Integer-valued APARCH processes in the analysis of time series of counts

The Asymmetric Power Arch representation for the volatility was introduced by Ding et al.(1993) in order to account for asymmetric responses in the volatility in the analysis of continuous-valued financial time series like, for instance, the log-return series of foreign exchange rates, stock indices or share prices. As reported by Brannas and Quoreshi (2010), asymmetric responses in volatility are also observed in time series of counts such as the number of intra-day transactions in stocks. In this work, an asymmetric power autoregressive conditional Poisson model is introduced for the analysis of time series of counts exhibiting asymmetric overdispersion. Basic probabilistic and statistical properties are summarized and parameter estimation is discussed. A simulation study is presented to illustrate the proposed model. Finally, an empirical application to a set of data concerning the daily number of stock transactions is also presented to attest for its practical applicability in data analysis.

Optimal alarms systems and its application to financial time series

This thesis focuses on the application of optimal alarm systems to non linear time series models. The most common classes of models in the analysis of real-valued and integer-valued time series are described. The construction of optimal alarm systems is covered and its applications explored. Considering models with conditional heteroscedasticity, particular attention is given to the Fractionally Integrated Asymmetric Power ARCH, FIAPARCH(p; d; q) model and an optimal alarm system is implemented, following both classical and Bayesian methodologies. Taking into consideration the particular characteristics of the APARCH(p; q) representation for financial time series, the introduction of a possible counterpart for modelling time series of counts is proposed: the INteger-valued Asymmetric Power ARCH, INAPARCH(p; q). The probabilistic properties of the INAPARCH(1; 1) model are comprehensively studied, the conditional maximum likelihood (ML) estimation method is applied and the asymptotic properties of the conditional ML estimator are obtained. The final part of the work consists on the implementation of an optimal alarm system to the INAPARCH(1; 1) model. An application is presented to real data series.