Aspects of non-linear time series analysis

Some difficulties of non-linear time series modelling are discussed. The importance of likelihood plots and information matrix calculation is demonstrated. A constructive use of reparametrization is illustrated. Possible non-invertibility of two bilinear models fitted to real data is discussed. The relation between the variance of error term and the variance of the estimate of the bilinear term is also studied. We have examined numerical solutions for a recursive relation for computing the rn-step-ahead conditional density of a non-linear autoregressive model by using the Chapman-Kolmogorov formula. The stationary marginal probability density function of the model is approximated by the rn-step-ahead conditional density, for sufficiently large m. The advantage of incorporating the matrix squaring procedure is also studied. The conditional mean (regression function) and conditional variance of some non-linear models are studied briefly .The importance of non-parametric estimation of regression functions in non-linear time series analysis is demonstrated. A comparison of likelihood ratio tests using the newly available asymptotic results with the non-likelihood ratio approach such as the modified Petruccelli-Davies's test and Tsay's test is studied. The 5% empirical critical values for the some cases of likelihood ratio test is also obtained. The• effect of outliers on the tests is briefly studied. The importance of the profile likelihood plots in locating the threshold and estimating the delay parameter is demonstrated. The blowfly data (raw and transformed) analysed. The importance of influential data is also discussed. The non-monotonicity of the conditional variance of the error of a rn-step non-linear least squares predictor is discussed. We have also studied methods of evaluating the conditional variance for non-linear autoregressive models and illustrated these with both real and simulated data. Bias correction is included. Moreover, the possibility of combinations of forecasts is explored. The performances of linear bilinear and SETAR models in predicting the sunspot numbers are compared.