New Introduction to Multiple Time Series Analysis

by Lutkepohl, Helmut

• ISBN13: 9783540262398
• ISBN10: 3540262393
• Format: Paperback
• Copyright: 2006-06-15
• Publisher: Textstream

Summary

This reference work and graduate level textbook considers a wide range of models and methods for analyzing and forecasting multiple time series. The models covered include vector autoregressive, cointegrated, vector autoregressive moving average, multivariate ARCH and periodic processes as well as dynamic simultaneous equations and state space models. Least squares, maximum likelihood, and Bayesian methods are considered for estimating these models. Different procedures for model selection and model specification are treated and a wide range of tests and criteria for model checking are introduced. Causality analysis, impulse response analysis and innovation accounting are presented as tools for structural analysis.The book is accessible to graduate students in business and economics. In addition, multiple time series courses in other fields such as statistics and engineering may be based on it. Applied researchers involved in analyzing multiple time series may benefit from the book as it provides the background and tools for their tasks. It bridges the gap to the difficult technical literature on the topic.

Table of Contents

Introduction	1	(13)
Objectives of Analyzing Multiple Time Series	1	(1)
Some Basics	2	(2)
Vector Autoregressive Processes	4	(1)
Outline of the Following Chapters	5	(8)
Part I Finite Order Vector Autoregressive Processes
Stable Vector Autoregressive Processes	13	(56)
Basic Assumptions and Properties of VAR Processes	13	(18)
Stable VAR(p) Processes	13	(5)
The Moving Average Representation of a VAR Process	18	(6)
Stationary Processes	24	(2)
Computation of Autocovariances and Autocorrelations of Stable VAR Processes	26	(5)
Forecasting	31	(10)
The Loss Function	32	(1)
Point Forecasts	33	(6)
Interval Forecasts and Forecast Regions	39	(2)
Structural Analysis with VAR Models	41	(25)
Granger-Causality, Instantaneous Causality, and Multi-Step Causality	41	(10)
Impulse Response Analysis	51	(12)
Forecast Error Variance Decomposition	63	(3)
Remarks on the Interpretation of VAR Models	66	(1)
Exercises	66	(3)
Estimation of Vector Autoregressive Processes	69	(66)
Introduction	69	(1)
Multivariate Least Squares Estimation	69	(13)
The Estimator	70	(3)
Asymptotic Properties of the Least Squares Estimator	73	(4)
An Example	77	(3)
Small Sample Properties of the LS Estimator	80	(2)
Least Squares Estimation with Mean-Adjusted Data and Yule-Walker Estimation	82	(5)
Estimation when the Process Mean Is Known	82	(1)
Estimation of the Process Mean	83	(2)
Estimation with Unknown Process Mean	85	(1)
The Yule-Walker Estimator	85	(2)
An Example	87	(1)
Maximum Likelihood Estimation	87	(7)
The Likelihood Function	87	(2)
The ML Estimators	89	(1)
Properties of the ML Estimators	90	(4)
Forecasting with Estimated Processes	94	(8)
General Assumptions and Results	94	(2)
The Approximate MSE Matrix	96	(2)
An Example	98	(2)
A Small Sample Investigation	100	(2)
Testing for Causality	102	(7)
A Wald Test for Granger-Causality	102	(1)
An Example	103	(1)
Testing for Instantaneous Causality	104	(2)
Testing for Multi-Step Causality	106	(3)
