# Univariate and Multivariate Methods PEARSON. Addison Wesley

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1 Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal

2 CHAPTER 1 CHAPTER 2 CHAPTER 3 Preface 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 Sample Mean Sample Autocovariance Function Sample Autocorrelation Function Sample Partial Autocorrelation Function 2.6 Moving Average and Autoregressive Representations of Time Series Processes 2.7 Linear Difference Equations Exercises Stationary Time Series Models 3.1 Autoregressive Processes The First-Order Auloregressive AR( 1) Process The Second-Order Autoregressive AR(2) Process The General pth-order Autoregressive AR(p) Process 3.2 Moving Average Processes The First-Order Moving Average MA( 1) Process The Second-Order Moving Average MA(2) Process The General gth-order Moving Average MA(<y) Process XIX IX

3 3.3 The Dual Relationship Between AR(p) and MA(ij) Processes 3.4 Autoregressive Moving Average ARMA(p, q) Processes The General Mixed ARMA(p, q) Process The ARMA( 1, 1) Process Exercises CHAPTER 4 Nonstationary Time Series Models Nonstationarity in the Mean Deterministic Trend Models Stochastic Trend Models and Differencing Autoregressive Integrated Moving Average (ARIMA) Models The General ARIMA Model The Random Walk Model The ARIMA(0, 1, 1) or IMA( 1,1) Model Nonstationarity in the Variance and the Autocovariance Variance and Autocovariance of the ARIMA Models Variance Stabilizing Transformations 83 Exercises 86 CHAPTER 5 Forecasting 5.1 Introduction 5.2 Minimum Mean Square Error Forecasts Minimum Mean Square Error Forecasts for ARMA Models 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 Exercises CHAPTER 6 Model Identification 6.1 Steps for Model Identification 6.2 Empirical Examples 6.3 The Inverse Autocorrelation Function (IACF)

4 XI 6.4 Extended Sample Autocorrelation Function and Other Identification Procedures The Extended Sample Autocorrelation Function (ESACF) Other Identification Procedures 133 Exercises 134 CHAPTER 7 Parameter Estimation, Diagnostic Checking, and Model Selection The Method of Moments Maximum Likelihood Method Conditional Maximum Likelihood Estimation Unconditional Maximum Likelihood Estimation and Backcasting Method Exact Likelihood Functions Nonlinear Estimation Ordinary Least Squares (OLS) Estimation in Time Series Analysis Diagnostic Checking Empirical Examples for Series W1-W Model Selection Criteria 156 Exercises 158 CHAPTER 8 Seasonal Time Series Models 8.1 General Concepts 8.2 Traditional Methods Regression Method Moving Average Method 8.3 Seasonal ARIMA Models 8.4 Empirical Examples Exercises CHAPTER 9 Testing for a Unit Root Introduction Some Useful Limiting Distributions Testing for a Unit Root in the AR( 1) Model Testing the AR(1) Model without a Constant Term Testing the AR( 1) Model with a Constant Term Testing the AR( 1) Model with a Linear Time Trend Testing for a Unit Root in a More General Model 196

5 xii Contents 9.5 Testing for a Unit Root in Seasonal Time Series Models Testing the Simple Zero Mean Seasonal Model Testing the General Multiplicative Zero Mean Seasonal Model 207 Exercises 211 CHAPTER 10 Intervention Analysis and Outlier Detection Intervention Models Examples of Intervention Analysis Time Series Outliers Additive and Innovational Outliers Estimation of the Outlier Effect When the Timing of the Outlier Is Known Detection of Outliers Using an Iterative Procedure Examples of Outlier Analysis Model Identification in the Presence of Outliers 230 Exercises 235 CHAPTER 11 Fourier Analysis General Concepts Orthogonal Functions Fourier Representation of Finite Sequences Fourier Representation of Periodic Sequences Fourier Representation of Nonperiodic Sequences: The Discrete-Time Fourier Transform Fourier Representation of Continuous-Time Functions Fourier Representation of Periodic Functions Fourier Representation of Nonperiodic Functions: The Continuous-Time Fourier Transform The Fast Fourier Transform 258 Exercises 261 CHAPTER 12 Spectral Theory of Stationary Processes The Spectrum The Spectrum and Its Properties The Spectral Representation of Autocovariance Functions: The Spectral Distribution Function Wold's Decomposition of a Stationary Process The Spectral Representation of Stationary Processes 272

6 xiii 12.2 The Spectrum of Some Common Processes The Spectrum and the Autocovariance Generating Function The Spectrum of ARMA Models The Spectrum of the Sum of Two Independent Processes The Spectrum of Seasonal Models The Spectrum of Linear Filters The Filter Function Effect of Moving Average Effect of Differencing Aliasing 285 Exercises 286 CHAPTER 13 Estimation of the Spectrum Periodogram Analysis The Periodogram Sampling Properties of the Periodogram Tests for Hidden Periodic Components The Sample Spectrum The Smoothed Spectrum Smoothing in the Frequency Domain: The Spectral Window Smoothing in the Time Domain: The Lag Window Some Commonly Used Windows Approximate Confidence Intervals for Spectral Ordinates ARMA Spectral Estimation 318 Exercises 321 CHAPTER 14 Transfer Function Models Single-Input Transfer Function Models General Concepts Some Typical Impulse Response Functions The Cross-Correlation Function and Transfer Function Models The Cross-Correlation Function (CCF) The Relationship between the Cross-Correlation Function and the Transfer Function Construction of Transfer Function Models Sample Cross-Correlation Function Identification of Transfer Function Models Estimation of Transfer Function Models 332

