## A STUDY OF THRESHOLD NONLINEAR AUTOREGRESSIVE MODELS

TITLE PAGE
APPROVAL PAGE
CERTIFICATION
DEDICATION
ACKNOWLEDGMENT
ABSTRACT

CHAPTER ONE
INTRODUCTION
1.1       Background of the Research
1.2       Statement of Problem
1.3       Objectives of the Research
1.4       Significance of the Study
1.5       Scope of the Study

CHAPTER TWO
LITERATURE REVIEW
2.1       Linear Models
2.1.1    Autoregressive (AR) Process
2.1.2    Moving Average (MA) process
2.1.3    Mixed models (ARMA/ARIMA) process
2.2       Non-linear models
2.3       Review of Related Studies

CHAPTER THREE
METHODOLOGY
3.1       Nonlinear Model Specification
3.2       Linear Model Specification

CHAPTER FOUR
MODEL ESTIMATION
4.1 Data for model formulation and testing
4.2 Data sources and description
4.3 Test for nonlinearity and threshold nonlinearity
4.3.1 Detection of nonlinearity graphically
4.3.2 Detection of nonlinearity:
4.3.3 Detection of Threshold Nonlinearity (Likelihood Ratio Test)
4.4 Estimation of a TAR model
4.5Model diagnostics via residual analysis
4.6 Estimation of a Linear Model
4.7 Model diagnostics via residual analysis
4.7 Comparison of the SETAR(2;3,2) and ARIMA(2,0,0) models via Residual analysis
4.8 Prediction of logged Nigerian monthly crude oil price

CHAPTER FIVE
CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion
5.2 Recommendation
Reference

ABSTRACT

This work examines Threshold Autoregressive models (TAR) on nonlinear time series. Linear models, such as ARIMA, reach their limitations with nonlinearities in the data. A self-exciting threshold autoregressive model was fitted along with an alternative ARIMA model. A simple log transform was applied to obtain stationarity and variance stabilization before the models were fit. Keenan, Tsay and likelihood ratio tests depict nonlinearity in the time series. The results of the residual analysis and the minimum Akaike Information criterion(MAIC) in the model estimation indicated that SETAR(2;3,2) with delay 1 is a better fit for Nigerian crude oil price than ARIMA(2,0,0).

CHAPTER ONE

INTRODUCTION

1.1            Background of the Research

The linear Gaussian models such as the autoregressive (AR), autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models as introduced by Box and Jenkins in the 1970s, (Box, Jenkins and Reinsel, 2008) and (Allard, 1998) have gained enormous popularity in time series literature.

Among the characteristics of ARMA processes is the property that the expectation of current values of the process can be linearly expressed as a function of its past values, the so-called linearity property. Although linear Gaussian models remain at the forefront of academic and applied researches, they have been found to leave certain structural and behavioural aspects of time series data unexplained (Watier and Richardson, 1995).

As a result, some researchers have proposed and formulated models for estimating those time series that exhibit nonlinear behaviour (Granger and Andersen, 1978; Tong, 1978; and Tong and Lim, 1980). Such models are called “Nonlinear models”. Nonlinear time series models provide a wider range of possible dynamics in time series data than do linear models. In recent years, statistical research in nonlinear time series analysis has grown rapidly. Among these nonlinear times series models are the threshold autoregressive (TAR) models, smooth transition autoregressive (STAR) models, bilinear models, the neural networks, and so on.

1.2            Statement of Problem

Generally, linear time series models have been widely applied in several fields of research; especially, given that many climates, economic and financial data analyst assume that virtually all time series are linear. This is however contrary to what is obtainable in reality (Futia, 1981). Nonlinear time series process has a wealth of structural and dynamic behaviour that cannot be taken care of by the popular class of ARMA models. Some of the features or properties of non linear.....

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