Time Series Analysis: Univariate and Multivariate Methods

With its broad coverage of methodology, this comprehensive book is a useful learning and reference tool for those in applied sciences where analysis and research of time series is useful. Its plentiful examples show the operational details and purpose of a variety of univariate and multivariate time series methods. Numerous figures, tables and real-life time series data sets illustrate the models and methods useful for analyzing, modeling, and forecasting data collected sequentially in time. The text also offers a balanced treatment between theory and applications.   Time Series Analysis is a thorough introduction to both time-domain and frequency-domain analyses of univariate and multivariate time series methods, with coverage of the most recently developed techniques in the field.

Table of Contents:
1: Overview 1.1 Introduction 1.2 Examples and Scope of This Book   2: Fundamental Concepts 2.1 Stochastic Processes 2.2 The Autocovariance and Autocorrelation Functions 2.3 The Partial Autocorrelation Function 2.4 White Noise Processes 2.5 Estimation of the Mean, Autocovariances, and Autocorrelations 2.5.1 Sample Mean 2.5.2 Sample Autocovariance Function 2.5.3 Sample Autocorrelation Function 2.5.4 Sample Partial Autocorrelation Function 2.6 Moving Average and Autoregressive Representations of Time Series Processes 2.7 Linear Difference Equations   3: Stationary Time Series Models 3.1 Autoregressive Processes 3.1.1 The First-Order Autoregressive AR(1) Process 3.1.2 The Second-Order Autoregressive AR(2) Process 3.1.3 The General pth-Order Autoregressive AR(p) Process 3.2 Moving Average Processes 3.2.1 The First-Order Moving Average MA(1) Process 3.2.2 The Second-Order Moving Average MA(2) Process 3.2.3 The General qth-Order Moving Average MA(q) Process 3.3 The Dual Relationship Between AR(p) and MA(q) Processes 3.4 Autoregressive Moving Average ARMA(p, q) Processes 3.4.1 The General Mixed ARMA(p, q) Process 3.4.2 The ARMA(1, 1) Process   4: Nonstationary Time Series Models 4.1 Nonstationarity in the Mean 4.1.1 Deterministic Trend Models 4.1.2 Stochastic Trend Models and Differencing 4.2 Autoregressive Integrated Moving Average (ARIMA) Models 4.2.1 The General ARIMA Model 4.2.2 The Random Walk Model 4.2.3 The ARIMA(0, 1, 1) or IMA(1, 1) Model 4.3 Nonstationarity in the Variance and the Autocovariance 4.3.1 Variance and Autocovariance of the ARIMA Models 4.3.2 Variance Stabilizing Transformations   5: Forecasting 5.1 Introduction 5.2 Minimum Mean Square Error Forecasts 5.2.1 Minimum Mean Square Error Forecasts for ARMA Models 5.2.2 Minimum Mean Square Error Forecasts for ARIMA Models 5.3 Computation of Forecasts 5.4 The ARIMA Forecast as a Weighted Average of Previous Observations 5.5 Updating Forecasts 5.6 Eventual Forecast Functions 5.7 A Numerical Example   6: Model Identification 6.1 Steps for Model Identification 6.2 Empirical Examples 6.3 The Inverse Autocorrelation Function (IACF) 6.4 Extended Sample Autocorrelation Function and Other Identification Procedures 6.4.1 The Extended Sample Autocorrelation Function (ESACF) 6.4.2 Other Identification Procedures   7: Parameter Estimation, Diagnostic Checking, and Model Selection 7.1 The Method of Moments 7.2 Maximum Likelihood Method 7.2.1 Conditional Maximum Likelihood Estimation 7.2.2 Unconditional Maximum Likelihood Estimation and Backcasting Method 7.2.3 Exact Likelihood Functions 7.3 Nonlinear Estimation 7.4 Ordinary Least Squares (OLS) Estimation in Time Series Analysis 7.5 Diagnostic Checking 7.6 Empirical Examples for Series W1—W7 7.7 Model Selection Criteria     8: Seasonal Time Series Models 8.1 General Concepts 8.2 Traditional Methods 8.2.1 Regression Method 8.2.2 Moving Average Method 8.3 Seasonal ARIMA Models 8.4 Empirical Examples   9: Testing for a Unit Root 9.1 Introduction 9.2 Some Useful Limiting Distributions 9.3 Testing for a Unit Root in the AR(1) Model 9.3.1 Testing the AR(1) Model without