New Introduction to Multiple Time Series Analysis 

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Comment from the Stata technical groupIncorporating recent advances, New Introduction to Multiple Time Series Analysis by Helmut Lütkepohl builds on the seminal Introduction to Multiple Time Series Analysis to create what is sure to become the specialty's standard textbook because of its style and depth of coverage. Like its predecessor, this book provides the most complete coverage of stationary vector autoregressive (VAR) and vector autoregressive moving average (VARMA) models of any book. Incorporating more than six chapters of new material, New Introduction to Multiple Time Series Analysis also provides extensive coverage of the vector errorcorrection model (VECM) for cointegrated processes, structural VARs, structural VECMs, cointegrated VARMA processes, and multivariate models for conditionally heteroskedastic processes. For each model, Lütkepohl provides a thorough discussion of six important topics:
Written in a clear, buildingblock style, the book begins with first principles and neatly adds pieces to the foundation as needed.  
Table of contentsView table of contents >> 1 Introduction
1.1 Objectives of Analyzing Multiple Time Series
1.2 Some Basics 1.3 Vector Autoregressive Processes 1.4 Outline of the Following Chapters Part 1 Finite Order Vector Autoregressive Processes
2 Stable Vector Autoregressive Processes
2.1 Basic Assumptions and Properties of VAR Processes
2.1.1 Stable VAR(p) Processes
2.2 Forecasting2.1.2 The Moving Average Representation of a VAR Process 2.1.3 Stationary Processes 2.1.4 Computation of Autocovariates and Autocorrelations of Stable VAR Processes
2.2.1 The Loss Function
2.3 Structural Analysis with VAR Models2.2.2 Point Forecasts 2.2.3 Interval Forecasts and Forecast Regions
2.3.1 GrangerCausality, Instantaneous Causality, and
MultiStep Causality
2.4 Exercises2.3.2 Impulse Response Analysis 2.3.3 Forecast Error Variance Decomposition 2.3.4 Remarks on the Interpretation of VAR Models 3 Estimation of Vector Autoregressive Processes
3.1 Introduction
3.2 Multivariate Least Squares Estimation
3.2.1 The Estimator
3.3 Least Squares Estimation with MeanAdjusted Data and YuleWalker Estimation3.2.2 Asymptotic Properties of the Least Squares Estimator 3.2.3 An Example 3.2.4 Small Sample Properties of the LS Estimator
3.3.1 Estimation when the Process Mean is Known
3.4 Maximum Likelihood Estimation3.3.2 Estimation of the Process Mean 3.3.3 Estimation with Unknown Process Mean 3.3.4 The YuleWalker Estimator 3.3.5 An Example
3.4.1 The Likelihood Function
3.5 Forecasting with Estimated Processes3.4.2 The ML Estimators 3.4.3 Properties of the ML Estimators
3.5.1 General Assumptions and Results
3.6 Testing for Causality3.5.2 The Approximate MSE Matrix 3.5.3 An Example 3.5.4 A Small Sample Investigation
3.6.1 A Wald Test for GrangerCausality
3.7 The Asymptotic Distributions of Impulse Responses and
Forecast Error Variance Decompositions3.6.2 An Example 3.6.3 Testing for Instantaneous Causality 3.6.4 Testing for MultiStep Causality
3.7.1 The Main Results
3.8 Exercises3.7.2 Proof of Proposition 3.6 3.7.3 An Example 3.7.4 Investigating the Distributions of the Impulse Responses by Simulation Techniques
3.8.1 Algebraic Problems
3.8.2 Numerical Problems 4 VAR Order Selection and Checking the Model
Adequacy
4.1 Introduction
4.2 A Sequence of Tests for Determining the VAR Order
4.2.1 The Impact of the Fitted VAR Order on the Forecast
MSE
4.3 Criteria for VAR Order Selection4.2.2 The Likelihood Ratio Test Statistic 4.2.3 A Testing Scheme for VAR Order Determination 4.2.4 An Example
4.3.1 Minimizing the Forecast MSE
