Time series and panel data econometrics

✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ TIME SERIES AND PANEL DATA ECONOMETRICS ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ Time Series and Panel Data Econometrics M HASHEM PESARAN ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © M Hashem Pesaran 2015 The moral rights of the author have been asserted First Edition published in 2015 Impression: All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2015936093 ISBN 978–0–19–873691–2 (HB) 978–0–19–875998–0 (PB) Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ To my wife and in memory of my parents ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ Preface T his book is concerned with recent developments in time series and panel data techniques for the analysis of macroeconomic and financial data It provides a rigorous, nevertheless user-friendly, account of the time series techniques dealing with univariate and multivariate time series models, as well as panel data models An overview of econometrics as a subject is provided in Pesaran (1987a) and updated in Geweke, Horowitz, and Pesaran (2008) It is distinct from other time series texts in the sense that it also covers panel data models and attempts at a more coherent integration of time series, multivariate analysis, and panel data models It builds on the author’s extensive research in the areas of time series and panel data analysis and covers a wide variety of topics in one volume Different parts of the book can be used as teaching material for a variety of courses in econometrics It can also be used as a reference manual It begins with an overview of basic econometric and statistical techniques and provides an account of stochastic processes, univariate and multivariate time series, tests for unit roots, cointegration, impulse response analysis, autoregressive conditional heteroskedasticity models, simultaneous equation models, vector autoregressions, causality, forecasting, multivariate volatility models, panel data models, aggregation and global vector autoregressive models (GVAR) The techniques are illustrated using Microfit (Pesaran and Pesaran (2009)) with applications to real output, inflation, interest rates, exchange rates, and stock prices The book assumes that the reader has done an introductory econometrics course It begins with an overview of the basic regression model, which is intended to be accessible to advanced undergraduates, and then deals with more advanced topics which are more demanding and suited to graduate students and other interested scholars The book is organized into six parts: Part I: Chapters to present the classical linear regression model, describe estimation and statistical inference, and discuss the violation of the assumptions underlying the classical linear regression model This part also includes an introduction to dynamic economic modelling, and ends with a chapter on predictability of asset returns Part II: Chapters to 11 deal with asymptotic theory and present the maximum likelihood and generalized method of moments estimation frameworks Part III: Chapters 12 and 13 provide an introduction to stochastic processes and spectral density analysis Part IV: Chapters 14 to 18 focus on univariate time series models and cover stationary ARMA models, unit root processes, trend and cycle decomposition, forecasting and univariate volatility models Part V: Chapters 19 to 25 consider a variety of reduced form and structural multivariate models, rational expectations models, as well as VARs, vector error corrections, cointegrating VARs, VARX models, impulse response analysis, and multivariate volatility models ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ viii Preface Part VI: Chapters 26 to 33 considers panel data models both when the time dimension (T) of the panels is short, as well as when panels with N (the cross-section dimension) and T are large These chapters cover a wide range of panel data models, starting with static panels with homogenous slopes and graduating to dynamic panels with slope heterogeneity, error crosssection dependence, unit roots, and cointegration There are also chapters dealing with the aggregation of large dynamic panels and the theory and practice of GVAR modelling This part of the book focuses more on large N and T panels which are less covered in other texts, and draws heavily on my research in this area over the past 20 years starting with Pesaran and Smith (1995) Appendices A and B present background material on matrix algebra, probability and distribution theory, and Appendix C provides an overview of Bayesian analysis This book has evolved over many years of teaching and research and brings together in one place a diverse set of research areas that have interested me It is hoped that it will also be of interest to others I have used some of the chapters in my teaching of postgraduate students at Cambridge University, University of Southern California, UCLA, and University of Pennsylvania Undergraduate students at Cambridge University have also been exposed to some of the introductory material in Part I of the book It is impossible to name all those who have helped me with the preparation of this volume But I would like particularly to name two of my Cambridge Ph.D students, Alexander Chudik and Elisa Tosetti, for their extensive help, particularly with the material in Part VI of the book The book draws heavily from my published and unpublished research In particular: Chapter is based on Pesaran (2010) Chapter 25 draws from Pesaran and Pesaran (2010) Chapter 32 is based on Pesaran (2003) and Pesaran and Chudik (2014) where additional technical details and proofs are provided Chapter 31 is based on Breitung and Pesaran (2008) and provides some updates and extensions Chapter 33 is based on Chudik and Pesaran (2015b) I would also like to acknowledge all my coauthors whose work has been reviewed in this volume In particular, I would like to acknowledge Ron Smith, Bahram Pesaran, Allan Timmermann, Kevin Lee, Yongcheol Shin, Vanessa Smith, Cheng Hsiao, Michael Binder, Richard Smith, Alexander Chudik, Takashi Yamagata, Tony Garratt, Til Schermann, Filippo di Mauro, Stéphane Dées, Alessandro Rebucci, Adrian Pagan, Aman Ullah, and Martin Weale It goes without saying that none of them is responsible for the material presented in this volume Finally, I would like to acknowledge the helpful and constructive comments and suggestions from two anonymous referees which provided me with further impetus to extend the coverage of the material included in the book and to improve its exposition over the past six months Ron Smith has also provided me with detailed comments and suggestions over a number of successive drafts I am indebted to him for helping me to see the wood from the trees over the many years that we have collaborated with each other Hashem Pesaran Cambridge and Los Angeles January 2015 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ Contents List of Figures List of Tables Part I Introduction to Econometrics Relationship Between Two Variables 1.1 1.2 1.3 1.4 Introduction The curve fitting approach The method of ordinary least squares Correlation coefficients between Y and X 1.4.1 Pearson correlation coefficient 1.4.2 Rank correlation coefficients 1.4.3 Relationships between Pearson, Spearman, and Kendall correlation coefficients Decomposition of the variance of Y Linear statistical models Method of moments applied to bivariate regressions The likelihood approach for the bivariate regression model 1.9 Properties of the OLS estimators 1.5 1.6 1.7 1.8 1.9.1 Estimation of σ 1.10 The prediction problem 1.10.1 Prediction errors and their variance 1.10.2 Ex ante predictions 1.11 Exercises Multiple Regression 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Introduction The classical normal linear regression model The method of ordinary least squares in multiple regression The maximum likelihood approach Properties of OLS residuals Covariance matrix of βˆ The Gauss–Markov theorem Mean square error of an estimator and the bias-variance trade-off Distribution of the OLS estimator The multiple correlation