146 research outputs found

    Pairwise versus mutual independence: visualisation, actuarial applications and central limit theorems

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    Accurately capturing the dependence between risks, if it exists, is an increasingly relevant topic of actuarial research. In recent years, several authors have started to relax the traditional 'independence assumption', in a variety of actuarial settings. While it is known that 'mutual independence' between random variables is not equivalent to their 'pairwise independence', this thesis aims to provide a better understanding of the materiality of this difference. The distinction between mutual and pairwise independence matters because, in practice, dependence is often assessed via pairs only, e.g., through correlation matrices, rank-based measures of association, scatterplot matrices, heat-maps, etc. Using such pairwise methods, it is possible to miss some forms of dependence. In this thesis, we explore how material the difference between pairwise and mutual independence is, and from several angles. We provide relevant background and motivation for this thesis in Chapter 1, then conduct a literature review in Chapter 2. In Chapter 3, we focus on visualising the difference between pairwise and mutual independence. To do so, we propose a series of theoretical examples (some of them new) where random variables are pairwise independent but (mutually) dependent, in short, PIBD. We then develop new visualisation tools and use them to illustrate what PIBD variables can look like. We showcase that the dependence involved is possibly very strong. We also use our visualisation tools to identify subtle forms of dependence, which would otherwise be hard to detect. In Chapter 4, we review common dependence models (such has elliptical distributions and Archimedean copulas) used in actuarial science and show that they do not allow for the possibility of PIBD data. We also investigate concrete consequences of the 'nonequivalence' between pairwise and mutual independence. We establish that many results which hold for mutually independent variables do not hold under sole pairwise independent. Those include results about finite sums of random variables, extreme value theory and bootstrap methods. This part thus illustrates what can potentially 'go wrong' if one assumes mutual independence where only pairwise independence holds. Lastly, in Chapters 5 and 6, we investigate the question of what happens for PIBD variables 'in the limit', i.e., when the sample size goes to infi nity. We want to see if the 'problems' caused by dependence vanish for sufficiently large samples. This is a broad question, and we concentrate on the important classical Central Limit Theorem (CLT), for which we fi nd that the answer is largely negative. In particular, we construct new sequences of PIBD variables (with arbitrary margins) for which a CLT does not hold. We derive explicitly the asymptotic distribution of the standardised mean of our sequences, which allows us to illustrate the extent of the 'failure' of a CLT for PIBD variables. We also propose a general methodology to construct dependent K-tuplewise independent (K an arbitrary integer) sequences of random variables with arbitrary margins. In the case K = 3, we use this methodology to derive explicit examples of triplewise independent sequences for which no CLT hold. Those results illustrate that mutual independence is a crucial assumption within CLTs, and that having larger samples is not always a viable solution to the problem of non-independent data

    Essays on well-being: a UK analysis

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    Forward uncertainty quantification with special emphasis on a Bayesian active learning perspective

