Linear models and time series analysis regression, ANOVA, ARMA and GARCH

Linear Models and Time-Series Analysis The Wiley Series in Probability and Statistics is well established and authoritative It covers many topics of current research interest in both pure and applied statistics and probability theory Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches This series provides essential and invaluable reading for all statisticians, whether in academia, industry, government, or research Series Editors: David J Balding, University College London, UK Noel A Cressie, University of Wollongong, Australia Garrett Fitzmaurice, Havard School of Public Health, USA Harvey Goldstein, University of Bristol, UK Geof Givens, Colorado State University, USA Geert Molenberghs, Katholieke Universiteit Leuven, Belgium David W Scott, Rice University, USA Ruey S Tsay, University of Chicago, USA Adrian F M Smith, University of London, UK Related Titles Quantile Regression: Estimation and Simulation, Volume by Marilena Furno, Domenico Vistocco Nonparametric Finance by Jussi Klemela February 2018 Machine Learning: Topics and Techniques by Steven W Knox February 2018 Measuring Agreement: Models, Methods, and Applications by Pankaj K Choudhary, Haikady N Nagaraja November 2017 Engineering Biostatistics: An Introduction using MATLAB and WinBUGS by Brani Vidakovic October 2017 Fundamentals of Queueing Theory, 5th Edition by John F Shortle, James M Thompson, Donald Gross, Carl M Harris October 2017 Reinsurance: Actuarial and Statistical Aspects by Hansjoerg Albrecher, Jan Beirlant, Jozef L Teugels September 2017 Clinical Trials: A Methodologic Perspective, 3rd Edition by Steven Piantadosi August 2017 Advanced Analysis of Variance by Chihiro Hirotsu August 2017 Matrix Algebra Useful for Statistics, 2nd Edition by Shayle R Searle, Andre I Khuri April 2017 Statistical Intervals: A Guide for Practitioners and Researchers, 2nd Edition by William Q Meeker, Gerald J Hahn, Luis A Escobar March 2017 Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, 2nd Edition by Katsuto Tanaka March 2017 Probability and Conditional Expectation: Fundamentals for the Empirical Sciences by Rolf Steyer, Werner Nagel March 2017 Theory of Probability: A critical introductory treatment by Bruno de Finetti February 2017 Simulation and the Monte Carlo Method, 3rd Edition by Reuven Y Rubinstein, Dirk P Kroese October 2016 Linear Models, 2nd Edition by Shayle R Searle, Marvin H J Gruber October 2016 Robust Correlation: Theory and Applications by Georgy L Shevlyakov, Hannu Oja August 2016 Statistical Shape Analysis: With Applications in R, 2nd Edition by Ian L Dryden, Kanti V Mardia July 2016 Matrix Analysis for Statistics, 3rd Edition by James R Schott June 2016 Statistics and Causality: Methods for Applied Empirical Research by Wolfgang Wiedermann (Editor), Alexander von Eye (Editor) May 2016 Time Series Analysis by Wilfredo Palma February 2016 Linear Models and Time-Series Analysis Regression, ANOVA, ARMA and GARCH Marc S Paolella Department of Banking and Finance University of Zurich Switzerland This edition first published 2019 © 2019 John Wiley & Sons Ltd 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, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions The right of Dr Marc S Paolella to be identified as the author of this work has been asserted in accordance with law Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office 9600 Garsington Road, Oxford, OX4 2DQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com Wiley also publishes its books in a variety of electronic formats and by print-on-demand Some content that appears in standard print versions of this book may not be available in other formats Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make This work is sold with the understanding that the publisher is not engaged in rendering professional services The advice and strategies contained herein may not be suitable for your situation You should consult with a specialist where appropriate Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages ® ® MATLAB is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This work’s use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB software Library of Congress Cataloging-in-Publication Data Names: Paolella, Marc S., author Title: Linear models and time-series analysis : regression, ANOVA, ARMA and GARCH / Dr Marc S Paolella Description: Hoboken, NJ : John Wiley & Sons, 2019 | Series: Wiley series in probability and statistics | Identifiers: LCCN 2018023718 (print) | LCCN 2018032640 (ebook) | ISBN 9781119431855 (Adobe PDF) | ISBN 9781119431985 (ePub) | ISBN 9781119431909 (hardcover) Subjects: LCSH: Time-series analysis | Linear models (Statistics) Classification: LCC QA280 (ebook) | LCC QA280 P373 2018 (print) | DDC 515.5/5–dc23 LC record available at https://lccn.loc.gov/2018023718 Cover Design: Wiley Cover Images: Images courtesy of Marc S Paolella Set in 10/12pt WarnockPro by SPi Global, Chennai, India 10 ® v Contents Preface xiii Part I Linear Models: Regression and ANOVA 1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.4.6 1.4.7 1.5 1.6 1.7 1.A 1.B 1.C Regression, Correlation, and Causality Ordinary and Generalized Least Squares Ordinary Least Squares Estimation Further Aspects of Regression and OLS Generalized Least Squares 12 The Geometric Approach to Least Squares 17 Projection 17 Implementation 22 Linear Parameter Restrictions 26 Formulation and Estimation 27 Estimability and Identifiability 30 Moments and the Restricted GLS Estimator 32 Testing With h = 34 Testing With Nonzero h 37 Examples 37 Confidence Intervals 42 Alternative Residual Calculation 47 Further Topics 51 Problems 56 Appendix: Derivation of the BLUS Residual Vector Appendix: The Recursive Residuals 64 Appendix: Solutions 66 Fixed Effects ANOVA Models 77 2.1 2.2 2.3 Introduction: Fixed, Random, and Mixed Effects Models 77 Two Sample t-Tests for Differences in Means 78 The Two Sample t-Test with Ignored Block Effects 84 The Linear Model 60 vi Contents 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 2.5 2.5.1 2.5.2 2.5.3 2.5.4 One-Way ANOVA with Fixed Effects 87 The Model 87 Estimation and Testing 88 Determination of Sample Size 91 The ANOVA Table 93 Computing Confidence Intervals 97 A Word on Model Assumptions 103 Two-Way Balanced Fixed Effects ANOVA 107 The Model and Use of the Interaction Terms 107 Sums of Squares Decomposition Without Interaction 108 Sums of Squares Decomposition With Interaction 113 Example and Codes 117 Introduction to Random and Mixed Effects Models 127 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.6.1 