Time Series WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg Editors Emeriti: Vic Burnett, J. Stuart Hunter, Joseph B. Kadane, Jozef L. Teugels A complete list of the titles in this series appears at the end of this volume. Time Series Applications to Finance with R and S-Plus® Second Edition Ngai Hang Chan The Chinese University of Hong Kong Department of Statistics Shatin, Hong Kong ®WILEY A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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HA30.3.C47 2010 332.0Γ51955—dc22 2010016632 Printed in the United States of America. 10 9 8 7 6 5 4 3 21 To Pat and our children, Calvin and Dennis Contents List of Figures xiii List of Tables xvii Preface xix Preface to the First Edition xxi 1 Introduction 1 1.1 Basic Description 1 1.2 Simple Descriptive Techniques 5 1.2.1 Trends 5 1.2.2 Seasonal Cycles 8 1.3 Transformations 9 1.4 Example 9 1.5 Conclusions 13 1.6 Exercises 13 2 Probability Models 15 2.1 Introduction 15 vii viii CONTENTS 2.2 Stochastic Processes 15 2.3 Examples 17 2.4 Sample Correlation Function 18 2.5 Exercises 20 3 Autoregressive Moving Average Models 23 3.1 Introduction 23 3.2 Moving Average Models 23 3.3 Autoregressive Models 25 3.3.1 Duality between Causality and Stationarity* 26 3.3.2 Asymptotic Stationarity 28 3.3.3 Causality Theorem 28 3.3.4 Covariance Structure of AR Models 29 3.4 ARMA Models 32 3.5 A RIM A Models 33 3.6 Seasonal A RIM A 35 3.7 Exercises 36 4 Estimation in the Time Domain 39 4.1 Introduction 39 4.2 Moment Estimators 39 4.3 Autoregressive Models 40 44 Moving Average Models 42 4.5 ARMA Models 43 4.6 Maximum Likelihood Estimates 44 4.7 Partial ACF 47 4.8 Order Selections* 49 4.9 Residual Analysis 53 4.10 Model Building 53 4.11 Exercises 54 5 Examples in SPLUS and R 59 5.1 Introduction 59 5.2 Example 1 59 5.3 Example 2 62 5.4 Exercises 69 6 Forecasting 71 CONTENTS ix 6.1 Introduction 71 6.2 Simple Forecasts 72 6.3 Box and Jenkins Approach 73 6.4 Treasury Bill Example 75 6.5 Recursions* 79 6.6 Exercises 80 7 Spectral Analysis 83 7.1 Introduction 83 7.2 Spectral Representation Theorems 83 7.3 Periodogram 87 7.4 Smoothing of Periodogram* 89 7.5 Conclusions 92 7.6 Exercises 93 8 Nonstationarity 97 8.1 Introduction 97 8.2 Nonstationarity in Variance 97 8.3 Nonstationarity in Mean: Random Walk with Drift 98 8.4 Unit Root Test 100 8.5 Simulations 102 8.6 Exercises 103 9 Heteroskedasticity 105 9.1 Introduction 105 9.2 ARCH 106 9.3 GARCH 109 9.4 Estimation and Testing for ARCH 111 9.5 Example of Foreign Exchange Rates 113 9.6 Exercises 120 10 Multivariate Time Series 123 10.1 Introduction 123 10.2 Estimation of μ and Τ 127 10.3 Multivariate ARM A Processes 127 10.3.1 Causality and Invertibility 128 10.3.2 Identifiability 129 10.4 Vector AR Models 130 χ CONTENTS 10.5 Example of Inferences for VAR 133 10.6 Exercises 141 11 State Space Models 143 11.1 Introduction 143 11.2 State Space Representation 143 11.3 Kalman Recursions 146 11.4 Stochastic Volatility Models 148 11.5 Example of Kalman Filtering of Term Structure 150 11.6 Exercises 157 12 Multivariate GARCH 159 12.1 Introduction 159 12.2 General Model 160 12.2.1 Diagonal Form 161 12.2.2 Alternative Matrix Form 162 12.3 Quadratic Form 162 12.3.1 Single-Factor GARCH(1,1) 162 12.3.2 Constant-Correlation Model 163 12.4 Example of Foreign Exchange Rates 163 12.4.1 The Data 164 12.4.2 Multivariate GARCH in SPLUS 164 12.4.3 Prediction 174 12.4.4 Predicting Portfolio Conditional Standard Deviations 174 12.4.5 BEKK Model 175 12.4.6 Vector-Diagonal Models 176 12.4.7 ARMA in Conditional Mean 177 12.5 Conclusions 178 12.6 Exercises 178 13 Cointegrations and Common Trends 179 13.1 Introduction 179 13.2 Definitions and Examples 180 13.3 Error Correction Form 183 13.4 Granger's Representation Theorem 185 13.5 Structure of Cointegrated Systems 189 13.6 Statistical Inference for Cointegrated Systems 190 13.6.1 Canonical Correlations 190