Time Series Analysis
Univariate and Multivariate Methods
2nd Edition
Published on 6. September 2005
Book
Hardback
640 pages
978-0-321-32216-6 (ISBN)
Description
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.
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.
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Content
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
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