3.1.6.2 3.2 3.2.1 3.2.1.1 3.2.1.2 3.2.2 3.3 3.3.1 3.3.1.1 3.3.1.2 3.3.1.3 3.3.2 3.3.2.1 3.3.2.2 3.3.2.3 3.4 3.A One-Factor Balanced Random Effects Model 128 Model and Maximum Likelihood Estimation 128 Distribution Theory and ANOVA Table 131 Point Estimation, Interval Estimation, and Significance Testing 137 Satterthwaite’s Method 139 Use of SAS 142 Approximate Inference in the Unbalanced Case 143 Point Estimation in the Unbalanced Case 144 Interval Estimation in the Unbalanced Case 150 Crossed Random Effects Models 152 Two Factors 154 With Interaction Term 154 Without Interaction Term 157 Three Factors 157 Nested Random Effects Models 162 Two Factors 162 Both Effects Random: Model and Parameter Estimation 162 Both Effects Random: Exact and Approximate Confidence Intervals 167 Mixed Model Case 170 Three Factors 174 All Effects Random 174 Mixed: Classes Fixed 176 Mixed: Classes and Subclasses Fixed 177 Problems 177 Appendix: Solutions 178 Part II Time-Series Analysis: ARMAX Processes 185 The AR(1) Model 187 4.1 4.2 4.3 Moments and Stationarity 188 Order of Integration and Long-Run Variance 195 Least Squares and ML Estimation 196 Contents 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.4 4.5 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.6.5 4.6.6 4.7 4.8 OLS Estimator of a 196 Likelihood Derivation I 196 Likelihood Derivation II 198 Likelihood Derivation III 198 Asymptotic Distribution 199 Forecasting 200 Small Sample Distribution of the OLS and ML Point Estimators 204 Alternative Point Estimators of a 208 Use of the Jackknife for Bias Reduction 208 Use of the Bootstrap for Bias Reduction 209 Median-Unbiased Estimator 211 Mean-Bias Adjusted Estimator 211 Mode-Adjusted Estimator 212 Comparison 213 Confidence Intervals for a 215 Problems 219 Regression Extensions: AR(1) Errors and Time-varying Parameters 223 5.1 5.2 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.4.1 5.3.4.2 5.4 5.5 5.5.1 5.5.2 5.6 5.6.1 5.6.2 5.6.3 5.6.3.1 5.6.3.2 5.6.4 The AR(1) Regression Model and the Likelihood 223 OLS Point and Interval Estimation of a 225 Testing a = in the ARX(1) Model 229 Use of Confidence Intervals 229 The Durbin–Watson Test 229 Other Tests for First-order Autocorrelation 231 Further Details on the Durbin–Watson Test 236 The Bounds Test, and Critique of Use of p-Values 236 Limiting Power as a → ±1 239 Bias-Adjusted Point Estimation 243 Unit Root Testing in the ARX(1) Model 246 Null is a = 248 Null is a < 256 Time-Varying Parameter Regression 259 Motivation and Introductory Remarks 260 The Hildreth–Houck Random Coefficient Model 261 The TVP Random Walk Model 269 Covariance Structure and Estimation 271 Testing for Parameter Constancy 274 Rosenberg Return to Normalcy Model 277 Autoregressive and Moving Average Processes 281 AR(p) Processes 281 Stationarity and Unit Root Processes 282 Moments 284 Estimation 287 Without Mean Term 287 Starting Values 290 6.1 6.1.1 6.1.2 6.1.3 6.1.3.1 6.1.3.2 vii viii Contents 6.1.3.3 6.1.3.4 6.2 6.2.1 6.2.2 6.3 6.A With Mean Term 292 Approximate Standard Errors 293 Moving Average Processes 294 MA(1) Process 294 MA(q) Processes 299 Problems 301 Appendix: Solutions 302 ARMA Processes 311 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.2 7.3 7.3.1 7.3.2 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.5 7.5.1 7.5.2 7.5.3 7.6 7.7 7.7.1 7.7.2 7.7.3 7.8 7.A 7.B Basics of ARMA Models 311 The Model 311 Zero Pole Cancellation 312 Simulation 313 The ARIMA(p, d, q) Model 314 Infinite AR and MA Representations 315 Initial Parameter Estimation 317 Via the Infinite AR Representation 318 Via Infinite AR and Ordinary Least Squares 318 Likelihood-Based Estimation 322 Covariance Structure 322 Point Estimation 324 Interval Estimation 328 Model Mis-specification 330 Forecasting 331 AR(p) Model 331 MA(q) and ARMA(p, q) Models 335 ARIMA(p, d, q) Models 339 Bias-Adjusted Point Estimation: Extension to the ARMAX(1, q) model 339 Some ARIMAX Model Extensions 343 Stochastic Unit Root 344 Threshold Autoregressive Models 346 Fractionally Integrated ARMA (ARFIMA) 347 Problems 349 Appendix: Generalized Least Squares for ARMA Estimation 351 Appendix: Multivariate AR(p) Processes and Stationarity, and General Block Toeplitz Matrix Inversion 357 Correlograms 8.1 8.1.1 8.1.2 8.1.3 8.1.3.1 8.1.3.2 8.1.3.3 359 Theoretical and Sample Autocorrelation Function 359 Definitions 359 Marginal Distributions 365 Joint Distribution 371 Support 371 Asymptotic Distribution 372 Small-Sample Joint Distribution Approximation 375 Contents 8.1.4 8.2 8.2.1 8.2.2 8.2.2.1 8.2.2.2 8.2.2.3 8.3 8.A Conditional Distribution Approximation 381 Theoretical and Sample Partial Autocorrelation Function 384 Partial Correlation 384 Partial Autocorrelation Function 389 TPACF: First Definition 389 TPACF: Second Definition 390 Sample Partial Autocorrelation Function 392 Problems 396 Appendix: Solutions 397 ARMA Model Identification 405 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Introduction 405 Visual Correlogram Analysis 407 Significance Tests 412 Penalty Criteria 417 Use of the Conditional SACF for Sequential Testing 421 Use of the Singular Value Decomposition 436 Further Methods: Pattern Identification 439 Part III Modeling Financial Asset Returns 443 10.1 10.2 10.2.1 10.2.2 10.2.3 10.2.4 10.3 10.3.1 10.3.2 10.3.3 10.4 10.5 10.6 10.6.1 10.6.2 10.6.3 10.6.4 10.6.5 10.6.6 445 Introduction 445 Gaussian GARCH and Estimation 450 Basic Properties 451 Integrated GARCH 452 Maximum Likelihood Estimation 453 Variance Targeting Estimator 459 Non-Gaussian ARMA-APARCH, QMLE, and Forecasting 459 Extending the Volatility, Distribution, and Mean Equations 459 Model Mis-specification and QMLE 464 Forecasting 467 Near-Instantaneous Estimation of NCT-APARCH(1,1) 468 S𝛼,𝛽 -APARCH and Testing the IID Stable Hypothesis 473 Mixed Normal GARCH 477 Introduction 477 The MixN(k)-GARCH(r, s) Model 478 Parameter Estimation and Model Features 479 Time-Varying Weights 482 Markov Switching Extension 484 Multivariate Extensions 484 11 Risk Prediction and Portfolio Optimization 10 11.1 Univariate GARCH Modeling 487 Value at Risk and Expected Shortfall Prediction 487 ix x Contents 11.2 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.3 11.3.1 11.3.2 11.3.3 11.3.4 11.3.5 MGARCH Constructs Via Univariate GARCH 493 Introduction 493 The Gaussian CCC and DCC Models 494 Morana Semi-Parametric DCC Model 497 The COMFORT Class 499 Copula Constructions 503 Introducing Portfolio Optimization 504 Some Trivial Accounting 504 Markowitz and DCC 510 Portfolio Optimization Using Simulation 513 The Univariate Collapsing Method 516 The ES Span 521 12 Multivariate t Distributions 12.1 12.2 