The Asymptotic Distributions of Impulse Responses and Forecast Error Variance Decompositions	109	(21)
The Main Results	109	(7)
Proof of Proposition 3.6	116	(2)
An Example	118	(8)
Investigating the Distributions of the Impulse Responses by Simulation Techniques	126	(4)
Exercises	130	(5)
Algebraic Problems	130	(2)
Numerical Problems	132	(3)
VAR Order Selection and Checking the Model Adequacy	135	(58)
Introduction	135	(1)
A Sequence of Tests for Determining the VAR Order	136	(10)
The Impact of the Fitted VAR Order on the Forecast MSE	136	(2)
The Likelihood Ratio Test Statistic	138	(5)
A Testing Scheme for VAR Order Determination	143	(2)
An Example	145	(1)
Criteria for VAR Order Selection	146	(11)
Minimizing the Forecast MSE	146	(2)
Consistent Order Selection	148	(3)
Comparison of Order Selection Criteria	151	(2)
Some Small Sample Simulation Results	153	(4)
Checking the Whiteness of the Residuals	157	(17)
The Asymptotic Distributions of the Autocovariances and Autocorrelations of a White Noise Process	157	(4)
The Asymptotic Distributions of the Residual Autocovariances and Autocorrelations of an Estimated VAR Process	161	(8)
Portmanteau Tests	169	(2)
Lagrange Multiplier Tests	171	(3)
Testing for Nonnormality	174	(7)
Tests for Nonnormality of a Vector White Noise Process	174	(3)
Tests for Nonnormality of a VAR Process	177	(4)
Tests for Structural Change	181	(8)
Chow Tests	182	(2)
Forecast Tests for Structural Change	184	(5)
Exercises	189	(4)
Algebraic Problems	189	(2)
Numerical Problems	191	(2)
VAR Processes with Parameter Constraints	193	(44)
Introduction	193	(1)
Linear Constraints	194	(27)
The Model and the Constraints	194	(1)
LS, GLS, and EGLS Estimation	195	(5)
Maximum Likelihood Estimation	200	(1)
Constraints for Individual Equations	201	(1)
Restrictions for the White Noise Covariance Matrix	202	(2)
Forecasting	204	(1)
Impulse Response Analysis and Forecast Error Variance Decomposition	205	(1)
Specification of Subset VAR Models	206	(6)
Model Checking	212	(5)
An Example	217	(4)
VAR Processes with Nonlinear Parameter Restrictions	221	(1)
Bayesian Estimation	222	(8)
Basic Terms and Notation	222	(1)
Normal Priors for the Parameters of a Gaussian VAR Process	223	(2)
The Minnesota or Litterman Prior	225	(2)
Practical Considerations	227	(1)
An Example	227	(1)
Classical versus Bayesian Interpretation of α in Forecasting and Structural Analysis	228	(2)
Exercises	230	(7)
Algebraic Exercises	230	(1)
Numerical Problems	231	(6)
Part II Cointegrated Processes
Vector Error Correction Models	237	(32)
Integrated Processes	238	(5)
VAR Processes with Integrated Variables	243	(1)
Cointegrated Processes, Common Stochastic Trends, and Vector Error Correction Models	244	(12)
Deterministic Terms in Cointegrated Processes	256	(2)
Forecasting Integrated and Cointegrated Variables	258	(3)
Causality Analysis	261	(1)
Impulse Response Analysis	262	(3)
Exercises	265	(4)
Estimation of Vector Error Correction Models	269	(56)
Estimation of a Simple Special Case VECM	269	(17)
Estimation of General VECMs	286	(19)
LS Estimation	287	(4)
EGLS Estimation of the Cointegration Parameters	291	(3)
ML Estimation	294	(5)
Including Deterministic Terms	299	(1)
Other Estimation Methods for Cointegrated Systems	300	(2)
An Example	302	(3)
Estimating VECMs with Parameter Restrictions	305	(4)
