Linear Models and Time-Series Analysis. Regression, ANOVA, ARMA and GARCH. Wiley Series in Probability and Statistics

• ID: 4517479
• Book
• 650 Pages
• John Wiley and Sons Ltd
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A comprehensive and timely edition on an emerging new trend in time series

Linear Models and Time–Series Analysis: Regression, ANOVA, ARMA and GARCH sets a strong foundation, in terms of distribution theory, for the linear model (regression and ANOVA), univariate time series analysis (ARMAX and GARCH), and some multivariate models associated primarily with modeling financial asset returns (copula–based structures and the discrete mixed normal and Laplace). It builds on the author′s previous book, Fundamental Statistical Inference: A Computational Approach, which introduced the major concepts of statistical inference. Attention is explicitly paid to application and numeric computation, with examples of Matlab code throughout. The code offers a framework for discussion and illustration of numerics, and shows the mapping from theory to computation.

The topic of time series analysis is on firm footing, with numerous textbooks and research journals dedicated to it. With respect to the subject/technology, many chapters in Linear Models and Time–Series Analysis cover firmly entrenched topics (regression and ARMA). Several others are dedicated to very modern methods, as used in empirical finance, asset pricing, risk management, and portfolio optimization, in order to address the severe change in performance of many pension funds, and changes in how fund managers work.

• Covers traditional time series analysis with new guidelines
• Provides access to cutting edge topics that are at the forefront of financial econometrics and industry
• Includes latest developments and topics such as financial returns data, notably also in a multivariate context
• Written by a leading expert in time series analysis
• Extensively classroom tested
• Includes a tutorial on SAS
• Supplemented with a companion website containing numerous Matlab programs
• Solutions to most exercises are provided in the book

Linear Models and Time–Series Analysis: Regression, ANOVA, ARMA and GARCH is suitable for advanced masters students in statistics and quantitative finance, as well as doctoral students in economics and finance. It is also useful for quantitative financial practitioners in large financial institutions and smaller finance outlets.

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I Linear Models: Regression and ANOVA 5

