Some observation driven models for time series

Abstract

We begin by reviewing generalized state-space models and the two categories into which they are typically divided, parameter driven and observation driven models. Since the models considered throughout the remainder of the thesis are observation driven, several examples of processes of this type are given. In Chapter 2 stationarity properties for two families of observation driven models are derived using results from Meyn and Tweedie (1993). The first family of models, BIN models, were developed by Rydberg and Shephard (1999) to analyze the number of trades occurring within a given time interval. We also consider a class of GLARMA models for modeling time series of counts. We show that a particular variant of a GLARMA model is uniformly ergodic. This enables us to use a procedure known as exact sampling to sample from the stationary distribution. In Chapter 3 we develop stationarity properties for a process used by Rydberg and Shephard (1998) for modeling stock prices. In the final Chapter, we return to the GLARMA models of Chapter 2. We calculate the maximum likelihood estimates of the model parameters and derive their asymptotic distribution. We also look at simulations as well as fit this model to a data set of asthma counts in order to determine how the theory applies in practice.