4.4 Checking the Whiteness of the Residuals4.3.2 Consistent Order Selection 4.3.3 Comparison of Order Selection Criteria 4.3.4 Some Small Sample Simulation Results
4.4.1 The Asymptotic Distributions of the Autocovariances
and Autocorrelations of a White Noise Process
4.5 Testing for Normality4.4.2 The Asymptotic Distributions of the Residual Autocovariances and Autocorrelations of an Estimated VAR Process 4.4.3 Portmanteau Tests 4.4.4 Lagrange Multiplier Tests
4.5.1 Tests for Nonnormality of a Vector White Noise Process
4.6 Tests for Structural Change4.5.2 Tests for Nonnormality of a VAR Process
4.6.1 Chow Tests
4.7 Exercises4.6.2 Forecast Tests for Structural Change
4.7.1 Algebraic Problems
4.7.2 Numerical Problems 5 VAR Processes with Parameter Constraints
5.1 Introduction
5.2 Linear Constraints
5.2.1 The Model and the Constraints
5.3 VAR Processes with Nonlinear Parameter Restrictions5.2.2 LS, GLS, and EGLS Estimation 5.2.3 Maximum Likelihood Estimation 5.2.4 Constraints for Individual Equations 5.2.5 Restrictions for the White Noise Covariance Matrix 5.2.6 Forecasting 5.2.7 Impulse Response Analysis and Forecast Error Variance Decompositions 5.2.8 Specification of Subset VAR Models 5.2.9 Model Checking 5.2.10 An Example 5.4 Bayesian Estimation
5.4.1 Basic Terms and Notation
5.5 Exercises5.4.2 Normal Priors for the Parameters of a Gaussian VAR Process 5.4.3 The Minnesota or Litterman Prior 5.4.4 Practical Considerations 5.4.5 An Example 5.4.6 Classical versus Bayesian Interpretation of α in Forecasting and Structural Analysis
5.5.1 Algebraic Exercises
5.5.2 Numerical Problems Part 2 Cointegrated Processes
6 Vector Error Correction Models
6.1 Integrated Processes
6.2 VAR Processes with Integrated Variables 6.3 Cointegrated Processes, Common Stochastic Trends, and Vector Error Correction Models 6.4 Deterministic Terms in Cointegrated Processes 6.5 Forecasting Integrated and Cointegrated Variables 6.6 Causality Analysis 6.7 Impulse Response Analysis 6.8 Exercises 7 Estimation of Vector Error Correction Models
7.1 Estimation of a Simple Special Case VECM
7.2 Estimation of General VECMs
7.2.1 LS Estimation
7.3 Estimating VECMs with Parameter Restrictions7.2.2 EGLS Estimation of the Cointegration Parameters 7.2.3 ML Estimation 7.2.4 Including Deterministic Terms 7.2.5 Other Estimation Methods for Cointegrated Systems 7.2.6 An Example
7.3.1 Linear Restrictions for the Cointegration Matrix
7.4 Bayesian Estimation of Integrated Systems7.3.2 Linear Restrictions for the ShortRun and Loading Parameters 7.3.3 An Example
7.4.1 The Model Setup
7.5 Forecasting Estimated Integrated and Cointegrated Systems7.4.2 The Minnesota or Litterman Prior 7.4.3 An Example 7.6 Testing for GrangerCausality
7.6.1 The Noncausality Restrictions
7.7 Impulse Response Analysis7.6.2 Problems Related to Standard Wald Tests 7.6.3 A Wald Test Based on a Lag Augmented VAR 7.6.4 An Example 7.8 Exercises
7.8.1 Algebraic Exercises
7.8.2 Numeric Exercises 8 Specification of VECMs
8.1 Lag Order Specification
8.2 Testing for the Rank of Cointegration
8.2.1 A VECM without Deterministic Terms
8.3 Subset VECMs8.2.2 A Nonzero Mean Term 8.2.3 A Linear Trend 8.2.4 A Linear Trend in the Variables and Not in the Cointegration Relations 8.2.5 Summary of Results and Other Deterministic Terms 8.2.6 An Example 8.2.7 Prior Adjustment of Deterministic Terms 8.2.8 Choice of Deterministic Terms 8.2.9 Other Approaches to Testing for teh Cointegrating Rank 8.4 Model Diagnostics
8.4.1 Checking for Residual Autocorrelation
8.5 Exercises8.4.2 Testing for Nonnormality 8.4.3 Tests for Structural Change
8.5.1 Algebraic Exercises
8.5.2 Numerical Exercises Part 3 Structural and Conditional Models
9 Structural VARs and VECMs
9.1 Structural Vector Autoregressions
9.1.1 The AModel