7 xiv Contents Diagnostic Checking of Transfer Function Models An Empirical Example Forecasting Using Transfer Function Models Minimum Mean Square Error Forecasts for Stationary Input and Output Scries Minimum Mean Square Error Forecasts for Nonstationary Input and Output Series An Example Bivariate Frequency-Domain Analysis Cross-Covariance Generating Functions and the Cross-Spectrum Interpretation of the Cross-Spectral Functions Examples Estimation of the Cross-Spectrum The Cross-Spectrum and Transfer Function Models Construction of Transfer Function Models through Cross-Spectrum Analysis Cross-Spectral Functions of Transfer Function Models Multiple-Input Transfer Function Models 361 Exercises 363 CHAPTER 15 Time Series Regression and GARCH Models Regression with Autocorrelated Errors ARCH and GARCH Models Estimation of GARCH Models Maximum Likelihood Estimation Iterative Estimation Computation of Forecast Error Variance Illustrative Examples 376 Exercises 380 CHAPTER 16 Vector Time Series Models Covariance and Correlation Matrix Functions Moving Average and Autoregressive Representations of Vector Processes The Vector Autoregressive Moving Average Process Covariance Matrix Function for the Vector AR(1) Model Vector AR(p) Models Vector MA(1) Models Vector MA(q) Models Vector ARMA( 1, 1) Models 398

8 xv 16.4 Nonstationary Vector Auloregressive Moving Average Models Identification of Vector Time Series Models Sample Correlation Matrix Function 401 [ Partial Autoregression Matrices Partial Lag Correlation Matrix Function 408 j 16.6 Model Fitting and Forecasting 414 i 16.7 An Empirical Example 416 [ Model Identification Parameter Estimation Diagnostic Checking Forecasting Further Remarks Spectral Properties of Vector Processes 421 Supplement 16.A Multivariate Linear Regression Models 423 Exercises 426 CHAPTER 17 More on Vector Time Series Unit Roots and Cointegration in Vector Processes Representations of Nonstationary Cointegrated Processes Decomposition of Z, Testing and Estimating Cointegration Partial Process and Partial Process Correlation Matrices Covariance Matrix Generating Function Partial Covariance Matrix Generating Function Partial Process Sample Correlation Matrix Functions An Empirical Example: The U.S. Hog Data Equivalent Representations of a Vector ARMA Model Finite-Order Representations of a Vector Time Series Process Some Implications 457 Exercises 460 CHAPTER 18 State Space Models and the Kalman Filter State Space Representation The Relationship between State Space and ARMA Models State Space Model Fitting and Canonical Correlation Analysis Empirical Examples The Kalman Filter and Its Applications 478 Supplement 18.A Canonical Correlations 483 Exercises 487

9 xvi Contents CHAPTER 19 Long Memory and Nonlinear Processes Long Memory Processes and Fractional Differencing Fractionally Integrated ARMA Models and Their ACF Practical Implications of the ARFIMA Processes Estimation of the Fractional Difference Nonlinear Processes Cumulants, Polyspectrum, and Tests for Linearity and Normality Some Nonlinear Time Series Models Threshold Autoregressive Models Tests for TAR Models Modeling TAR Models 502 Exercises 506 CHAPTER 20 Aggregation and Systematic Sampling in Time Series Temporal Aggregation of the ARIMA Process The Relationship of Autocovariances between the Nonaggregate and Aggregate Series Temporal Aggregation of the IMA(J, q) Process Temporal Aggregation of the AR(p) Process Temporal Aggregation of the ARIMA(p, d, q) Process The Limiting Behavior of Time Series Aggregates The Effects of Aggregation on Forecasting and Parameter Estimation Hilbert Space The Application of Hilbert Space in Forecasting The Effect of Temporal Aggregation on Forecasting Information Loss Due to Aggregation in Parameter Estimation Systematic Sampling of the ARIMA Process The Effects of Systematic Sampling and Temporal Aggregation on Causality Decomposition of Linear Relationship between Two Time Series An Illustrative Underlying Model The Effects of Systematic Sampling and Temporal Aggregation on Causality The Effects of Aggregation on Testing for Linearity and Normality Testing for Linearity and Normality The Effects of Temporal Aggregation on Testing for Linearity and Normality 537

10 xvii 20.6 The Effects of Aggregation on Testing for a Unit Root The Model of Aggregate Scries The Effects of Aggregation on the Distribution of the Test Statistics The Effects of Aggregation on the Significance Level and the Power of the Test Examples General Cases and Concluding Remarks Further Comments 549 Exercises 550 References 553 Appendix 565 Time Series Data Used for Illustrations 565 Statistical Tables 565 Author Index 601 Subject Index 605

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