Linear Restrictions for the Cointegration Matrix	305	(2)
Linear Restrictions for the Short-Run and Loading Parameters	307	(2)
An Example	309	(1)
Bayesian Estimation of Integrated Systems	309	(6)
The Model Setup	310	(1)
The Minnesota or Litterman Prior	310	(2)
An Example	312	(3)
Forecasting Estimated Integrated and Cointegrated Systems	315	(1)
Testing for Granger-Causality	316	(5)
The Noncausality Restrictions	316	(1)
Problems Related to Standard Wald Tests	317	(1)
A Wald Test Based on a Lag Augmented VAR	318	(2)
An Example	320	(1)
Impulse Response Analysis	321	(2)
Exercises	323	(2)
Algebraic Exercises	323	(1)
Numerical Exercises	324	(1)
Specification of VECMs	325	(32)
Lag Order Selection	325	(2)
Testing for the Rank of Cointegration	327	(16)
A VECM without Deterministic Terms	328	(2)
A Nonzero Mean Term	330	(1)
A Linear Trend	331	(1)
A Linear Trend in the Variables and Not in the Cointegration Relations	331	(1)
Summary of Results and Other Deterministic Terms	332	(3)
An Example	335	(2)
Prior Adjustment for Deterministic Terms	337	(4)
Choice of Deterministic Terms	341	(1)
Other Approaches to Testing for the Cointegrating Rank	342	(1)
Subset VECMs	343	(2)
Model Diagnostics	345	(6)
Checking for Residual Autocorrelation	345	(3)
Testing for Nonnormality	348	(1)
Tests for Structural Change	348	(3)
Exercises	351	(6)
Algebraic Exercises	351	(1)
Numerical Exercises	352	(5)
Part III Structural and Conditional Models
Structural VARs and VECMs	357	(30)
Structural Vector Autoregressions	358	(10)
The A-Model	358	(4)
The B-Model	362	(2)
The AB-Model	364	(3)
Long-Run Restrictions a la Blanchard-Quah	367	(1)
Structural Vector Error Correction Models	368	(4)
Estimation of Structural Parameters	372	(5)
Estimating SVAR Models	372	(4)
Estimating Structural VECMs	376	(1)
Impulse Response Analysis and Forecast Error Variance Decomposition	377	(6)
Further Issues	383	(1)
Exercises	384	(3)
Algebraic Problems	384	(1)
Numerical Problems	385	(2)
Systems of Dynamic Simultaneous Equations	387	(32)
Background	387	(1)
Systems with Unmodelled Variables	388	(7)
Types of Variables	388	(2)
Structural Form, Reduced Form, Final Form	390	(3)
Models with Rational Expectations	393	(1)
Cointegrated Variables	394	(1)
Estimation	395	(5)
Stationary Variables	396	(2)
Estimation of Models with I(1) Variables	398	(2)
Remarks on Model Specification and Model Checking	400	(1)
Forecasting	401	(5)
Unconditional and Conditional Forecasts	401	(4)
Forecasting Estimated Dynamic SEMs	405	(1)
Multiplier Analysis	406	(2)
Optimal Control	408	(3)
Concluding Remarks on Dynamic SEMs	411	(1)
Exercises	412	(7)
Part IV Infinite Order Vector Autoregressive Processes
Vector Autoregressive Moving Average Processes	419	(28)
Introduction	419	(1)
Finite Order Moving Average Processes	420	(3)
VARMA Processes	423	(6)
The Pure MA and Pure VAR Representations of a VARMA Process	423	(3)
A VAR(1) Representation of a VARMA Process	426	(3)
The Autocovariances and Autocorrelations of a VARMA (p, q) Process	429	(3)
Forecasting VARMA Processes	432	(2)
Transforming and Aggregating VARMA Processes	434	(8)
Linear Transformations of VARMA Processes	435	(5)
Aggregation of VARMA Processes	440	(2)
Interpretation of VARMA Models	442	(2)
Granger-Causality	442	(2)
Impulse Response Analysis	444	(1)
Exercises	444	(3)
