Multivariate Time Series Analysis and Applications

William W.S. Wei, PhD, is a Professor of Statistics at Temple University in Philadelphia, Pennsylvania, USA. He has been a Visiting Professor at many universities including Nankai University in China, National University of Colombia in Colombia, Korea University in Korea, National Chiao Tung University, National Sun Yat-Sen University, and National Taiwan University in Taiwan, and Middle East Technical University in Turkey. His research interests include time series analysis, forecasting methods, high dimensional problems, statistical modeling, and their applications.

• About the author xiiiPreface xvAbout the Companion website xvii1 Fundamental Concepts and Issues in Multivariate Time Series Analysis 11.1 Introduction 11.2 Fundamental concepts 31.2.1 Correlation and partial correlation matrix functions 31.2.2 Vector white noise process 71.2.3 Moving average and autoregressive representations of vector processes 7Projects 9References 92 Vector Time Series Models 112.1 Vector moving average processes 112.2 Vector autoregressive processes 142.2.1 Granger causality 182.3 Vector autoregressive moving average processes 182.4 Nonstationary vector autoregressive moving average processes 212.5 Vector time series model building 212.5.1 Identification of vector time series models 212.5.2 Sample moments of a vector time series 222.5.2.1 Sample mean and sample covariance matrices 222.5.2.2 Sample correlation matrix function 232.5.2.3 Sample partial correlation matrix function and extended cross-correlation matrices 242.5.3 Parameter estimation, diagnostic checking, and forecasting 242.5.4 Cointegration in vector time series 252.6 Seasonal vector time series model 262.7 Multivariate time series outliers 272.7.1 Types of multivariate time series outliers and detections 272.7.2 Outlier detection through projection pursuit 292.8 Empirical examples 322.8.1 First model of US monthly retail sales revenue 322.8.2 Second model of US monthly retail sales revenue 432.8.3 US macroeconomic indicators 472.8.4 Unemployment rates with outliers 58Software code 65Projects 100References 1013 Multivariate Time Series Regression Models 1053.1 Introduction 1053.2 Multivariate multiple time series regression models 1053.2.1 The classical multiple regression model 1053.2.2 Multivariate multiple regression model 1063.3 Estimation of the multivariate multiple time series regression model 1083.3.1 The Generalized Least Squares (GLS) estimation 1083.3.2 Empirical Example I – U.S. retail sales and some national indicators 1093.4 Vector time series regression models 1143.4.1 Extension of a VAR model to VARX models 1143.4.2 Empirical Example II – VARX models for U.S. retail sales and some national indicators 1153.5 Empirical Example III – Total mortality and air pollution in California 120Software code 129Projects 137References 1374 Principle Component Analysis of Multivariate Time Series 1394.1 Introduction 1394.2 Population PCA 1404.3 Implications of PCA 1414.4 Sample principle components 1424.5 Empirical examples 1454.5.1 Daily stock returns from the first set of 10 stocks 1454.5.1.1 The PCA based on the sample covariance matrix 1474.5.1.2 The PCA based on the sample correlation matrix 1504.5.2 Monthly Consumer Price Index (CPI) from five sectors 1524.5.2.1 The PCA based on the sample covariance matrix 1534.5.2.2 The PCA based on the sample correlation matrix 154Software code 157Projects 160References 1615 Factor Analysis of Multivariate Time Series 1635.1 Introduction 1635.2 The orthogonal factor model 1635.3 Estimation of the factor model 1655.3.1 The principal component method 1655.3.2 Empirical Example I – Model 1 on daily stock returns from the second set of 10 stocks 1665.3.3 The maximum likelihood method 1695.3.4 Empirical Example II – Model 2 on daily stock returns from the second set of 10 stocks 1735.4 Factor rotation 1755.4.1 Orthogonal rotation 1765.4.2 Oblique rotation 1765.4.3 Empirical Example III – Model 3 on daily stock returns from the second set of 10 stocks 1775.5 Factor scores 1785.5.1 Introduction 1785.5.2 Empirical Example IV – Model 4 on daily stock returns from the second set of 10 stocks 1795.6 Factor models with observable factors 1815.7 Another empirical example – Yearly U.S. sexually transmitted diseases (STD) 1835.7.1 Principal components analysis (PCA) 1835.7.1.1 PCA for standardized Zt 1835.7.1.2 PCA for unstandardized Zt 1865.7.2 Factor analysis 1865.8 Concluding remarks 193Software code 194Projects 200References 2016 Multivariate GARCH Models 2036.1 Introduction 2036.2 Representations of multivariate GARCH models 2046.2.1 VEC and DVEC models 2046.2.2 Constant Conditional Correlation (CCC) models 2066.2.3 BEKK models 2076.2.4 Factor models 2086.3 O-GARCH and GO-GARCH models 2096.4 Estimation of GO-GARCH models 2106.4.1 The two-step estimation method 