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    Uncertainty quantification (UQ) in its broadest sense aims at quantitatively studying all sources of uncertainty arising from both computational and real-world applications. Although many subtopics appear in the UQ field, there are typically two major types of UQ problems: forward and inverse uncertainty propagation. The present study focuses on the former, which involves assessing the effects of the input uncertainty in various forms on the output response of a computational model. In total, this thesis reports nine main developments in the context of forward uncertainty propagation, with special emphasis on a Bayesian active learning perspective. The first development is concerned with estimating the extreme value distribution and small first-passage probabilities of uncertain nonlinear structures under stochastic seismic excitations, where a moment-generating function-based mixture distribution approach (MGF-MD) is proposed. As the second development, a triple-engine parallel Bayesian global optimization (T-PBGO) method is presented for interval uncertainty propagation. The third contribution develops a parallel Bayesian quadrature optimization (PBQO) method for estimating the response expectation function, its variable importance and bounds when a computational model is subject to hybrid uncertainties in the form of random variables, parametric probability boxes (p-boxes) and interval models. In the fourth research, of interest is the failure probability function when the inputs of a performance function are characterized by parametric p-boxes. To do so, an active learning augmented probabilistic integration (ALAPI) method is proposed based on offering a partially Bayesian active learning perspective on failure probability estimation, as well as the use of high-dimensional model representation (HDMR) technique. Note that in this work we derive an upper-bound of the posterior variance of the failure probability, which bounds our epistemic uncertainty about the failure probability due to a kind of numerical uncertainty, i.e., discretization error. The fifth contribution further strengthens the previously developed active learning probabilistic integration (ALPI) method in two ways, i.e., enabling the use of parallel computing and enhancing the capability of assessing small failure probabilities. The resulting method is called parallel adaptive Bayesian quadrature (PABQ). The sixth research presents a principled Bayesian failure probability inference (BFPI) framework, where the posterior variance of the failure probability is derived (not in closed form). Besides, we also develop a parallel adaptive-Bayesian failure probability learning (PA-BFPI) method upon the BFPI framework. For the seventh development, we propose a partially Bayesian active learning line sampling (PBAL-LS) method for assessing extremely small failure probabilities, where a partially Bayesian active learning insight is offered for the classical LS method and an upper-bound for the posterior variance of the failure probability is deduced. Following the PBAL-LS method, the eighth contribution finally obtains the expression of the posterior variance of the failure probability in the LS framework, and a Bayesian active learning line sampling (BALLS) method is put forward. The ninth contribution provides another Bayesian active learning alternative, Bayesian active learning line sampling with log-normal process (BAL-LS-LP), to the traditional LS. In this method, the log-normal process prior, instead of a Gaussian process prior, is assumed for the beta function so as to account for the non-negativity constraint. Besides, the approximation error resulting from the root-finding procedure is also taken into consideration. In conclusion, this thesis presents a set of novel computational methods for forward UQ, especially from a Bayesian active learning perspective. The developed methods are expected to enrich our toolbox for forward UQ analysis, and the insights gained can stimulate further studies

    Application of complex network theory for flood estimation under current and future climate.

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    Understanding the nature and unravelling the extents of connections between various components in hydrologic systems have always remained a fundamental challenge in hydrologic research community. The complexity of hydrologic system stems from various aspects, including the constitution of numerous components and sub-components, their direct or indirect internal connections, interactions with climate and ecosystem, and the complicated dynamics due to natural evolution and human activities, among others. Furthermore, the impacts of evidenced warming climate and anthropogenic activities on hydrologic system force us to tackle a host of new considerations into detecting, attributing, and predicting hydrologic processes under nonstationary condition. All these factors pose a significant challenge to system modellers, hydrologists and water resource researchers and emphasize the importance of the application of latest and innovative scientific theories for hydrologic research. In catchment modelling and management applications, a proper understanding of the connectivity of different components is important, for example, rainfall, soil type, land use, slope and finally the catchment response, i.e., floods. Considering hydrologic stations or catchments as networks can lead to new insights and is emerging as an attractive field of research within the scientific community. The main objective of this thesis is to present the strength of network theory in solving hydrological problems by developing and implementing network-based numerical models. The research is divided into four main parts. In the first part, a network-based framework is developed to delineate homogeneous neighbours for ungauged catchment to be used with regional flood frequency analysis (RFFA). The developed framework clearly demonstrates the strength of network theory in identifying meaningful homogeneous region thus improving the accuracy of regional flood quantile estimations. In the second part, a network theory based RFFA approach has been integrated with nonstationary climate conditions and its ability in predicting future flood peaks under warming climate is demonstrated. The developed approach shows clear advantages over the existing nonstationary frequency distribution-based approaches. The third part focuses on investigating the temporal and spatial connectivity patterns of hydrologic variables. By assuming individual locations or timesteps as the nodes of a network, constructed time series are used to form network metrics and to define the strength of connections between nodes and the nature of network structure. Our findings indicate the utility and effectiveness of network theory in exploring and analysing different temporal and spatial connections in hydrology. Finally, the fourth part explores the joint dependence of extreme rainfall events in the space based on the theory of networks, where a novel measure to quantify the possibility of concurrence in complex systems is developed. Results show a weaker spatial dependence under warmer temperatures, however, a stronger dependence at El Niño and La Niña periods

