Unit Roots, Cointegration, and Structural Change Time series analysis has undergone many changes in recent years with the advent of unit roots and cointegration. Maddala and Kim present a comprehensive review of these important developments and examine structural change. The volume provides an analysis of unit root tests, problems with unit root testing, estimation of cointegration systems, cointegration tests, and econometric estimation with integrated regressors. The authors also present the Bayesian approach to these problems and bootstrap methods for small-sample inference. The chapters on structural change discuss the problems of unit root tests and cointegration under structural change, outliers and robust methods, the Markov switching model, and Harvey's structural time series model. Unit Roots, Cointegration, and Structural Change is a major contribution to Themes in Modern Econometrics, of interest both to specialists and graduate and upper-undergraduate students. G. S. MADDALA is University Eminent Scholar at the Ohio State University and one of the most distinguished econometricians writing today. His many acclaimed publications include Limited Dependent and Qualitative Variables in Econometrics (Cambridge, 1983) and Econometrics (McGraw-Hill, 1977) and Introduction to Econometrics (MacMillan, 1988, 1992). IN-MOO KIM is Professor of Economics at Sung Kyun Kwan University, Seoul, Korea. UNIT ROOTS COINTEGRATION AND STRUCTURAL CHANGE G. S. Maddala The Ohio State University In-Moo Kim Sung Kyun Kwan University CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521582575 © Cambridge University Press 1998 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1998 Sixth printing 2004 A catalogue record for this publication is available from the British Library ISBN 978-0-521-58257-5 hardback ISBN 978-0-521-58782-2 paperback Transferred to digital printing 2007 To my parents G. S. Maddala To Jong Han, Jung Youn, and So Youn In-Moo Kim Contents Figures page xii Tables xiii Preface xvii Part I Introduction and basic concepts 1 1 Introduction 3 References 6 2 Basic concepts 8 2.1 Stochastic processes 8 2.2 Some commonly used stationary models 11 2.3 Box-Jenkins methods 17 2.4 Integrated variables and cointegration 20 2.5 Spurious regression 28 2.6 Deterministic trend and stochastic trend 29 2.7 Detrending methods 32 2.8 VAR, ECM, and ADL 34 2.9 Unit root tests 37 2.10 Cointegration tests and ECM 39 2.11 Summary 41 References 42 Part II Unit roots and cointegration 45 3 Unit roots 47 3.1 Introduction 47 3.2 Unit roots and Wiener processes 49 3.3 Unit root tests without a deterministic trend 60 3.4 DF test with a linear deterministic trend 65 vn viii Contents 3.5 Specification of deterministic trends 72 3.6 Unit root tests for a wide class of errors 74 3.7 Sargan-Bhargava and Bhargava tests 82 3.8 Variance ratio tests 86 3.9 Tests for TSP versus DSP 87 3.10 Forecasting from TS versus DS models 89 3.11 Summary and conclusions 92 References 92 4 Issues in unit root testing 98 4.1 Introduction 98 4.2 Size distortion and low power of unit root tests 100 4.3 Solutions to the problems of size and power 103 4.4 Problem of overdifferencing: MA roots 116 4.5 Tests with stationarity as null 120 4.6 Confirmatory analysis 126 4.7 Frequency of observations and power of unit root tests 129 4.8 Other types of nonstationarity 131 4.9 Panel data unit root tests 133 4.10 Uncertain unit roots and the pre-testing problem 139 4.11 Other unit root tests 140 4.12 Median-unbiased estimation 141 4.13 Summary and conclusions 145 References 146 5 Estimation of cointegrated systems 155 5.1 Introduction 155 5.2 A general CI system 155 5.3 A two-variable model: Engle-Granger methods 156 5.4 A triangular system 160 5.5 System estimation methods 165 5.6 The identification problem 173 5.7 Finite sample evidence 175 5.8 Forecasting in cointegrated systems 184 5.9 Miscellaneous other problems 187 5.10 Summary and conclusions 191 References 191 6 Tests for cointegration 198 6.1 Introduction 198 6.2 Single equation methods: residual-based tests 198 Contents ix 6.3 Single equation methods: ECM tests 203 6.4 Tests with cointegration as null 205 6.5 Multiple equation methods 211 6.6 Cointegration tests based on LCCA 222 6.7 Other tests for cointegration 226 6.8 Miscellaneous other problems 228 6.9 Of what use are cointegration tests? 