1 The Linear Model 6

1.1 Regression, Correlation, and Causality 6

1.2 Ordinary and Generalized Least Squares 11

1.2.1 Ordinary Least Squares (OLS) Estimation 11

1.2.2 Further Aspects of Regression and OLS 12

1.2.3 Generalized Least Squares (GLS) 16

1.3 The Geometric Approach to Least Squares 21

1.3.1 Projection 21

1.3.2 Implementation 27

1.4 Linear Parameter Restrictions 31

1.4.1 Formulation and Estimation 32

1.4.2 Estimability and Identiability 35

1.4.3 Moments and the Restricted GLS Estimator 38

1.4.4 Testing With h — 0 39

1.4.5 Testing With Nonzero h 42

1.4.6 Examples 43

1.4.7 Con—dence Intervals 48

1.5 Alternative Residual Calculation 52

1.6 Further Topics 56

1.7 Problems 62

1.A Appendix: Derivation of the BLUS Residual Vector 66

1.B Appendix: The Recursive Residuals 71

1.C Appendix: Solutions 72

2 Fixed E—ects ANOVA Models 82

2.1 Introduction: Fixed, Random, and Mixed E—ects Models 82

2.2 Two Sample t Test for Di—erences in Means 84

2.3 The Two Sample t Test with Ignored Block E—ects 90

2.4 One Way ANOVA with Fixed E—ects 91

2.4.1 The Model 91

2.4.2 Estimation and Testing 93

2.4.3 Determination of Sample Size 95

2.4.4 The ANOVA Table 97

2.4.5 Computing Con—dence Intervals 101

2.4.6 A Word on Model Assumptions 108

2.5 Two Way Balanced Fixed E—ects ANOVA 110

2.5.1 The Model and Use of the Interaction Terms 111

2.5.2 Sums of Squares Decomposition Without Interaction 112

2.5.3 Sums of Squares Decomposition With Interaction 117

2.5.4 Example and Codes 121

3 Introduction to Random and Mixed E—ects Models 131

3.1 One–Factor Balanced Random E—ects Model 132

3.1.1 Model and Maximum Likelihood Estimation 133

3.1.2 Distribution Theory and ANOVA Table 136

3.1.3 Point Estimation, Interval Estimation, and Signi—cance Testing 140

3.1.4 Satterthwaite′s Method 143

3.1.5 Use of SAS 145

3.1.6 Approximate Inference in the Unbalanced Case 146

3.1.6.1 Point Estimation in the Unbalanced Case 147

3.1.6.2 Interval Estimation in the Unbalanced Case 152

3.2 Crossed Random E—ects Models 155

3.2.1 Two Factors 156

3.2.1.1 With Interaction Term 156

3.2.1.2 Without Interaction Term 159

3.2.2 Three Factors 160

3.3 Nested Random E—ects Models 164

3.3.1 Two Factors 165

3.3.1.1 Both E—ects Random: Model and Parameter Estimation 165

3.3.1.2 Both E—ects Random: Exact and Approximate C.I.s 170

3.3.1.3 Mixed Model Case 173

3.3.2 Three Factors 176

3.3.2.1 All E—ects Random 176

3.3.2.2 Mixed: Classes Fixed 178

3.3.2.3 Mixed: Classes and Subclasses Fixed 179

3.4 Problems 179

3.A Appendix: Solutions 180

II Time Series Analysis: ARMAX Processes 186

4 The AR(1) Model 187

4.1 Moments and Stationarity 188

4.2 Order of Integration and Long—Run Variance 194

4.3 Least Squares and ML Estimation 195

4.3.1 OLS Estimator of a 195

4.3.2 Likelihood Derivation I 196

4.3.3 Likelihood Derivation II 197

4.3.4 Likelihood Derivation III 197

4.3.5 Asymptotic Distribution 199

4.4 Forecasting 200

4.5 Small Sample Distribution of the OLS and ML Point Estimators 202

4.6 Alternative Point Estimators of a 206

4.6.1 Use of the Jackknife for Bias Reduction 206

4.6.2 Use of the Bootstrap for Bias Reduction 208

4.6.3 Median–Unbiased Estimator 209

4.6.4 Mean–Bias Adjusted Estimator 210

4.6.5 Mode–Adjusted Estimator 211

4.6.6 Comparison 212

4.7 Con—dence Intervals for a 213

4.8 Problems 217

5 Regression Extensions: AR(1) Errors and TVP 219

5.1 The AR(1) Regression Model and the Likelihood 219

5.2 OLS Point and Interval Estimation of a 222

5.3 Testing a — 0 in the ARX(1) Model 224

5.3.1 Use of Con—dence Intervals 225

5.3.2 The Durbin–Watson Test 226

5.3.3 Other Tests for First–Order Autocorrelation 227

5.3.4 Further Details on the Durbin–Watson Test 231

5.3.4.1 The Bounds Test, and Critique of Use of p–Values 232

5.3.4.2 Limiting Power as a Ñ —1 235

5.4 Bias–Adjusted Point Estimation 239

5.5 Unit Root Testing in the ARX(1) Model 242

5.5.1 Null is a — 1 244

5.5.2 Null is a 1 250

5.6 Time–Varying Parameter (TVP) Regression 254

5.6.1 Motivation and Introductory Remarks 254

5.6.2 The Hildreth–Houck Random Coe—cient (HHRC) Model 255

5.6.3 The TVP Random Walk Model 262

5.6.3.1 Covariance Structure and Estimation 263

5.6.3.2 Testing for Parameter Constancy 267

5.6.4 Rosenberg Return to Normalcy Model 269

6 Autoregressive and Moving Average Processes 272

6.1 AR(p) Processes 272

6.1.1 Stationarity and Unit Root Processes 273

6.1.2 Moments 276

6.1.3 Estimation 278

6.1.3.1 Without Mean Term 278

6.1.3.2 Starting Values 281

6.1.3.3 With Mean Term 283

6.1.3.4 Approximate Standard Errors 284

6.2 Moving Average Processes 286

6.2.1 MA(1) Process 286

6.2.2 MA(q) Processes 289

6.3 Problems 292

6.A Appendix: Solutions 293

7 Autoregressive Moving Average Processes 301

7.1 Basics of ARMA Models 301

7.1.1 The Model 301

7.1.2 Zero Pole Cancellation 302

7.1.3 Simulation 304

7.1.4 The ARIMA(p; d; q) Model 304

7.2 In—nite AR and MA Representations 305

7.3 Initial Parameter Estimation 308

7.3.1 Via the In—nite AR Representation 309

7.3.2 Via In—nite AR and Ordinary Least Squares 310

7.4 Likelihood–Based Estimation 312

7.4.1 Covariance Structure 312

7.4.2 Point Estimation 315

7.4.3 Interval Estimation 319

7.4.4 Model Mis–speci—cation 321

7.5 Forecasting 323

7.5.1 ARppq Model 323

7.5.2 MA(q) and ARMA(p; q) Models 327

7.5.3 ARIMA(p; d; q) Models 330

7.6 Bias–Adjusted Point Estimation: Extension to the ARMAXp1; qq model 331

7.7 Some ARIMAX Model Extensions 335

7.7.1 Stochastic Unit Root 335

7.7.2 Threshold Autoregressive Models 337

7.7.3 Fractionally Integrated ARMA (ARFIMA) 339

7.8 Problems 341

7.A Appendix: Generalized Least Squares for ARMA Estimation 343

7.B Appendix: Multivariate AR(p) Processes and Stationarity, and General Block Toeplitz Matrix Inversion 349

