We quantify the performance of approximations to stochastic filtering by the Kullback- Leibler divergence to the optimal Bayesian filter. Using a two-state Markov process that drives a Brownian measurement process as prototypical test case, we compare two stochastic filtering approximations: a static low-pass filter as baseline, and machine learning of Volterra expansions using nonlinear Vector Auto-Regression (nVAR). We highlight the crucial role of the chosen performance metric, and present two solutions to the specific challenge of predicting a likelihood bounded between 0 and 1.

Gegenstand dieser Dissertation sind die Auswirkungen von Unsicherheit, die hohe Aufmerksamkeit unter Akademikern und Politikern erregt hat. Der erste Aufsatz beschäftigt sich mit den Folgen von Unsicherheit, die von Finanzmärkten ausgeht. Zu diesem Zweck belege ich die empirische Relevanz von Finanzmarktunsicherheit mithilfe von SVAR Methoden. Anschließend benutze ich das von Gertler and Karadi (2011) entwickelte DSGE-Modell, um den Transmissionsmechanismus aufzudecken. Im Einklang mit der empirischen Evidenz impliziert das Modell einen Rückgang der Wirtschaftsleistung als Reaktion auf einen Anstieg der Finanzmarktunsicherheit. Dieses Ergebnis entsteht hauptsächlich aufgrund einer Verschärfung der endogenen Leverage-Beschränkung, die den finanziellen Akzelerator auslöst. Im zweiten Aufsatz schlage ich eine asymptotische Perturbationsmethode vor, um DSGE Modelle mit endogener Portfolioentscheidung zu lösen. Im Gegensatz zu existierenden Verfahren kann sie benutzt werden, um Approximationen höheren Grades von Bruttovermögenswerten zu ermitteln. Der vorgeschlagene Lösungsalgorithmus wird evaluiert, indem ich ein Lucas-Tree-Modell mit Portfolioentscheidung löse. Der Schwerpunkt liegt dabei auf den Folgen von struktureller Heterogenität in der Unsicherheit zwischen den Ländern. Die vorgeschlagene Methode erfasst diese Asymmetrie und kann zu einer Verbesserung von der Qualität der Approximation führen. Der dritte Aufsatz untersucht die Folgen von globalen Unsicherheitsschocks für die Bankenportfolios und die makroökonomischen Aggregate. Zu diesem Zweck benutze ich ein Zwei-Länder DSGE Modell mit endogener Portfolioentscheidung und Bilanzrestriktionen im Bankensektor. Die Bankenportfolios sind charakterisiert durch einen Home Bias, der mit den Daten konsistent ist. Außerdem führt ein Anstieg der Finanzmarktunsicherheit zum Rückgang der grenzüberschreitenden Bruttoanlagen und der Wirtschaftsleistung weltweit. Dies entspricht den Entwicklungen während der globalen Finanzkrise. / This thesis examines the macroeconomic implications of uncertainty which has attracted attention within both the academic literature and policy community. The first essay investigates the effects of uncertainty originating in financial markets. To this end, I first document empirical relevance of financial uncertainty using SVAR methods. Second, I employ the DSGE framework developed by Gertler and Karadi (2011) to uncover the underlying transmission mechanism. In line with the empirical evidence, the model generates a decline in economic activity in response to an increase in financial uncertainty. This outcome arises mainly because of tightening of leverage constraints which in turn triggers the financial accelerator mechanism. In the second essay, I propose an asymptotic perturbation method to solve DSGE models with endogenous portfolio choice. In contrast to existing local techniques, it can be used to compute a higher-order approximation of gross asset holdings. I evaluate the proposed method by solving a Lucas tree model with portfolio choice. The focus lies on implications of cross-country structural heterogeneity in economic uncertainty for international asset holdings. The proposed method accounts for these asymmetries and can consequently lead to an improvement in quality of the approximation. Finally, the third essay examines the consequences of global uncertainty shocks for banking portfolios and macroeconomic aggregates. To this end, I employ a two-country DSGE model with balance-sheet constrained financial intermediaries and endogenous portfolio choice. Countries are assumed to be ex-ante asymmetric, which allows me to consider both developed and emerging economies. The model implies a home bias in banking assets that is consistent with the data. Moreover, an increase in financial uncertainty leads to a decline in cross-border portfolios and a worldwide reduction in economic activity, which is consistent with dynamics observed during the global financial crisis.

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.