The Heavy Tailed Phenomenom of Doubly Stochastic Time Series Models

Specialeforsvar: Beinta Biskopstø Joensen

Titel: The Heavy Tailed Phenomenom of Doubly Stochastic Time Series Models

Resume: The aim of this thesis is to study the doubly stochastic time series model given by an autoregressive model of order one, the space equation, with the autoregressive coefficient itself given by an autoregressive model, the state equation. The model is described by four parameters; mu the intercept of the autoregressive coefficient of the space process, a the autoregressive coefficient of the state process and the variances of the noise processes. Conditions for stationarity, ergodicity and existence of moments are obtained and the parameters values satisfying the conditions are estimated from simulations. The properties of the stationary distribution is studied, providing results about the tail, moments and autocorrelation. When a equals zero, the well know AR(1)-ARCH(1) representation of the model is obtained, providing a broad range of theoretical results. When a is diferent from zero the main approach is simulations. The recursive Kalman
algorithm is obtained providing estimates of the conditional expectation and variance. Further, it can be used to calculate the maximum likelihood estimates of the mode. A simulation study shows that the optimisation often works quite well. Even in the case of infinite variance models, reasonable estimation is obtained. The obtained results are used to fit the model to real data, considering both disaster data and log returns.

Vejleder: Olivier Wintenberger
Censor: Mette Havning