    Impact of oil prices on stock market performance

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    The study investigates the impact of oil prices on stock market performance in ten countries, including Canada. The volatility in oil prices and the accompanying swings in stock market performance raised the question of what, if any, is the relationship between these variables. The research seeks to address six strands of the phenomenon. The study evaluates the impact of oil prices on stock performance at the stock market’s aggregate and sector market levels. It establishes the effects of macroeconomic variables on stock market performance. Furthermore, it evaluates the role of the business cycle in the oil price shocks and stock market interface. Lastly, it examines the influence of oil prices on stock market performance in net oil-importing and oil-exporting countries. The empirical investigation uses monthly data from January 2003 to December 2020 and quarterly data from 1990Q1 to 2020Q4. Primary and secondary data were analysed using statistical tools and econometric modelling. The investigation employs the impulse response function, EGARCH and Markov switching models. The thesis concludes that the relationship between oil prices and stock market performance is time-varying, asymmetrical, heterogeneous and complex as several sector or country-specific factors drive the relationship. Specifically, the findings suggest that the response of the stock market sectors to oil price shocks differs substantially, depending on their degree of oil dependence and multiple transmission mechanisms. The findings further indicate that stock returns-generating processes in a net oil-exporting country like Canada exhibited a high degree of persistence in conditional variance, and the modelling of asymmetry was positive. Positive shocks from macroeconomic variables impact the country’s stock market more than negative shocks of the same magnitude. Two structural breaks are identified in the Canadian economy between 1990 and 2020. The data was further divided into two subsamples to reflect the two possible states for an economy, the bear and bull periods. Empirical analysis revealed that GDP, exchange rate, inflation rate, interest rate, and oil prices are significant drivers of the country’s stock market performance in economic contraction. During the expansion era, all the variables considered in the study, excluding GDP, significantly drive stock market performance. Hence, oil prices and stock market relationships tend to improve more during the economic expansion period than during the contraction era. Further analysis affirmed that the impact of oil price shocks is only significant in the top two net oil-importing countries. These findings convey information that guides policymakers in formulating macroeconomic policies, investors and portfolio managers in risk diversification relating to decision-making and investment strategies. The study investigates the impact of oil prices on stock market performance in ten countries, including Canada. The volatility in oil prices and the accompanying swings in stock market performance raised the question of what, if any, is the relationship between these variables. The research seeks to address six strands of the phenomenon. The study evaluates the impact of oil prices on stock performance at the stock market’s aggregate and sector market levels. It establishes the effects of macroeconomic variables on stock market performance. Furthermore, it evaluates the role of the business cycle in the oil price shocks and stock market interface. Lastly, it examines the influence of oil prices on stock market performance in net oil-importing and oil-exporting countries. The empirical investigation uses monthly data from January 2003 to December 2020 and quarterly data from 1990Q1 to 2020Q4. Primary and secondary data were analysed using statistical tools and econometric modelling. The investigation employs the impulse response function, EGARCH and Markov switching models. The thesis concludes that the relationship between oil prices and stock market performance is time-varying, asymmetrical, heterogeneous and complex as several sector or country-specific factors drive the relationship. Specifically, the findings suggest that the response of the stock market sectors to oil price shocks differs substantially, depending on their degree of oil dependence and multiple transmission mechanisms. The findings further indicate that stock returns-generating processes in a net oil-exporting country like Canada exhibited a high degree of persistence in conditional variance, and the modelling of asymmetry was positive. Positive shocks from macroeconomic variables impact the country’s stock market more than negative shocks of the same magnitude. Two structural breaks are identified in the Canadian economy between 1990 and 2020. The data was further divided into two subsamples to reflect the two possible states for an economy, the bear and bull periods. Empirical analysis revealed that GDP, exchange rate, inflation rate, interest rate, and oil prices are significant drivers of the country’s stock market performance in economic contraction. During the expansion era, all the variables considered in the study, excluding GDP, significantly drive stock market performance. Hence, oil prices and stock market relationships tend to improve more during the economic expansion period than during the contraction era. Further analysis affirmed that the impact of oil price shocks is only significant in the top two net oil-importing countries. These findings convey information that guides policymakers in formulating macroeconomic policies, investors and portfolio managers in risk diversification relating to decision-making and investment strategies