a Constant Term 9.3.2 Testing the AR(1) Model with a Constant Term 9.3.3 Testing the AR(1) Model with a Linear Time Trend 9.4 Testing for a Unit Root in a More General Model 9.5 Testing for a Unit Root in Seasonal Time Series Models 9.5.1 Testing the Simple Zero Mean Seasonal Model 9.5.2 Testing the General Multiplicative Zero Mean Seasonal Model   10: Intervention Analysis and Outlier Detection 10.1 Intervention Models 10.2 Examples of Intervention Analysis 10.3 Time Series Outliers 10.3.1 Additive and Innovational Outliers 10.3.2 Estimation of the Outlier Effect When the Timing of the Outlier Is Known 10.3.3 Detection of Outliers Using an Iterative Procedure 10.4 Examples of Outlier Analysis 10.5 Model Identification in the Presence of Outliers   11: Fourier Analysis 11.1 General Concepts 11.2 Orthogonal Functions 11.3 Fourier Representation of Finite Sequences 11.4 Fourier Representation of Periodic Sequences 11.5 Fourier Representation of Nonperiodic Sequences: The Discrete-Time Fourier Transform 11.6 Fourier Representation of Continuous-Time Functions 11.6.1 Fourier Representation of Periodic Functions 11.6.2 Fourier Representation of Nonperiodic Functions: The Continuous-Time Fourier Transform 11.7 The Fast Fourier Transform   12: Spectral Theory of Stationary Processes 12.1 The Spectrum 12.1.1 The Spectrum and Its Properties 12.1.2 The Spectral Representation of Autocovariance Functions: The Spectral Distribution Function 12.1.3 Wold’s Decomposition of a Stationary Process 12.1.4 The Spectral Representation of Stationary Processes 12.2 The Spectrum of Some Common Processes 12.2.1 The Spectrum and the Autocovariance Generating Function 12.2.2 The Spectrum of ARMA Models 12.2.3 The Spectrum of the Sum of Two Independent Processes 12.2.4 The Spectrum of Seasonal Models 12.3 The Spectrum of Linear Filters 12.3.1 The Filter Function 12.3.2 Effect of Moving Average 12.3.3 Effect of Differencing 12.4 Aliasing   13: Estimation of the Spectrum 13.1 Periodogram Analysis 13.1.1 The Periodogram 13.1.2 Sampling Properties of the Periodogram 13.1.3 Tests for Hidden Periodic Components 13.2 The Sample Spectrum 13.3 The Smoothed Spectrum 13.3.1 Smoothing in the Frequency Domain: The Spectral Window 13.3.2 Smoothing in the Time Domain: The Lag Window 13.3.3 Some Commonly Used Windows 13.3.4 Approximate Confidence Intervals for Spectral Ordinates 13.4 ARMA Spectral Estimation   14: Transfer Function Models 14.1 Single-Input Transfer Function Models 14.1.1 General Concepts 14.1.2 Some Typical Impulse Response Functions 14.2 The Cross-Correlation Function and Transfer Function Models 14.2.1 The Cross-Correlation Function (CCF) 14.2.2 The Relationship between the Cross-Correlation Function and the Transfer Function 14.3 Construction of Transfer Function Models 14.3.1 Sample Cross-Correlation Function 14.3.2 Identification of Transfer Function Models 14.3.3 Estimation of Transfer Function Models 14.3.4 Diagnostic Checking of Transfer Function Models 14.3.5 An Empirical Example 14.4 Forecasting Using Transfer Function Models 14.4.1 Minimum Mean Square Error Forecasts for Stationary Input and Output Series 14.4.2 Minimum Mean Square Error Forecasts for Nonstationary Input and Output Series 14.4.3 An Example 14.5 Bivariate Frequency-Domain Analysis 14.5.1 Cross-Covariance Generating Functions and the Cross-Spectrum 14.5.2 Interpretation of the Cross-Spectral Functions 14.5.3Examples 14.5.4 Estimation of the Cross-Spectrum 14.6 The Cross-Spectrum and Transfer Function Models 14.6.1 Construction of Transfer Function Models through Cross-Spectrum Analysis 14.6.2 Cross-Spectral Functions of Transfer Function Models 14.7 Multiple-Input Transfer Function Models   15: Time Series Regression and GARCH Models 15.1 Regression with Autocorrelated Errors 15.2 ARCH and GARCH Models 15.3 Estimation of GARCH Models 15.3.1 Maximum Likelihood Estimation 15.3.2 Iterative Estimation 15.4 Computation of Forecast Error Variance 15.5 Illustrative