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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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.

Tipologia del documento
Tesi di dottorato
Simoni, Anna
Dottorato di ricerca
Settore disciplinare
Settore concorsuale
Parole chiave
Functional data, Inverse Problems, Tikhonov and Hilbert Scale regularization, g-prior, Posterior Consistency, Instrumental Regression, Asset Pricing Functional
Data di discussione
10 Giugno 2009

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