Table of Contents Overview 1.1 Introduction 1.2 Examples and Scope of This Book 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 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 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 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 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 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 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 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 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 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 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 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 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 14.5.3 Examples 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 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 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 1.