About the author xiii

Preface xv

About the Companion website xvii

1 Fundamental Concepts and Issues in Multivariate Time Series Analysis 1

1.1 Introduction 1

1.2 Fundamental concepts 3

1.2.1 Correlation and partial correlation matrix functions 3

1.2.2 Vector white noise process 7

1.2.3 Moving average and autoregressive representations of vector processes 7

Projects 9

References 9

2 Vector Time Series Models 11

2.1 Vector moving average processes 11

2.2 Vector autoregressive processes 14

2.2.1 Granger causality 18

2.3 Vector autoregressive moving average processes 18

2.4 Nonstationary vector autoregressive moving average processes 21

2.5 Vector time series model building 21

2.5.1 Identification of vector time series models 21

2.5.2 Sample moments of a vector time series 22

2.5.2.1 Sample mean and sample covariance matrices 22

2.5.2.2 Sample correlation matrix function 23

2.5.2.3 Sample partial correlation matrix function and extended cross-correlation matrices 24

2.5.3 Parameter estimation, diagnostic checking, and forecasting 24

2.5.4 Cointegration in vector time series 25

2.6 Seasonal vector time series model 26

2.7 Multivariate time series outliers 27

2.7.1 Types of multivariate time series outliers and detections 27

2.7.2 Outlier detection through projection pursuit 29

2.8 Empirical examples 32

2.8.1 First model of US monthly retail sales revenue 32

2.8.2 Second model of US monthly retail sales revenue 43

2.8.3 US macroeconomic indicators 47

2.8.4 Unemployment rates with outliers 58

Software code 65

Projects 100

References 101

3 Multivariate Time Series Regression Models 105

3.1 Introduction 105

3.2 Multivariate multiple time series regression models 105

3.2.1 The classical multiple regression model 105

3.2.2 Multivariate multiple regression model 106

3.3 Estimation of the multivariate multiple time series regression model 108

3.3.1 The Generalized Least Squares (GLS) estimation 108

3.3.2 Empirical Example I - U.S. retail sales and some national indicators 109

3.4 Vector time series regression models 114

3.4.1 Extension of a VAR model to VARX models 114

3.4.2 Empirical Example II - VARX models for U.S. retail sales and some national indicators 115

3.5 Empirical Example III - Total mortality and air pollution in California 120

Software code 129

Projects 137

References 137

4 Principle Component Analysis of Multivariate Time Series 139

4.1 Introduction 139

4.2 Population PCA 140

4.3 Implications of PCA 141

4.4 Sample principle components 142

4.5 Empirical examples 145

4.5.1 Daily stock returns from the first set of 10 stocks 145

4.5.1.1 The PCA based on the sample covariance matrix 147

4.5.1.2 The PCA based on the sample correlation matrix 150

4.5.2 Monthly Consumer Price Index (CPI) from five sectors 152

4.5.2.1 The PCA based on the sample covariance matrix 153

4.5.2.2 The PCA based on the sample correlation matrix 154

Software code 157

Projects 160

References 161

5 Factor Analysis of Multivariate Time Series 163

5.1 Introduction 163

5.2 The orthogonal factor model 163

5.3 Estimation of the factor model 165

5.3.1 The pri