12.3 12.4 12.5 12.5.1 12.5.2 12.5.3 12.5.4 12.5.5 12.6 12.6.1 12.6.2 12.6.3 12.6.4 12.6.5 12.7 12.A 12.B 525 Multivariate Student’s t 525 Multivariate Noncentral Student’s t 530 Jones Multivariate t Distribution 534 Shaw and Lee Multivariate t Distributions 538 The Meta-Elliptical t Distribution 540 The FaK Distribution 541 The AFaK Distribution 542 FaK and AFaK Estimation: Direct Likelihood Optimization 546 FaK and AFaK Estimation: Two-Step Estimation 548 Sums of Margins of the AFaK 555 MEST: Marginally Endowed Student’s t 556 SMESTI Distribution 557 AMESTI Distribution 558 MESTI Estimation 561 AoNm -MEST 564 MEST Distribution 573 Some Closing Remarks 574 ES of Convolution of AFaK Margins 575 Covariance Matrix for the FaK 581 13 Weighted Likelihood 13.1 13.2 13.3 13.4 587 Concept 587 Determination of Optimal Weighting 592 Density Forecasting and Backtest Overfitting 594 Portfolio Optimization Using (A)FaK 600 14 Multivariate Mixture Distributions 14.1 14.1.1 14.1.2 14.1.3 611 The Mixk Nd Distribution 611 Density and Simulation 612 Motivation for Use of Mixtures 612 Quasi-Bayesian Estimation and Choice of Prior 614 Bibliography Rockinger, M and Jondeau, E (2002) Entropy Densities with an Application to Autoregressive Conditional Skewness and Kurtosis Journal of Econometrics, 106:119–142 Romano, J P and Wolf, M (2001) Subsampling Intervals in Autoregressive Models with Linear Time Trend Econometrica, 69:1283–1314 Rombouts, J V K and Stentoft, L (2009) Bayesian Option Pricing Using Mixed Normal Heteroskedasticity Models CREATES Research Papers 2009-07, School of Economics and Management, University of Aarhus Rombouts, J V K and Stentoft, L (2011) Multivariate Option Pricing with Time Varying Volatility and Correlations Journal of Banking & Finance, 35(9):2267–2281 Rosenberg, B (1973) The Analysis of a Cross-Section of Time Series by Stochastically Convergent Parameter Regression Annals of Economic and Social Measurement, 2:399–428 Rosenkrantz, W A (1997) Introduction to Probability and Statistics for Scientists and Engineers McGraw-Hill, New York Rounvinez, C (1997) Going Greek with VaR Risk, 10(2):57–65 Roussas, G G (1997) A Course in Mathematical Statistics Academic Press, San Diego, 2nd edition Rousseau, P and Wachtel, P (2002) Inflation Thresholds and the Finance-Growth Nexus Journal of International Money and Finance, 21:777–793 Roy, S N (1953) On a Heuristic Method of Test Construction and its use in Multivariate Analysis Annals of Mathematical Statistics, 24(2):220–238 Rubin, H (1950) Note on Random Coefficients In Koopmans, T C., editor, Statistical Inference in Dynamic Economic Models: Cowles Commission for Research in Economics, Monograph No 10, pages 419–421 John Wiley & Sons, New York Ruppert, D (2004) Statistics and Finance: An Introduction Springer, New York Russell, B (2009) The ABC of Relativity Routledge, Taylor & Francis Group, London Originally published: A B C of Relativity, George Allen & Unwin, London, 1925 Ruud, P A (2000) An Introduction to Classical Econometric Theory Oxford University Press, Oxford Sahai, H and Ageel, M I (2000) The Analysis of Variance: Fixed, Random and Mixed Models Springer, New York Sahai, H and Ojeda, M M (2004) Analysis of Variance for Random Models Volume I: Balanced Data Theory, Methods, Applications and Data Analysis Birkhäuser, Boston Sahai, H and Ojeda, M M (2005) Analysis of Variance for Random Models Volume II: Unbalanced Data Theory, Methods, Applications and Data Analysis Birkhäuser, Boston Saikkonen, P and Lütkepohl, H (2002) Testing for a Unit Root in a Time Series with a Level Shift at Unknown Time Econometric Theory, 18:313–348 Samworth, R (2005) Small Confidence Sets for the Mean of a Spherically Symmetric Distribution Journal of the Royal Statistical Society, Series B, 67:343–361 Santos, A A P and Moura, G V (2014) Dynamic Factor Multivariate GARCH Model Computational Statistics & Data Analysis, 76:606–617 Santos, A A P., Nogales, F J., and Ruiz, E (2013) Comparing Univariate and Multivariate Models to Forecast Portfolio Value-at-Risk Journal of Financial Econometrics, 11(2):400–441 Sargan, J D and Bhargava, A (1983) Testing Residuals from Least Squares Regression for being Generated by the Gaussian Random Walk Econometrica, 51:153–174 SAS/STAT 9.2 User’s Guide (2008) SAS Institute Inc., Cary, NC, USA Satterthwaite, F E (1946) An Approximate Distribution of Estimates of Variance Components Biometrics Bulletin, 2:110–114 865 866 Bibliography Sawa, T (1972) Finite Sample Properties of the k-Class Estimator Econometrica, 40(4):653–680 Sawa, T (1978) The Exact Moments of the Least Squares Estimator for the Autoregressive Model Journal of Econometrics, 8:159–172 Scheffé, H (1953) A Method of Judging all Contrasts in the Analysis of Variance Biometrika, 40:87–104 Scheffé, H (1959) The Analysis of Variance John Wiley & Sons, New York Scherrer, A., Larrieu, N., Owezarski, P., Borgnat, P., and Abry, P (2007) Non-Gaussian and Long Memory Statistical Characterizations for Internet Traffic with Anomalies IEEE Transactions on Dependable and Secure Computing, 4(1):56–70 Schlattmann, P (2009) Medical Applications of Finite Mixture Models Springer, Heidelberg Schoenberg, I J (1938) Metric Spaces and Completely Monotone Functions Annals of Mathematics, Second Series, 39(4):811–841 Scholz, M., Nielsen, J P., and Sperlich, S (2012) Nonparametric Prediction of Stock Returns Guided by Prior Knowledge Graz economics papers, University of Graz, Department of Economics Scholz, M., Nielsen, J P., and Sperlich, S (2015) Nonparametric Prediction of Stock Returns Based on Yearly Data: The Long-Term View Insurance: Mathematics and Economics, 65:143–155 Schott, J R (2002) Testing for Elliptical Symmetry in Covariance-Matrix Based Analyses Statistics and Probability Letters, 60(4):395–404 Schott, J R (2005) Matrix Analysis for Statistics John Wiley & Sons, New York, 2nd edition Schur, I (1917) Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind Journal für die reine und angewandte Mathematik, 147:205–232 Schwert, G W (1989a) Tests for Unit Roots: A Monte Carlo Investigation Journal of Business and Economics Statistics, 7:147–159 Schwert, W G (1989b) Why Does Stock Market Volatility Change Over Time? Journal of Finance, 44:1115–1153 Searle, S R (1971) Linear Models John Wiley & Sons, New York Searle, S R (1982) Matrix Algebra Useful for Statistics John Wiley & Sons, New York Searle, S R., Casella, G., and McCulloch, C E (1992) Variance Components John Wiley & Sons, New York Searle, S R and Gruber, M H J (2017) Linear Models John Wiley & Sons, Hoboken, NJ, 2nd edition Seber, G A F and Lee, A J (2003) Linear