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# multiple time series models

(Parte **1** de 7)

Quantitative Applications in the Social Sciences

1.Analysis of Variance,2nd Edition Iversen/

Norpoth 2.Operations Research Methods Nagel/Neef 3.Causal Modeling,2nd Edition Asher 4.Tests of Significance Henkel 5.Cohort Analysis,2nd Edition Glenn 6.Canonical Analysis and Factor

Comparison Levine 7.Analysis of Nominal Data,2nd Edition

Reynolds 8.Analysis of Ordinal Data

Hildebrand/Laing/Rosenthal 9.Time Series Analysis,2nd Edition Ostrom 10.Ecological Inference Langbein/Lichtman 1.Multidimensional Scaling Kruskal/Wish 12.Analysis of Covariance Wildt/Ahtola 13.Introduction to Factor Analysis

Kim/Mueller 14.Factor Analysis Kim/Mueller 15.Multiple Indicators Sullivan/Feldman 16.Exploratory Data Analysis Hartwig/Dearing 17.Reliability and Validity Assessment

Carmines/Zeller 18.Analyzing Panel Data Markus 19.Discriminant Analysis Klecka 20.Log-Linear Models Knoke/Burke 21.Interrupted Time Series Analysis

McDowall/McCleary/Meidinger/Hay 2.Applied Regression Lewis-Beck 23.Research Designs Spector 24.Unidimensional Scaling McIver/Carmines 25.Magnitude Scaling Lodge 26. Multiattribute Evaluation

Edwards/Newman 27.Dynamic Modeling

Huckfeldt/Kohfeld/Likens 28.Network Analysis Knoke/Kuklinski 29.Interpreting and Using Regression Achen 30.Test Item Bias Osterlind 31.Mobility Tables Hout 32.Measures of Association Liebetrau 3.Confirmatory Factor Analysis Long 34.Covariance Structure Models Long 35.Introduction to Survey Sampling Kalton 36.Achievement Testing Bejar 37.Nonrecursive Causal Models Berry 38.Matrix Algebra Namboodiri 39.Introduction to Applied Demography

Rives/Serow 40.Microcomputer Methods for Social

Scientists,2nd Edition Schrodt 41.Game Theory Zagare 42.Using Published Data Jacob 43.Bayesian Statistical Inference Iversen 4.Cluster Analysis Aldenderfer/Blashfield 45.Linear Probability,Logit,and Probit Models Aldrich/Nelson

46.Event History Analysis Allison 47.Canonical Correlation Analysis Thompson 48.Models for Innovation Diffusion Mahajan/

Peterson 49.Basic Content Analysis,2nd Edition

Weber 50.Multiple Regression in Practice Berry/

Feldman 51.Stochastic Parameter Regression Models

Newbold/Bos 52.Using Microcomputers in Research

Madron/Tate/Brookshire 53.Secondary Analysis of Survey Data

Kiecolt/Nathan 54.Multivariate Analysis of Variance

Bray/Maxwell 5.The Logic of Causal Order Davis 56.Introduction to Linear Goal Programming

Ignizio 57.Understanding Regression Analysis

Schroeder/Sjoquist/Stephan 58.Randomized Response Fox/Tracy 59.Meta-Analysis Wolf 60.Linear Programming Feiring 61.Multiple Comparisons Klockars/Sax 62.Information Theory Krippendorff 63.Survey Questions Converse/Presser 64.Latent Class Analysis McCutcheon 65.Three-Way Scaling and Clustering

Arabie/Carroll/DeSarbo 6.Q Methodology McKeown/Thomas 67.Analyzing Decision Making Louviere 68.Rasch Models for Measurement Andrich 69.Principal Components Analysis Dunteman 70.Pooled Time Series Analysis Sayrs 71.Analyzing Complex Survey Data, 2nd Edition Lee/Forthofer 72.Interaction Effects in Multiple Regression, 2nd Edition Jaccard/Turrisi 73.Understanding Significance Testing Mohr 74.Experimental Design and Analysis Brown/

Melamed 75.Metric Scaling Weller/Romney 76.Longitudinal Research,2nd Edition

Menard 7.Expert Systems Benfer/Brent/Furbee 78.Data Theory and Dimensional Analysis

