenous constraints suggested by theory, such as monotonicity of price response, can be accommodated. Our formulation is particularly suited to estimating reaction to pricing, where individual-level monotonicity is justified theoretically and empirically, but linearity is typically not. The method is illustrated in a conjoint application in which all covariates are splined simultaneously and in three panel data sets, each of which has a single price spline. Empirical results indicate that piecewise linear splines with a modest number of knots fit these data well, substantially better than heterogeneous linear and log-linear a priori specifications. In terms of price response specifically, we find that although aggregate market-level curves can be nearly linear or log-linear, individuals often deviate widely from either. Using splines, hold-out prediction improvement over the standard heterogeneous probit model ranges from 6% to 14% in the scanner applications and exceeds 20% in the conjoint study. Moreover, “optimal” profiles in conjoint and aggregate price response curves in the scanner applications can differ markedly under the standard and the spline-based models.