Regression Analysis John Wiley & Sons, Hoboken, NJ, 2nd edition Segnon, M., Lux, T., and Gupta, R (2017) Modeling and Forecasting the Volatility of Carbon Dioxide Emission Allowance Prices: A Review and Comparison of Modern Volatility Models Renewable and Sustainable Energy Reviews, 69:692–704 Semrl, P (1996) On a Matrix Version of Cochran’s Statistical Theorem Linear Algebra and its Applications, 237–238:477–487 Seneta, E (2004) Fitting the Variance-Gamma Model to Financial Data Journal of Applied Probability, 41, Stochastic Methods and Their Applications:177–187 Sentana, E (1995) Quadratic ARCH models The Review of Economic Studies, 62(4):639–661 Severini, T A (2005) Elements of Distribution Theory Cambridge University Press, Cambridge Shaman, P and Stine, R A (1988) The Bias of Autoregressive Coefficient Estimators Journal of the American Statistical Association, 83(403):842–848 Shao, J and Tu, D (1995) The Jackknife and Bootstrap Springer, New York Shaw, W T and Lee, K T A (2008) Bivariate Student t Distributions with Variable Marginal Degrees of Freedom and Independence Journal of Multivariate Analysis, 99:1276–1287 Bibliography Shi, S and Song, Y (2016) Identifying Speculative Bubbles Using an Infinite Hidden Markov Model Journal of Financial Econometrics, 14(1):159–184 Shipley, B (2016) Cause and Correlation in Biology Cambridge University Press, Cambridge Shively, P A (2001) Trend-Stationary GNP: Evidence from a new exact Pointwise Most Powerful Invariant Unit Root Test Journal of Applied Econometrics, 16:537–551 Shively, P A (2003) The Nonlinear Dynamics of Stock Prices The Quarterly Review of Economics and Finance, 43(3):505–517 Shively, T S (1988a) An Analysis of Tests for Regression Coefficient Stability Journal of Econometrics, 39:367–386 Shively, T S (1988b) An Exact Test for a Stochastic Coefficient in a Time Series Regression Model Journal of Time Series Analysis, 9(1):81–88 Shumway, R H and Stoffer, D S (2000) Time Series Analysis and Its Applications Springer, New York Shumway, T (1997) The Delisting Bias in CRSP Data Journal of Finance, 52(1):327–340 Silva, E S and Hassani, H (2015) On the Use of Singular Spectrum Analysis for Forecasting U.S Trade Before, During and After the 2008 Recession International Economics, 141:34–49 Silvennoinen, A and Teräsvirta, T (2009) Multivariate GARCH Models In Andersen, T G., Davis, R A., Kreiss, J.-P., and Mikosch, T., editors, Handbook of Financial Time Series, pages 201–229 Springer, Berlin Singh, B., Nagar, A L., Choudhry, N K., and Raj, B (1976) On the Estimation of Structural Change: A Generalization of the Random Coefficients Regression Model International Economic Review, 17(2):340–361 Skovgaard, I M (1987) Saddlepoint Expansions for Conditional Distributions Journal of Applied Probability, 24:275–287 Slim, S., Koubaa, Y., and BenSaïda, A (2016) Value-at-Risk Under Lévy GARCH models: Evidence from Global Stock Markets Journal of International Financial Markets, Institutions & Money, 46:30–53 Small, J P (1993) The Limiting Power of Point Optimal Autocorrelation Tests Communications in Statistics—Theory and Methods, 22(2):3907–3916 Smetanina, E (2017) Real-Time GARCH Journal of Financial Econometrics, 15(4):561–601 So, M K P and Choi, C Y (2009) A Multivariate Factor Threshold Stochastic Volatility Model Journal of Forecasting, 28:712–735 So, M K P and Yip, I W H (2012) Multivariate GARCH Models with Correlation Clustering Journal of Forecasting, 31(5):443–468 Sobreira, N and Nunes, L C (2016) Tests for Multiple Breaks in the Trend with Stationary or Integrated Shocks Oxford Bulletin of Economics and Statistics, 78(3):394–411 Sollis, R., Newbold, P., and Leybourne, S J (2000) Stochastic Unit Roots Modelling of Stock Price Indices Applied Financial Economics, 10(3):311–315 Solnik, B and Longin, F (2001) Extreme Correlation of International Equity Markets Journal of Finance, 2(LVI):649–676 Song, D.-K., Park, H.-J., and Kim, H.-M (2014) A Note on the Characteristic Function of Multivariate t Distribution Communications for Statistical Applications and Methods, 21(1):81–91 Sortino, F A and van der Meer, R (1991) Downside Risk The Journal of Portfolio Management, 17(4):27–31 Sowell, F (1992) Maximum Likelihood Estimation of Stationary Univariate Fractionally Integrated Time Series Models Journal of Econometrics, 53(1–3):165–188 Spiegelhalter, D (2017) Trust in Numbers Journal of the Royal Statistical Society, Series A, 180:949–965 867 868 Bibliography Srivastava, M S (1987) Asymptotic Distribution of Durbin-Watson Statistic Economics Letters, 24(2):157–160 Stapleton, J H (1995) Linear Statistical Models John Wiley & Sons, New York Stein, E M and Shakarchi, R (2005) Real Analysis Measure Theory, Integration and Hilbert Spaces Princeton University Press Stigler, S M (1981) Gauss and the Invention of Least Squares Annals of Statistics, 9(3):465–474 Stivers, A (2018) Equity Premium Predictions with Many Predictors: A Risk-Based Explanation of the Size and Value Factors Journal of Empirical Finance, 45:126–140 Stock, J H (1994) Unit Roots, Structural Breaks and Trends In Engle, R F and McFadden, D L., editors, Palgrave Handbook of Econometrics, Volume 4, chapter 46, pages 2739–2841 Elsevier, Amsterdam Stock, J H and Watson, M W (1998) Median Unbiased Estimation of Coefficient Variance in a Time-Varying Parameter Model Journal of the American Statistical Association, 93(441):349–358 Stolbov, M (2013) The Finance-Growth Nexus Revisited: From Origins to a Modern Theoretical Landscape Economics: The Open-Access, Open-Assessment E-Journal, 7(2) Stoyanov, S., Samorodnitsky, G., Rachev, S., and Ortobelli, S (2006) Computing the Portfolio Conditional Value-at-Risk in the alpha-stable Case Probability and Mathematical Statistics, 26:1–22 Stroup, W W and Mulitze, D K (1991) Nearest Neighbor Adjusted Best Linear Unbiased Prediction The American Statistician, 45(3):194–200 Stuart, A and Ord, J K (1994) Kendall’s Advanced Theory of Statistics, Volume 1, Distribution Theory Edward Arnold, London, 6th edition Stuart, A., Ord, J K., and Arnold, S F (1999) Kendall’s Advanced Theory of Statistics, Volume 2A, Classical Inference and the Linear Model Edward Arnold, London, 6th edition Su, Y (2012) Smooth Test for Elliptical Symmetry In 2012 International Conference on Machine Learning and Cybernetics, volume 4, pages 1279–1284 Sucarrat, G., Pretis, F., and Reade, J (2017) gets: General-to-Specific (GETS) Modelling and Indicator Saturation Methods R package version 0.12 Available at: https://CRAN.R-project.org/package=gets Suh, S (2016) A Combination Rule for Portfolio Selection with Transaction Costs