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Title: Essays in robust control
Author: Thamotheram, Craig P.
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2012
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My thesis consists of three chapters on issues related to the macroeconomic literature of robust control. I detail the salient properties of a RBC model with robust preferences and stochastic-volatility in technological growth. Using this as a framework, I study how to numerically approximate the worst case distribution and how agents check to see whether or not the misspecifications they consider are reasonable. This literature allows for agents to distrust the underlying statistical model governing the economy and seek policies that are ’robust’ to this model misspecification. This is accomplished in the context of a fictitious two player dynamic game between an optimising agent and her “evil” alter ego who is attempting to distort the underlying probability distribution to push the agent into unfavourable regions. The evil agent is constrained to the extent that she can distort the distribution by a parameter on the relative entropy (Kullback-Leibler divergence) between her chosen distortion and the underlying probability distribution. This parameter is somewhat arbitrary and the literature proposes a simple log-likelihood ratio test to pin it down by placing restrictions on the ability of the agent to statistically distinguish between the time series generated under the approximating and distorted model. The solution to the “evil” agents distorted distribution has no known closed form in the non-linear non-Gaussian model used but can be approximated with perturbation methods. Standard Monte Carlo methods then allow the distribution to be sampled from. The first chapter constitutes an introduction to the model used as the base unit for economic analysis in the following chapters. It introduces robustness as a concept to incorporate aversion to model misspecification. Details a very simple RBC model with the addition of stochastic volatility in technology and presents its solution methodology. Finally, the economic dynamics of the model are tested in order to validate that the model is reasonable representation of the macroeconomy and does indeed have interesting non-linearities, as well as that the non-Guassianity introduced by stochastic volatility is economically significant. This is done by considering the models moments and impulse response functions to volatility shocks. The second chapter explores the trade-offs involved in approximating and drawing from the worst case distribution using perturbation methods, a novel method using a simple completing the square argument and two Monte Carlo techniques, importance sampling and MCMC. Naturally, these techniques engender a certain degree of Monte Carlo variation in the resulting approximations. I explore the accuracy of these approximations using summary statistics and ? inaccuracy, a measure of distance between two distributions. Linear and Gaussian models of robust control allow for a analytic solution of the worst case distribution that remains Gaussian. I use this as guidance to check if a simple translated and spread Gaussian distribution approximates the worst case distribution to a similar level of accuracy as that recorded for the Monte Carlo methods. Additionally, I apply the formal empirical test of the finiteness of the variance of the importance sampling weights of ? to check the validity of using IS. This chapter serves as a practical guide for those wishing to apply these techniques as to the trade-offs involved with the various methods. The third chapter explores the statistical distinguishability between the two time series using sequential Monte Carlo methods, namely the particle filter, to evaluate the likelihoods. In comparison to the majority of the literature on robust control, the likelihoods that result from the non-linear and non-Gaussian model studied are non-analytic. Thus, the evaluation of their likelihood introduces a second source of variation in the calculation of detection error. I design a simple yet novel approach that explicitly acknowledges this in the calculation of the detection error. In order to choose the parameters for this new approach I present Monte Carlo evidence on the base unit of its construction. Also, I explore the properties of DEP across three dimensions. First, I test to see if DEP varies across the state space as well. In order to do this, and give the best chance for a difference to be found, I calculate the DEP conditional upon a given initial state. Second, for differing values of the robustness parameter I calculate the DEP in the spirit of ?, where the simulated series are generated from the unconditional distribution and the particle filter for each model also takes its initial state values from their respective unconditional distributions. Finally, for a given value of the robustness parameter I test to see if the simulation method for the worst case distribution affects the evolution of DEP, testing perturbation versus completing the square.
Supervisor: Not available Sponsor: Economic and Social Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: TJ Mechanical engineering and machinery