    Mathematical Modeling of Biological Systems

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    Mathematical modeling is a powerful approach supporting the investigation of open problems in natural sciences, in particular physics, biology and medicine. Applied mathematics allows to translate the available information about real-world phenomena into mathematical objects and concepts. Mathematical models are useful descriptive tools that allow to gather the salient aspects of complex biological systems along with their fundamental governing laws, by elucidating the system behavior in time and space, also evidencing symmetry, or symmetry breaking, in geometry and morphology. Additionally, mathematical models are useful predictive tools able to reliably forecast the future system evolution or its response to specific inputs. More importantly, concerning biomedical systems, such models can even become prescriptive tools, allowing effective, sometimes optimal, intervention strategies for the treatment and control of pathological states to be planned. The application of mathematical physics, nonlinear analysis, systems and control theory to the study of biological and medical systems results in the formulation of new challenging problems for the scientific community. This Special Issue includes innovative contributions of experienced researchers in the field of mathematical modelling applied to biology and medicine

    Development and application of statistical learning methods in insurance and finance

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    This thesis deals with the development and application of statistical learning methods in insurance and finance. Firstly, we focus on an insurance-linked financial instrument type namely catastrophe bond. Given the intricacies, and the over-the-counter nature of the market where these instruments are traded, we introduce a flexible statistical learning model called random forest. We use real data in order to predict the spread of a new catastrophe bond at issuance and identify the importance of various variables in their ability to predict the spread in a purely predictive framework. Finally, we develop and implement a series of robustness checks to ensure repeatability of prediction performance and predictors’ importance results. Secondly, we explore a decision-making problem which is faced in an abundance of interdisciplinary settings referring to the combination of different experts’ opinions on a given topic. Focusing on the case where opinions are expressed in a probabilistic manner, we suggest employing a finite mixture modelling methodology to capture various sources of heterogeneity in experts’ opinions, and assist the decision maker to test their very own judgement on opinions weights allocation too. An application in an actuarial context is presented where different actuaries report their opinions about a quantile-based risk measure to decide on the level of reserves they need to hold for regulatory purposes. Finally, we focus on the problem of regression analysis for multivariate count data in order to capture the dependence structures between multiple count response variables based on explanatory variables, which is encountered across several disciplines. In particular, we introduce a multivariate Poisson-Generalized Inverse Gaussian regression model with varying dispersion and shape for modelling different types of insurance claims and their associated counts and we provide a real-data application in non-life insurance

