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

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.

Conformation and Stereodynamic of Hindered Aromatic System

The preparation of conformationally hindered molecules and their study by DNMR and computational methods are my thesis’s core. In the first chapter, the conformations and the stereodynamics of symmetrically ortho-disubstituted aryl carbinols and aryl ethers are described. In the second chapter, the structures of axially chiral atropisomers of hindered biphenyl carbinols are studied. In the third chapter, the steric barriers and the -barrier of 1,8-di-aylbiphenylenes are determined. Interesting atropisomers are found in the cases of arylanthrones, arylanthraquinones and arylanthracenes and are reported in the fourth chapter. By the combined use of dynamic NMR, ECD spectroscopy and DFT computations, the conformations and the absolute configurations of 2-Naphthylalkylsulfoxides are studied in the fifth chapter. In the last chapter, a new synthetic route to ,’-arylated secondary or tertiary alcohols by lithiated O-benzyl-carbamates carrying an N-aryl substituent and DFT calculations to determinate the cyclic intermediate are reported. This work was done in the research group of Prof. Jonathan Clayden, at the University of Manchester.