coefficient xxvii xxix 3 6 8 10 12 13 14 18 19 20 21 22 24 24 24 27 28 30 31 34 36 37 39 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1050 Subject Index generalized method of moments (GMM) (cont.) asymptotic normality 230–1 consistency 230 generalized instrumental variable estimator 235–41 generalized R2 for IV regressions 239 Sargan’s general misspecification test 239–40 Sargan’s test of residual serial correlation for IV regressions 240–1 two-stage least squares 238–9 and instrumental variables see instrumental variables and GMM misspecification test 234–5 optimal weighting matrix 232 panel cointegration 852 population moment conditions 226–8, 235 RE models, estimation 500–1 short T dynamic panel data models 689 two-step and iterated estimators 233–4, 689 utilization of 225 German DAX index 142 Germany inflation persistence 894, 895 output growth (VAR models) 513, 516, 518 GIRF see generalized impulse response function (GIRF) GIVE see generalized instrumental variable estimator (GIVE) global financial crisis (2008) 142, 145, 411, 925, 926 global imbalances and exchange rate misalignment 928 global vector autoregressive (GVAR) modelling 563, 933–5 see also vector autoregressive (VAR) models approximating a global factor model 909–11 approximating factor augmented stationary high dimensional VARs 911–14 and Asian financial crisis (1997) 900 benefits of 900 dimensionality curse 903–8 empirical applications 923–32 forecasting 917–21 forecasting applications 924–5 global finance applications 925–7 global macroeconomic applications 927–32 impulse response analysis 915–17 large-scale VAR reduced form data representation 901–3 long-run properties 921–2 panel cointegration 841 permanent/transitory component decomposition 922 sectoral/other applications 932 specification tests 923 theoretical justification of approach 909–14 theory and practice 900–35 two-step approach of 901 GLS see generalized least squares (GLS) GMM see generalized method of moments (GMM) Goldfeld–Quandt test, heteroskedasticity 89, 90 goodness of fit 358 gradient methods 958–9 method of steepest ascent 959 Newton-Raphson 958–9 Granger causality 513–17 and Granger non-causality 516–17, 576 granularity condition 753 Great Depression (1929) 146 Grunfeld’s investment equation 437, 441 G-test of Phillips and Sul 737 GVAR see global vector autoregressive (GVAR) modelling habit formation, aggregation of life-cycle consumption decision rules under 887–92 Hannan–Quinn criterion (HQC), model selection 123, 250 Hausman test panel data models with strictly exogenous regressors 659–63, 673 slope homogeneity, testing for 735–7 spatial panel econometrics 804 heterogeneous panel data models, large 703–49, 746–9 see also panel data models with strictly exogenous regressors; short Tdynamic panel data models Bayesian analysis 730–1 dynamic heterogeneous panels 723–4 fixed effects (FE) specification 710 heterogeneous panels with strictly exogenous regressors 704–6 large sample bias of pooled estimators in dynamic models 724–8 mean group estimator 717–23, 728–30 multifactor error structure, large heterogeneous panels with 763–72 pooled estimators in heterogeneous panels 706–13 spatial panel econometrics 811–13 Swamy estimator/test 713–17, 719–23, 737–8 testing for slope homogeneity see slope homogeneity, testing for heteroskedasticity 83–93, 92 additive specification 87, 91 in cross-section regressions 83 diagnostic checks and tests 89–92 efficient estimation of regression coefficients in presence of 86 errors 83, 113 F-test 90, 91 Gauss–Markov theorem 83, 86 general models 86–9 Goldfeld–Quandt test 89, 90 graphical checks and tests 89 maximum likelihood estimation 87, 88, 89 mean-variance specification 87, 91 models with serially correlated/heteroskedastic errors 115–18 multiple regression 30 multiplicative specification 86–7, 90 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ Subject Index OLS estimators, using 84, 85, 86, 89, 91 panel data models with strictly exogenous regressors 661, 668 parametric tests 89, 90–2 regression models with heteroskedastic disturbances 83–5 heteroskedasticity autocorrelation consistent (HAC) estimator 233 heteroskedasticity-consistent variance (HCV) estimators 85, 117, 118 higher-order lags 535–6, 566 histogram 77, 143 Hodrick–Prescott (HP) filter 358–60, 922, 928 Holder’s inequality 982 homoskedasticity 10, 25, 26, 30 household consumption expenditure, cross-sectional regressions 83 housing 844–8, 930–1, 932 hypothesis testing, regression models 51–82, 79–82 alternative hypothesis 52, 53 Chow test (stability of regression coefficients) 77 coefficient of multiple correlation and F-test 65–6 composite hypotheses 51 confidence intervals 52, 59 critical or rejection region of test 51 error types 52–3 F-test see F-statistic/test implications of misspecification of regression model on hypothesis testing 74–5 Jarque–Bera’s test of normality of regression residuals 75–6 joint confidence region 66–7 linear restrictions see linear restrictions maintained hypothesis 52 versus model selection 247–8 models with serially correlated/heteroskedastic errors 115–18 multicollinearity problem 67–72 multiple models 58–9 non-parametric estimation of density function 77–9 null hypothesis see null hypothesis predictive failure test 76–7 relationship between different ways of testing β = 55–8 simple hypotheses 51, 53–5 size of test 52–3 stability of regression coefficients, testing 77 statistical hypothesis and statistical testing 51–2 testing significance of dependence between ϒ and X 55–8 t-test see t-statistic/test idempotent matrix 30, 946 IID see independently identically distributed (IID) random variables impulse response analysis 584–608, 605–8 Blanchard and Quah (1989) model 603 in cointegrating VARs 596–7 empirical distribution of impulse response functions and persistence profiles 597 forecast error variance decompositions 592–5 Gali’s IS-LM model 603–4 generalized impulse response function 589–90 GVAR models 915–17 see also global vector autoregressive (GVAR) modelling identification of a single structural block in a structural model 590–1 identification of monetary policy shocks 604–5 identification of short-run effects in structural VAR models 598–600 macro and aggregated idiosyncratic shocks 878–81 multiple regression 43–4 multivariate systems 585 orthogonalized impulse response function 586–9 persistence profiles for cointegrating relations 597 structural systems with permanent and transitory shocks 600–2 SVARs 600–1, 603 1051 traditional impulse response functions 584–5 in VARX models 595–7 independently identically distributed (IID) random variables see also random variables aggregation in large panels 861 asymptotic theory 177, 180 maximum likelihood (ML) estimation 196, 200, 203 inequalities Cauchy–Schwarz 981–2 Chebyshev 980 Holder 982 Jensen 982–3 infinite moving average process 270, 271, 272, 347 infinite vector moving average process 537 inflation global 927–8 persistence of see inflation persistence rates of 860–1 variance-inflation factor (VIF) 70 inflation persistence aggregation 892–6 data 893 estimation results 894–5 micro model of consumer prices 893–4 sources 895–6 information and processing costs 154–5 innovation error 275 instrumental variables and GMM 225, 807 Ahn and Schmidt model 685–6 Anderson and Hsiao model 681 Arellano and Bond model 682–5 Arellano and Bover models (with time-invariant regressors) 686–8 Blundell and Bond model 688–91 over-identifying restrictions, testing for 691 spatial panel econometrics 807–10 instrumental variables (IV) 117 integrated GARCH (IGARCH) hypothesis 623–4, 625 intercept terms, regression equations 30, 33, 75 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1052 Subject Index interest rates time series 25 trend-cycle decomposition 556–9 International Monetary Fund (IMF) 923 interval forecasts 388, 389, 423–4 investors rationality 137, 155 risk-averse 151–3, 392 risk-neutral 148–51 irrationality, individual 137 Italy, inflation persistence 894 Jackknife procedure 314, 778 Japan, output growth (VAR models) 513, 515, 518 Jarque–Bera’s test, normality of regression residuals 75–6, 141 JA-test (non-nested) 252 Jensen’s