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This thesis has used bivariate time series models to investigate the long-run causal relationships between climatic variables. The cointegration approach, widely used in econometrics, has been shown to provide more reliable estimates for detection and attribution of trends in global mean temperature. The traditional ordinary least squares (OLS) and total least squares (TLS) esti- mates from a static regression model are critically compared with the maximum likelihood (ML) estimates from a cointegrating vector autoregressive (VAR) model. Using synthetic data, generated by a simple stochastic model of the climate-carbon system, the estimates are compared against a known true value and evaluated in terms of key desirable statistical properties. Results show that the OLS estimates are strongly negatively biased, TLS estimates are less biased than OLS and posi- tively biased compared to the VAR-ML estimates. TLS estimates are much more uncertain than those from the other approaches. VAR-ML estimates are less biased and more e cient compared to estimates from the traditional approaches. Comparison has also been made using real historic global mean temperature data and climate model simulations from Coupled Model Intercomparison Project 5 (CMIP5) archive, and similar conclusions were found. All CMIP5 model runs were found to have cointegrating relationship with historical observed temperature. Another issue addressed in this thesis is the Granger causality between paleoclimate temperature and CO2. Di erent extensions of the VAR model were used to assess Granger causality between the two variables. This research has shown that two-way causality (feedback) is occurring between temperature and CO2, particularly during the glacial epochs. Impulse-response analysis was also carried out to quantify dynamic interactions between the variables. This showed that each variable reacted positively to a shock in another. For example, a 100ppmv increase in CO2 can induce an increase of up to 4 C in temperature and a 1 C increase in temperature induces up to 2.3ppmv increase in CO2 during glacial periods in particular. A shock to CO2 during the warmer interglacial periods was seen to induce an explosive increase in both temperature and CO2.

We present sampling-based methodologies for the estimation of structural time series in the presence of outliers and structural shifts. We start by considering a simple structural model: a local level model, in the presence of outliers and level shifts. The existence of shocks is accounted for by including a product of intervention variables in the measurement and transition equations. These factors are composed of the product of an indicator variable and a parameter for the magnitude of the intervention variable, defining the size of the shocks. The Gibbs sampler is the Markov chain Monte Carlo method used for estimating the intervention model. Our contribution is in the use of a uniform prior distribution for the size of intervention variables. We show that this choice provides advantages over the usual multinomial and normal prior assumptions. The methodology is extended to a basic structural model. Using this model formulation, we consider 4 types of shocks: outliers, level, slope and seasonal shifts. The use of simulation based methods for this range of different breaks in structural models is not dealt with in the existing literature. By using the Gibbs sampler, we simultaneously estimate all the hyperparameters, detect the position of the shocks and estimate their size. Finally, we consider the local level model in the presence of outliers and level shifts for the case where one of the hyperparameters is equal to zero. In this situation, simulation based methods usually assume a multinomial prior distribution for the size of the intervention variables. We use a uniform prior, and present a two stages sampling scheme. In this two stage process the Gibbs sampler is first run on an auxiliary data set which has the same shocks as the original data set. For all the methods presented, performance is assessed by Monte Carlo studies and empirical applications to real data sets.

This thesis examines structural breaks in time series regressions where both regressors and errors may exhibit long range dependence. Statistical properties of methods for detecting and estimating structural breaks are analysed and asymptotic distribution of estimators and test statistics are obtained. Valid bootstrap methods of approximating the limiting distribution of the relevant statistics are also developed to improve on the asymptotic approximation in finite samples or to deal with the problem of unknown asymptotic distribution. The performance of the asymptotic and bootstrap methods are compared through Monte Carlo experiments. A background of the concepts of structural breaks, long memory and bootstrap is offered in Introduction where the main contribution of the thesis is also outlined. Chapter 1 proposes a fluctuation-type test procedure for detecting instability of slope coefficients. A first-order bootstrap approximation of the distribution of the test statistic is proposed. Chapter 2 considers estimation and testing of the time of the structural break. Statistical properties of the estimator are examined under a range of assumptions on the size of the break. Under the assumption of shrinking break, a bootstrap approximation of the asymptotic test procedure is proposed. Chapter 3 addresses a drawback of the assumption of fixed size of break. Under this assumption, the asymptotic distribution of the estimator of the breakpoint depends on the unknown underlying distribution of data and thus it is not available for inference purposes. The proposed solution is a bootstrap procedure based on a specific type of deconvolution.

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In this thesis, the problem of counting gauge invariant operators in certain supersymmetric theories is discussed. These objects have a very important role in supersymmetric gauge theories, since they can be used to describe the space of zero-energy solutions, called moduli space, of such theories. In order to approach the counting problem, a technique is used based on a function known in Algebraic Geometry as the Hilbert series. For the examined theories, this can be considered a a partition function counting gauge invariant operators in the field theory according to their charges under quantum global symmetries. In the first part of the thesis, particular focus will be given to the application of the Hilbert series to conformal Chern-Simons theories living on the world-volume of M2-branes probing different toric Calabi-Yau 4-fold singularities. It will be shown how the Hilbert series can be combined with the brane tiling formalism to characterise the mesonic moduli space of vacua of a given theory through its generators and the relations they satisfy. Then, toric duality for these theories will be presented, with special attention to the role played by Hilbert series in making such feature manifest between two or more theories. Finally, Chern-Simons theories living on M2-branes probing cones over smooth toric Fano 3-folds and their mesonic Hilbert series will be presented. In the second part, it will be shown how the Hilbert series can be applied to counting gauge invariant operators in supersymmetric generalisations of Quantum Chromodynamics, known as SQCD theories. The discussion will hinge on a specific class of theories, with N multiplets transforming in the fundamental and anti-fundamental and one in the adjoint representation of the gauge group. For each classical group, the Hilbert series of the moduli space will be used to determine the dimension on the spaces, their generators and to argue that they are all Calabi-Yau manifolds.