Nonparametric estimation of an instrumental regression: A quasi-Bayesian approach based on regularized posterior |
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Authors: | Jean-Pierre Florens Anna Simoni |
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Institution: | 1. Toulouse School of Economics, Université de Toulouse 1 - Capitole, 21, Allée de Brienne, 31000 Toulouse, France;2. Department of Decision Sciences and IGIER, Università Bocconi, Via Sarfatti, 25 - 20136 Milano, Italy |
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Abstract: | We propose a quasi-Bayesian nonparametric approach to estimating the structural relationship φ among endogenous variables when instruments are available. We show that the posterior distribution of φ is inconsistent in the frequentist sense. We interpret this fact as the ill-posedness of the Bayesian inverse problem defined by the relation that characterizes the structural function φ. To solve this problem, we construct a regularized posterior distribution, based on a Tikhonov regularization of the inverse of the marginal variance of the sample, which is justified by a penalized projection argument. This regularized posterior distribution is consistent in the frequentist sense and its mean can be interpreted as the mean of the exact posterior distribution resulting from a Gaussian prior distribution with a shrinking covariance operator. |
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Keywords: | C11 C14 C30 |
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