inequality 982–3 joint confidence region, hypothesis testing 66–7 joint hypothesis problem, and dynamic stochastic equilibrium formulations 153 Jordan decomposition 954 J-test (non-nested) 252 Kaiser criterion 447 Kalman filter RE models 500 and state space models 361–4 Keane and Runkle method (short T dynamic panel data models) 691–2 Kendall’s τ correlation 5, hypothesis testing 57, 58 kernel (lag window) 78, 79, 114, 321, 813 Keynesian theory 242 Khinchine’s theorem 177–8, 204 King and Watson method (rational expectations models) 485–6 Kolmogorov’s theorem 178–9 Kolmogorov–Smirnov statistic 619, 625 KPSS test statistic 346 Kronecker matrix 433, 471 Kronecker product and vec operator 635, 948–50 Kuipers score 397, 398 Kullback–Leibler information criterion (KLIC) 204 kurtosis (tail-fatness) 75, 141, 145, 151, 621 coefficients 142–3, 146 labour market 931 labour productivity, cross-section regression of output growth 83 lag operators 129, 518, 960–1 stochastic processes 269, 278 lagged values 26, 80, 151, 207, 285, 306, 426, 521, 548, 566, 571, 735 aggregation of large panels 868, 872 autocorrelated disturbances 101, 103, 108, 112, 117 cointegration analysis 535 conditional correlation of asset returns, modelling 619 cross-sectional dependence, in panels 776, 777, 782, 783 dynamic economic modelling 126, 128 forecasting 378, 381–2 generalized method of moments 228–9 GVAR models 903 see also global vector autoregressive models heterogeneous panel data models, large 723, 733 impulse response analysis 585 multiple regression 26, 41, 46 multivariate RE models 467, 468, 470, 473, 490, 493, 496 short Tdynamic panel data models 677, 682, 685, 686, 691 two variables, relationship between 19, 21 vector autoregressive models 517–18 volatility 416, 418 Lagrange multiplier (LM) test ARCH/GARCH effects, testing for 417 cross-sectional dependence, in panels 784, 785 heteroskedasticity 91 maximum likelihood estimation 195, 218 principal components 446 procedure 212, 213–14 of residual serial correlation 112–13 Lasso (Least Absolute Shrinkage and Selection Operator) regressions 261–2, 914 Latin America 929, 930 law of large numbers 177–80 dependent and heterogeneously distributed observations 182–5 strong 178, 179 uniform strong 179–80 weak 178, 181 least squares criterion least squares cross-validation method 78 least squares dummy variable (LSDV) 644–5 Lehman Brothers, collapse 160 L’Hopital’s rule, asymmetric loss function 375 likelihood approach see also maximum likelihood estimation bivariate regressions 13–14, 29 likelihood function 195–7 likelihood ratio approach 212, 213, 218 log-likelihood ratio statistics for tests of residual serial correlation 105–6 likelihood-based tests 212–22 Lagrange multiplier test procedure see Lagrange multiplier test Likelihood ratio test procedure 212, 213, 218 Wald test procedure 195, 212, 214–22 quasi-maximum likelihood estimator 773–4, 802 testing whether is diagonal 439–41 transformed 692–5 Linberg–Feller’s theorem 181–2 Lindberg condition 182 linear panels, with strictly exogenous regressors 634–5 linear regression classical normal linear regression model see classical normal linear regression model forecast uncertainty sources 387 generalized model 94 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ Subject Index maximum likelihood estimation 218 non-linear in variables 47–8 non-nested tests for linear regression models 250–3 with normal errors 196–7, 218 population moment conditions 226 rival models 245–6 linear restrictions see also hypothesis testing, regression models estimation of cointegrating relations under 545–6 exactly identified case, cointegrating relations 545 general, testing 64–5 over-identified case, cointegrating relations 545–6 system estimation subject to, in multivariate analysis 434–6 testing F-test 65–6 general linear restrictions 64–5 joint tests 62–4 in multivariate analysis 438–9 on regression coefficients 59–62 linear statistical models 10–12 classical normal linear regression model see classical normal linear regression model linear-quadratic (LQ ) decision problem 391 LINEX function 375, 379 liquidity, and predictability 160 LM test see Lagrange multiplier (LM) test logit versus probit models 246–7 log-likelihood function autocorrelated disturbances 102 bimodal function 108 Cochrane–Orcutt (C-O) iterative method 106 cointegration analysis 532, 533, 534, 535, 541, 544 dependent observations 209–10 Gaussian errors, ML estimation with 421, 422 log-likelihood ratio statistics for tests of residual serial correlation 105–6 log-likelihood ratio statistics, over-identifying of restrictions on cointegrating relations 546–7 non-nested tests, linear regression models 257 panel data models with strictly exogenous regressors 650 reduced rank regression 461 state space models 364 Student’s t-distributed errors, ML estimation with 421, 422 VAR models 512, 513, 517 VARX models 564–5, 579 long memory processes, unit root tests 346–51 and cross-sectional aggregation 349–51 fractionally integrated 348–9 spectral density of long memory processes 348 Long Term Capital, downfall (1998) 160 long-run relationships see also cointegration analysis analysis of long-run 921–2 bounds testing approaches to analysis of 526–7 concept 779 dynamic economic modelling of long-run and short-run effects 125–6 GVARs, long-run properties 921–2 identification in a cointegrating VARX 572–3 identification of long-run effects 530–2 identification of long-run relationships 921 long-run identification problem 531 persistence profiles 922 structural modelling estimation of cointegrating relations under general linear restrictions 545–6 identification of cointegrating relations 544–5 log-likelihood ratio statistics, over-identifying of restrictions on cointegrating relations 546–7 VARX modelling 574–80 testing for number of cointegrating factors 921 1053 losses, forecasting asymmetric loss function 375–6 losses associated with point forecasts and forecast optimality 373–6 quadratic loss function 373–5 test statistics of forecast accuracy based on loss differential 394–6 Lp mixingales 185, 328 Lucas critique 859 Lyapounov’s inequality 169, 179 Lyapounov’s theorem 181 MA processes see moving average (MA) processes macroeconomics aggregation 859 business cycle synchronization 928–9 China, rising role in world economy 928–9 EMU membership, impact 929–30 fiscal and monetary policy, effects 931 global imbalances and exchange rate misalignment 928 global inflation 927–8 GVAR models 927–32 housing 930–1 labour market 931 panel unit root testing 835 small open economy models 905 United States as dominant economy 928 volatility, in macro-econometric modelling 411 weather shocks 932 marginal density 198 marginal utility of consumption 152 market collapse (2000) 142 Markov chain Monte Carlo (MCMC) methods 502 Markov’s inequality 171 martingale difference process 133 asymptotic theory 184, 186 cointegration analysis 542 RE models 488–9, 500 unit root tests 327–8 martingale process 133, 326–7 mathematics complex numbers 939–40 difference equations 961–4 eigenvalues 946 eigenvectors 946 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1054 Subject Index mathematics (cont.) Fourier analysis 941–2 Kronecker product and vec operator 948–50 lag operators 960–1 mathematical expectations and moments of random variables 969–70 matrices and matrix operations see matrices mean value theorem 956 numerical optimization techniques 957–60 spectral radius 952–3 Taylor’s theorem 957 trigonometric functions 940–1 matrices 942–5 see also mathematics calculus 954–6 covariance 103 decompositions 953–4 determinant 944–5 diagonal 946 Fisher’s information matrix 88, 201 generalized inverses 948 idempotent 30, 946 inner product form 206 inverse of 947–8 matrix operations 943–4 Moore–Penrose inverse 906, 948 multicollinearity and prediction problem 72–4 Newey–West heteroskedasticity and autocorrelation consistent variance 113 norms 951–2 orthogonal 946 outer product form and inner product form 201 partitioned 950–1 positive definite matrices and quadratic forms 945 projection 30 rank 944 residual 42 special 945–6 trace 944 triangular 945 max ADF unit root test 345 maximum