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We investigate conditions for the ergodicity of threshold autoregressive time series by embedding the time series in a general state Markov chain and apply a FosterLyapunov drift condition to demonstrate ergodicity of the Markov chain. We are particularly interested in demonstrating V uniform ergodicity where the test function V () is a function of a norm on the statespace. In this dissertation we provide conditions under which the general state space chain may be approximated by a simpler system, whether deterministic or stochastic, and provide conditions on the simpler system which imply V uniform ergodicity of the general state space Markov chain and thus the threshold autoregressive time series embedded in it. We also examine conditions under which the general state space chain may be classified as transient. Finally, in some cases we provide conditions under which central limit theorems will exist for the V uniformly ergodic general state space chain.

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The State Space Model (SSM) encompasses the class of multivariate linear models, in particular, regression models with fixed, time-varying and random parameters, time series models, unobserved components models and combinations thereof. The well-known Kalman Filter (KF) provides a unifying tool for conducting statistical inferences with the SSM. A major practical problem with the KF concerns its initialization when either the initial state or the regression parameter (or both) in the SSM are diffuse. In these situations, it is common practice to either apply the KF to a transformation of the data which is functionally independent of the diffuse parameters or else initialize the KF with an arbitrarily large error covariance matrix. However neither approach is entirely satisfactory. The data transformation required in the first approach can be computationally tedious and furthermore it may not preserve the state space structure. The second approach is theoretically and numerically unsound. Recently however, De Jong (1991) has developed an extension of the KF, called the Diffuse Kalman Filter (DKF) to handle these diffuse situations. The DKF does not require any data transformation. The thesis contributes further to the theoretical and computational aspects of con ducting statistical inferences using the DKF. First, we demonstrate the appropriate initialization of the DKF for the important class of time-invariant SSM’s. This result is useful for maximum likelihood statistical inference with the SSM. Second, we derive and compare alternative pseudo-likelihoods for the diffuse SSM. We uncover some interesting characteristics of the DKF and the diffuse likelihood with the class of ARMA models. Third, we propose an efficient implementation of the DKF, labelled the collapsed DKF (CDKF). The latter is derived upon sweeping out some columns of the pertinent matrices in the DKF after an initial number of iterations. The CDKF coincides with the KF in the absence of regression effects in the SSM. We demonstrate that in general the CDKF is superior in practicality and performance to alternative algorithms proposed in the literature. Fourth, we consider maximum likelihood estimation in the SSM using an EM (Expectation-Maximization) approach. Through a judicious choice of the complete data, we develop an CDKF-EM algorithm which does not require the evaluation of lag one state error covariance matrices for the most common estimation exercise required for the SSM, namely the estimation of the covariance matrices of the disturbances in the SSM. Last we explore the topic of diagnostic testing in the SSM. We discuss and illustrate the recursive generation of residuals and the usefulness of the latters in pinpointing likely outliers and points of structural change.

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Entropy measures have been widely used to quantify the complexity of theoretical and experimental dynamical systems. In this thesis, two novel entropy measures are developed based on using coarse quantization to classify and compare dynamical features within a time series; quantized dynamical entropy (QDE) and a quantized approximation of sample entropy (QASE). Following this, comprehensive guidelines for the quantification of complexity are presented based on a detailed investigation of the performance characteristics of the two developed measures and three existing measures; permutation entropy, sample entropy and fuzzy entropy. The sensitivity of the considered entropy measures to changes in dynamics was assessed using the case study of characterizing bipedal walking gait dynamics. Based on the analysis conducted, it was found that sample entropy and fuzzy entropy, while computationally inefficient, provide the best overall performance. In instances where computational efficiency is vital, QDE and QASE serve as viable alternatives to existing methods.