We describe procedures for Bayesian estimation and testing in both cross sectional and longitudinal\ud data smooth coefficient models (with and without endogeneity problems). The smooth\ud coefficient model is a generalization of the partially linear or additive model wherein coefficients\ud on linear explanatory variables are treated as unknown functions of an observable covariate. In\ud the approach we describe, points on the regression lines are regarded as unknown parameters and\ud priors are placed on differences between adjacent points to introduce the potential for smoothing\ud the curves. The algorithms we describe are quite simple to implement - estimation, testing and\ud smoothing parameter selection can be carried out analytically in the cross-sectional smooth coefficient\ud model, and estimation in the hierarchical models only involves simulation from standard\ud distributions.\ud We apply our methods by fitting several hierarchical models using data from the National Longitudinal\ud Survey of Youth (NLSY). We explore the relationship between ability and log wages and\ud flexibly model how returns to schooling vary with measured cognitive ability. In a generalization of\ud this model, we also permit endogeneity of schooling and describe simulation-based methods for inference\ud in the presence of the endogeneity problem. We find returns to schooling are approximately\ud constant throughout the ability support and that simpler (and often used) parametric specifications\ud provide an adequate description of these relationships
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