Some Special Classes Of Multivariate Generalized Autoregressive Conditional Heteroscadasticity (Garch) Models

ABSTRACT

In this research we focused on Multivariate Generalised Autoregressive Conditional Heteroskedasticity models for volatility series using response vector of variances.  The work aimed at developing alternative multivariate GARCH models characterised by either autoregressive or moving average process. Isolated Multivariate Generalised Conditional Heteroskedasticity, ISO-MGARCH (p,0) models and Isolated Multivariate Generalised Conditional Heteroskedasticity, ISO-MGARCH(0,q) models are identified from MGARCH (p,q) model under specific conditions. To ascertain the models applicability, the isolated univariate and multivariate GARCH (2,0) models were fitted to volatility measures of Nigeria average, urban and rural consumer price indices from January 1995 to December 2019 after the series were subjected to stationarity checks using the positive definiteness property of the sub-autocovariance or autocorrelation matrices of individual vector processes as components of the cross-covariance or cross-autocorrelation matrix to ascertain the stationarity of the series,. The volatility series were also subjected to autocorrelation and partial autocorrelation checks, as applicable to stationary autoregressive moving average process, where single autoregressive and moving average models are identified under certain conditions. This justified the isolation of pure autoregressive and pure moving average MGARCH models. Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and Schwarz’s Information criterion (SIC) compare the isolated multivariate GARCH models with the existing univariate GARCH models, and its simulated values, the results revealed the same comparative advantage in capturing volatility series.

TABLE OF CONTENTS

                                                                                                                                    Page

Title Page                                                                                                                    i

Declaration                                                                                                                  ii

Certification                                                                                                                iii

Dedication                                                                                                                  iv

Acknowledgments                                                                                                      v

Table of Contents                                                                                                       vi

List of Tables                                                                                                              viii

List of Figures                                                                                                             ix

Abstract                                                                                                                      x

CHAPTER 1: INTRODUCTION

1.1       Background of the Study                                                                               1

1.2       Statement of the Problem                                                                               9

1.3       Objective of the Work                                                                                    9

1.4       Justification of the Study                                                                               10

1.5       Scope of Study                                                                                               11

1.6       Significance of the Study                                                                               11

CHAPTER 2: REVIEW OF RELATED LITERATURE                                  12

2.1       Review of Models                                                                                          12

2.2       Empirical Review                                                                                            21

CHAPTER 3: METHODOLOGY                                                                          28

3.1      Cross-covariances                                                                                           28

3.2       Auto-correlations and Cross-auto-correlations                                               31

3.3       Positive Definitness of Auto-correlations and Cross-

            auto-correlation matrices                                                                                 33

3.4       Univariate Case                                                                                               34

3.4.1        Testing for ARCH Effects                                                                             34

3.4.2    Model estimation                                                                                            36

3.4.3    Post estimation test                                                                                         36

3.5       Multivariate Case                                                                                            36

3.6        Volatility Measure                                                                                         38

3. 7      Conditions for Model Identification                                                              42

3.7.1    Proof                                                                                                               42

3.8       Model Selection Criteria                                                                                 44

3.8.1    Akaike Information Criterion (AIC):                                                             44

3.8.2    Bayesian Information Criterion (BIC):                                                           45

3.8.3    Schwarz’s Information Criterion (SIC):                                                         45

CHAPTER 4: DATA ANALYSIS AND DISCUSSION OF RESULTS                       46

4.1        Numerical Verification                                                                                  46

4.2       Components of the Autocorrelation and Cross-autocorrelations                   47

4.3       Positive Definiteness of 3x3 Component Autocorrelation Matrices              49

4.4       Positive Definiteness of 9x9 Autocorrelation Matrix                                     49

4.5       Graphical Analysis                                                                                          53

4.6       Univariate GARCH (p,0) Model Estimates                                                   55

4.7       Isolated Multivariate GARCH (p,0) Model Estimates                                   57

CHAPTER 5: SUMMARY AND CONCLUSION                                               64

5.1       Summary and Conclusion                                                                               64

5.2       Recommendation                                                                                            66

            References                                                                                                      67

            Appendices                                                                                                     71

LIST OF TABLES

4.1       Parameter Estimates of the Univariate GARCH (2,0) Models          56

4.2       Parameter Estimates of the Multivariate GARCH(2,0) Models         57

4.3       The parameter estimates for 3000 Data Points Simulated Values      59

4.4       Information Criteria                                                                            62

4.5       Information Criteria for Simulated Values                                         62

LIST OF FIGURES

4.1       Return series of Average CP                                                                          53

4.2       Return Series of Urban CPI                                                                            53

4.3       Return Series of Rural CPI                                                                             54

4.4       Autocorrelation Function of Average Consumer Price Index                        54

4.5       Partial Autocorrelation Function of Average Consumer Price Index             54

4.6       Autocorrelation function the residual of  MGARCH model                    61