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

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Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance Alma Mater Studiorum - Universit`adi Bologna DOTTORATO DI RICERCA IN ECONOMIA Ciclo XX Settore scienti¯co disciplinare di a®erenza: SECS-P/05 - ECONOMETRIA. Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance Presentata da: Anna SIMONI Coordinatore Dottorato: Relatore: Andrea ICHINO Sergio PASTORELLO Esame Finale Anno 2009 Contents 1 Introduction 1 2 Regularized Posterior in linear ill-Posed Inverse Problems 6 2.1 Introduction . 6 2.2 The Model . 8 2.2.1 Sampling Probability Measure . 8 2.2.2 Prior Speci¯cation and Identi¯cation . 8 2.2.3 Construction of the Bayesian Experiment . 10 2.3 Solution of the Ill-Posed Inverse Problem . 10 2.3.1 Tikhonov Regularized Posterior distribution . 11 2.3.2 Tikhonov regularization in the Prior Variance Hilbert scale . 12 2.4 Asymptotic Analysis . 13 2.4.1 Speed of convergence with Tikhonov regularization in the Prior Vari- ance Hilbert Scale . 16 2.5 The case with unknown operator K ...................... 17 2.5.1 Asymptotic Analysis . 18 2.6 The case with di®erent operator for each observation . 20 2.6.1 Marginalization of the Bayesian experiment . 21 2.6.2 Asymptotic Analysis . 23 2.7 Conclusions . 24 2.8 Appendix A: Proofs . 24 2.9 Appendix B: Examples . 35 2.10 Appendix C: Monte Carlo Simulations . 38 3 On the Regularization Power of the Prior Distribution in Linear ill-Posed Inverse Problems 43 3.1 Introduction . 43 3.2 Asymptotic Analysis . 47 3.3 A particular case . 50 3.4 g as an hyperparameter . 52 3.5 Conclusion . 54 3.6 Appendix A: proofs . 54 3.7 Appendix B: Numerical Implementation . 55 4 Nonparametric Estimation of an Instrumental Regression: a Bayesian Approach Based on Regularized Posterior 57 4.1 Introduction . 58 4.2 The Model . 59 4.3 Bayesian Analysis . 62 4.3.1 The Student t Process . 66 i ii 4.3.2 Asymptotic Analysis . 67 4.3.3 Independent Priors . 69 4.4 The Unknown Operator Case . 72 4.4.1 Unknown Finite Dimensional Parameter . 72 4.4.2 Unknown In¯nite Dimensional Parameter . 76 4.5 Numerical Implementation . 78 4.5.1 Data driven method for choosing ® ................... 82 4.6 Conclusions . 85 4.7 Appendix A . 86 5 Bayesian Nonparametric Estimation of Asset Pricing Functionals 96 5.1 Introduction . 96 5.2 Rational Expectations Asset Pricing Model . 100 5.2.1 Lucas' (1978) Model . 100 5.2.2 Martingale Property . 102 5.2.3 Integral Equations of Second Kind and Characterization of the Op- erator . 103 5.3 Bayesian Econometric Analysis . 105 5.3.1 Nonparametric Estimation of the Transition Density . 105 5.3.2 Construction of the Bayesian experiment . 107 5.3.3 Analysis of the Posterior Distribution . 111 5.3.4 Tikhonov Regularized Posterior Distribution . 113 5.3.5 Tikhonov regularization in the Prior Variance Hilbert scale . 114 5.4 Asymptotic Analysis . 114 5.4.1 Speed of convergence with classical Tikhonov regularization . 115 5.4.2 Speed of convergence with Tikhonov regularization in the Prior Vari- ance Hilbert Scale . 117 5.4.3 Comparison with the classical estimation of the pricing functional . 118 5.5 A g-prior with Regularizing Power . 119 5.6 Prior on the Variance Parameter . 121 5.6.1 Conjugate model . 121 5.7 Conclusions . 124 5.8 Appendix A: Proofs . 125 5.9 Appendix B: Numerical Implementation . 134 References . 135 List of Figures 2.1 Figure 2.1 . 39 2.2 Figure 2.2 . 41 2.3 Figure 2.3 . 42 3.1 Figure 3.1 . 56 4.1 Figure 4.1 . 80 4.2 Figure 4.2 . 81 4.3 Figure 4.3 . 82 4.4 Figure 4.4 . 84 4.5 Figure 4.5 . 85 5.1 Figure 5.1 . 135 5.2 Figure 5.2 . 136 iii Chapter 1 Introduction Recent econometric theory has developed an increasing interest in nonparametric estima- tion of structural parameters of economic models. The (possibly functional) parameter of interest describes the economic agent's behavior or the equilibrium market's characteris- tics. In other words, structural econometrics has not for purpose the estimation of objects directly linked to the data's distribution F , like the density or the hazard function, or the estimation of some characteristics of the data's conditional distribution, such as the conditional expectation. The structural parameters have not a statistical interpretation and, in general, they are not simple transformations of the sampling distribution of the data. The relationship between structural parameters and the sampling distribution F is, in most of the cases, only implicitly de¯ned through a functional equation. Hence, the structural parameter is characterized as solution of this functional equation. When the dimension of the structural parameter is ¯nite, the functional equation reduces to a matrix equation. Several authors, see, among others, Florens (2003), Hall and Horowitz (2005), Linton and Mammen (2005), Carrasco, Florens and Renault (2007), have developed a general frame- work for structural functional inference in connection with the