eigenvalue statistic, cointegration analysis 540–1 maximum likelihood (ML) estimation 195–224, 222–4 see also likelihood approach; quasi-maximum likelihood estimator (QMLE) of AR(1) processes 309–12 of AR(p) processes 312–13 asymptotic distribution of estimator 318 asymptotic properties of estimators 203–9, 210–12 autocorrelated disturbances 101 bivariate regression model 14 cointegration analysis 539, 549 commodity price models 930 consistency for ML estimators 204 DCC model 615–17 DSGE models 489 first-order conditions 215 fixed-effects estimator, derivation as a ML estimator 645 Gaussian 421, 616, 765 and GMM 225 heterogeneous and dependent observations 209–12 heterogeneous panel data models, large 716 heteroskedasticity 87, 88, 89 likelihood function 195–7 likelihood-based tests 212–22 test procedure 213–14 Wald test procedure 195, 212, 214–22 log-likelihood function for dependent observations 209–10 MA(1) processes 303–6 multiple regression 28–9 pseudo-true values 244 random effects model 649–50 rational expectations models 498–500 reduced rank regression 462 regularity conditions/preliminary results 200–3 spatial panel econometrics 802 with Student’s t-distributed errors and returns 421–3, 616–17 SURE models 436–7 weak and strict exogeneity 197–200 weekly returns, volatilities and conditional correlations in 622–3 MCMC (Markov chain Monte Carlo) methods 502 mean, hypothesis testing 52 mean group estimator (MGE) 717–23 of dynamic heterogeneous panels 728–30 pooled 731–4 relationship with Swamy estimator 719–3 small sample bias 730 spatial panel econometrics 811 mean lag 127–8 mean square error, of estimator 36 mean squared forecast error (MSFE) criteria decision-based forecast evaluation framework 390, 392, 394 defined 373 iterated and direct multi-step methods 383 and quadratic cost functions 391–2 mean-square error criteria (MSE) 234 mean-variance specification, heteroskedasticity 87, 91 method of moments bivariate regressions 12–13 estimator 228 generalized see generalized method of moments (GMM) MA(1) processes, estimation 302–3 Microfit 107, 110, 111, 559 Microfit 5.0 142, 308, 342, 359 MULTI.BAT (Microfit batch file) 68 Middle East and North Africa (MENA) 929 misleading inferences 26 misspecification asymptotic theory 191 forecast combination 385 implications for OLS estimators 44–6 inclusions of irrelevant regressors 46 omitted variable problem 45 of regression model, implications on hypothesis testing 74–5 Sargan’s general misspecification test 239–40 test 234–5 ML estimation see maximum likelihood (ML) estimation ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ Subject Index model selection 242–64, 262–4 see also Akaike information criterion (AIC), linear regression models; Schwarz Bayesian criterion (SBC), non-nested tests, model selection Bayesian analysis 259–61, 989–90 combination of models, Bayesian approach to 259–61 consistency properties of criteria 250 criteria 249–50 formulation of econometric models 243–4 versus hypothesis testing 247–8 Lasso regressions 261–2, 914 models with different transformations of dependent variable Bera–McAleer test statistic 253 double-length regression test statistic 254–5 PE test statistic 253 Sargan and Vuong’s likelihood criteria 257–8 simulated Cox’s non-nested test statistics 256–7 probit versus logit models 246–7 pseudo-true values 244–7 rival linear regression models 245–6 moment conditions see also method of moments exact numbers 228–9 excess of 229–31 population moment conditions 226–8, 235 monetary policy shocks, identification 604–5 money market equilibrium (MME) 580 Monte Carlo investigations see also aggregation aggregation in large panels 860, 881–7 design 882–3 estimation using aggregate and disaggregate data 883–4 results 884–7 cointegration analysis 543, 547 cross-sectional dependence, in panels 765, 771, 775, 778, 783, 785 forecasting 396 and GMM 233, 234 heterogeneous panel data models, large 730, 731, 743 Markov chain Monte Carlo (MCMC) methods 502 max ADF unit root test 345 model combination 261 multivariate analysis 453, 455–6, 457 non-nested tests, linear regression models 252, 257 panel cointegration 843, 852, 853 panel unit root testing 834, 838, 839 short T dynamic panel data models 689, 691, 700 spatial panel econometrics 812 spurious regression problem 26 Wald test procedure 214 Moore–Penrose inverse matrix 906, 948 Moran’s I test 784 moving average error model 121 moving average (MA) processes 269–72, 276–7, 595 autocorrelated disturbances 98 forecasting 381–2 infinite 270, 271, 272, 347 MA(1) processes, estimation maximum likelihood (ML) estimation 303–6 method of moments 302–3 regression equations with MA(q) error processes, estimation 306–8 MA(q) error processes, estimation of regression equations with 306–8 MSFE see mean squared forecast error (MSFE) criteria multicollinearity problem 24 hypothesis testing 67–74 and prediction problem 72–4 seriousness, measuring 70 multinomial distribution 977 multi-period returns 138 multiple correlation coefficient 24, 39–41 multiple regression 24–50, 48–50 ceteris paribus assumption 43, 44 1055 classical normal linear regression model 24–7, 41—2 covariance matrix of regression coefficients βˆ 31–3 distribution of OLS estimator 37–9 disturbances of regression equation 24–5 Frisch-Waugh-Lovell theorem 43, 48 Gauss–Markov theorem 14, 17, 18, 24, 34–6, 83 heteroskedasticity 30 homoskedasticity 25, 26, 30 impulse response analysis 43–4 interpretation of coefficients 43–4 irrelevant regressors, inclusion 46 linear regressions that are non-linear in variables 47–8 maximum likelihood approach 28–9 mean square error of an estimator and bias-variance trade off 36 multiple correlation coefficient 24, 39–41 ordinary least squares method 24, 27–8, 30–1, 37–9 orthogonality 25, 26, 30 partitioned regression 24, 41–3 properties of OLS residuals 30–1 multiplicative specification, heteroskedasticity 86–7, 90 multi-step ahead forecasting 373, 379–80 multivariate analysis 431–66, 464–6 canonical correlation analysis 458–61 common factor models 448–58 determining number of factors 454–8 distributions 967–8, 977–9 endogenous variables, system of equations with 441–5 forecasting 392, 517–18 generalized least squares estimator 432–4 heteroskedasticity 85 hypothesis testing 65–6 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1056 Subject Index multivariate analysis (cont.) impulse response systems 585 iterated instrumental variables estimator 444–5 linear/non-linear restrictions, testing of 438–9 LR statistic for testing whether is diagonal 439–41 maximum likelihood estimation of SURE models 436–7 normal distributions 27 principal components (PC) 446–8 and cross-section average estimators of factors 450–4 reduced rank regression 461–3 seemingly unrelated regression equations 431–41 spectral density 518–20 system estimation subject to linear restrictions 434–6 two- and three-stage least squares 431, 442–4, 444 multivariate generalized autoregressive conditional heteroskedastic (MGARCH) 609 multivariate normal distribution 978 Mundell-Flemming trilemma 927 Nadaraya-Watson kernal 814 National Bureau of Economic Research (NBER) 360 National Longitudinal Surveys (NLS), of Labor Market Experience 633 negative exponential utility (finance application) 392–4 neoclassical investment model 482 net present value (NPV) 150 new Keynesian Phillips curve (NKPC) 475, 476, 494, 928 Newey–West heteroskedasticity and autocorrelation consistent (HAC) variance matrix 113 Newey–West robust variance estimator 113–15 Newey–West SHAC estimator 813 Newton-Raphson method 305, 364, 546, 733, 958–9 Nickell bias 679 Nielson Datasets 633 Nikkei 225 (NK) index 142, 621 NKPC (new Keynesian Phillips curve) 475, 476, 494 non-autocorrelated errors 10, 25, 26 non-linear restrictions, testing 438–9 non-nested tests, linear regression models Encompassing test 253 globally and partially non-nested models 248 hypotheses 51 JA-test 252 J-test 252 N-test 251 NT-test 251–2 simulated Cox’s non-nested test statistics 256–7 W-test 252 non-parametric approaches see also parametric tests cointegration analysis 548–9 hypothesis testing 77–9 spatial panel econometrics 813–14 non-spherical