Jacoby 79.Regression Diagnostics Fox 80.Computer-Assisted Interviewing Saris 81.Contextual Analysis Iversen 82.Summated Rating Scale Construction

Spector 83.Central Tendency and Variability Weisberg 84.ANOVA:Repeated Measures Girden 85.Processing Data Bourque/Clark 86.Logit Modeling DeMaris

87.Analytic Mapping and Geographic

Databases Garson/Biggs 8.Working With Archival Data

Elder/Pavalko/Clipp 89.Multiple Comparison Procedures

Toothaker 90.Nonparametric Statistics Gibbons 91.Nonparametric Measures of Association

Gibbons 92.Understanding Regression Assumptions

Berry 93.Regression With Dummy Variables Hardy 94.Loglinear Models With Latent Variables

Hagenaars 95.Bootstrapping Mooney/Duval 96.Maximum Likelihood Estimation Eliason 97.Ordinal Log-Linear Models Ishii-Kuntz 98.Random Factors in ANOVA Jackson/

Brashers 9.Univariate Tests for Time Series Models

Cromwell/Labys/Terraza 100.Multivariate Tests for Time Series Models

Cromwell/Hannan/Labys/Terraza 101. Interpreting Probability Models: Logit,

Probit,and Other Generalized Linear Models Liao 102.Typologies and Taxonomies Bailey 103.Data Analysis:An Introduction

Lewis-Beck 104.Multiple Attribute Decision Making

Yoon/Hwang 105.Causal Analysis With Panel Data Finkel 106.Applied Logistic Regression Analysis, 2nd Edition Menard 107.Chaos and Catastrophe Theories Brown 108.Basic Math for Social Scientists:

Concepts Hagle 109.Basic Math for Social Scientists:

Problems and Solutions Hagle 110.Calculus Iversen 1. Regression Models: Censored, Sample

Selected,or Truncated Data Breen 112.Tree Models of Similarity and Association

James E.Corter 113.Computational Modeling Taber/Timpone 114.LISREL Approaches to Interaction Effects in Multiple Regression Jaccard/Wan 115.Analyzing Repeated Surveys Firebaugh 116.Monte Carlo Simulation Mooney 117.Statistical Graphics for Univariate and

Bivariate Data Jacoby 118.Interaction Effects in Factorial Analysis of Variance Jaccard

119.Odds Ratios in the Analysis of

Contingency Tables Rudas 120.Statistical Graphics for Visualizing

Multivariate Data Jacoby 121.Applied Correspondence Analysis

Clausen 122.Game Theory Topics Fink/Gates/Humes 123.Social Choice:Theory and Research

Johnson 124.Neural Networks Abdi/Valentin/Edelman 125.Relating Statistics and Experimental

Design:An Introduction Levin 126.Latent Class Scaling Analysis Dayton 127.Sorting Data:Collection and Analysis

Coxon 128.Analyzing Documentary Accounts

Hodson 129.Effect Size for ANOVA Designs

Cortina/Nouri 130.Nonparametric Simple Regression:

Smoothing Scatterplots Fox 131.Multiple and Generalized Nonparametric

Regression Fox 132.Logistic Regression:A Primer Pampel 133.Translating Questionnaires and Other

Research Instruments:Problems and Solutions Behling/Law 134.Generalized Linear Models:A United

Approach Gill 135.Interaction Effects in Logistic Regression

Jaccard 136.Missing Data Allison 137.Spline Regression Models Marsh/Cormier 138.Logit and Probit:Ordered and

Multinomial Models Borooah 139. Correlation: Parametric and

Nonparametric Measures Chen/Popovich 140.Confidence Intervals Smithson 141.Internet Data Collection Best/Krueger 142.Probability Theory Rudas 143.Multilevel Modeling Luke 144.Polytomous Item Response Theory

Models Ostini/Nering 145.An Introduction to Generalized Linear

Models Dunteman/Ho 146.Logistic Regression Models for Ordinal

Response Variables O’Connell 147.Fuzzy Set Theory:Applications in the

Social Sciences Smithson/Verkuilen 148.Multiple Time Series Models Brandt/Williams

Quantitative Applications in the Social Sciences A SAGE PUBLICATIONS SERIES

Series/Number 07–148

Patrick T. Brandt University of Texas at Dallas

John T. Williams University of California, Riverside

Copyright 2007 by Sage Publications, Inc.

