Analysis of dependence structure between the Rand/U.S Dollar exchange rate and the gold/platinum prices by KAJINGULU MALANDALA submitted in accordance with the requirements for the degree of MASTER OF SCIENCE in the subject Statistics at the UNIVERSITY OF SOUTH AFRICA SUPERVISOR: PROF J. O. OLAOMI APRIL 2018 i Abstract Copulas functions are a flexible tool for modelling the dependence structure between variables. The joint and marginal distributions of Copulas are not constrained by the assumptions of normality. This study examines the dependence structure between the gold, platinum prices and the ZAR/U.S.D exchange rate using Copulas. The study found that marginal distributions of Copulas follows the ARMA (1, 1)-EGARCH (1, 1) and ARMA(1, 1)-APARCH (1, 1) models under different error terms including the normal, the student-t and the skew student-t error terms. It used the Normal, the Student-t, the Gumbel, the rotated Gumbel, the Clayton, the rotated Clayton, the Plackett, the Joe Clayton and the Normal time varying Copulas to analyse the dependence structure between returns prices of gold, platinum and ZAR/U.S.D exchange rate. The results showed evidence of a positive strong dependence between the returns prices of gold, platinum and returns on the Rand/U.S.D exchange rate for constant and time varying Copulas. The result also showed a co-movement of exchange rates and gold and platinum prices during a rise or declining prices of gold and platinum. The results imply that fluctuations in gold and platinum prices generate Rand/U.S.D exchange rate volatility. Keys words: Copulas, ARMA, EGARCH, APARCH, dependence structure, exchange rate, commodity prices. ii Dedication I would like to dedicate this dissertation to my loving and supportive wife: Patricia Nyembwe and my children, Emmanuel, Gabriel, Abigael and Dora, for their continuous supports and care. I would also like to dedicate this dissertation to my mother and father for their prayers and supports. To God Almighty for the grace and favour, which he renews every morning in my life. Lord Jesus, You are my Rock and my Refuge. iv Acknowledgment I would like to acknowledge those who have contributed and supported me during my studies. My sincere gratitude to Prof. John Olaomi for his supervision and contributions. Your support has been unconditional. Thank you for your guidance, contribution, motivation and knowledge. I appreciate your efforts; you have been instrumental in helping me to achieve this milestone. I am grateful to Dr. Rapoo, the chairperson of the department of Statistics for her support and encouragements. I would like to express my gratitude to all Staff members at the department of Statistics of the University of South Africa for their support and appreciation. I would also like to thank my wife, Dr. Patricia Nyembwe, My children, Emmanuel, Gabriel, Abigail and Dora, for their love and support. My appreciation goes to my parents, brothers, sisters for their love, and to my family members and colleagues for their supports and prayers. v Contents Declaration.............................................................................................................................. iii Dedication ................................................................................................................................ iv Acknowledgment ...................................................................................................................... v List of tables............................................................................................................................. xi List of figures .......................................................................................................................... xii List of abreviations .............................................................................................................. xiii Chapter One: Introduction ..................................................................................................... 1 1.1 Background of the Study ........................................................................................ 1 1.2 Gap in Knowledge .................................................................................................. 4 1.3 Research questions ................................................................................................. 5 1.4 Aim and Objectives of the Study............................................................................ 6 1.5 Importance of the Study ......................................................................................... 6 1.6 Outline of the Study................................................................................................ 6 Chapter Two: Literature review ............................................................................................ 8 2.1 Commodity and Currency ...................................................................................... 8 2.2 Exchange Rate (ER) ............................................................................................... 9 Nominal Exchange Rate (NER) .............................................................................. 9 Nominal Effective Exchange Rate (NEER) ............................................................ 9 Real Exchange Rate (RER) ................................................................................... 10 Real Effective Exchange Rate (REER) ................................................................. 10 2.3 Classification of Exchange Rate ........................................................................... 10 2.3.1 Exchange Rate Regimes ....................................................................................... 13 2.4 Factors Affecting the Exchange Rate Volatility ................................................... 13 2.5 Exchange Rate Regimes and Exchange Rate Fluctuations .................................. 15 2.6 History of the South African Rand Exchange Rate .............................................. 16 2.7 Rand-U.S.D Exchange Rate: The Place of the Prices of Gold and Platinum ....... 18 2.8 Behaviour of Commodity Prices .......................................................................... 19 2.9 Independence ........................................................................................................ 20 vi 2.10 Dependence .......................................................................................................... 21 2.10.1 Measures of Dependence ...................................................................................... 21 2.10.2 Correlation ............................................................................................................ 21 2.10.3 Concordance Measure .......................................................................................... 23 Characteristics of concordances measures are: (references) ............................................ 23 2.11 Rank Correlation .......................................................................................................... 23 2.11.1 Kendall Tau .......................................................................................................... 23 2.11.2 The Spearman Rho ............................................................................................... 24 2.12 Definitions of Time Series Concepts .................................................................... 24 2.12.1 Time Series ........................................................................................................... 24 2.12.2 Autocovariance and Autocorrelation .................................................................... 25 2.12.3 Stationary .............................................................................................................. 25 2.12.4 White Noise Process ............................................................................................. 25 2.13 GARCH Models ................................................................................................... 26 2.13.1 Some Advantages of GARCH model: .................................................................. 27 2.13.2 Some of the limitations of the GARCH model: .................................................... 27 2.14 Previous Work on Currencies and Commodity Prices ......................................... 27 Chapter Three: Methodology ............................................................................................... 35 3.1 Data....................................................................................................................... 35 3.2 Model for marginal distribution ........................................................................... 36 3.2.1 Autoregressive Process ......................................................................................... 36 3.2.2 Moving Average Models ......................................................................................... 36 3.2.3 APARCH (p, q) .................................................................................................... 37 3.2.4 EGARCH (p, q) ..................................................................................................... 37 3.2.5 Distribution of Error Terms .................................................................................. 38 3.2.5.1 The Normal Distribution ...................................................................................... 38 3.2.5.2 The Student-t Distribution ...................................................................................