disturbances, regression models with 94 normal equations, OLS problem normal linear regression model see classical normal linear regression model normality assumptions asymptotic normality 205, 230–1 departures from normality 142 Jarque–Bera’s test, normality of regression residuals 75–6 multiple regression 25, 27, 28 normal distributions 974–5 n-step ahead forecast error 592 N-test (non-nested) 251 NT-test (non-nested) 251–2 null hypothesis see also hypothesis testing, regression models autocorrelated disturbances 114, 118 autocovariances, estimation 301–2 cointegration analysis 540 Dickey–Fuller (DF) unit root tests 332 fixed effects, testing for 659 forecasting 398, 402, 406 and GMM 234 heteroskedasticity 90 hypothesis testing 52, 53, 54, 57, 58, 61, 63, 64 Lagrange multiplier (LM) test 214, 218 model selection 248 panel unit root testing 819, 822–5, 826, 827, 830 returns of assets, predictability 141 sphericity 787 stationarity, testing for 345 vector autoregressive models 512 numerical optimization techniques direct search methods 959–60 gradient methods 958–9 grid search methods 957 OECD (Organisation for Economic Co-operation and Development) 580, 633 oil shocks 513, 930 OLS estimator 37–9, 96 see also ordinary least squares (OLS) analysis/regression ARDL models 122, 123, 127 asymptotic theory 192 autocorrelated disturbances 96, 113 biased 199 compared to GLS 96 distribution 37–9 estimation of α 18–19 implications of misspecification for 44–6 inconsistency of estimator of dynamic models with serially correlated errors 315–17 Phillips–Hansen fully modified 527–9 pooled 636–9, 652 properties 14–19 single-equation 434 stochastic transformation 115 unbiased 14 omitted variable problem, misspecification 45 one-sided moving average process, versus two-sided representation 269 one-step ahead forecast 373 optimal weighting matrix, generalized method of moments 232 optimality, forecast 373–6 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ Subject Index ordinary least squares (OLS) analysis/regression ARCH/GARCH effects, testing for 417, 418 cointegration analysis 527, 532, 549–50 common factor models 449 estimator see OLS estimator fully modified OLS (FM-OLS) approach 527, 850, 854 and GMM 229, 238 heteroskedasticity 84, 85, 86, 89, 91 hypothesis testing 53 method 4–5, 27–8 in multiple regression 24, 27–8, 30–1, 37–9 non-nested tests, linear regression models 252, 253 orthogonality 30 Pesaran–Timmermann (PT) market-timing test 398 properties of residuals 30–1 regressions, second generation panel unit root tests 835–6 residuals 30–1, 112 vector autoregressive models 510 orthogonality 4, 10, 234, 304, 501, 697, 946 multiple regression 25, 26, 30 orthogonalized forecast error variance decomposition 592–3 orthogonalized impulse response function 586–9 output gap relationship 578, 580 output growths, VAR models Germany 513, 516 Japan 513, 515 United States 513, 514 overlapping returns 138 panel cointegration 855–8 see also panel unit root testing with cross-sectional dependence 853–5 cross-unit cointegration 836–7 estimation of cointegrating relations in panels 850–5 general considerations 839–43 multiple cointegration, tests for 849–50 residual-based approaches 843–9 spurious regression 843–8 system estimators 852–3 tests 848–50 panel corrected standard errors (PCSE) 835 panel data models aggregation of large panels see under aggregation cross-sectional dependence see cross-sectional dependence, in panels dynamic 200, 699–700 large heterogeneous see heterogeneous panel data models, large non-linear unobserved effects models 699–700 short T dynamic models see short Tdynamic panel data models spatial panel econometrics see spatial panel econometrics with strictly exogenous regressors see panel data models with strictly exogenous regressors unit roots and cointegration in panels see panel cointegration; panel unit root testing panel data models with strictly exogenous regressors 633–75, 674–5, 676 see also seemingly unrelated regression equations (SURE) models cross-sectional regression 650–3 estimation of the variance of pooled OLS, FE and RE estimators of β (robust to heteroskedasticity and serial correlation) 653–6 fixed effects versus random effects 653 specification 639–45 testing for 659–63 between group estimator of β 650–3 Hausman’s misspecification test 659–63, 673 heterogeneous panels 704–5 linear panels with strictly exogenous regressors 634–5 non-linear unobserved effects 670–1 pooled OLS estimator 636–9 random effects specification 646–50 1057 relation between FE, RE and cross-sectional estimators 652–3 relation between pooled OLS and RE estimators 652 time invariant effects, estimation FEF-IV estimation 667–70 HT estimation procedure 665–7 time-specific effects 657–9 time-specific formulation 635 unbalanced panels 671–3 unit-specific formulation 634 Panel Study of Income Dynamics (PSID) 633 panel unit root testing 817–38, 855–8 see also panel cointegration asymptotic power of tests 825–6 cross-sectional dependence 833–4 Dickey–Fuller (DF) unit root tests 817, 818, 819, 821, 822, 830 distribution of tests under null hypothesis 822–5 finite sample properties of tests 838–9 first generation panel unit root tests 821–33 GLS regressions, tests based on 834–5 heterogeneous trends 826–8 measuring proportion of cross-units with unit roots 832–3 model and hypotheses to test 818–20 OLS regressions, tests based on 835–6 other approaches to 830–2 and panel cointegration see panel cointegration second generation panel unit root tests 833–6 short-run dynamics 828–30 Panel VARs (PVAR) models 695, 852, 901, 902 parametric tests see also non-parametric approaches cointegration analysis 548 heteroskedasticity 89, 90–2 hypothesis testing 77 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1058 Subject Index partial adjustment model 120, 123–4, 125, 129 partitioned matrices 950–1 partitioned regression 24, 41–3 Parzen kernel 114 Parzen window 320, 346 PC see principal components (PC) PE test statistic 253 Pearson correlation coefficient 6, penalized regression techniques 242, 262 percentiles, statistical models of returns 140–1 persistence profiles 597, 922 impulse response analysis 596, 597 Pesaran and Yamagata -test 738–41 Pesaran–Timmermann (PT) market-timing test 397–8 generalized PT test for serially dependent outcomes 399–400 regression approach to derivation of 398–9 relationship to Kuipers score 398 Phillips–Hansen fully modified OLS estimator 527–9 Phillips–Perron (PP) test 339–1 point and interval forecasts 423–4 Poisson distribution 974 polynomial distributed lag models 120–1 pooled mean group (PMG) estimators 732, 733, 734 population moment conditions 226–8, 235 prediction/predictability see also forecasting of asset returns see returns of assets errors and variance 20–1 ex ante predictions 21–2 multi-category variables, predictability tests for 400–6 prediction problem 19–22 predictive distribution 376 predictive failure test 76–7 stochastic volatility models 419 stock market predictability and market efficiency 147–53 price-dividend ratio 150–1 prices and returns see also returns of assets multi-period returns 138 overlapping returns 138 single period returns 137–8 principal components (PC) 446–8 and cross-section average estimators of factors 450–4 dynamic panels, estimators for 774–5 estimators 764–5, 774–5 probability, convergence in 167–8 probability and statistics Brownian motion 983–4 characteristic function 972–3 Cochran’s theorem/related results 979–80 correlation versus independence 971–2 covariance and correlation 970–1 cumulative distribution 966 density function 966 mathematical expectations and moments of random variables 969–70 probability distribution 966 probability limits involving unit root processes 984 probability space and random variables 965 useful inequalities 980–3 useful probability distributions 973–9 probability forecasts/probability event forecasts 376–8, 424 estimation of probability forecast densities 378 versus interval forecasts 388 probability integral transforms (PIT) 624, 626 probit versus logit models 246–7 profitable opportunities, exploiting in practice 159–61 projection matrix, OLS residuals 30 pseudo-true values 