    Optimazation of marine sediments characterization via statistical analysis

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    The task of geotechnical site characterization includes defining the layout of ground units and establishing their relevant engineering properties. This is an activity in which uncertainties of different nature (inherent, experimental, of interpretation…) are always present and in which the amount and characteristics of available data are highly variable. Probabilistic methodologies are applied to assess and manage uncertainties. A Bayesian perspective of probability, that roots probability on belief, is well suited for geotechnical characterization problems, as it has flexibility to handle different kind of uncertainties and highly variable datasets –in quality and quantity. This thesis addresses different topics of geotechnical site characterization from a probabilistic perspective, with emphasis on offshore investigation, on the Cone Penetration Test (CPTu) and on Bayesian methodologies.The first topic addresses soil layer delineation based on CPT(u) data. The starting point is the recognition that layer delineation is problem-oriented and has a strong subjective component. We propose a novel CPTu record analysis methodology which aims to: a) elicit the heuristics that intervene in layer delineation, facilitating communication and coherence in interpretation b) facilitate probabilistic characterization of the identified layers c) is simple and intuitive to use. The method is based on sequential distribution fitting in conventionally accepted classification spaces (Soil Behavior Type charts). The proposed technique is applied at different sites, illustrating how it can be related to borehole observations, how it compares with alternative methodologies and how it can be extended to create cross-site profiles. The second topic addresses strain-rate corrections of dynamic CPTu data. Dynamic CPTu impact on the seafloor and are very agile characterization tools. However, they require transformation to equivalent quasi-static results that can be conventionally interpreted. Up to now the necessary corrections are either too vague or require the acquisition of paired dynamic and quasi-static CPTu records (i.e., same location’s acquisition). A Bayesian methodology is applied to derive strain-rate coefficients in a more general setting, one in which some quasi-static CPTu records are available in the study area, but they need not be paired to any converted dynamic CPTu. Application to a case study offshore Nice shows that the results match those obtained using paired tests. Furthermore, strain rate correction coefficients and transformed quasi-static profiles are expressed in probabilistic terms.The third topic addressed is the optimization of soil unit weight prediction from CPTu readings. A Bayesian Mixture Analysis is applied to a global database to identify hidden soil classes within it. The goal is to improve the accuracy of regressions between geotechnical parameters obtained by exploiting the database. The method is applied to predict soil unit weight from CPTu data, a problem that has intrinsic practical interest but it is also representative of difficulties faced by a larger class of problems in geotechnical regression. Results highlight a decrease of systematic transformation uncertainty and an improve of accuracy of soil unit weight prediction from CPTu at new sites. In a final application we present a probabilistic earthquake-induced landslide susceptibility map of the South-West Iberian margin. A simplified geotechnical pixel-based slope stability model is considered to whole study area within which the key stability model parameters are treated as random variables. Site characterization at the regional scale combines a global database with available geotechnical data through a Bayesian scheme. Outputs (landslide susceptibility maps) are derived from a reliability-based design procedure (Montecarlo simulations) providing a robust landslide susceptibility prediction at the site according to Receiver Operating Curve (ROC).La caracterización geotécnica de un emplazamiento incluye la definición de la disposición de las unidades de suelo y el establecimiento de sus propiedades de ingeniería relevantes. Es una actividad en la que siempre están presentes incertidumbres y en la que la cantidad y las caracteristicas de los datos disponibles son muy variables. Para evaluar y gestionar las incertidumbres se aplican metodologías probabilísticas. Una perspectiva bayesiana de la probabilidad es muy adecuada para la caracterización geotécnica, ya que tiene flexibilidad para manejar incertidumbres y datos muy variables. Esta tesis aborda diferentes temas de caracterización geotécnica desde una perspectiva probabilística, con énfasis en la investigación en alta mar, en el ensayo de penetración de cono (CPTu) y en las metodologías bayesianas El primer tema aborda la delineación de la capa de suelo basada en los datos CPT(u). El punto de partida es el reconocimiento de que la delineación de capas tiene un fuerte componente subjetivo. Proponemos una novedosa metodología de análisis de registros CPTu que tiene como objetivo: a) expresar la heurística que interviene en la delineación de capas, facilitando la comunicación en la su interpretación b) facilitar la caracterización probabilística de las capas identificadas c) uso sencillo e intuitivo. El método se basa en el ajuste de distribuciones secuenciales en espacios de clasificación (tablas de comportamiento del suelo). La técnica propuesta se aplica en diferentes emplazamientos, ilustrando cómo puede relacionarse con sondeos, cómo se compara con metodologías alternativas y cómo puede ampliarse para crear perfiles entre emplazamientos. El segundo tema aborda las correcciones de la velocidad de deformación de los datos del CPTu dinámico (que impactan en el fondo marino y son herramientas de caracterización muy ágiles). Sin embargo, requieren una transformación a resultados equivalentes que puedan ser interpretados convencionalmente. Hasta ahora las correcciones necesarias son vagas o requieren la adquisición de CPTu dinámicos y cuasi-estáticos emparejados. Se aplica una metodologia bayesiana para derivar los coeficientes de velocidad de deformación en un entorno más general, en el que se dispone de algunos registros de CPTu cuasi­estáticos en la zona de estudio, pero no es necesario emparejarlos con ningún CPTu dinámico convertido. La aplicación a un estudio de caso en el mar de Niza muestra que los resultados coinciden con los obtenidos mediante pruebas emparejadas. El tercer tema abordado es la optimización de la predicción del peso unitario del suelo a partir de las lecturas del CPTu. Se aplica un análisis de mezclas bayesiano a una base de datos global para identificar las clases de suelo ocultas en ella. El objetivo es mejorar la precisión de las regresiones entre los parámetros geotécnicos obtenidos explotando la base de datos. El método se aplica a la predicción del peso unitario del suelo a partir de los datos del CPTu. Los resultados destacan una disminución de la incertidumbre sistemática de la transformación y una mejora de la precisión de la predicción del peso unitario del suelo a partir de CPTu en nuevos sitios. En una aplicación final presentamos un mapa probabilistico de susceptibilidad a los deslizamientos de tierra inducidos por terremotos en el margen suroeste de la Península Ibérica. Se considera un modelo geotécnico simplificado de estabilidad de laderas basado en píxeles para toda el área de estudio, dentro del cual los parámetros clave del modelo de estabilidad se tratan como variables aleatorias. La caracterización a escala regional combina una base de datos global con los datos geotécnicos disponibles mediante un esquema bayesiano. Mapas de susceptibilidad a los corrimientos de tierra se derivan de un procedimiento de diseño basado en la fiabilidad que proporciona una predicción robusta de la susceptibilidad a deslizamientos de tierra en el sitio de acuerdo con la curva operativa del receptor (ROC).Postprint (published version