233 6.10 Conclusions 241 References 242 7 Econometric modeling with integrated regressors 249 7.1 1(1) regressors not cointegrated 249 7.2 1(1) regressors cointegrated 250 7.3 Unbalanced equations 251 7.4 Lagged dependent variables: the ARDL model 252 7.5 Uncertain unit roots 254 7.6 Uncertain unit roots and cointegration 256 7.7 Summary and conclusions 258 References 258 Part III Extensions of the basic model 261 8 The Bayesian analysis of stochastic trends 263 8.1 Introduction to Bayesian inference 264 8.2 The posterior distribution of an autoregressive parameter 266 8.3 Bayesian inference on the Nelson-Plosser data 268 8.4 The debate on the appropriate prior 271 8.5 Classical tests versus Bayesian tests 277 8.6 Priors and time units of measurement 277 8.7 On testing point null hypotheses 278 8.8 Further comments on prior distributions 284 8.9 Bayesian inference on cointegrated systems 287 8.10 Bayesian long-run prediction 290 8.11 Conclusion 291 References 292 9 Fractional unit roots and fractional cointegration 296 9.1 Some definitions 296 9.2 Unit root tests against fractional alternatives 298 9.3 Estimation of ARFIMA models 300 9.4 Estimation of fractionally cointegrated models 302 Contents 9.5 Empirical relevance of fractional unit roots 303 9.6 Summary and conclusions 305 References 306 10 Small sample inference: bootstrap methods 309 10.1 Introduction 309 10.2 A review of the bootstrap approach 309 10.3 The AR(1) model 322 10.4 Bootstrapping unit root tests 325 10.5 The moving block bootstrap and extensions 328 10.6 Issues in bootstrapping cointegrating regressions 332 10.7 Miscellaneous other applications 335 10.8 Conclusions 336 References 336 11 Cointegrated systems with 1(2) variables 342 11.1 Determination of the order of differencing 342 11.2 Cointegration analysis with 1(2) and 1(1) variables 348 11.3 Empirical applications 355 11.4 Summary and conclusions 358 References 359 12 Seasonal unit roots and seasonal cointegration 362 12.1 Effect of seasonal adjustment 364 12.2 Seasonal integration 365 12.3 Tests for seasonal unit roots 366 12.4 The unobserved component model 371 12.5 Seasonal cointegration 375 12.6 Estimation of seasonally cointegrated systems 376 12.7 Empirical evidence 378 12.8 Periodic autoregression and periodic integration 379 12.9 Periodic cointegration and seasonal cointegration 381 12.10 Time aggregation and systematic sampling 381 12.11 Conclusion 382 References 383 Part IV Structural change 387 13 Structural change, unit roots, and cointegration 389 13.1 Tests for structural change 390 13.2 Tests with known break points 390 13.3 Tests with unknown break points 391 Contents xi 13.4 A summary assessment 398 13.5 Tests for unit roots under structural change 399 13.6 The Bayesian approach 402 13.7 A summary assessment of the empirical work 407 13.8 Effect of structural change on cointegration tests 410 13.9 Tests for structural change in cointegrated relationships 411 13.10 Miscellaneous other issues 414 13.11 Practical conclusions 416 References 418 14 Outliers and unit roots 425 14.1 Introduction 425 14.2 Different types of outliers in time series models 425 14.3 Effects of outliers on unit root tests 428 14.4 Outlier detection 437 14.5 Robust unit root tests 440 14.6 Robust estimation of cointegrating regressions 445 14.7 Outliers and seasonal unit roots 448 14.8 Conclusions 448 References 449 15 Regime switching models and structural time series models 454 15.1 The switching regression model 454 15.2 The Markov switching regression model 455 15.3 The Hamilton model 457 15.4 On the usefulness of the MSR model 460 15.5 Extensions of the MSR model 463 15.6 Gradual regime switching models 466 15.7 A model with parameters following a random walk 469 15.8 A general state-space model 470 15.9 Derivation of the Kalman filter 472 15.10 Harvey's structural time series model (1989) 475 15.11 Further comments on structural time series models 477 15.12 Summary and conclusions 479 References 479 16 Future directions 486 References 488 Appendix 1 A brief guide to asymptotic theory 490 Author index 492 Subject index 500 Figures 2.1 Correlogram of an AR(2) model 16 2.2 Examples of two AR(1) processes with a drift 23 2.3 The variances of xt and yt 24 2.4 The autocorrelations of Xt and yt 25 2.5 Cointegrated and independent 1(1) variables 27 2.6 ARIMA(O,1,1) and its components 31 3.1 Random walk and step function 53 8.1 Marginal posterior distributions of p when p = 1 272 xn Tables 2.1 Regression of integrated variables 32 3.1 Critical values for Dickey-Fuller tests 64 3.2 Asymptotic distributions of the t-ratios for different DGPs 71 3.3 Critical values for the Schmidt-Phillips LM test 85 3.4 Nelson and Plosser's results 89