Examples   16: Vector Time Series Models 16.1 Covariance and Correlation Matrix Functions 16.2 Moving Average and Autoregressive Representations of Vector Processes 16.3 The Vector Autoregressive Moving Average Process 16.3.1 Covariance Matrix Function for the Vector AR(1) Model 16.3.2 Vector AR(p) Models 16.3.3 Vector MA(1) Models 16.3.4 Vector MA(q) Models 16.3.5 Vector ARMA(1, 1) Models 16.4 Nonstationary Vector Autoregressive Moving Average Models 16.5 Identification of Vector Time Series Models 16.5.1 Sample Correlation Matrix Function 16.5.2 Partial Autoregression Matrices 16.5.3 Partial Lag Correlation Matrix Function 16.6 Model Fitting and Forecasting 16.7 An Empirical Example 16.7.1 Model Identification 16.7.2 Parameter Estimation 16.7.3 Diagnostic Checking 16.7.4 Forecasting 16.7.5 Further Remarks 16.8 Spectral Properties of Vector Processes Supplement 16.A Multivariate Linear Regression Models   17: More on Vector Time Series 17.1 Unit Roots and Cointegration in Vector Processes 17.1.1 Representations of Nonstationary Cointegrated Processes 17.1.2 Decomposition of Zt 17.1.3 Testing and Estimating Cointegration 17.2 Partial Process and Partial Process Correlation Matrices 17.2.1 Covariance Matrix Generating Function 17.2.2 Partial Covariance Matrix Generating Function 17.2.3 Partial Process Sample Correlation Matrix Functions 17.2.4 An Empirical Example: The U.S. Hog Data 17.3 Equivalent Representations of a Vector ARMA Model 17.3.1 Finite-Order Representations of a Vector Time Series Process 17.3.2 Some Implications   18: State Space Models and the Kalman Filter 18.1 State Space Representation 18.2 The Relationship between State Space and ARMA Models 18.3 State Space Model Fitting and Canonical Correlation Analysis 18.4 Empirical Examples 18.5 The Kalman Filter and Its Applications Supplement 18.A Canonical Correlations   19: Long Memory and Nonlinear Processes 19.1 Long Memory Processes and Fractional Differencing 19.1.1 Fractionally Integrated ARMA Models and Their ACF 19.1.2 Practical Implications of the ARFIMA Processes 19.1.3 Estimation of the Fractional Difference 19.2 Nonlinear Processes 19.2.1 Cumulants, Polyspectrum, and Tests for Linearity and Normality 19.2.2 Some Nonlinear Time Series Models 19.3 Threshold Autoregressive Models 19.3.1 Tests for TAR Models 19.3.2 Modeling TAR Models   20: Aggregation and Systematic Sampling in Time Series 20.1 Temporal Aggregation of the ARIMA Process 20.1.1 The Relationship of Autocovariances between the Nonaggregate and Aggregate Series 20.1.2 Temporal Aggregation of the IMA(d, q) Process 20.1.3 Temporal Aggregation of the AR(p) Process 20.1.4 Temporal Aggregation of the ARIMA(p, d, q) Process 20.1.5 The Limiting Behavior of Time Series Aggregates 20.2 The Effects of Aggregation on Forecasting and Parameter Estimation 20.2.1 Hilbert Space 20.2.2 The Application of Hilbert Space in Forecasting 20.2.3 The Effect of Temporal Aggregation on Forecasting 20.2.4 Information Loss Due to Aggregation in Parameter Estimation 20.3 Systematic Sampling of the ARIMA Process 20.4 The Effects of Systematic Sampling and Temporal Aggregation on Causality 20.4.1 Decomposition of Linear Relationship between Two Time Series 20.4.2 An Illustrative Underlying Model 20.4.3 The Effects of Systematic Sampling and Temporal Aggregation on Causality 20.5 The Effects of Aggregation on Testing for Linearity and Normality 20.5.1 Testing for Linearity and Normality 20.5.2 The Effects of Temporal Aggregation on Testing for Linearity and Normality 20.6 The Effects of Aggregation on Testing for a Unit Root 20.6.1 The Model of Aggregate Series 20.6.2 The Effects of Aggregation on the Distribution of the Test Statistics 20.6.3 The Effects of Aggregation on the Significance Level and the Power of the Test 20.6.4Examples 20.6.5 General Cases and Concluding Remarks 20.7 Further Comments   References   Appendix Time Series Data Used for Illustrations Statistical Tables Author Index   Subject Index