International Review of Finance, 16(3):393–420 Sutradhar, B C (1986) On the Characteristic Function of Multivariate Student t-Distribution Canadian Journal of Statistics, 14(4):329–337 Swamy, P A V B (1971) Statistical Inference in Random Coefficient Regression Models Springer, New York Swamy, P A V B., Conway, R K., and LeBlanc, M R (1988) The Stochastic Coefficients Approach to Econometric Modeling Part I: A Critique of Fixed Coefficients Models Journal of Agricultural Economics Research, 40(2):2–10 Swamy, P A V B., Hall, S G., Tavlas, G S., and von zur Muehlen, P (2017) On The Interpretation of Instrumental Variables in the Presence of Specification Errors: A Reply Econometrics, 5(3):1–3 Article 32 Swamy, P A V B and Tavlas, G S (1995) Random Coefficient Models: Theory and Applications Journal of Economic Surveys, 9(2):165–196 Swamy, P A V B and Tavlas, G S (2001) Random Coefficient Models In Baltagi, B H., editor, A Companion to Theoretical Econometrics, chapter 19, pages 410–428 Blackwell Publishing, Oxford Swamy, P A V B., Tavlas, G S., and Hall, S G (2015) On the Interpretation of Instrumental Variables in the Presence of Specification Errors Econometrics, 3(1):55–64 Bibliography Tamhane, A C and Dunlop, D D (2000) Statistics and Data Analysis: From Elementary to Intermediate Prentice Hall, Upper Saddle River, NJ Tanaka, K (1996) Time Series Analysis: Nonstationary and Noninvertible Distribution Theory John Wiley & Sons Ltd, New York Tanizaki, H (2000) Bias Correction of OLSE in the Regression Model with Lagged Dependent Variables Journal of Computational Statistics & Data Analysis, 34:495–511 Tashman, A (2010) A Regime-switching Approach to Model-based Stress Testing Journal of Risk Model Validation, 3:89–101 Tashman, A and Frey, R J (2009) Modeling Risk in Arbitrage Strategies Using Finite Mixtures Quantitative Finance, 9:495–503 Tay, A S and Wallis, K F (2000) Density Forecasting: A Survey Journal of Forecasting, 19(4):124–143 Tayefi, M and Ramanathan, T V (2012) An Overview of FIGARCH and Related Time Series Models Austrian Journal of Statistics, 41(3):175–196 Taylor, J W (2008) Estimating Value at Risk and Expected Shortfall Using Expectiles Journal of Financial Econometrics, 6(2):231–252 Taylor, J W and Yu, K (2016) Using Auto-Regressive Logit Models to Forecast the Exceedance Probability for Financial Risk Management Journal of the Royal Statistical Society, Series A, Statistics in Society, 179(4):1069–1092 Taylor, S (1986) Modelling Financial Time Series John Wiley & Sons, New York Temme, N M (1982) The Uniform Asymptotic Expansion of a Class of Integrals Related to Cumulative Distribution Functions SIAM Journal of Mathematical Analysis, 13:239–253 Teräsvirta, T (1994) Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models Journal of the American Statistical Association, 89:208–218 Teräsvirta, T (1998) Modelling Economic Relationships with Smooth Transition Regressions In Ullah, A and Giles, D E A., editors, Handbook of Applied Economic Statistics, pages 507–552 Marcel Dekker, New York Teräsvirta, T (2009) An Introduction to Univariate GARCH Models In Andersen, T G., Davis, R A., Kreiß, J.-P., and Mikosch, T., editors, Handbook of Financial Time Series, pages 17–42 Springer, Berlin Teräsvirta, T., Tjøstheim, D., and Granger, C W J (2010) Modelling Nonlinear Economic Time Series Oxford University Press, Oxford Teräsvirta, T and Zhao, Z (2011) Stylized Facts of Return Series, Robust Estimates and Three Popular Models of Volatility Applied Financial Economics, 21:67–94 Theil, H (1965) The Analysis of Disturbances in Regression Analysis Journal of the American Statistical Association, 60:1067–1079 Theil, H (1968) A Simplification of the BLUS Procedure for Analyzing Regression Disturbances Journal of the American Statistical Association, 63:242–251 Theil, H (1971) Principles of Econometrics John Wiley & Sons, New York Theil, H and Goldberger, A S (1961) On Pure and Mixed Statistical Estimation in Economics International Economic Review, 2(1):65–78 Theodossiou, P (1998) Financial Data and the Skewed Generalized T Distribution Management Science, 44(12):1650–1661 Thiel, H and Mennes, L B M (1959) Multiplicative Randomness in Time Series Regression Analysis Mimeographed Report No 5901 Thode, Jr., H C (2002) Testing for Normality Marcel Dekker, New York 869 870 Bibliography Tiao, G C and Box, G E P (1981) Modelling Multiple Time Series with Applications Journal of the American Statistical Association, 76:802–816 Tillman, J A (1975) The Power of the Durbin–Watson Test Econometrica, 43:959–974 Timmermann, A (2000) Density Forecasting in Economics and Finance Journal of Forecasting, 19(4):231–234 Tjøstheim, D (1986) Some Doubly Stochastic Time Series Journal of Time Series Analysis, 7:51–72 Tong, H (1978) On a Threshold Model In Chen, C H., editor, Pattern Recognition and Signal Processing, pages 575–586 Sijhoff and Noordhoff, Alpen aan den Rijn Tong, H (1983) Threshold Models in Non-linear Time Series Analysis Lecture Notes in Statistics, No 21 Springer, New York Tong, H (1990) Non-linear Time Series: A Dynamical System Approach Oxford University Press, Oxford Tong, H (2007) Birth of the Threshold Time Series Model Statistica Sinica, 17:8–14 Tong, H (2011) Threshold Models in Time Series Analysis—30 Years On (with discussion) Statistics and Its Interface, 4:107–136 Tong, H and Lim, K S (1980) Threshold Autoregression, Limit Cycles and Cyclical Data (with discussion) Journal of the Royal Statistical Society, Series B, 42(3):245–292 Trench, W F (2003) Introduction to Real Analysis Prentice Hall, Upper Saddle River, NJ Tsay, R S (1998) Testing and Modeling Multivariate Threshold Models Journal of the American Statistical Association, 93(443):1188–1202 Tsay, R S (2010) Analysis of Financial Time Series John Wiley & Sons, Hoboken, NJ, 3rd edition Tsay, R S (2012) An Introduction to Analysis of Financial Data with R John Wiley & Sons, Hoboken, NJ Tsay, R S (2014) Multivariate Time Series Analysis: With R and Financial Applications John Wiley & Sons, Hoboken, NJ Tse, Y K and Tsui, A K C (2002) A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model With Time-Varying Correlations Journal of Business and Economic Statistics, 20(3):351–362 Tunnicliffe Wilson, G (1979) Some Efficient Computational Procedures for High Order ARMA Models Journal of Statistical Computation and Simulation, 8:301–309 Ullah, A., Srivastava, V K., and Roy, N (1995) Moments of the Function of Non-Normal Random Vector with Applications to Econometric Estimators and Test Statistics Econometric Reviews, 14(4):459–471 Uppuluri, V R R and Carpenter, J A (1969) The