Generalised Dynamic Nonlinear Time Series Regression and Forecasting: Theory with Applications

Peng, Rong (2021) Generalised Dynamic Nonlinear Time Series Regression and Forecasting: Theory with Applications. University of Southampton, Doctoral Thesis, 166pp.

Record type: Thesis (Doctoral)

Abstract

This thesis aims to develop a series of nonlinear time series models for analysing count data, especially to overcome the “curse of dimensionality” for high and ultra-high dimensions. This is of particular needs for big data analysis in applications to discrete-valued outcome events, such as financial market direction, infected patients number in epidemiology and etc., where the nature of data is often unknown.

In contrast to time series for continuous responses, where numerous related studies are available, literature paid scant attention to discrete-valued time series estimation and forecasting. Existing studies are developed based on the extension of classic AutoRegressive Moving Averge model (ARMA). To better capture the relationship between response and exogenous variables, we have proposed a semi-parametric procedure called the “ Generalised Model Averaging MArginal nonlinear Regressions (GMAMaR) and showed the uniform consistency for local maximum likelihood estimation of one dimensional non-parametric local linear estimation. The asymptotic properties of the procedure are established under mild conditions on the time series observations that are of β-mixing property. This model has overcame the “curse of dimensionality” by taking the advantage of
cheap computational cost of low dimensional estimation and the idea of model averaging to approximate the true estimates.

In particular, to deal with the popular binary classification problem, we study a special case of logistic regression, namely “Model Averaging MArginal nonlinear LOgistic Regressions (MAMaLOR). This is the case where binary outcome is considered. The performance of our proposed model is superior when compared to conventional method with numerical examples.

We notice another problem when facing big data that only a few of them are truly useful in explaining the responses out of hundreds and thousands exogenous variables. Thus, we propose a penalise maximum likelihood estimation for variable
selection combined with our developed model by utilising adaptive LASSO as a tool. A new computational procedure is also suggested to solve the proposed penalised likelihood estimation. By extracting important information from data, the
performance of our proposed methods is improved significantly both in estimation and in prediction.

Last but not least, with the on-going event of COVID-19 in the UK, we further consider the spatial effects along with temporal dependency. The idea is thus to extend time series analysis to the domain of spatio-temporal modelling. We utilise
proposed model to investigate impacts of micro variables of the implementation of lockdown on the daily number of confirmed cases. The results are consistent with the consensus of epidemiology studies, and deeper understandings of how to
adapt and prioritise the policies in the combat of epidemic are also provided.

To conclude, the proposed series of nonlinear time series models show great potential in the context of discrete-valued events. While providing a more accurate estimation and prediction, the models also offer a better interpretability and deeper understanding of the relationships between response and potential factors. We hope to demonstrate that this thesis thus contribute to the development of this area, and could be further extended to the area of sptio-temporal and other areas
of applications.