Resumen de Modeling financial returns with skew-slash innovations
Financial returns often present a complex relation with previous observations, along with a slight skewness and high kurtosis, which can not typically be captured via Gaussian distributions. As a consequence, we need to develop flexible models that are able to capture these features. To respond to this problem, several families of distributions have been proposed. For example, Azzalini (1985) presented the Skew-Normal distribution, which is able to capture the underlying skewness. Also, Lange and Sinsheimer (1993) show us a way to pick up the kurtosis by means of the Slash distribution, a much needed feature in this type of study. Other distributions such as the Skew-Slash proposed by Wang and Genton (2006) allow us to capture both skewness and heavy tails. In this thesis, we begin by proposing the use of a Generalized Autoregressive Conditional Heteroskedastic (GARCH) process, introduced by Bollerslev (1986), with Skew-Slash innovations to model univariate financial time series of returns. In this case, we derive formulae for the higher order moments of this distribution, which show that this distribution can incorporate both moderate skewness and high kurtosis. We also obtain the Maximum Likelihood estimations and we propose a Bayesian inference procedure for the GARCH model with Skew-Slash innovations, and illustrate the performance of our proposed methodology using simulations, as well as a real data example using the logreturns of the Standard & Poor's index from January 3rd, 2000 to December 28th, 2013. Afterwards, for the multivariate case, we propose to use an extension of the GARCH process, such as the Dynamic Conditional Correlation model, introduced by Tse and Tsui (2002), with multivariate Skew-Slash innovations for financial returns in a Bayesian framework, and it is illustrated using a Markov Chain Montecarlo (MCMC) within Gibbs algorithm performed on simulated data, as well as real data drawn from the daily closing prices of the Dow Jones and NASDAQ indices from January 2nd, 1996 until December 29th, 2006 on a first example, and the daily log-returns of the DAX, CAC40, and Nikkei indices between October 10th, 1991 and December 30th, 1996 in a second example.