8 Correlograms 351

8.1 Theoretical and Sample Autocorrelation Function 351

8.1.1 De—nitions 351

8.1.2 Marginal Distributions 356

8.1.3 Joint Distribution 361

8.1.3.1 Support 361

8.1.3.2 Asymptotic Distribution 364

8.1.3.3 Small–Sample Joint Distribution Approximation 366

8.1.4 Conditional Distribution Approximation 371

8.2 Theoretical and Sample Partial Autocorrelation Function 374

8.2.1 Partial Correlation 374

8.2.2 Partial Autocorrelation Function 379

8.2.2.1 TPACF: First De—nition 379

8.2.2.2 TPACF: Second De—nition 380

8.2.2.3 Sample Partial Autocorrelation Function 382

8.3 Problems 386

8.A Appendix: Solutions 386

9 ARMA Model Identi—cation 394

9.1 Introduction 394

9.2 Visual Correlogram Analysis 396

9.3 Signi—cance Tests 400

9.4 Penalty Criteria 405

9.5 Use of the Conditional SACF for Sequential Testing 408

9.6 Use of the Singular Value Decomposition 422

9.7 Further Methods: Pattern Identi—cation 426

III Modeling Financial Asset Returns 429

10 Univariate GARCH Modeling 430

10.1 Introduction 430

10.2 Gaussian GARCH and Estimation 434

10.2.1 Basic Properties 435

10.2.2 Integrated GARCH 437

10.2.3 Maximum Likelihood Estimation 437

10.2.4 Variance Targeting Estimator 442

10.3 Non–Gaussian ARMA–APARCH, QMLE, and Forecasting 443

10.3.1 Extending the Volatility, Distribution, and Mean Equations 443

10.3.2 Model Mis–Speci—cation and QMLE 447

10.3.3 Forecasting 449

10.4 Near–Instantaneous Estimation of NCT–APARCH(1,1) 450

10.5 S—;—–APARCH and Testing the IID Stable Hypothesis 454

10.6 Mixed Normal GARCH 458

10.6.1 Introduction 458

10.6.2 The MixN(k)–GARCH(r; s) Model 460

10.6.3 Parameter Estimation and Model Features 461

10.6.4 Time–Varying Weights 464

10.6.5 Markov Switching Extension 466

10.6.6 Multivariate Extensions 467

11 Risk Prediction and Portfolio Optimization 468

11.1 Value at Risk and Expected Shortfall Prediction 468

11.2 MGARCH Constructs Via Univariate GARCH 474

11.2.1 Introduction 474

11.2.2 The Gaussian CCC and DCC Models 476

11.2.3 Morana Semi—Parametric DCC Model 479

11.2.4 The COMFORT Class 481

11.2.5 Copula Constructions 486

11.3 Introducing Portfolio Optimization 486

11.3.1 Some Trivial Accounting 486

11.3.2 Markowitz and DCC 493

11.3.3 Portfolio Optimization Using Simulation 496

11.3.4 The Univariate Collapsing Method 499

11.3.5 The ES Span 504

12 Multivariate t Distributions 507

12.1 Multivariate Student′s t 507

12.2 Multivariate Noncentral Student′s t 512

12.3 Jones Multivariate t Distribution 515

12.4 Shaw and Lee Multivariate t Distributions 518

12.5 The Meta–Elliptical t Distribution 521

12.5.1 The FaK Distribution 521

12.5.2 The AFaK Distribution 523

12.5.3 FaK and AFaK Estimation: Direct Likelihood Optimization 524

12.5.4 FaK and AFaK Estimation: Two–Step Estimation 528

12.5.5 Sums of Margins of the AFaK 533

12.6 MEST: Marginally Endowed Student′s t 534

12.6.1 SMESTI Distribution 535

12.6.2 AMESTI Distribution 536

12.6.3 MESTI Estimation 540

12.6.4 AoNm–MEST 542

12.6.5 MEST Distribution 545

12.7 Some Closing Remarks 551

12.A ES of Convolution of AFaK Margins 552

12.B Covariance Matrix for the FaK 556

13 Weighted Likelihood 562

13.1 Concept 562

13.2 Determination of Optimal Weighting 566

13.3 Density Forecasting and Backtest Over—tting 569

13.4 Portfolio Optimization Using (A)FaK 576

