Covariance Matrix Estimation and Particle Filtering Methods for Parameter Estimation in Stochastic Volatility Models

摘要

The primary focus of the first chapter of this thesis is to estimate high dimensional covariance matrices in a Bayesian set-up. Unless the number of observations is very large, estimating a covariance matrix with the positive definiteness constraint is often inefficient. We propose an effective regularization scheme that exploits the method proposed in Giordani et al.(2012). We reparameterise the covariance matrix through the Cholesky decomposition of the correlation matrix and estimate Cholesky elements by maximizing the log-posterior using a stochastic optimization algorithm. We investigate the performance of proposed method under different loss functions and also apply it to three real data examples.From the second chapter onwards, we concentrate on the estimation of volatility models which can be represented by state space models. Particle filters provide an approach to inference for such models where observations arrive sequentially in time. It was originally developed for online filtering and prediction of nonlinear or non-Gaussian state space models. The literature to learn about model parameters using the particle filtering is rather limited. This thesis makes a contribution to the literature on the stochastic volatility models by implementing particle markov chain monte carlo methods (pMCMC), specifically particle Gibbs with backward simulation methods, to infer about the model parameters. The asymmetry (or leverage which capture the correlations between the innovations of the asset returnsand those of the latent volatility processes) effect is one of the main issues in financial econometrics and it will be explored in this thesis. The chapters of this thesis provide distinct but complementary contributions to the literature of parameter estimation of stochastic volatility models.In multivariate time series analysis, multivariate stochastic volatility model with leverage and cross leverage effects are considered and extended to heavy tailed errors. This model has some desirable properties, both in terms of statistics and empirical perspectives. The deviance information criterion (DIC) is used as model selection criteria. Applied to U.S., UK and Germany stock prices, the model generated some interesting results, including that cross leverage effects are significantly high in absolute value and heavy tailed errors are preferred over Gaussian errors.