2106.4.2 The weighted scatter estimation method 2116.5 Properties of the weighted scatter estimator 2136.5.1 Asymptotic distribution and statistical inference 2136.5.2 Combining information from different weighting functions 2146.6 Empirical examples 2156.6.1 U.S. weekly interest over time on six exercise items 2156.6.1.1 Choose a best VAR/VARMA model 2166.6.1.2 Finding a VARMA-ARCH/GARCH model 2186.6.1.3 The fitted values from VAR(1)-ARCH(1) model 2216.6.2 Daily log-returns of the SP 500 index and three financial stocks 2226.6.3 The analysis of the Dow Jones Industrial Average of 30 stocks 225Software code 229Projects 234References 2347 Repeated Measurements 2377.1 Introduction 2377.2 Multivariate analysis of variance 2397.2.1 Test treatment effects 2397.2.2 Empirical Example I – First analysis on body weight of rats under three different treatments 2417.3 Models utilizing time series structure 2437.3.1 Fixed effects model 2437.3.2 Some common variance–covariance structures 2477.3.3 Empirical Example II – Further analysis on body weight of rats under three different treatments 2507.3.4 Random effects and mixed effects models 2527.4 Nested random effects model 2537.5 Further generalization and remarks 2547.6 Another empirical example – the oral condition of neck cancer patients 255Software code 257Projects 258References 2588 Space–Time Series Models 2618.1 Introduction 2618.2 Space–time autoregressive integrated moving average (STARIMA) models 2628.2.1 Spatial weighting matrix 2628.2.2 STARIMA models 2658.2.3 STARMA models 2668.2.4 ST-ACF and ST-PACF 2678.3 Generalized space–time autoregressive integrated moving average (GSTARIMA) models 2728.4 Iterative model building of STARMA and GSTARMA models 2738.5 Empirical examples 2738.5.1 Vehicular theft data 2738.5.2 The annual U.S. labor force count 2798.5.3 U.S. yearly sexually transmitted disease data 281Software code 289Projects 298References 2989 Multivariate Spectral Analysis of Time Series 3019.1 Introduction 3019.2 Spectral representations of multivariate time series processes 3049.3 The estimation of the spectral density matrix 3099.3.1 The smoothed spectrum matrix 3099.3.2 Multitaper smoothing 3139.3.3 Smoothing spline 3159.3.4 Bayesian method 3169.3.5 Penalized Whittle likelihood 3179.3.6 VARMA spectral estimation 3189.4 Empirical examples of stationary vector time series 3209.4.1 Sample spectrum 3209.4.2 Bayesian method 3259.4.3 Penalized Whittle likelihood method 3279.4.4 Example of VAR spectrum estimation 3279.5 Spectrum analysis of a nonstationary vector time series 3299.5.1 Introduction 3299.5.2 Spectrum representations of a nonstationary multivariate process 3319.5.2.1 Time-varying autoregressive model 3329.5.2.2 Smoothing spline ANOVA model 3339.5.2.3 Piecewise vector autoregressive model 3349.5.2.4 Bayesian methods 3369.6 Empirical spectrum example of nonstationary vector time series 337Software code 341Projects 434References 43510 Dimension Reduction in High-Dimensional Multivariate Time Series Analysis 43710.1 Introduction 43710.2 Existing methods 43810.2.1 Regularization methods 43910.2.1.1 The lasso method 43910.2.1.2 The lag-weighted lasso method 44010.2.1.3 The hierarchical vector autoregression (HVAR) method 44010.2.2 The space–time AR (STAR) model 44210.2.3 The model-based cluster method 44310.2.4 The factor analysis 44310.3 The proposed method for high-dimension reduction 44410.4 Simulation studies 44610.4.1 Scenario 1 44610.4.2 Scenario 2 44910.4.3 Scenario 3 44910.5 Empirical examples 45210.5.1 The macroeconomic time series 45210.5.2 The yearly U.S. STD data 45710.6 Further discussions and remarks 45910.6.1 More on clustering 45910.6.2 Forming aggregate data through both time domain and frequency domain clustering 46110.6.2.1 Example of time domain clustering 46110.6.2.2 Example of frequency domain clustering 46110.6.2.2.1 Clustering using similarity measures 46310.6.2.2.2 Clustering by subjective observation 46310.6.2.2.3 Hierarchical clustering 46310.6.2.2.4 Nonhierarchical clustering using the K-means method 46310.6.3 The specification of aggregate matrix and its associated aggregate dimension 46610.6.4 Be aware of other forms of aggregation 46610.A Appendix: Parameter Estimation Results of Various Procedures 46710.A.1 Further details of the macroeconomic time series 46710.A.1.1 VAR(1) 46710.A.1.2 Lasso 46810.A.1.3 Componentwise 47010.A.1.4 Own-other 47110.A.1.5 Elementwise 47310.A.1.6 The factor model 47510.A.1.7 The model-based cluster 47510.A.1.8 The proposed method 47710.A.2 Further details of the STD time series 47810.A.2.1 VAR 47810.A.2.2 Lasso 47810.A.2.3 Componentwise 47910.A.2.4 Own-other 48110.A.2.5 Elementwise 48210.A.2.6 The STAR model 48410.A.2.7 The factor model 48610.A.2.8 The model-based cluster 48710.A.2.9 The proposed method 488Software code 490Projects 505References 506Author index 509Subject index 515