The Asymmetric Alpha-Power Skew-t Distribution

In this paper, we propose a new asymmetric and heavy-tail model that generalizes both the skew-t and power-t models. Properties of the model are studied in detail. The score functions and the elements of the observed information matrix are given. The process to estimate the parameters in model is discussed by using the maximum likelihood approach. Also, the observed information matrix is shown to be non-singular at the whole parametric space. Two applications to real data sets are reported to demonstrate the usefulness of this new model.

The Stochastic Stationary Root Model

We propose and study the stochastic stationary root model. The model resembles the cointegrated VAR model but is novel in that: (i) the stationary relations follow a random coefficient autoregressive process, i.e., exhibhits heavy-tailed dynamics, and (ii) the system is observed with measurement error. Unlike the cointegrated VAR model, estimation and inference for the SSR model is complicated by a lack of closed-form expressions for the likelihood function and its derivatives. To overcome this, we introduce particle filter-based approximations of the log-likelihood function, sample score, and observed Information matrix. These enable us to approximate the ML estimator via stochastic approximation and to conduct inference via the approximated observed Information matrix. We conjecture the asymptotic properties of the ML estimator and conduct a simulation study to investigate the validity of the conjecture. Model diagnostics to assess model fit are considered. Finally, we present an empirical application to the 10-year government bond rates in Germany and Greece during the period from January 1999 to February 2018.