We consider the filtering problem in linear state space models with heavy tailed process and measurement noise. Our work is based on Student's t distribution, for which we give a number of useful results. The derived filtering algorithm is a generalization of the ubiquitous Kalman filter, and reduces to it as special case. Both Kalman filter and the new algorithm are compared on a challenging tracking example where a maneuvering target is observed in clutter. / MC Impulse

We document how sampling from a conditional Student's t distribution is implemented in stochvol. Moreover, a simple example using EUR/CHF exchange rates illustrates how to use the augmented sampler. We conclude with results and implications. (author's abstract)

Over the last twenty years or so the Dynamic Volatility literature has produced a wealth of univariate and multivariate GARCH type models. While the univariate models have been relatively successful in empirical studies, they suffer from a number ofweaknesses, such as unverifiable parameter restrictions, existence of moment conditions and the retention of Normality. These problems are naturally more acute in the multivariate GARCH type models, which in addition have the problem of overparameterization. This dissertation uses the Student's t distribution and follows the Probabilistic Reduction (PR) methodology to modify and extend the univariate and multivariate volatility models viewed as alternative to the GARCH models. Its most important advantage is that it gives rise to internally consistent statistical models that do not require ad hoc parameter restrictions unlike the GARCH formulations. Chapters 1 and 2 provide an overview of my dissertation and recent developments in the volatility literature. In Chapter 3 we provide an empirical illustration of the PR approach for modeling univariate volatility. Estimation results suggest that the Student's t AR model is a parsimonious and statistically adequate representation of exchange rate returns and Dow Jones returns data. Econometric modeling based on the Student's t distribution introduces an additional variable - the degree of freedom parameter. In Chapter 4 we focus on two questions relating to the `degree of freedom' parameter. A simulation study is used to examine:(i) the ability of the kurtosis coefficient to accurately capture the implied degrees of freedom, and (ii) the ability of Student's t GARCH model to estimate the true degree of freedom parameter accurately. Simulation results reveal that the kurtosis coefficient and the Student's t GARCH model (Bollerslev, 1987) provide biased and inconsistent estimators of the degree of freedom parameter. Chapter 5 develops the Students' t Dynamic Linear Regression (DLR) }model which allows us to explain univariate volatility in terms of: (i) volatility in the past history of the series itself and (ii) volatility in other relevant exogenous variables. Empirical results of this chapter suggest that the Student's t DLR model provides a promising way to model volatility. The main advantage of this model is that it is defined in terms of observable random variables and their lags, and not the errors as is the case with the GARCH models. This makes the inclusion of relevant exogenous variables a natural part of the model set up. In Chapter 6 we propose the Student's t VAR model which deals effectively with several key issues raised in the multivariate volatility literature. In particular, it ensures positive definiteness of the variance-covariance matrix without requiring any unrealistic coefficient restrictions and provides a parsimonious description of the conditional variance-covariance matrix by jointly modeling the conditional mean and variance functions. / Ph. D.

An effective risk management program requires reliable risk measurement. Failure to assess inherited risks in mortgage-backed securities in the U.S. market contributed to the financial crisis of 2007–2008, which has prompted government regulators to pay greater attention to controlling risk in banks, investment funds, credit unions, and other financial institutions to prevent bankruptcy and financial crisis in the future. In order to calculate risk in a reliable manner, this thesis has focused on the statistical modeling of expected return and volatility. The primary aim of this study is to propose a framework, based on the probabilistic reduction approach, to reliably quantify market risk using statistical models and historical data. Particular emphasis is placed on the importance of the validity of the probabilistic assumptions in risk measurement by demonstrating how a statistically misspecified model will lead the evaluation of risk astray. The concept of market risk is explained by discussing the narrow definition of risk in a financial context and its evaluation and implications for financial management. After highlighting empirical evidence and discussing the limitations of the ARCH-GARCH-type volatility models using exchange rate and stock market data, we proposed Student's t Autoregressive models to estimate expected return and volatility to measure risk, using Value at Risk (VaR) and Expected Shortfall (ES). The misspecification testing analysis shows that our proposed models can adequately capture the chance regularities in exchange rates and stock indexes data and give a reliable estimation of regression and skedastic functions used in risk measurement. According to empirical findings, the COVID-19 pandemic in the first quarter of 2020 posed an enormous risk to global financial markets. The risk in financial markets returned to levels prior to the COVID-19 pandemic in 2021, after COVID-19 vaccine distribution started in developed countries. / Doctor of Philosophy / Reliable risk measurement is necessary for any effective risk management program. Hence, the primary purpose of this dissertation was to propose a framework to quantify market risk using statistical models and historical data, with a particular emphasis placed on checking the validity of probabilistic assumptions underlying models. After discussing the concept of market risk and its evaluation methods in financial management, we explored the empirical evidence in financial data and highlighted some limitations of other well-known modeling approaches. In order to ameliorate limitations, this study proposed Student's t Autoregressive models to estimate the conditional mean and the conditional variance of the financial variables and use them to measure risk via two popular methods: Value at Risk (VaR) and Expected Shortfall (ES). Further investigation shows that our proposed models can adequately model exchange rates and stock indexes data and give reliable estimations to use in risk measurement. We used our model to quantify risk in global financial markets in recent years. The results show that the COVID-19 pandemic posed an enormous risk to global financial markets in the first quarter of 2020. In 2021, the level of risk in financial markets returned to levels before the COVID-19 pandemic, after COVID-19 vaccine distribution started in developed countries.