9.2 Structural Vector Error Correction Models9.1.2 The BModel 9.1.3 The ABModel 9.1.4 LongRun Restrictions à la BlanchardQuah 9.3 Estimation of Structural Parameters
9.3.1 Estimating SVAR Models
9.4 Impulse Response Analysis and Forecast Error Variance9.3.2 Estimating Structural VECMs 9.5 Further Issues 9.6 Exercises
9.6.1 Algebraic Problems
9.6.2 Numerical Problems 10 Systems of Dynamic Simultaneous Equations
10.1 Background
10.2 Systems with Unmodelled Variables
10.2.1 Types of Variables
10.3 Estimation10.2.2 Structural Form, Reduced Form, Final Form 10.2.3 Models with Rational Expectations 10.2.4 Cointegrated Variables
10.3.1 Stationary Variables
10.4 Remarks on Model Specification and Model Checking10.3.2 Estimation of Models with I(1) Variables 10.5 Forecasting
10.5.1 Unconditional and Conditional Forecasts
10.6 Multiplier Analysis10.5.2 Forecasting Estimated Dynamic SEMs 10.7 Optimal Control 10.8 Concluding Remarks on Dynamic SEMs 10.9 Exercises Part 4 Infinite Order Vector Autoregressive Processes
11 Vector Autoregressive Moving Average Processes
11.1 Introduction
11.2 Finite Order Moving Average Processes 11.3 VARMA Processes
11.3.1 The Pure MA and Pure VAR Representations of a
VARMA Process
11.4 The Autocovariances and Autocorrelations of a
VARMA(p, q) Process11.3.2 A VAR(1) Representation of a VARMA Process 11.5 Forecasting VARMA Processes 11.6 Transforming and Aggregating VARMA Processes
11.6.1 Linear Transformations of VARMA Processes
11.7 Interpretation of VARMA Models11.6.2 Aggregation of VARMA Processes
11.7.1 GrangerCausality
11.8 Exercises11.7.2 Impulse Reponse Analysis 12 Estimation of VARMA Models
12.1 The Identification Problem
12.1.1 Nonuniqueness of VARMA Representations
12.2 The Gaussian Likelihood Function12.1.2 Final Equations Form and Echelong Form 12.1.3 Illustrations
12.2.1 The Likelihood Function of an MA(1) Process
12.3 Computation of the ML Estimates12.2.2 The MA(q) Case 12.2.3 The VARMA(1,1) Case 12.2.4 The General VARMA(p, q) Case
12.3.1 The Normal Equations
12.4 Asymptotic Properties of the ML Estimators12.3.2 Optimization Algorithms 12.3.3 The Information Matrix 12.3.4 Preliminary Estimation 12.3.5 An Illustration
12.4.1 Theoretical Results
12.5 Forecasting Estimated VARMA Processes12.4.2 A Real Data Example 12.6 Estimated Impulse Responses 12.7 Exercises 13 Specification and Checking the Adequacy of VARMA
13.1 Introduction
13.2 Specification of the Final Equations Form
13.2.1 A Specification Procedure
13.3 Specification of Echelon Forms13.2.2 An Example
13.3.1 A Procedure for Small Systems
13.4 Remarks on Other Specification Strategies for VARMA Models13.3.2 A Full Search Procedure Based on Linear Least Squares Computations 13.3.3 HannanKavalieris Procedure 13.3.4 Poskitt’ Procedure 13.5 Model Checking
13.2.1 LM Tests
13.6 Critique of VARMA Model Fitting13.5.2 Residual Autocorrelations and Portmanteau Tests 13.5.3 Prediction Tests for Structural Change 13.7 Exercises 14 Cointegrated VARMA Processes
14.1 Introduction
14.2 The VARMA Framework for I(1) Variables
14.2.1 Levels of VARMA Models
14.3 Estimation14.2.2 The Reverse Echelon Form 14.2.3 The Error Correction Echelong Form
14.3.1 Estimation of ARMA_{RE} Models
14.4 Specification of ECARMA_{RE} Models14.3.2 Estimation of ECARMA_{RE} Models
14.4.1 Specification of Kronecker Indices
14.5 Forecasting Cointegrated VARMA Processes14.4.2 Specification of the Cointegrating Rank 14.6 An Example 14.7 Exercises
14.7.1 Algebraic Exercises
14.7.2 Numerical Exercises 15 Fitting Finite Order VAR Models to Infinite Order Processes
15.1 Background
15.2 Multivariate Least Squares Estimation 15.3 Forecasting
15.3.1 Theoretical Results
15.4 Impulse Response Analysis and Forecast Error
Variance Decompositions15.3.2 An Example
