Modelos de espaço de estados não Gaussianos: distribuições de caudas pesadas

Abstract

This thesis contains three papers that expand the knowledge about a new family of state space model proposed by Santos et al. (2010) called non-Gaussian state space model (NGSSM). This family of models is very interesting because, besides containing a significant set of probability distributions, the likelihood function can be written in an exact form. Consequently, there is the possibility of performing inference about theparameters without the need of numerical methods, such as the particle filter. In the first paper it is shown that besides Weibull and Pareto proposed in the Santos et al. (2010) paper, five other heavy tailed distributions are contained in the NGSSM. They are: Log-normal, log-gamma, Fréchet, Lévy, Skew GED. To evaluate classical and Bayesian estimators for heavy tailed models of the NGSSM Monte Carlosimulations are performed. The results demonstrate empirically that the estimators are not asymptotically biased and they are consistent. The heavy tailed models are estimated for the series of the most important stock exchange indexes of America, such as S&P 500, NASDAQ, IBOVESPA, INMEX, MERVAL, IPSA. The results are compared with the GARCH models and it is observed that the Weibull model of NGSSM shows better results for all time series studied. In the second paper, it is evaluated the behavior of the maximum likelihood esti-mator of the parameters of the heavy tailed models when the time series is small. It isobserved that the parameter w is always overestimated, regardless the model and the maximization algorithm used. Obtaining a suitable estimator for w is critical, because when this parameter is overestimated the variability of the time series is underestimated. Penalty functions are proposed for the likelihood function and, consequently, penalizedmaximum likelihood estimators are proposed and evaluated. The results demonstrate that the estimators proposed reduce significantly the bias when compared with the bias obtained by the maximum likelihood estimator. In the third paper it is evaluated the behavior of the asymptotic confidence interval of the parameters of the heavy tailed models when the time series is small. It is observed that the confidence intervals for the parameter w are inadequate, either using the maximum likelihood estimator or penalized maximum likelihood estimator. Thus bootstrapconfidence intervals are proposed and evaluated. The results show that the bootstrap confidence interval with bias correction obtained from the parametric bootstrap has coverage rates very close to the nominal level used in the empirical study.