Quantitative Analyst/Statistician, Risk ManagementAIG International, NY, NY
Bayesian Time Series: Financial Models and Spectral Analysis
This dissertation studies models for Bayesian time series analysis in two areas: harmonic components models for Bayesian spectral analysis, and stochastic volatility models for time domain analysis of financial time series. Developments in Bayesian spectral analysis and parameter estimation during the 1980s demonstrated connections between Bayesian frequency estimation and traditional Fourier transform/periodogram methods. However, accurate estimation of frequencies under the Bayesian framework has been inaccessible as computations are difficult. This thesis addresses this problem, and develops efficient MCMC sampling approaches to posterior computation. This involves novel methods for exploring highly multimodal posterior distributions for frequency parameters in harmonic models, which are illustrated in various applied contexts. Motivated by problems in geological time series, we then investigate issues of uncertain timing in frequency estimation. This leads to development of a harmonic model with uncertain timing to illustrate the use of Bayesian MCMC simulation methods as a general approach for complex models in Bayesian spectral analysis. We explore this in various data analyses, investigating both questions of raw errors in timing and connections with time deformations. Volatility plays a central role in modern finance especially in the pricing of derivative securities. Two active research areas on time-varying volatility are ARCH/GARCH type models and stochastic volatility models. ARCH type models are empirically successful but they lack economic intuition. Stochastic volatility models are statistically elegant and have strong connection to continuous-time finance models. Yet further development and comparison of the model with ARCH type models were hampered by difficulties in estimation. This thesis addresses these issues and proposes efficient Bayesian MCMC sampling procedure for a log-AR(1) stochastic volatility model. This is extended to include new simulation-based model diagnostic methods for in-sample model adequacy check, and the result is compared with popular EGARCH models. Finally, we propose a new model which combines both historical volatility and implied volatility under one model framework, which can be used for both forecasting and testing of the hypothesis of the existence of stochastic volatility.