Time Series Analysis

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
      • 14.5.3 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
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
      • 20.6.4 Examples
      • 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