14 Multivariate Mixture Distributions 587

14.1 The MixkNd Distribution 587

14.1.1 Density and Simulation 588

14.1.2 Motivation for Use of Mixtures 588

14.1.3 Quasi–Bayesian Estimation and Choice of Prior 591

14.1.4 Portfolio Distribution and Expected Shortfall 596

14.2 Model Diagnostics and Forecasting 598

14.2.1 Assessing Presence of a Mixture 598

14.2.2 Component Separation and Univariate Normality 600

14.2.3 Component Separation and Multivariate Normality 604

14.2.4 Mixed Normal Weighted Likelihood and Density Forecasting 606

14.2.5 Density Forecasting: Optimal Shrinkage 607

14.2.6 Moving Averages of — 614

14.3 MCD for Robustness and Mix2Nd Estimation 620

14.4 Some Thoughts on Model Assumptions and Estimation 622

14.5 The Multivariate Laplace and MixkLapd Distributions 623

14.5.1 The Multivariate Laplace and EM Algorithm 624

14.5.2 The MixkLapd and EM Algorithm 628

14.5.3 Estimation via MCD Split and Forecasting 632

14.5.4 Estimation of Parameter b 635

14.5.5 Portfolio Distribution and Expected Shortfall 636

14.5.6 Fast Evaluation of the Bessel Function 637

IV Appendices 640

A Distribution of Quadratic Forms 641

A.1 Distribution and Moments 641

A.1.1 Probability Density and Cumulative Distribution Functions 642

A.1.2 Positive Integer Moments 643

A.1.3 Moment Generating Functions 645

A.2 Basic Distributional Results 649

A.3 Ratios of Quadratic Forms in Normal Variables 652

A.3.1 Calculation of the CDF 652

A.3.2 Calculation of the PDF 654

A.3.2.1 Numeric Di—erentiation 654

A.3.2.2 Use of Geary′s formula 655

A.3.2.3 Use of Pan′s formula 656

A.3.2.4 Saddlepoint Approximation 658

A.4 Problems 660

A.A Appendix: Solutions 662

B Moments of Ratios of Quadratic Forms 669

B.1 For X — Nn 0; —2I— and B — I 669

B.2 For X — Np0;—q 682

B.3 For X — Np—; Iq 687

B.4 For X — Np—;—q 693

B.5 Useful Matrix Algebra Results 698

B.6 Saddlepoint Equivalence Result 702

C Some Useful Multivariate Distribution Theory 707

C.1 Student′s t Characteristic Function 707

C.2 Sphericity and Ellipticity 713

C.2.1 Introduction 713

C.2.2 Sphericity 715

C.2.3 Ellipticity 723

C.2.4 Testing Ellipticity 744

D SAS 748

D.1 Introduction to SAS 750

D.1.1 Background 750

D.1.2 Working with SAS on a PC 750

D.1.3 Introduction to the Data Step and the Program Data Vector 752

D.2 Basic Data Handling 759

D.2.1 Method 1 761

D.2.2 Method 2 763

D.2.3 Method 3 764

D.2.4 Creating Data Sets from Existing Data Sets 765

D.2.5 Creating Data Sets from Procedure Output 766

D.3 Advanced Data Handling 769

D.3.1 String Input and Missing Values 769

D.3.2 Using set with first.var and last.var 770

D.3.3 Reading in text —les 774

D.3.4 Skipping over Headers 775

D.3.5 Variable and Value Labels 776

D.4 Generating Charts, Tables and Graphs 777

D.4.1 Simple Charting and Tables 778

D.4.2 Date and Time Formats/Informats 782

D.4.3 High Resolution Graphics 783

D.4.3.1 The GPLOT Procedure 783

D.4.3.2 The GCHART Procedure 787

D.4.4 Linear Regression and Time Series Analysis 788

D.5 The SAS Macro Processor 791

D.5.1 Introduction 791

D.5.2 Macro Variables 792

D.5.3 Macro Programs 794

D.5.4 A Useful Example 796

D.5.4.1 Method 1 797

D.5.4.2 Method 2 799

D.6 Problems 800

D.A Appendix: Solutions 802

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