    Estimating Dependences and Risk between Gold Prices and S&P500: New Evidences from ARCH,GARCH, Copula and ES-VaR models

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    This thesis examines the correlations and linkages between the stock and commodity in order to quantify the risk present for investors in financial market (stock and commodity) using the Value at Risk measure. The risk assessed in this thesis is losses on investments in stock (S&P500) and commodity (gold prices). The structure of this thesis is based on three empirical chapters. We emphasise the focus by acknowledging the risk factor which is the non-stop fluctuation in the prices of commodity and stock prices. The thesis starts by measuring volatility, then dependence which is the correlation and lastly measure the expected shortfalls and Value at risk (VaR). The research focuses on mitigating the risk using VaR measures and assessing the use of the volatility measures such as ARCH and GARCH and basic VaR calculations, we also measured the correlation using the Copula method. Since, the measures of volatility methods have limitations that they can measure single security at a time, the second empirical chapter measures the interdependence of stock and commodity (S&P500 and Gold Price Index) by investigating the risk transmission involved in investing in any of them and whether the ups and downs in the prices of one effect the prices of the other using the Time Varying copula method. Lastly, the third empirical chapter which is the last chapter, investigates the expected shortfalls and Value at Risk (VaR) between the S&P500 and Gold prices Index using the ES-VaR method proposed by Patton, Ziegel and Chen (2018). Volatility is considered to be the most popular and traditional measure of risk. For which we have used ARCH and GARCH model in our first empirical chapter. However, the problem with volatility is that it does not take into account the direction of an investments’ movement: volatility of stocks is that they suddenly jump higher and investors are not distressed with gains. When we talk about investors for them the risk is about the odds of losing money, after my research and findings VaR is based on the common-sense fact. Hence, investors care about the odds of big losses, VaR answers the question, what is my worst-case scenario? Or simply how much I could lose in a really bad month? The results of the thesis demonstrated that measuring volatility (ARCH GARCH) alone was not sufficient in measuring the risk involved in an investment therefore methodologies such as correlation and VAR demonstrates better results. In terms of measuring the interdependence, the Time Varying Copula is used since the dynamic structure of the de- pendence between the data can be modelled by allowing either the copula function or the dependence parameter to be time varying. Lastly, hybrid model further demonstrates the average return on a risky asset for which Expected Shortfall (ES) along with some quantile dependence and VaR (Value at risk) is utilised. Basel III Accord which is applied in coming years till 2019 focuses more on ES unlike VaR, hence there is little existing work on modelling ES. The thesis focused on the results from the model of Patton, Ziegel and Chen (2018) which is based on the statistical decision theory. Patton, Ziegel and Chen (2018), overcame the problem of elicitability for ES by using ES and VaR jointly and propose the new dynamic model of risk measure. This research adds to the contribution of knowledge that measuring risk by using volatility is not enough for measuring risk, interdependence helps in measuring the dependency of one variable over the other and estimations and inference methods proposed by Patton, Ziegel and Chen (2018) using simulations proposed in ES-VaR model further concludes that ARCH and GARCH or other rolling window models are not enough for determining the risk forecasts. The results suggest, in first empirical chapter we see volatility between Gold prices and S&P500. The second empirical chapter results suggest conditional dependence of the two indexes is strongly time varying. The correlation between the stock is high before 2008. The results further displayed slight stronger bivariate upper tail, which signifies that the conditional dependence of the indexes is influence by positive shocks. The last empirical chapter findings proposed that measuring forecasts using ES-Var model proposed by Patton, Ziegel and Chen (2018) does outer perform forecasts based on univariate GARCH model. Investors want to 10 protect themselves from high losses and ES-VaR model discussed in last chapter would certainly help them to manage their funds properly
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