All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.

For information:

Sage Publications, Inc. 2455 Teller Road Thousand Oaks, California 91320 E-mail: order@sagepub.com

Sage Publications Ltd. 1 Oliver’s Yard 5 City Road LondonEC1Y1SP UnitedKingdom

Sage Publications India Pvt. Ltd. B-42, Panchsheel Enclave PostBox 4109 New Delhi 110017 India

Printed in the United States of America Library of Congress Cataloging-in-Publication Data

Brandt, Patrick T. Multiple time series models / Patrick T. Brandt, John T. Williams. p. cm. — (Quantitative applications in the social sciences, vol. 148)

Includes bibliographical references and index. ISBN 1-4129-0656-3; 978-1-4129-0656-2 (pbk.) 1. Times series analysis—Mathematical models. I. Williams, John T. I. Title. II. Series:

Sage university papers series. Quantitative applications in the social sciences. HA30.3.B732007 519.5′5—dc22 2006010016

Acquisitions Editor: Lisa Cuevas Shaw Associate Editor: Margo Beth Crouppen Editorial Assistant: Karen Greene Production Editor: Melanie Birdsall Copy Editor: QuADS Prepress (P) Ltd. Typesetter: C&M Digitals (P) Ltd Indexer: Ellen Slavitz Cover Designer: Janet Foulger

List of Figures vii List of Tables viii Series Editor’s Introduction ix Preface xi

1. Introduction to Multiple Time Series Models 1

1.1 Simultaneous Equation Approach 4 1.2 ARIMAApproach 6 1.3 Error Correction or LSE Approach 7 1.4 Vector Autoregression Approach 9 1.5 Comparison and Summary 12

2. Basic Vector Autoregression Models 14

2.1 Dynamic Structural Equation Models 15 2.2 Reduced Form Vector Autoregressions 18 2.3 Relationship of a Dynamic Structural Equation

Model to a Vector Autoregression Model 20 2.4 Working With This Model 2 2.5 Specification and Analysis of VAR Models 23

2.5.1 Estimation of VAR 24 2.5.2 Lag Length Specification 24 2.5.3 Testing Serial Correlation in the Residuals 28 2.5.4 Granger Causality 32 2.5.5 Interpreting Granger Causality 34 2.5.6 Testing Other Restrictions in a VAR Model 36 2.5.7 Impulse Response and Moving

Average Response Analysis 36 2.5.8 Error Bands for Impulse Responses 41 2.5.9 Innovation Accounting or Decomposition of Forecast Error Variance 45

2.6 Other Specification Issues 48

2.6.1 Should Differencing Be Used for Trending Data? 49 2.6.2 Data Transformations and Whitening 49

2.7 Unit Roots and Error Correction in VARs 50

2.7.1 Error Correction Representation of Unit Root Data 50 2.7.2 Error Correction as a VAR Model 52 2.7.3 VARVersusVECM(ECM) 54

2.8 Criticisms of VAR 56

3. Examples of VAR Analyses 59 3.1 Public Mood and Macropartisanship 59

3.1.1 Testing for Unit Roots 61 3.1.2 Specifying the Lag Length 62 3.1.3 Estimation of the VAR 63 3.1.4 Granger Causality Testing 65 3.1.5 Decomposition of the Forecast Error Variance 6 3.1.6 Impulse Response Analysis 68

3.2 Effective Corporate Tax Rates 71

3.2.1 Data 72 3.2.2 Testing for Unit Roots 73 3.2.3 Specifying the Lag Length 73 3.2.4 Granger Causality Testing 74 3.2.5 Impulse Response Analysis 7 3.2.6 Decomposition of the Forecast Error Variance 79 3.2.7 A Further Robustness Check 81

3.3 Conclusion 82

Appendix: Software for Multiple Time Series Models 85 Notes 89 References 92 Index 96 About the Authors 9

