Bootstrap methods and parameter estimation in time series threshold modelling

The aim of this thesis is to investigate of bootstrap methods (Efron, 1979), in the the performance estimation of parameter estimates in non-linear time series models, in particular SETAR models (Tong, 1993). First and higher order SETAR models in known and unknown thresholds cases are considered. To assess the performance of bootstrap methods, we first give an extensive simulation study (by using simulated normal errors), in chapters 3 and 4, to investigate large and small sample behaviours of the true sampling distributions of parameter estimates of SETAR models and how they are affected by sample size. First and higher order SETAR models in the known and unknown threshold cases are considered. An introduction to the bootstrap methods (Efron, 1979 ) is given in chapter 5. The effect of sample size on the bootstrap distributions of parameter estimates of first and higher order SETAR models in the known and unknown threshold cases ( for given order, delay and number of thresholds ) are also investigated in this chapter, via simulation and by using the same models used in the simulated normal errors 'true distribution' case ( chapters 3 & 4). The results are compared with simulated normal case in order to assess the bootstrap results. Tong and Lim (1980) method is used for fitting SETAR models to bootstrap samples, which is also used in the initial fit. Moreover, applications of bootstrap to celebrated data sets, namely, the logarithmically transformed lynx data covering the period (182-1934); and the sunspot numbers covering the period (1700- 1920), are attempted. The cyclical behaviours of bootstrap models are also examined. Finally, in chapter 5, an attempt is also made to study the problem of non-linear properties of the skeleton of a non-linear autoregressive process (Jones, 1976) via simulation and we study in particular a limit cycle behaviour