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5. Time Series Analysis: Univariate and Multivariate Methods (Classic Version)

# Time Series Analysis: Univariate and Multivariate Methods (Classic Version), 2nd edition

• William W.S. Wei

2nd edition

Time Series Analysis: Univariate and Multivariate Methods (Classic Version)

ISBN-13: 9780134995366

Includes: Paperback
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• Paperback

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## Overview

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.

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

8.2.1 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

10.3.2 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

References

Appendix

Time Series Data Used for Illustrations

Statistical Tables

Author Index

Subject Index

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