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 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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 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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. .. 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... 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