3.1 Quarterly Macropartisanship and Public Mood, 1958:4–1996:4 60

3.2 Impulse Response Analysis for Macropartisanship and Public Mood 68

3.3 Impulse Response Analysis for Macropartisanship and Public Mood 70

3.4 ECTR Data 72

3.5 Moving Average Responses for Four-Variable VAR With 90% Error Bands, 1977–1994 79

3.6 Moving Average Responses for Four-Variable VAR With 90% Error Bands, 1953–1994 83 vii

1.1 Comparison of Time Series Modeling Approaches 13 3.1 Augmented Dickey-Fuller Test Results 62

3.2 AIC and BIC Values for Macropartisanship andPublicMoodVAR 63

3.3 Likelihood Ratio Tests for Lag Length 64

3.4 VAR Estimates for the 1-Lag Model of Public Mood and Macropartisanship 64

3.5 Granger Causality Tests for Public Mood and Macropartisanship Based on the VAR(1) Model 65

3.6 Decomposition of the Forecast Error Variance for the VAR(2) Model of Public Mood and Macropartisanship 67

3.7 Unit Root Tests for ECTR Example Variables 73

3.8 AIC and BIC Lag Length Diagnostics for theECTRVARModel 74

3.9 Likelihood Ratio Tests for Lag Length 75

3.10 Exogeneity Tests for ECTR and Corporate Political Action Committees, 1977–1994 76

3.1 Exogeneity Tests for ECTR, Real Investment, and Real Income: 1953–1994, 1960–1994, 197–1994 7

3.12 Decomposition of Error Variance for VAR 80 viii

Social and economic scientists have long been fascinated by and taken advantage of time series data. A first systematic exploitation of the richness of such data is William Playfair’s The Commercial and Political Atlas, published 220 years ago and containing 43 time series graphs. By plotting the national debt of England against time, for example, Playfair could easily identify the impact of major historical events such as the accession of Queen Anne in 1701, the Spanish War of the 1730s, and the American (Revolutionary) War that began in 1775.

Playfair also used graphs with more than one time series. The graph below charts the curve of imports and that of the exports against time, clearly demonstrating that there is a relationship between the two as well as between them and time, not to mention the main purpose of Playfair’s defining import-export balance against and in favor of England. Charts like this show that the two time series may not be independent processes.

The usefulness of time series graphs notwithstanding, Playfair’s presentation leaves many questions unanswered. What caused imports or exports to rise and fall? Did the amount of imports have an impact on the amount of stropmIfoeniL stropxEfoeniL stropmI stropxE

BALANCE in FAVOR of ENGLAND

Exports and Imports To and From Denmark and Norway From 1700 to 1780

The bottom line is divided into years, the right-hand line into £10,0 each. BALANCE AGAINST

SOURCE: w.unc.edu/~nielsen/soci208/m2/m2033.jpg ix exports and vice versa? Perhaps most important and interesting, what changed the regime of import-export balance from against to in favor of England? Answering these questions requires a proper analysis of the dynamic simultaneous processes, and Brandt and Williams’s Multiple Time Series Models, seen by my predecessor Michael Lewis-Beck as a worthwhile project, presents methods for such analyses.

The authors discuss the assumptions and specificities of four main approaches for time series data: autoregressive integrated moving average models, simultaneous equation systems, error correction models, and vector autoregression (VAR) models. They then focus on the details such as the specification and estimation of inference in VAR as well as tests for Granger causality and assessment of dynamic causal relationships via impulse response functions and measures of uncertainty before offering two complete examples of the VAR model of multiple time series data. A welcome addition to the series, this book complements the existing volumes of Time Series Analysis (No. 9), Univariate Tests for Time Series Models (No. 9), and Multivariate Tests for Time Series Models (No. 100).

—TimFutingLiao Series Editor

AboutThisBook

This project is several years in the making. John first proposed writing this book to me in 1999 when he was teaching time series analysis and I was his teaching assistant at the Inter-University Consortium for Political and Social Research (ICPSR) Summer Program at the University of Michigan. The project sat idle until John revived the idea in 2002—although we had discussed it in the interim.

Regrettably, John passed away in September 2004 while we were working on this book. Prior to this tragic event, we had actually completed a fair amount of the outline and plan for the book. John’s influence pervades this book—from the way the ideas are presented to the general outlook of how to ‘‘do’’ multiple time series analysis. I have tried to stay as close as possible to the original conception of the project that John and I devised.

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