Inverse of a Matrix Occurring in First-Order Moving-Average Models Sankhya, Series A, 31(1):79–82 van Belle, G (2008) Statistical Rules of Thumb John Wiley & Sons, Hoboken, NJ, 2nd edition van der Leeuw, J (1994) The Covariance Matrix of ARMA Errors in Closed Form Journal of Econometrics, 63(2):397–405 van der Weide, R (2002) GO-GARCH: A Multivariate Generalized Orthogonal GARCH Model Journal of Applied Econometrics, 17:549–564 van Dijk, D., Teräsvirta, T., and Franses, P H (2002) Smooth Transition Autoregressive Models—A Survey of Recent Developments Econometric Reviews, 21:1–47 Vandebril, R., Van Barel, M., and Mastronardi, N (2008) Matrix Computations and Semiseparable Matrices Volume I: Linear Systems The Johns Hopkins University Press, Baltimore Vargas, G A (2006) An Asymmetric Block Dynamic Conditional Correlation Multivariate GARCH Model The Philippine Statistician, 55(1–2):83–102 Bibliography Vaynman, I and Beare, B K (2014) Stable Limit Theory for the Variance Targeting Estimator In Chang, Y., Fomby, T B., and Park, J Y., editors, Advances in Econometrics: Essays in Honor of Peter C B Phillips, Volume 33, chapter 18, pages 639–672 Emerald Group Publishing Limited, Bingley, UK Vecchio, A (2003) A Bound for the Inverse of a Lower Triangular Toeplitz Matrix SIAM Journal on Matrix Analysis and Applications, 24(4):1167–1174 Vervaat, W (1979) On a Stochastic Difference Equation and a Representation of Non-Negative Infinitely Divisible Random Variables Advances in Applied Probability, 11:750–783 Vinod, H D (1973) Generalization of the Durbin–Watson Statistic for Higher Order Autoregressive Processes Communications in Statistics, 2:115–144 Virbickaite, A., Ausin, M C., and Galeano, P (2016) A Bayesian Non-Parametric Approach to Asymmetric Dynamic Conditional Correlation Model with Application to Portfolio Selection Computational Statistics & Data Analysis, 100:814–829 Vlaar, P J G and Palm, F C (1993) The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroscedasticity, and Jumps Journal of Business & Economic Statistics, 11(3):351–360 von Neumann, J (1941) Distribution of the Ratio of the Mean Square Successive Difference to the Variance Annals of Mathematical Statistics, 12:367–395 Vrontos, I D., Dellaportas, P., and Politis, D (2003) A Full-Factor Multivariate GARCH Model Econometrics Journal, 6(2):312–334 Wald, A (1947) A Note on Regression Analysis Annals of Mathematical Statistics, 18:586–589 Walker, G (1931) On Periodicity in Series of Related Terms Proceedings of the Royal Society of London A, 131:518–532 Wallis, W A (1980) The Statistical Research Group, 1942–1945 Journal of the American Statistical Association, 75(370):320–330 Wan, A T K., Zou, G., and Banerjee, A (2007) The Power of Autocorrelation Tests Near the Unit Root in Models with Possibly Mis-Specified Linear Restrictions Economics Letters, 94:213–219 Wang, C.-S and Zhao, Z (2016) Conditional Value-at-Risk: Semiparametric Estimation and Inference Journal of Econometrics, 195(1):86–103 Wang, M and Li, Y (2011) Pricing of Convertible Bond Based on GARCH Model In Wu, D D., editor, Quantitative Financial Risk Management, pages 77–86 Springer, Berlin Wang, Y., Wu, C., and Yang, L (2016) Forecasting Crude Oil Market Volatility: A Markov Switching Multifractal Volatility Approach International Journal of Forecasting, 32:1–9 Watson, G N (1922) A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge Watson, M W (1994) Vector Autoregressions and Cointegration In Engle, R F and McFadden, D L., editors, Palgrave Handbook of Econometrics, Volume 4, chapter 47, pages 2843–2915 Elsevier, Amsterdam Watson, M W and Engle, R F (1985) Testing for Regressoin Coefficient Stability with a Stationary AR(1) Alternative The Review of Economics and Statistics, 67:341–346 Wells, C (1996) The Kalman Filter in Finance Kluwer Academic Publishing, Dordrecht West, B T., Welch, K B., and Galecki, A T (2015) Linear Mixed Models: A Practical Guide Using Statistical Software CRC Press, Boca Raton, 2nd edition West, M and Harrison, J (1997) Bayesian Forecasting and Dynamic Models Springer, New York, 2nd edition 871 872 Bibliography White, H (1980) A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity Econometrica, 48:817–838 White, H (1982) Maximum Likelihood Estimation of Misspecified Models Econometrica, 50(1):1–25 White, H (1994) Estimation, Inference, and Specification Analysis Cambridge University Press, New York White, H., Kim, T.-H., and Manganelli, S (2015) VAR for VaR: Measuring Tail Dependence Using Multivariate Regression Quantiles Journal of Econometrics, 187(1):169–188 Williams, J S (1962) A Confidence Interval for Variance Components Biometrika, 49:278–281 Winkelmann, R (2008) Econometric Analysis of Count Data Springer, Berlin, 5th edition Winker, P and Maringer, D (2009) The Convergence of Estimators Based on Heuristics: Theory and Application to a GARCH Model Computational Statistics, 24(3):533–550 Wong, C S and Li, W K (2001) On a Logistic Mixture Autoregressive Model Biometrika, 88:833–846 Wooldridge, J M (2009) Introductory Econometrics: A Modern Approach South-Western: Cengage Learning, Mason, OH, 4th edition Wooldridge, J M (2010) Econometric Analysis of Cross Section and Panel Data MIT Press, Cambridge, MA, 2nd edition Wright, R (2017) Why Buddhism is True: The Science and Philosophy of Meditation and Enlightenment Simon & Schuster, New York Wu, L., Meng, Q., and Velazquez, J C (2015) The Role of Multivariate Skew-Student Density in the Estimation of Stock Market Crashes European Journal of Finance, 21(13–14):1144–1160 Wu, P and Crato, N (1995) New Tests for Stationarity and Parity Reversion: Evidence on New Zeland Real Exchange Rates Empirical Economics, 20:559–613 Yadav, P K., Pope, P F., and Paudyal, K (1994) Threshold Autoregressive Modeling in Finance: The Price Differences of Equivalent Assets Mathematical Finance, 4(2):205–221 Yajima, Y (1985) On Estimation of Long Memory Time Series Models Australian Journal of Statistics, 27(3):303–320 Yakowitz, S J and Spragins, J D (1968) On the Identifiability of Finite Mixtures Annals of Mathematical Statistics, 39(1):209–214 Yamamoto, T (1976) Asymptotic Mean Square Prediction Error for an Autoregressive Model with Estimated Coefficients Applied Statistics, 25:123–127 Yamamoto, Y and Perron, P (2013) Estimating and Testing Multiple Structural Changes in Linear Models Using Band Spectral Regressions Econometrics Journal, 16(3):400–429 Yang, F and Leon-Gonzalez, R (2010) Bayesian Estimation and Model Selection in the Generalized Stochastic Unit Root Model Studies in Nonlinear Dynamics & Econometrics, 14(4) Article Yang, R.