inverse problem literature. A complete list of references can be found in Carrasco, Florens and Renault (2007). Economic theory can provide information about the shape (like convexity, concavity) or the di®erentiability of the parameter of interest, but never provides a parametric form for it. Hence, it is really suitable to not restrict inference to parametric classes. Nonpara- metric estimation reduces the risk of mispeci¯cation but, at the same time, rises problems of continuity, uniqueness and existence of the solution of the corresponding functional equation, so that some care must be taken in solving it. Such a problem is known as the ill-posedness of the inverse problem we want to solve. The lack of continuity of the solution entails a strong sensibility of the solution to the estimation errors of certain elements of the functional equation. The noises in the functional equation arise because some elements in it can be (and usually are) imperfectly known (like for instance the sampling distribution) and they are replaced by consistent estimators. Hence, small estimation errors can be strongly ampli¯ed in the estimated parameter. Classical econometric literature deals with these problems by proposing di®erent tech- niques of stabilization of the solution. Classical stabilization techniques consist in replac- ing the non-continuous estimator with an approximation of it that is continuous and that converges in the sense that, as the noise level in the functional equation tends to zero, the approximated estimator tends to the true one. My work develops bayesian nonparametric methods to estimate structural economic quan- tities. It analyzes the role played by the prior distribution in solving these problems of continuity and existence of the solution of the functional equation. 1 2 Bayesian analysis considers an inverse problem in a di®erent way with respect to the classical analysis since it restates the functional equation in a larger space of probabil- ity distributions. This reformulation of an inverse problem was due to Franklin (1970) [33]. From a Bayesian point of view all the quantities in the structural model are random functions. Hence, the structural parameter of interest having generated the data is the realization of some random process. I substantially look for an estimation of such process and I take the posterior distribution of the parameter as the estimator. When the dimension of the problem is ¯nite the ill-posedness is principally due to a problem of multicollinearity. In this case, classical and Bayesian approaches are strongly related since the ridge regression (that is a classical method proposed for dealing with multicollinearity) has a Bayesian interpretation. Therefore, in ¯nite dimension we can re- move the ill-posedness by incorporating the available prior information. On the contrary, I prove that in in¯nite dimensional problems, a general prior distribution does not get ride of the ill-posedness of the problem since general prior covariance matrices do not have the regularization properties that have in the ¯nite dimensional case. In particular, being covariance matrices impossible to continuously inverse we still need some regularization technique and the bayesian approach only lies in changing the nature of the problem. Nevertheless, an exciting result of my work is that there exists a class of prior distribu- tions, or more precisely, of covariance operators, that are able to solve for the ill-posedness as in the ¯nite dimensional case. I believe that a bayesian approach for solving functional equations is suitable for many reasons that I discuss in the following. (i) It is very important to incorporate, in the estimation procedure, the information that we may have a-priori on the structural pa- rameter that we want to recover, for instance shape information or other constraints given by economic theory. This can be easily done through a Bayesian procedure. We could choose a prior mean function of the form suggested by the economic theory. Otherwise, we can specify a prior covariance operator that restricts the space in which the solution can lie to a subspace of functions satisfying the constraints we want to impose. The im- portance of incorporating the prior information that is available is not to underestimate, above all in nonparametric estimation where the parameter is often weakly identi¯ed by data because the amount of data is small with respect to the dimension of the parameter to estimate (although it is identi¯ed from a mathematical point of view). Moreover, in the most applied research, for instance in ¯nance or in consulting studies, people are strongly inclined to exploit all the prior information that they can have, like the opinion of some expert. (ii). The fact to get a posterior distribution of the structural parameter of interest rep- resents a big advantage with respect to classical estimation procedures.
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