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.

The Effect of Preservative Interventions on the Chemical- Physical and Structural Characteristics of Panel Painting

This work studies the impact of two traditional Romanian treatments, Red Petroleum and Propolis, in terms of real efficiency and consequence on the wooden artifacts. The application of these solutions is still a widely adopted and popular technique in preservative conservation but the impact of these solutions is not well known. It is important to know the effect of treatments on chemical-physical and structural characteristics of the artifacts, not only for understanding the influence on present conditions but also for foreseeing the future behavior. These treatments with Romanian traditional products are compared with a commercial antifungal product, Biotin R, which is utilized as reference to control the effectiveness of Red Petroleum and Propolis. Red Petroleum and Propolis are not active against mould while Biotin R is very active. Mould attack is mostly concentrated in the painted layer, where the tempera, containing glue and egg, enhance nutrition availability for moulds. Biotin R, even if is not a real insecticide but a fungicide, was the most active product against insect attack of the three products, followed by Red Petroleum, Propolis and untreated reference. As for colour, it did not change so much after the application of Red Petroleum and Biotin R and the colour difference was almost not perceptible. On the contrary, Propolis affected the colour a lot. During the exposure at different RH, the colour changes significantly at 100% RH at equilibrium and this is mainly due to the mould attack. Red Petroleum penetrates deeply into wood, while Propolis does not penetrate and remains only on the surface. However, Red Petroleum does not interact chemically with wood substance and it is easy volatilized in oven-dry condition. On the contrary Propolis interacts chemically with wood substance and hardly volatilized, even in oven-dry condition and consequently Propolis remains where it penetrated, mostly on the surface. Treatment by immersion has impact on wood physical parameters while treatment by brushing does not have significant impact. Especially Red Petroleum has an apparent impact on moisture content (MC) due to the penetration of solution, while Propolis does not penetrate so much and remains only on surface therefore Propolis does not have so much impact as Red Petroleum. However, if the weight of the solution penetrated in wood is eliminated, there is not significant difference in MC between treated and untreated samples. Considering physical parameters, dimensional stability is an important parameter. The variation of wood moisture content causes shrinkages/swelling of the wood that polychrome layer can only partially follow. The dimension of wooden supports varied under different moisture conditioning; the painted layer cannot completely follow this deformation, and consequently a degradation and deterioration caused by detachment, occurs. That detachment affects the polychrome stratification of the panel painting and eventually the connections between the different layer compositions of the panel painting.