-C (2010) Towards Understanding and Use of Mixed-Model Analysis of Agricultural Experiments Canadian Journal of Plant Science, 90(5):605–627 Yoon, G (2003) A Simple Model that Generates Stylized Facts of Returns UCSD Economics Working Paper No 2003-04 Young, P C (2011) Recursive Estimation and Time-Series Analysis Springer, Berlin, 2nd edition Yule, G U (1927) On a Method for Investigating Periodicities in Disturbed Series with Special Reference to Wolfer’s Sunspot Numbers Philosophical Transactions of the Royal Society of London A, 226:267–298 Zaman, A (2002) Maximum Likelihood Estimates for the Hildreth–Houck Random Coefficients Model Econometrics Journal, 5(1):237–262 Bibliography Zeileis, A (2006) Object-Oriented Computation of Sandwich Estimators Journal of Statistical Software, 16:1–16 Zeisel, H (1989) On the Power of the Durbin–Watson Test Under High Autocorrelation Communications in Statistics—Theory and Methods, 18:3907–3916 Zellner, A (2001) Keep it Sophisticatedly Simple In Zellner, A., Keuzenkamp, H A., and McAleer, M., editors, Simplicity, Inference and Modelling, pages 242–262 Cambridge University Press, Cambridge Zhang, K and Chan, L (2009) Efficient Factor GARCH Models and Factor-DCC Models Quantitative Finance, 9(1):71–91 Zhigljavsky, A (2010) Singular Spectrum Analysis for Time Series: Introduction to this Special Issue Statistics and Its Interface, 3:255–258 Zhou, T and Chan, L (2008) Clustered Dynamic Conditional Correlation Multivariate GARCH Model In Song, I.-Y., Eder, J., and Nguyen, T M., editors, Data Warehousing and Knowledge Discovery: 10th International Conference, DaWaK 2008 Turin, Italy, September 2-5, 2008 Proceedings, pages 206–216 Zhu, D and Galbraith, J W (2010) A Generalized Asymmetric Student-t Distribution with Application to Financial Econometrics Journal of Econometrics, 157(2):297–305 Zhu, D and Galbraith, J W (2011) Modeling and Forecasting Expected Shortfall with the Generalized Asymmetric Student-t and Asymmetric Exponential Power Distributions Journal of Empirical Finance, 18(4):765–778 Zhu, L.-X and Neuhaus, G (2003) Conditional Tests for Elliptical Symmetry Journal of Multivariate Analysis, 84(2):284–298 Zhu, Q J., Bailey, D H., López de Prado, M., and Borwein, J M (2017) The Probability of Backtest Overfitting Journal of Computational Finance, 20(4):39–69 Zinde-Walsh, V (1988) Some Exact Formulae for Autogressive Moving Average Processes Econometric Theory, 4:384–402 Zivot, E (2018) Modeling Financial Time Series with R Announced, and presumably forthcoming Zivot, E and Wang, J (2006) Modeling Financial Time Series with S-PLUS Springer, New York 873 875 Index a Affine subspace 27 AIC 10, 313, 417 Ancillarity 696 ANOVA 77 Additive 108 ANCOVA 78 Balanced 88 Best linear unbiased predictor (BLUP) 148 Block 107 Classes 127 Control group 91 Crossed 152 Dunnett’s Method 103 Error variance 129 Expected mean squares 96 Fixed effects 77 Ignored block effects 84 Interaction 107 Intra-class variance 129 Intraclass correlation coefficient 139 Levels 127 Mixed model 78 Nested 152 One-way 31, 87 Pilot study 121 Random effects 77, 128 Repeated measures 118 Sample size determination 91 Sums of squares 93 Treatments 127 Two-way 107 Unbalanced 88 Variance components 127 ARFIMA model 347 Square summability 349 ARIMA model 314 Forecasting 339 Fractional 347 Seasonality 314 ARMA model 311 Bootstrap 329, 337 Confidence intervals 328 Covariance structure 322 Forecasting 335 Identification 405 Infinite AR and MA representation 315 Invertibility 328 Mis-specification 330 Missing values 328 Square summability 349 Stationarity 328 Subset 416 Zero pole cancellation 259, 312 ARMAX model 311 Artificial intelligence Asian financial crisis 446 Autocorrelation 187 Autoregressive model 13, 188 Asymptotic Distribution of m.l.e 199 Bootstrap 209 conditional m.l.e 197 Confidence intervals 215 Covariance 191 Expected value 189 Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH, First Edition Marc S Paolella © 2019 John Wiley & Sons Ltd Published 2019 by John Wiley & Sons Ltd 876 Index Autoregressive model (contd.) Explosive process 192 Forecasting 200, 331 Information set 200 Jackknife 208 Latent equation 223 mean square prediction error 201 Multivariate 357 Observation equation 223 Order p 281 Random walk 192 SETAR 346 Smooth transition 347 Subset 315, 393, 435 Threshold 346 Threshold autoregressive stochastic unit root model 346 Unit root 192, 282 Variance 189 Vector AR(1) process 334 Yule Walker Equations 286, 292, 390 Estimator 201, 291, 292, 302, 318 b Backtest overfitting 519, 598, 641 Backtesting 490 Bartlett’s formula 373 Basel committee on banking supervision 638 Basu’s lemma 696 BDS test 475 BEKK 493 Bessel function 650, 663, 734 Bias-variance tradeoff 471, 637 Biased test 232 BIC 313, 417 Bilinear form 12 Black-Litterman model 29 Bootstrap 215, 329 Burn-in period 313 c Canonical reduction 702 Causality CAViaR 489 Characteristic function 734 characteristic generator 742 CIMITYM 635 Co-integration 195, 247, 806 Cochrane-Orcutt 238 Coefficient of multiple determination (R2 ) 15 Column space 17, 690 COMFORT 499 Common factor restrictions 224 Complexity 406 Conditional ACF (CACF) 422 Conditional autoregressive expectile (CARE) 490 constructed portfolio return series 516 Contagion 594, 613 Copula 503, 540 Correlogram Inverse 439 Modified 439 Sample ACF (SACF) 363 Sample partial ACF (SPACF) 392 Theoretical ACF (TACF) 359 Theoretical partial ACF (TPACF) 389, 390 Visual analysis 407 Cramer’s rule 391, 732 Credit scoring 56 Curse of dimensionality 600 d Dancing shadows Data generating process (d.g.p.) 193 Delta method 202 Delta-gamma hedging 673 Density forecasting 594 Density generator 748 Dimension (linear space) 17 Distribution AFaK 542 copula 503 doubly noncentral F 84 Elliptic 739 FaK 541 GAt 462, 626 generalized hyperbolic 734 GHyp 735 Index Identified 612 IGam 526, 530, 736 Jones multivariate t 534 MEST 556, 573 meta-elliptical 767 meta-elliptical Student’s t 541 MixGAt 576 MixN 611 Multivariate Laplace 649 multivariate noncentral t 530 multivariate Student’s t 525, 736 MVNCT 740 Noncentral t 462, 468, 526 normal mean-variance mixture 556 Shaw and Lee multivariate t 538 Stable Paretian 462, 473 Symmetric multivariate stable 748 DJIA 469, 473, 492, 494 Dot product 17 Durbin-Levinson algorithm 392 Durbin-Watson test 47, 227, 229, 230, 236, 249, 698, 701 Bounds test 236 Generalized 238, 808 Inconclusive region 236 Limiting power 239 e Elastic net 52 Elicitability 492 Ellipticity 739 EM algorithm 614 Equally-weighted portfolio 514 Equi-correlation 18 Estimability 31 Exchangeable 588 Exogeneity 11 Expected Mean Squares 96 