191, 244–7 PT test see Pesaran–Timmermann (PT) test of market timing pth -difference equations 961 purchasing power parity (PPP) 575, 578 quadratic cost functions 391–2 quadratic determinantal equation method (QDE) 473–6, 481, 499 quadratic loss function, forecasting 373–5 quadratic mean, convergence in 169, 170 quasi-maximum likelihood estimator (QMLE) 773–4, 802, 852 quasi-time demeaning data 648 QZ decomposition see generalized Schur decomposition R2 , adjusted 40–1 random coefficients, aggregation of stationary micro relations with 874–5 random effects (RE) specification fixed effects versus random effects 653 GLS estimator 646–9 ML estimation of random effects model 649–50 spatial panel econometrics 801, 803–7 random variables convergence in distribution 172–6 in probability 167–8 with probability (sure convergence) 168–9, 171 relationships among modes 170–2 in s-th mean 167, 169–70 independent 968 independently identically distributed see independently identically distributed (IID) random variables moments of 969–70 probability event forecasts 377 and probability space 965 Taylor series expansion of functions 177, 217 random walk model Beveridge–Nelson decomposition 364 cointegration analysis 523 difference stationary processes 324–5 pictorial examples 325 returns of assets and efficient market hypothesis 136, 149, 150, 151 variance ratio test 331 rank correlation coefficients 6–8 rational distributed lag models 121 rational expectations hypothesis (REH) 467 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ Subject Index rational expectations (RE) models 120, 129–34, 504–6 backward recursive solution 482–3 Bayesian analysis 501–3 bias of RE estimators, in short T dynamic panel data models 678–81 Blanchard and Kahn method 483–5 calibration and identification 496–8 containing expectations of exogenous variables 130 with current expectations of endogenous variables 130–1 DSGE models general framework 489–90 with lags 493–5 without lags 490–3 efficient market hypothesis 156, 157 with feedbacks 476–8 ’finite-horizon’ 482–3 with forward and backward components 472–6 with future expectations of endogenous variables 131–3 forward solution 468–70 method of undetermined coefficients 470–2 multivariate RE models 467–72 GMM estimation 500–2 higher-order case 479–82 identification, general treatment 495–8 King and Watson method 485–6 lagged values 467, 468, 470, 473, 490, 493, 496 martingale difference process 488–9 maximum likelihood estimation 498–500 multivariate 467–506 quadratic determinantal equation method 473–6, 481, 499 retrieving solution for yt 481–2 Sims method 486–8 rational hypothesis (REH) 129–30, 133 rationality, efficient market hypothesis 155 RE models see rational expectations (RE) models realized volatility (RV) 412 reduced rank hypothesis 461 reduced rank regression (RRR) 403, 461–3 regression coefficients efficient estimation of in presence of heteroskedasticity 86 linear restrictions, testing on 59–62 multiple, interpretation of 43–4 stability of (Chow test) 77 regression line 3, regression models with autocorrelated disturbances 98–106 adjusted residuals, R2 , and other statistics 103–4 AR(1) and AR(2) cases 99, 102–3 covariance matrix of exact ML estimators for AR(1) and AR(2) disturbances 103 estimation 99–100 higher-order error processes 100–1 log-likelihood ratio statistics for tests of residual serial correlation 105–6 with heteroskedastic disturbances 83–5 hypothesis testing see hypothesis testing, regression models implications of misspecification on hypothesis testing 74–5 multiple 58–9 with non-spherical disturbances 94 simple see simple regressions regressions absolute distance/minimum distance auxiliary 92, 253, 254 bivariate see bivariate regressions coefficients see regression coefficients cross-country growth 83 cross-sectional 83, 650–3 generalized R2 for IV regressions 239 GLS see generalized least squares (GLS) 1059 hypothesis testing in models see hypothesis testing, regression models interpretation of multiple regression coefficients 43–4 Lasso 261–2, 914 linear see linear regression MA(q) error processes, estimation of regression equations with 306–7 models see regression models multiple see multiple regression OLS see ordinary least squares (OLS) analysis/regression orthogonal partitioned 41–3 penalized regression techniques 242, 262 PT test, regression approach to derivation of 398–9 reverse 4, Spearman rank spurious 26, 843–8 stock return 147 three variable models 33, 59, 91 regularity conditions 200–3, 244 residual matrices 42 residual serial correlation, consequences 95 returns of assets see also efficient market hypothesis (EMH); weekly returns, volatilities and conditional correlations in and alternative versions of efficient market hypothesis 153–5 conditional correlation of, modelling see conditional correlation of asset returns, modelling 609–30 covariance of asset returns with marginal utility of consumption 152 cross-correlation of returns 145 daily returns 144, 145 empirical evidence 142–4 extent to which predictable 145 log-price change and relative price change 137 measures of departure from normality 141 monthly stock market returns 145–6 multi-period returns 138 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1060 Subject Index returns of assets (cont.) normality, departures from 142 overlapping returns 138 percentiles, critical values, and Value at Risk 140–1 predictability 136–61, 161–4 and prices 137–8 random walk model, stock prices 136, 149, 150, 151 S&P 500 index 142, 143, 146 single period returns 137–8 skewness 75, 141, 146 statistical models 139–41 statistical properties 142–4 stock return regressions 147 stylized facts 144 weekly returns 142 reverse regression 4, 6, 23, 56 Ridge regression 262 risk-averse investors 151–3, 392 Riskmetrics 623 RiskMetrics™ ( JP Morgan) method 412–13 risk-neutral investors 148–51 risk-return relationships, volatility 419–20 root mean squared forecast error (RMSFE) 574 S&P 500 (SP) index 142, 143, 146 conditional correlation of asset returns, modelling 621, 628 industry groups 423 Sargan and Vuong’s likelihood criteria 257–8 Sargan’s general misspecification test 239–40 saturation level, logistic function with 47 scatter diagrams Schur/generalized Schur decomposition 486, 953 Schwarz Bayesian criterion (SBC), model selection 123, 249–50, 338, 576, 712 vector autoregressive models 512, 513 seemingly unrelated regression equations (SURE) models 431, 440, 443 see also dynamic seemingly unrelated regression (DSUR) estimator cross-sectional dependence, in panels 751 maximum likelihood estimation 436–8 panel data models with strictly exogenous regressors 634 panel unit root tests 817 temporal heterogeneity 812 vector autoregressive models 510 serial correlation errors see serially correlated errors first and second order coefficients 145 Lagrange multiplier test of residual serial correlation 112–13 residual, consequences 95 Sargan’s test of residual serial correlation for IV regressions 240–1 testing for 111–13 serially correlated errors heteroskedastic 115–18 inconsistency of the OLS estimator of dynamic models with 315–17 when arising 94 serially dependent outcomes case of serial dependency in outcomes 400–6 generalized PT test for 399–400 Sharpe ratios 154, 160, 394 shocks aggregation in large panels 870–1, 878–9 credit supply 931 identification in a structured model 590–1 long memory processes, unit root tests 346–8 macro and aggregated idiosyncratic, impulse responses 878–81 monetary policy, identification 604–5 oil 513, 930 orthogonalized 592 permanent and transitory, structural systems with 600–2 structural 599, 915 system-wide 589, 597 variable-specific 590 weather 932 short T dynamic panel data models 676–702, 701–2 bias of the FE and RE estimators 678–81 dynamic, non-linear unobserved effects models 699–700 dynamic panels with short T and large N 676–7 instrumental variables and GMM 681–91 Keane and Runkle method 691–2 over-identifying restrictions, testing for 691 short dynamic panels with unobserved factor error structure 696–9 transformed likelihood approach 692–5 shrinkage (ridge) estimator, Bayesian 914, 992–3 Silverman rule of thumb 78 