Estimation of VARMA Models	447	(46)
The Identification Problem	447	(12)
Nonuniqueness of VARMA Representations	447	(5)
Final Equations Form and Echelon Form	452	(3)
Illustrations	455	(4)
The Gaussian Likelihood Function	459	(8)
The Likelihood Function of an MA(1) Process	459	(2)
The MA (q) Case	461	(2)
The VARMA (1,1) Case	463	(1)
The General VARMA (p, q) Case	464	(3)
Computation of the ML Estimates	467	(12)
The Normal Equations	468	(2)
Optimization Algorithms	470	(3)
The Information Matrix	473	(1)
Preliminary Estimation	474	(3)
An Illustration	477	(2)
Asymptotic Properties of the ML Estimators	479	(8)
Theoretical Results	479	(7)
A Real Data Example	486	(1)
Forecasting Estimated VARMA Processes	487	(3)
Estimated Impulse Responses	490	(1)
Exercises	491	(2)
Specification and Checking the Adequacy of VARMA Models	493	(22)
Introduction	493	(1)
Specification of the Final Equations Form	494	(4)
A Specification Procedure	494	(3)
An Example	497	(1)
Specification of Echelon Forms	498	(9)
A Procedure for Small Systems	499	(2)
A Full Search Procedure Based on Linear Least Squares Computations	501	(2)
Hannan-Kavalieris Procedure	503	(2)
Poskitt's Procedure	505	(2)
Remarks on Other Specification Strategies for VARMA Models	507	(1)
Model Checking	508	(3)
LM Tests	508	(2)
Residual Autocorrelations and Portmanteau Tests	510	(1)
Prediction Tests for Structural Change	511	(1)
Critique of VARMA Model Fitting	511	(1)
Exercises	512	(3)
Cointegrated VARMA Processes	515	(16)
Introduction	515	(1)
The VARMA Framework for I(1) Variables	516	(5)
Levels VARMA Models	516	(2)
The Reverse Echelon Form	518	(1)
The Error Correction Echelon Form	519	(2)
Estimation	521	(2)
Estimation of ARMARE Models	521	(1)
Estimation of EC-ARMARE Models	522	(1)
Specification of EC-ARMARE Models	523	(3)
Specification of Kronecker Indices	523	(2)
Specification of the Cointegrating Rank	525	(1)
Forecasting Cointegrated VARMA Processes	526	(1)
An Example	526	(2)
Exercises	528	(3)
Algebraic Exercises	528	(1)
Numerical Exercises	529	(2)
Fitting Finite Order VAR Models to Infinite Order Processes	531	(26)
Background	531	(1)
Multivariate Least Squares Estimation	532	(4)
Forecasting	536	(4)
Theoretical Results	536	(2)
An Example	538	(2)
Impulse Response Analysis and Forecast Error Variance Decompositions	540	(5)
Asymptotic Theory	540	(3)
An Example	543	(2)
Cointegrated Infinite Order VARs	545	(7)
The Model Setup	546	(3)
Estimation	549	(2)
Testing for the Cointegrating Rank	551	(1)
Exercises	552	(5)
Part V Time Series Topics
Multivariate ARCH and GARCH Models	557	(28)
Background	557	(2)
Univariate GARCH Models	559	(3)
Definitions	559	(2)
Forecasting	561	(1)
Multivariate GARCH Models	562	(7)
Multivariate ARCH	563	(1)
MGARCH	564	(3)
Other Multivariate ARCH and GARCH Models	567	(2)
Estimation	569	(7)
Theory	569	(2)
An Example	571	(5)
Checking MGARCH Models	576	(3)
ARCH-LM and ARCH-Portmanteau Tests	576	(1)
LM and Portmanteau Tests for Remaining ARCH	577	(1)
Other Diagnostic Tests	578	(1)
An Example	578	(1)
Interpreting GARCH Models	579	(3)
Causality in Variance	579	(1)
Conditional Moment Profiles and Generalized Impulse Responses	580	(2)
Problems and Extensions	582	(2)
Exercises	584	(1)
Periodic VAR Processes and Intervention Models	585	(26)
Introduction	585	(2)