Expected shortfall (ES) 487, 622, 663, 766 Span 521 Expectiles 490 Extreme value theory 489 f Fiducial inference 150 Filtered historical simulation 488 Final prediction error (FPE) 417 Forecasting 331 Four horsemen 613 Frisch-Waugh-Lovell theorem 11, 24, 57, 226 Functionally independent 11 g GARCH 446, 554 APARCH 460 ARCH 446 COMFORT 499 Constant conditional correlation (CCC) 494 Dynamic conditional correlation (DCC) 494 Dynamic conditional score 460 EVT 489 FIGARCH 460, 481 Integrated 453 Markov switching 484 Mixed normal 477 Quadratic ARCH 460 Variance targeting estimator 459 Varying correlations (VC) 494 YAARCH 446 Geary’s formula 682 General-to-Specific (GETS) 53 Generalized inverse 226, 748 Global financial crisis 448, 631, 658 Global warming Gram-Schmidt 17, 21, 23, 24 h Half life 207 Hannan-Quinn criterion (HQ) 417 Heteroskedastic and autocorrelation consistent (HAC) estimator 15 Heteroskedasticity 6, 262, 446 Hypothesis test LBI 231 POI 231 UMP 239 UMPI 35, 766 UMPU 229 877 878 Index m Hypothesis testing Composite normality 625 Neyman-Pearson 27 Significance 26, 412 i Idempotent matrix 20 Identifiability 31 Impulse indicator saturation 53 Independent components analysis (ICA) Inequality Cauchy-Schwarz 57 Information set 200, 451, 592 Inner product 17 Innovation process 188, 446 Interaction 107 j Jacobian 197, 536, 752 k Kalman filter 30, 47, 49, 260, 328 Kendall’s 𝜏 554 Kummer’s Transformation 717 l Lag operator 224, 281 Lagrange multipliers 29, 61 LASSO 52 Leading principle minor 283 Leverage effect 482, 612 Likelihood Concentrated 55, 263, 271 Likelihood ratio statistic 59 Linear model Dependent variable Endogenous variable Explanatory variables Generalized 55 Linear span 17 Link function 56 Logit 56 Long memory process 347 Long-run variance 195 484 Machine learning Elastic net LARS LASSO Mahalanobis distance 629 Robust 631 Maple 183 Matlab Nested functions 75 Mean reversion 247, 599 Mean-bias adjusted estimator 211 Median-unbiased estimator 211 Minimum covariance determinant 630, 645 Mixed model 28, 29 Mode-adjusted estimator 212 Moment generating function 733 Momentum effect 463 Moving average model 13, 294 Invertibility 296 Order q 299 Multicollinearity 52 Multifractal model 448 Multiple imputation 53, 328 Murder rate 5, 40, 42 n NASDAQ 491 News impact curve 460 Nonlinear time series models Norm (of a vector) 17 343 o Order of integration 195 I(0) 195 Orthogonal complement 18 Orthonormal 17 Orthonormal (basis) matrix 17 Overfitting 406 p Parsimony 294, 311, 312, 316, 339, 406, 408, 409, 416, 418, 429, 483 Partial correlation 384 Partially adaptive estimation 54 Index Partitioned inverse 57 Peaks over threshold 489 Pie chart 797 Pivotal quantity 138 Poincaré separation theorem 237, 727 Pooled variance estimator 81 Portfolio distribution 555, 620, 662 Portfolio optimization Equally weighted 512 FaK 600 Markowitz 510 Simulation 513 Univariate collapsing method (UCM) 516 Principle axis theorem 670 Principle components analysis (PCA) 485, 510 Probability integral transform 541 Probability of default 56 Probit 56 Profile log-likelihood 456, 661 Projection matrix 19, 23 Projection theorem 19, 389 Pseudo maximum likelihood estimator 246 Purchasing power parity 207 q Quadratic form 12, 206, 669, 767 Bilinear 669 Generalized 675 Ratio 366, 679 Moments 695 Quantile regression 4, 55 Quasi-Bayesian prior 616 Quasi-maximum likelihood 464 r Radial random variable 742 Random walk with drift 192 Realized predictive log-likelihood 592 normalized sum 593, 635 Regression Adaptive 187 Adjusted R2 10 Bonferroni method 44, 97 Coefficient of multiple determination (R2 ) Confidence intervals 42 Controlling for Design matrix 10 Forecasting 51 Gauss-Markov theorem Generalized least squares (g.l.s.) 13 Heteroskedasticity 6, 50 Least squares Locally disjoint broken trend model 41 Mallows’ Ck 10 Maximum modulus t intervals 44 Missing values 53 Model specification 52 Multicollinearity 52 Normal equations Omitted variables 187 Ordinary least squares Parameter constancy 53, 259 Partially adaptive estimation 53 Piecewise linear 41 Quantile 4, 55 Residuals BLUS 48 LUS 48, 50 Recursive 49, 64, 365, 397 Restricted generalized least squares 33 Restricted least squares 28, 58 Ridge 52 Robust Estimation 53 Sample splitting model 54 Scheffé’s method 45, 97, 102 Simple linear Structural break 41, 50, 53 Sums of squares Explained (ESS) Residual (RSS) 8, 24 Total (TSS) Threshold 54 Time series 40 Time-varying linear constraints 30 Time-varying parameters 53, 259 Hildreth-Houck 261 Random walk 269 Rosenberg Return to Normalcy 277 Weighted least squares 13 879 880 Index Religion Response function 56 Return to Normalcy 277 Returns Percentage log 344 RiskMetrics 469 Stylized facts 445, 474 Sub-prime crisis 448 Sufficiency 136, 198 Survivorship bias 494, 645 Synthetic assumption 464 t s S&P500 195, 473 Sample autocorrelation function 705 Sample autocorrelation function (SACF) 50 SAS 77, 98, 127, 137, 142, 454, 773 Satterthwaite’s method 140 Sausages 405 Shadows Sharpe ratio 519 Shrinkage 594, 617 Signed likelihood ratio statistic 415 Singular spectrum analysis (SSA) 315 Singular value decomposition 226, 436 Sortino ratio 520 Spearman’s 𝜌 554 Spherical 739 Sphericity 740 Spillover 594 Spurious correlation Spurious trend 193 Stable tail adjusted return ratio 520 State space representation 30, 53, 260, 328 Stationarity Non-stationary process 192 Strict 192 Trend 247 Up to order m 192 Weak 191 Stein’s lemma 767 Stochastic volatility (SV) 451, 503 Structural break 192 Studentized range distribution 99 Tail dependence 561, 613 Tail estimation 475 Tail index 628 Tea leaves 407 Time-varying parameters 192 Time-varying skewness 480 Transaction costs 504 proportional 508 u Unit root Stochastic 344, 345 structural breaks 254 Test 247 Dickey-Fuller 248 KPSS 256 Univariate collapsing method (UCM) v Value at risk (VaR) 487 Violations 490 Variation-free 11, 25 Vech 546 Vector autoregression 279 Vector error correction models 806 w Weighted likelihood 587 White noise 188 z Zero pole cancellation 312, 329, 330 517 ... Title: Linear models and time-series analysis : regression, ANOVA, ARMA and GARCH / Dr Marc S Paolella Description: Hoboken, NJ : John Wiley & Sons, 2019 | Series: Wiley series in probability and. .. theater, popular in Southeast Asia; see, e.g., Pigliucci and Kaplan (2006, p 2) Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH, First Edition Marc S Paolella © 2019 John Wiley... Part I (and Appendices A and B) addressing the linear (Gaussian) model and ANOVA, Part II detailing the ARMA and ARMAX univariate time-series paradigms (along with unit root testing and time-varying