simple regressions hypothesis testing 53–5 and multiple regressions 32, 39 Sims method, RE models 486–8 simulated annealing 579, 959–60 simultaneous equations model (SEM) 493, 590–1 single equation approaches cointegration analysis 525–8 panel cointegration 850–2 single period return 137–8 skewness 75, 141 slippage costs 160 slope heterogeneity 820 slope homogeneity, testing for 439, 734–45, 764 bias-corrected bootstrap tests for the AR(1) model 743–4 in earnings dynamics 744–6 extensions of the -tests 741–2 G-test of Phillips and Sul 737 Hausman-type tests for panels 735–7 Pesaran and Yamagata -test 738–41 standard F-test 735 Swamy’s test 737–8 Slutsky’s convergence theorems 173–6, 187, 207, 216 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ Subject Index small open economy (SOE) macroeconomic models 905 smoothing parameter 78 South Africa 929 Southern Oscillation Index (SOI) 932 spatial autoregressive (SAR) specification 800 spatial correlation 797–8 spatial error component (SEC) 801 spatial error models 800–1 spatial heteroskedasticity autocorrelation consistent (SHAC) estimator 813 spatial lag models 798–800 spatial lag operator 798 spatial moving average (SMA) 801 spatial panel econometrics 797–816, 815–16 dynamic panels with spatial dependence 810 estimation 802–10 fixed effects specification 802 heterogeneous panels 811–13 instrumental variables and GMM 807–10 maximum likelihood estimator 802 non-parametric approaches 813–14 random effects specification 803–7 spatial dependence in panels 798–802, 814–15 spatial error models 800–1 spatial lag models 798–800 spatial weights and spatial lag operator 798 temporal heterogeneity 812–13 testing for spatial dependence 814–15 weak cross-sectional dependence in spatial panels 801–2 Spearman rank regression 5, 6–7, 8, 785 spectral analysis 285–94, 292–4 distributed lag models, spectral density 291–2 properties of spectral density function 287–91 relation between f (ω) and autovariance generation function 289–91 spectral representation theorem 285–7 spectral decomposition 953 spectral density see also spectral analysis and autocovariance generating function 273 cointegration analysis 530 distributed lag models 291–2 estimation 319–21 of long memory processes 348 multivariate 518–20 properties of function 287–91 spectral representation theorem 286 standardized 331, 367 trend-cycle decomposition of unit root processes 367 weighting schemes for estimating 318 spectral radius 952–3 spectral representation theorem 285–7 spurious regression 26, 843–8 square summable sequence, stochastic processes 270 SSR (sum of squares of residuals) 63 state space models and Kalman filter 361–4 static factor model 448 stationary stochastic processes 267–8, 281 stationary time series processes 297–323, 321–3 asymptotic distribution of ML estimator 318 estimation of autocovariances 299–302 estimation of autoregressive (AR) processes 308–13 maximum likelihood estimation of AR(1) processes 309–12 maximum likelihood estimation of AR(p) processes 312–13 estimation of MA(1) processes maximum likelihood estimation 303–6 method of moments 302–3 regression equations with MA(q) error processes, estimation 306–8 estimation of mixed ARMA processes 317–18 1061 estimation of the mean 297–9 inconsistency of the OLS estimator of dynamic models with serially correlated errors 315–17 sample bias-corrected estimators of autocorrelation coefficient, ϕ, small 313–15 spectral density, estimation 318–21 testing for stationarity 345–346 Yule–Walker estimators 308–9 statistical aggregation 864–5 statistical fit 242, 247 statistical hypothesis and statistical testing 51–2 see also hypothesis testing, regression models item statistical inference, classical theory 51 steepest ascent, method of 959 s-th mean, convergence in 167, 169–70 stochastic equilibrium 268 stochastic orders Op (·) and op (·) 176–7 stochastic processes 267–84, 281–4 absolutely summable sequence 270, 272, 273, 274 autocovariance function 269, 271 autocovariance generating function 272–4 classical decomposition of time series 274–5 moving average 269–72, 276–7 see also moving average (MA) processes stationary 267–8, 281 trend-stationary processes 268, 275 white noise 268, 269 stochastic trend representation 368–9 stochastic volatility models 419 stock market crash (1929) 146 stock market crash (2008) 142, 145, 411, 925 stock market predictability and market efficiency 147–53 risk-averse investors 151–3 risk-neutral investors 148–51 stock prices, random walk model 136, 149 stock return 25 stock returns, monthly 145–6 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1062 Subject Index strict exogeneity 15, 26, 197–200 see also exogeneity; panel data models with strictly exogenous regressors; seemingly unrelated regression equations (SURE) models heterogeneous panel data models, large 704–6 unbiased 199 weak and strict 26, 197–200 strict stationarity, stochastic processes 268 strong law for asymptotically uncorrelated processes 184 strong law for mixing processes 184 strong law of large numbers 178, 179 structural time series approach 360–1 structural VARs (SVARs) 600–1, 603 structural VEC (SVEC) 601 Student t-distribution 618 Student’s t-distributed errors distributions 976 ML estimation with 421–3 subsampling procedure 837–8 sum of squares of residuals (SSR) 63 SURE models see seemingly unrelated regression equations (SURE) models Swamy estimator/test 713–17 relationship with mean group estimator 719–3 testing for slope homogeneity 737–8 Sylverster equations 470 tail-fatness see kurtosis (tail-fatness) Taylor series expansion of functions 177, 217 Taylor’s theorem 957 tests/testing asymptotic power of panel unit root tests 825–6 bootstrap tests of slope homogeneity for AR(1) model, bias-corrected 743–4 cointegration VAR models 540–3 VARX models 570–1, 571–2, 577–80 cointegration analysis 546–9 cross-sectional dependence (CD) tests 793–4 DCC model 618–19 error cross-sectional dependence 783–93 fixed effects specification 659–63 forecasting 400–6 F-test 65–6, 735 GARCH effects 418–19 Granger non-causality, block 516–17 G-test of Phillips and Sul 737 heteroskedasticity 89–92 hypothesis testing see hypothesis testing, regression models likelihood-based tests 212–22 linear restrictions 59–66, 438–9 linear versus log-linear consumption functions 259 long-run relationships 526–7 misspecification 234–5 multiple cointegration 849–50 non-nested tests see non-nested tests, linear regression models for over-identifying restrictions 691 panel unit root testing see panel unit root testing parametric tests 90–2, 548 power of a test 52 residual serial correlation 105–6 residual-based, cointegration analysis 525–6 small sample properties of test statistics 547–9 spatial dependence in panels 814–15 specification, GVAR models 923 unit root see Dickey–Fuller (DF) unit root tests; panel unit root testing; unit root processes and tests weak exogeneity 569 three variable models 33, 59, 91 three-stage least squares (3SLS) 443, 444 time domain techniques 267 time series analysis classical decomposition 274–5 cyclical component 275 financial and macro-economic time series 25 long-term trend 275 residual component 275 seasonal component 275 spurious regression problem 26, 843–8 stationary processes, estimation see stationary time series processes total impact effect, measuring 43 US macroeconomic time series 1959–2002 385 trace statistic, asymptotic distribution 541–3 transaction costs 160 transversality condition 132, 149, 484 trend and cycle decomposition 358–72 band-pass filter 358, 360 Hodrick–Prescott filter 358–60 interest rates 556–9 state space models and Kalman filter 361–4 structural time series approach 360–1 trend-cycle decomposition of unit root processes 364–9 trend-cycle decomposition of unit root processes see also trend and cycle decomposition Beveridge–Nelson decomposition 364–7 stochastic trend representation 368–9 Watson decomposition 367 trended variables 192 trend-stationary processes versus first difference stationary processes 328–9 stochastic processes 268, 275 trigonometric functions 940–1 t-statistics/test 41, 54, 68, 69, 116 panel unit root testing 821, 822, 823 Tukey window 320, 