Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...

We apply the local linear regression technique for estimation of functional-coefficient regression models for time series data. The models include threshold autoregressive models (Tong 1990) and functional-coefficient autoregressive models (Chen and Tsay 1993) as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better post-sample forecasting performance. We have also proposed a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined cross-validatory estimation for the expected forecasting errors. The proposed methodology is data-analytic and is of appreciable flexibility to analyze complex and multivariate nonlinear structures without suffering from the "curse of dimensionality". The asymptotic properties of the proposed estimators are investigated under the ff-mixing condition. Both simulated and real data examples are used for illustration. Key Words: ff-mixing; Asymptotic normali...

In this paper we investigate the multi-period forecast performance of a number of empirical selfexciting threshold autoregressive (SETAR) models that have been proposed in the literature for modelling exchange rates and GNP, amongst other variables. We take each of the empirical SETAR models in turn as the DGP to ensure that the `non-linearity' characterises the future, and compare the forecast performance of SETAR and linear autoregressive models on a number of quantitative and qualitative criteria. Our results indicate that non-linear models have an edge in certain states of nature but not in others, and that this can be highlighted by evaluating forecasts conditional upon the regime.

Abstract. The problem of testing for linearity and the number of regimes in the context of self-exciting threshold autoregressive (SETAR) models is reviewed. We describe least-squares methods of estimation and inference. The primary complication is that the testing problem is non-standard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is non-standard. Simulation methods to calculate asymptotic and bootstrap distributions are presented. As the sampling distributions are quite sensitive to conditional heteroskedasticity in the error, careful modeling of the conditional variance is necessary for accurate inference on the conditional mean. We illustrate these methods with two applications Ð annual sunspot means and monthly U.S. industrial production. We find that annual sunspots and monthly industrial production are SETAR(2) processes. Keywords. SETAR models; Thresholds; Non-standard asymptotic theory; Bootstrap

This thesis considers continuous time autoregressive processes defined by stochastic differential equations and develops some methods for modelling time series data by such processes. The first part of the thesis looks at continuous time linear autoregressive (CAR) processes defined by linear stochastic differential equations. These processes are well-understood and there is a large body of literature devoted to their study. I summarise some of the relevant material and develop some further results. In particular, I propose a new and very fast method of estimation using an approach analogous to the Yule–Walker estimates for discrete time autoregressive processes. The models so estimated may be used for preliminary analysis of the appropriate model structure and as a starting point for maximum likelihood estimation. A natural extension of CAR processes is the class of continuous time threshold autoregressive (CTAR) processes defined by piecewise linear stochastic differential

In this paper we propose a method for determining the number of regimes in threshold autoregressive models using smooth transition autoregression as a tool. As the smooth transition model is just an approximation to the threshold autoregressive one, no asymptotic properties are claimed for the proposed method. Tests available for testing the adequacy of a smooth transition autoregressive model are applied sequentially to determine the number of regimes. A simulation study is performed in order to find out the finite-sample properties of the procedure and to compare it with two other procedures available in the literature. We find that our method works reasonably well for both single and multiple threshold models. Key words: Model specification, model selection criterion, nonlinear modelling, sequential testing, switching regression.