Following the Probabilistic Reduction(PR) Approach, this paper proposes the Student’s Autoregressive (St-AR) Model, Student’s t Vector Autoregressive (St-VAR) Model and their heterogeneous versions, as an alternative to the various ARCH type models, to capture univariate and multivariate volatility. The St-AR and St-VAR models differ from the latter volatility models because they give rise to internally consistent statistical models that do not rely on ad-hoc specification and parameter restrictions, but model the conditional mean and conditional variance jointly. The univariate modeling is illustrated using the Real Effect Exchange Rate(REER) indices of three mainstream currencies in Asia (RMB, Hong Kong Dollar and Taiwan Dollar), while the multivariate volatility modeling is applied to investigate the relationship between the REER indices and stock price indices in mainland China, as well as the relationship between the stock prices in mainland China and Hong Kong. Following the PR methodology, the information gained in Mis-Specification(M-S) testing leads to respecification strategies from the original Normal-(V)AR models to the St-(V)AR models. The results from formal Mis-Specification (M-S) tests and forecasting performance indicate that the St-(V)AR models provide a more appropriate way to model volatility for certain types of speculative price data. / Ph. D.

此論文研究有什麼因子會影響台灣股票加權指數報酬率之偏態係數。過去的文獻顯示，交易量和報酬率為可能的因子。實證的結果確實發現，交易量和報酬率顯著地影響偏態係數。 / This study examines the determinants for conditional skewness of the return distribution of the Taiwan Stock Exchange Value-Weighted Index. Important driving factors that affect conditional skewness, based on the theory literature, include trading volumes and returns. To capture the skewness in the data, the family of time series model we consider focuses on the specifications of higher-order moments than mean and volatility that conventional models look at. With the specifications, we are able to test whether the factors, volumes and returns, can influence conditional skewnees of the return distribution. Our results suggest the significance of the factors using data from the Taiwan Stock Exchange Value-Weighted Index.

This thesis has used the Fama and French five-factor model (FF5M) and proposed an alternative model. The proposed model is named the Fama and French five-factor heteroscedastic student's model (FF5HSM). The model utilises an ARMA model for the returns with the FF5M factors incorporated and a GARCH(1,1) model for the volatility. The FF5HSM uses returns data from the FF5M's portfolio construction for the Japanese stock market and the five risk factors. The portfolio's capture different levels of market capitalisation, and the factors capture market risk. The ARMA modelling is used to address the autocorrelation present in the data. To deal with the heteroscedasticity in daily returns of stocks, a GARCH(1,1) model has been used. The order of the GARCH-model has been concluded to be reasonable in academic literature for this type of data. Another finding in earlier research is that asset returns do not follow the assumption of normality that a regular regression model assumes. Therefore, the skewed student's t-distribution has been assumed for the error terms. The result of the data indicates that the FF5HSM has a better in-sample fit than the FF5M. The FF5HSM addresses heteroscedasticity and autocorrelation in the data and minimises them depending on the portfolio. Regardingforecasting, both the FF5HSM and the FF5M are accurate models depending on what portfolio the model is applied on.