346 two variables, relationship between 3–23, 22–3 correlation coefficients between ϒ and X 5–8 curve fitting approach 3–4 decomposition of variance of ϒ 8–10 likelihood approach, bivariate regressions 13–14 linear statistical models 10–12 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ Subject Index method of moments, applied to bivariate regressions 12–13 OLS estimators, properties 14–19 ordinary least squares, method of 4–5 prediction problem 19–22 two-sided representation, versus one-sided moving average process 269 two-stage least squares (2SLS) 238–9, 431, 442– 3, 444 two-way fixed effects specification 657 type II errors 52 ϒ and X correlation coefficients between 5–8 testing significance of dependence between 55–8 unbiasedness/unbiased estimators 14 see also best linear unbiased estimator (BLUE); bias asymptotic unbiasedness 206 heteroskedasticity 84 multiple regression 32, 35, 44 panel unit root testing 830 unbounded memory 326 uncertainty forecast 373, 387–9 parameter 388 unconditional models 243 uncovered interest parity (UIP) 575, 580 undetermined coefficients method, RE models 470–2 uniform (or rectangular) kernel 114 uniform distributions 974 uniform mixing coefficient 183 uniform strong law of large numbers 179–80 unit root processes and tests 324–57, 351–7 see also cointegration analysis ADF–GLS unit root test 341–2 Dickey–Fuller unit root tests see Dickey–Fuller (DF) unit root tests difference stationary processes 324–5 long memory processes 346–7 Lp mixingales 328 martingale difference process 327–8 martingale process 326–7 max ADF unit root test 345 models with intercepts and a linear trend 340–1, 342 models with intercepts but without trend 340, 341–2 Phillips–Perron test 339–1 probability limits involving unit root processes 984 rational expectations models 132 related processes 326–8 short memory processes 346 stationarity, testing for 34 trend-cycle decomposition of unit root processes Beveridge–Nelson decomposition 358, 364–7, 368 stochastic trend representation 368–9 Watson decomposition 367 trend-stationary versus first difference stationary processes 328–9 unit roots and cointegration in panels see panel cointegration; panel unit root testing variance ratio test 329–32 vector autoregressive models 509 weighted symmetric tests of unit root 342–4 United Kingdom diffusion of house prices 761, 763 financial linkages between London and New York 932 long-run structural model for UK 574–80 United States as dominant economy 928 financial linkages between London and New York 932 house prices 844–8 monetary policy shocks 927 negative credit supply shocks 931 output growth (VAR models) 513, 514, 519 Value-at-Risk (VaR) analysis conditional correlation of asset returns 609 conditional correlation of asset returns, modelling 618 probability event forecasts 377 1063 statistical models of returns 140–1 VAR models see vector autoregressive (VAR) models variables see also exogeneity canonical 483, 485 common, introducing 907–8 dummy see dummy variables endogenous 130–3, 431, 441–5 exogenous 130 forcing 26, 132, 133, 468 instrumental see instrumental variables and GMM lagged dependent 112 linear regressions that are non-linear in 47–8 models with different transformations of dependent variable 253–9 multi-category, predictability tests for 400–6 omitted variable problem, misspecification 45 one-period lagged dependent 479 random see random variables relationship between two see two variables, relationship between three variable models 33, 59, 91 trended and non-trended 192 variance ratio test 329–32 variance-inflation factor (VIF) 70 VARMA processes 482, 551 VARX modelling see vector autoregressive process with exogenous variables (VARX) modelling VEC models see vector error correction (VEC) models vector autoregressive process with exogenous variables (VARX) modelling 563–83, 581–3, 596 efficient estimation 567–8 empirical application 574–80 estimation and testing of model 577–80 five cases 568 forecasting using 573–4 and GVAR modelling 901, 913 higher-order lags 566 ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 8/9/2015, SPi ✐ 1064 Subject Index vector autoregressive process with exogenous variables (VARX) modelling (cont.) identifying long-run relationships in a cointegrating VARX 572–3 impulse response analysis in models 595–7 long-run structural model for UK 574–80 testing for cointegration in 569–72 testing Hr against Hmy 571 testing Hr against Hr+1 570–1 testing Hr in presence of I(0) weakly exogenous regressors 571–2 testing weak exogeneity 569 weakly exogenous I(1) variables 563–6 vector autoregressive (VAR) models 520–2 see also autoregressive (AR) processes; cointegration analysis Beveridge–Nelson decomposition in 552–6 cointegration of VAR asymptotic distribution of trace statistic 541–3 impulse response analysis 596–7 maximum eigenvalue statistic 540–1 multiple cointegrating relations 529–30 testing for cointegration 540–3 trace statistic 541–3 treatment of trends 536–8 companion form of VAR(p) model 508 deterministic components 510–12 estimation 509–10 factor-augmented, aggregation of 872–7 forecasting with multivariate models 517–18 Granger causality 513–17 high dimensional VARs 900, 901, 911–14 large Bayesian 902 large-scale VAR reduced form data representation 901–3 multivariate spectral density 518–20 output growths 513, 514, 515, 516, 518, 519 panel cointegration 839 Panel VARs 695, 852, 902, 903 short-run effects in structural models, identification 598–600 stationary conditions for VAR (p) 508–9 SVARs 600–1, 603 testing for block Granger non-causality 516–17 unit root case 509 VAR order selection 512–13 VAR(1) model 507, 517–18, 519–20, 532, 878 VAR(p) model 508–9, 535, 536, 586, 598 vector error correction (VEC) models see also cointegration analysis estimation of short-run parameters 549–50 and GVAR modelling 924 small sample properties of test statistics 547 treatment of trends 536 and VARX models 567, 568, 569 volatility 426–8 conditional variance models 412–13 econometric approaches 413–17 Absolute GARCH-in-mean model 417 ARCH(1) and GARCH(1,1) specifications 414–15 exponential GARCH-in-mean model 416–17 higher-order GARCH models 415–16 estimation of ARCH and ARCH-in-mean models 420–3 ML estimation with Gaussian errors 421 ML estimation with Student’s t-distributed errors 421–3 forecasting with GARCH models 423–5 implied, market-based 411 intra-daily returns 411 measurement and modelling of 411–28 parameter variations and ARCH effects 420 and predictability 159–60 realized 412 RiskMetrics™ ( JP Morgan) method 412–13 risk-return relationships 419–20 stochastic models 419 testing for ARCH/GARCH effects 417–19 Wald test procedure 117, 125, 438, 526, 822 maximum likelihood (ML) estimation 195, 212, 214–22 Watson decomposition 367 weak law of large numbers (WLLN) 178, 181 weak stationarity, stochastic processes 268 weather shocks 932 weekly returns, volatilities and conditional correlations in 620–9 asset specific estimates 623–4 changing volatilities and correlations 626–9 devolatized returns, properties 621–2 ML estimation 622–3 post estimation evaluation of t-DCC model 624–5 recursive estimates and VaR diagnostics 625–6 weighted symmetric tests of unit root critical values 345 treatment of deterministic components 344 weighted symmetric estimates 342–4 white noise process 268, 269 Wiener processes 115, 335 window size (bandwidth) 78, 114, 116 Wold’s decomposition 275, 364 Wright’s demand equation 228–9 W-test (non-nested) 252 Yule–Walker equations/estimators 280, 308–9 zero concordance 58 zero mean 10, 25, 179 ✐ ✐ ✐ ... SPi ✐ TIME SERIES AND PANEL DATA ECONOMETRICS ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ ✐ ✐ ✐ ✐ ✐ OUP CORRECTED PROOF – FINAL, 10/9/2015, SPi ✐ Time Series and Panel Data Econometrics. .. distinct from other time series texts in the sense that it also covers panel data models and attempts at a more coherent integration of time series, multivariate analysis, and panel data models It... developments in time series and panel data techniques for the analysis of macroeconomic and financial data It provides a rigorous, nevertheless user-friendly, account of the time series techniques