# Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

Simoni, Anna (2009) Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Economia, 20 Ciclo. DOI 10.6092/unibo/amsdottorato/1211.
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## Abstract

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

Abstract
Tipologia del documento
Tesi di dottorato
Autore
Simoni, Anna
Supervisore
Dottorato di ricerca
Ciclo
20
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
Functional data, Inverse Problems, Tikhonov and Hilbert Scale regularization, g-prior, Posterior Consistency, Instrumental Regression, Asset Pricing Functional
URN:NBN
DOI
10.6092/unibo/amsdottorato/1211
Data di discussione
10 Giugno 2009
URI