15.4.1 Asymptotic Theory
15.5 Cointegrated Infinite Order VARs15.4.2 An Example
15.5.1 The Model Setup
15.6 Exercises15.5.2 Estimation 15.5.3 Testing for the Cointegrating Rank Part 5 Time Series Topics
16 Multivariate ARCH and GARCH Models
16.1 Background
16.2 Univariate GARCH Models
16.2.1 Definitions
16.3 Multivariate GARCH Models16.2.2 Forecasting
16.3.1 Multivariate ARCH
16.4 Estimation16.3.2 MGARCH 16.3.3 Other Multivariate ARCH and GARCH Models
16.4.1 Theory
16.5 Checking MGARCH Models16.4.2 An Example
16.5.1 ARCHLM and ARCHPortmanteau Tests
16.6 Interpreting GARCH Models16.5.2 LM and Portmanteau Tests for Remaining ARCH 16.5.3 Other Diagnostic Tests 16.5.4 An Example
16.6.1 Causality in Variance
16.7 Problems and Extensions16.6.2 Conditional Moment Profiles and Generalized Impulse Responses 16.8 Exercises 17 Periodic VAR Processes and Intervention
Models
17.1 Introduction
17.2 The VAR(p) Model with Time Varying Coefficients
17.2.1 General Properties
17.3 Periodic Processes17.2.2 ML Estimation
17.3.1 A VAR Representation with Time Invariant Coefficients
17.4 Intervention Models17.3.2 ML Estimation and Testing for Time Varying Coefficients 17.3.3 An Example 17.3.4 Bibliographical Notes and Extensions
17.4.1 Interventions in the Intercept Model
17.5 Exercises17.4.2 A Discrete Change in the Mean 17.4.3 An Illustrative Example 17.4.4 Extensions and References 18 State Space Models
18.1 Background
18.2 State Space Models
18.2.1 The Model Setup
18.3 The Kalman Filter18.2.2 More General State Space Models
18.3.1 The Kalman Filter Recursions
18.4 Maximum Likelihood Estimation of State Space Models18.3.2 Proof of the Kalman Filter Recursions
18.4.1 The LogLikelihood Function
18.5 A Real Data Example18.4.2 The Identification Problem 18.4.3 Maximization of the LogLikelihood Function 18.4.4 Asymptotic Properties of the ML Estimator 18.6 Exercises Appendix
A Vectors and Matrices
A.1 Basic Definitions
A.2 Basic Matrix Operations A.3 The Determinant A.4 The Inverse, the Adjoint, and Generalized Inverses
A.4.1 Inverse and Adjoint of a Square Matrix
A.5 The RankA.4.2 Generalized Inverses A.6 Eigenvalues and vectors — Characteristic Values and Vectors A.7 The Trace A.8 Some Special Matrices and Vectors
A.8.1 Idempotent and Nilpotent Matrices
A.9 Decomposition and Diagonalization of MatricesA.8.2 Orthogonal Matrices and Vectors and Orthogonal Complements A.8.3 Definite Matrices and Quadratic Forms
A.9.1 The Jordan Canonical Form
A.10 Partitioned MatricesA.9.2 Decomposition of Symmetric Matrices A.9.3 The Choleski Decomposition of a Positive Definite Matrix A.11 The Kronecker Product A.12 The vec and vech Operators and Related Matrices
A.12.1 The Operators
A.13 Vector and Matrix DifferentiationA.12.2 Elimination, Duplication, and Commutation Matrices A.14 Optimization of Vector Functions A.15 Problems B Multivariate Normal and Related Distributions
B.1 Multivariate Normal Distributions
B.2 Related Distributions C Stochastic Convergence and Asymptotic Distributions
C.1 Concepts of Stochastic Convergence
C.2 Order in Probability C.3 Infinite Sums of Random Variables C.4 Laws of Large Numbers and Central Limit Theorems C.5 Standard Asymptotic Properties of Estimators and Test Statistics C.6 Maximum Likelihood Estimation C.7 Likelihood Ratio, Lagrange Multiplier, and Wald Tests C.8 Unit Root Asymptotics
C.8.1 Univariate Processes
C.8.2 Multivariate Processes D Evaluating Properties of Estimators and Test Statistics by Simulation and Resampling Techniques
D.1 Simulating a Multiple Time Series with VAR Generation Process
D.2 Evaluating Distributions of Functions of Multiple Time Series by Simulation D.3 Resampling Methods References
Index of Notation
Author Index
Subject Index