The VAR(p) Model with Time Varying Coefficients	587	(4)
General Properties	587	(2)
ML Estimation	589	(2)
Periodic Processes	591	(13)
A VAR Representation with Time Invariant Coefficients	592	(3)
ML Estimation and Testing for Time Varying Coefficients	595	(7)
An Example	602	(2)
Bibliographical Notes and Extensions	604	(1)
Intervention Models	604	(5)
Interventions in the Intercept Model	605	(1)
A Discrete Change in the Mean	606	(2)
An Illustrative Example	608	(1)
Extensions and References	609	(1)
Exercises	609	(2)
State Space Models	611	(34)
Background	611	(2)
State Space Models	613	(12)
The Model Setup	613	(11)
More General State Space Models	624	(1)
The Kalman Filter	625	(6)
The Kalman Filter Recursions	626	(4)
Proof of the Kalman Filter Recursions	630	(1)
Maximum Likelihood Estimation of State Space Models	631	(6)
The Log-Likelihood Function	632	(1)
The Identification Problem	633	(1)
Maximization of the Log-Likelihood Function	634	(2)
Asymptotic Properties of the ML Estimator	636	(1)
A Real Data Example	637	(4)
Exercises	641	(4)
Appendix
A. Vectors and Matrices	645	(32)
A.1 Basic Definitions	645	(1)
A.2 Basic Matrix Operations	646	(1)
A.3 The Determinant	647	(2)
A.4 The Inverse, the Adjoint, and Generalized Inverses	649	(2)
A.4.1 Inverse and Adjoint of a Square Matrix	649	(1)
A.4.2 Generalized Inverses	650	(1)
A.5 The Rank	651	(1)
A.6 Eigenvalues and -vectors - Characteristic Values and Vectors	652	(1)
A.7 The Trace	653	(1)
A.8 Some Special Matrices and Vectors	653	(3)
A.5.1 Idempotent and Nilpotent Matrices	653	(1)
A.5.2 Orthogonal Matrices and Vectors and Orthogonal Complements	654	(1)
A.5.3 Definite Matrices and Quadratic Forms	655	(1)
A.9 Decomposition and Diagonalization of Matrices	656	(3)
A.9.1 The Jordan Canonical Form	656	(2)
A.9.2 Decomposition of Symmetric Matrices	658	(1)
A.9.3 The Choleski Decomposition of a Positive Definite Matrix	658	(1)
A.10 Partitioned Matrices	659	(1)
A.11 The Kronecker Product	660	(1)
A.12 The vec and vech Operators and Related Matrices	661	(3)
A.12.1 The Operators	661	(1)
A.12.2 Elimination, Duplication, and Commutation Matrices	662	(2)
A.13 Vector and Matrix Differentiation	664	(7)
A.14 Optimization of Vector Functions	671	(4)
A.15 Problems	675	(2)
B. Multivariate Normal and Related Distributions	677	(4)
B.1 Multivariate Normal Distributions	677	(1)
B.2 Related Distributions	678	(3)
C. Stochastic Convergence and Asymptotic Distributions	681	(26)
C.1 Concepts of Stochastic Convergence	681	(3)
C.2 Order in Probability	684	(1)
C.3 Infinite Sums of Random Variables	685	(4)
C.4 Laws of Large Numbers and Central Limit Theorems	689	(3)
C.5 Standard Asymptotic Properties of Estimators and Test Statistics	692	(1)
C.6 Maximum Likelihood Estimation	693	(1)
C.7 Likelihood Ratio, Lagrange Multiplier, and Wald Tests	694	(4)
C.8 Unit Root Asymptotics	698	(9)
C.8.1 Univariate Processes	698	(5)
C.8.2 Multivariate Processes	703	(4)
D. Evaluating Properties of Estimators and Test Statistics by Simulation and Resampling Techniques	707	(6)
D.1 Simulating a Multiple Time Series with VAR Generation Process	707	(1)
D.2 Evaluating Distributions of Functions of Multiple Time Series by Simulation	708	(1)
D.3 Resampling Methods	709	(4)
References	713	(20)
Index of Notation	733	(8)
Author Index	741	(6)
Subject Index	747