Resumen de Bayesian non-parametrics for time-varying volatility models
Understanding, modeling and predicting volatilities of financial time series has been extensively researched for more than 30 years and the interest in the subject is far from decreasing. The two most popular approaches to model volatility are based on the Autoregressive Conditional Heteroscedasticity (ARCH) type and Stochastic Volatility (SV) type models. Besides selecting the best model for the volatility, distributional assumptions for the returns are equally important. Traditionally, the errors have been assumed to be Gaussian or Student-t, however these assumptions are rather restrictive. An alternative approach is to use a mixture of distributions, which can approximate arbitrarily any distribution given a sufficient number of mixture components. Models with the mixture distribution for the errors outperform the Gaussian ones and do not require additional restrictions on the degrees of freedom parameter as the Student-t one. As for the inference and prediction, the Bayesian approach is especially well-suited for GARCH and SV models and provides some advantages compared to classical estimation techniques, as outlined by Ardia & Hoogerheide (2010). Therefore in this thesis we consider different Bayesian non-parametric specifications for the errors for GARCH and SV models. Also, we employ two Bayesian estimation approaches: Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC). Chapter 2 reviews the existing literature on the most relevant Bayesian inference methods for univariate and multivariate GARCH and SV models. The advantages and drawbacks of each procedure are outlined as well as the advantages of the Bayesian approach versus classical procedures. The chapter makes emphasis on Bayesian nonparametrics for time-varying volatility models that avoid imposing arbitrary parametric distributional assumptions. Finally, the chapter presents an alternative Bayesian estimation technique - Sequential Monte Carlo, that allows for an on-line type inference. The major part of the contents of this chapter resulted into a paper by Virbickaite et al. (2013), which has been accepted in the Journal of Economic Surveys. Chapter 3 considers an asymmetric dynamic conditional correlation (ADCC) model to estimate the time-varying correlations of financial returns where the individual volatilities are driven by GJR-GARCH models. This composite model takes into consideration the asymmetries in individual assets' volatilities, as well as in the correlations. The errors are modeled using a Dirichlet location-scale mixture of multivariate Normals allowing for a flexible return distribution in terms of skewness and kurtosis. This gives rise to a Bayesian non-parametric ADCC (BNP-ADCC) model, as opposed to a symmetric specification, called BNP-DCC. Then these two models are estimated using MCMC and compared by considering a sample of Apple Inc. and NASDAQ Industrial index daily returns. The obtained results reveal that for this particular data set the BNP-ADCC outperforms the BNP-DCC model. Finally, an illustrative asset allocation exercise is presented. The contents of this chapter resulted into a paper by Virbickaite, Ausín & Galeano (2014), which has been accepted in Computational Statistics and Data Analysis. Chapter 4 designs a Particle Learning (PL) algorithm for estimation of Bayesian nonparametric Stochastic Volatility models for financial data. The performance of this particle method is then compared with the standard MCMC methods for non-parametric SV models. PL performs as well as MCMC, and at the same time allows for on-line type inference. The posterior distributions are updated as new data is observed, which is prohibitively costly using MCMC. Further, a new non-parametric SV model is proposed that incorporates Markov switching jumps. The proposed model is estimated by using PL and tested on simulated data. Finally, the performance of the two nonparametric SV models, with and without Markov switching, is compared by using real financial time series. The results show that including a Markov switching specification provides higher predictive power in the tails of the distribution. The contents of this chapter resulted into a working paper by Virbickaite, Lopes, Ausín & Galeano (2014). Finally, Chapter 5 concludes and proposes general future research lines that could be viewed as natural extensions of the ideas presented in the thesis.
