Financial engineers have access to enormous quantities of data but need powerful methods for extracting quantitative information, particularly about volatility and risks. Key features of this textbook are: illustration of concepts with financial markets and economic data, R Labs with real-data exercises, and integration of graphical and analytic methods for modeling and diagnosing modeling errors. Despite some overlap with the author's undergraduate textbook Statistics and Finance: An Introduction, this book differs from that earlier volume in several important aspects: it is graduate-level; computations and graphics are done in R; and many advanced topics are covered, for example, multivariate distributions, copulas, Bayesian computations, VaR and expected shortfall, and cointegration.

The prerequisites are basic statistics and probability, matrices and linear algebra, and calculus.

Some exposure to finance

is helpful.

Notation . : xxi

1 Introduction . 1

1.1 Bibliographic Notes . 3

1.2 References . 4

2 Returns . 5

2.1 Introduction . 5

2.1.1 Net Returns . 5

2.1.2 Gross Returns . 6

2.1.3 Log Returns . 6

2.1.4 Adjustment for Dividends . 7

2.2 The Random Walk Model . 8

2.2.1 Random Walks . 8

2.2.2 Geometric Random Walks . 8

2.2.3 Are Log Prices a Lognormal Geometric Random Walk? 9

2.3 Bibliographic Notes . 10

2.4 References . 10

2.5 R Lab . 11

2.5.1 Data Analysis . 11

2.5.2 Simulations . 12

2.6 Exercises . 14

3 Fixed Income Securities . 17

3.1 Introduction . 17

3.2 Zero-Coupon Bonds . 18

3.2.1 Price and Returns Fluctuate with the Interest Rate . 18

3.3 Coupon Bonds . 19

3.3.1 A General Formula . 20

3.4 Yield to Maturity . 21

3.4.1 General Method for Yield to Maturity . 22

3.4.2 Spot Rates . 23

3.5 Term Structure . 24

3.5.1 Introduction: Interest Rates Depend Upon Maturity . 24

3.5.2 Describing the Term Structure . 24

3.6 Continuous Compounding . 29

3.7 Continuous Forward Rates . 30

3.8 Sensitivity of Price to Yield . 32

3.8.1 Duration of a Coupon Bond . 32

3.9 Bibliographic Notes . 33

3.10 References . 34

3.11 R Lab . 34

3.11.1 Computing Yield to Maturity . 34

3.11.2 Graphing Yield Curves . 36

3.12 Exercises . 36

4 Exploratory Data Analysis . : 41

4.1 Introduction . 41

4.2 Histograms and Kernel Density Estimation . 43

4.3 Order Statistics, the Sample CDF, and Sample Quantiles . 48

4.3.1 The Central Limit Theorem for Sample Quantiles . 49

4.3.2 Normal Probability Plots . 50

4.3.3 Half-Normal Plots . 54

4.3.4 Quantile{Quantile Plots . 57

4.4 Tests of Normality . 59

4.5 Boxplots . 61

4.6 Data Transformation . 62

4.7 The Geometry of Transformations . 66

4.8 Transformation Kernel Density Estimation . 70

4.9 Bibliographic Notes . 73

4.10 References . 73

4.11 R Lab . 74

4.11.1 European Stock Indices . 74

4.12 Exercises . 77

5 Modeling Univariate Distributions . 79

5.1 Introduction . 79

5.2 Parametric Models and Parsimony . 79

5.3 Location, Scale, and Shape Parameters . 80

5.4 Skewness, Kurtosis, and Moments . 81

5.4.1 The Jarque{Bera test . 86

5.4.2 Moments . 86

5.5 Heavy-Tailed Distributions . 87

5.5.1 Exponential and Polynomial Tails . 87

5.5.2 t-Distributions . 88

5.5.3 Mixture Models . 90

5.6 Generalized Error Distributions . 93

5.7 Creating Skewed from Symmetric Distributions . 95

5.8 Quantile-Based Location, Scale, and Shape Parameters . 97

5.9 Maximum Likelihood Estimation . 98

5.10 Fisher Information and the Central Limit Theorem for the

MLE . 98

5.11 Likelihood Ratio Tests . 101

5.12 AIC and BIC . 102

5.13 Validation Data and Cross-Validation . 103

5.14 Fitting Distributions by Maximum Likelihood . 106

5.15 Proˉle Likelihood . 115

5.16 Robust Estimation . 117

5.17 Transformation Kernel Density Estimation with a Parametric

Transformation . 119

5.18 Bibliographic Notes . 122

5.19 References . 122

5.20 R Lab . 123

5.20.1 Earnings Data . 123

5.20.2 DAX Returns . 125

5.21 Exercises . 126

6 Resampling . : 131

6.1 Introduction . 131

6.2 Bootstrap Estimates of Bias, Standard Deviation, and MSE . 132

6.2.1 Bootstrapping the MLE of the t-Distribution . 133

6.3 Bootstrap Conˉdence Intervals . 136

6.3.1 Normal Approximation Interval . 136

6.3.2 Bootstrap-t Intervals . 137

6.3.3 Basic Bootstrap Interval . 139

6.3.4 Percentile Conˉdence Intervals . 140

6.4 Bibliographic Notes . 144

6.5 References . 145

6.6 R Lab . 145

6.6.1 BMW Returns . 145

6.7 Exercises . 147

7 Multivariate Statistical Models . : 149

7.1 Introduction . 149

7.2 Covariance and Correlation Matrices . 149

7.3 Linear Functions of Random Variables . 151

7.3.1 Two or More Linear Combinations of Random Variables153

7.3.2 Independence and Variances of Sums . 154

7.4 Scatterplot Matrices . 155

7.5 The Multivariate Normal Distribution . 156

7.6 The Multivariate t-Distribution . 157

7.6.1 Using the t-Distribution in Portfolio Analysis . 160

7.7 Fitting the Multivariate t-Distribution by Maximum Likelihood160

7.8 Elliptically Contoured Densities . 162

7.9 The Multivariate Skewed t-Distributions . 164

7.10 The Fisher Information Matrix . 166

7.11 Bootstrapping Multivariate Data . 167

7.12 Bibliographic Notes . 169

7.13 References . 169

7.14 R Lab . 169

7.14.1 Equity Returns . 169

7.14.2 Simulating Multivariate t-Distributions . 171

7.14.3 Fitting a Bivariate t-Distribution . 172

7.15 Exercises . 173

8 Copulas . 175

8.1 Introduction . 175

8.2 Special Copulas . 177

8.3 Gaussian and t-Copulas . 177

8.4 Archimedean Copulas . 178

8.4.1 Frank Copula . 178

8.4.2 Clayton Copula . 180

8.4.3 Gumbel Copula . 181

8.5 Rank Correlation . 182

8.5.1 Kendall's Tau . 183

8.5.2 Spearman's Correlation Coe±cient . 184

8.6 Tail Dependence . 185

8.7 Calibrating Copulas . 187

8.7.1 Maximum Likelihood . 188

8.7.2 Pseudo-Maximum Likelihood. 188

8.7.3 Calibrating Meta-Gaussian and Meta-t-Distributions . 189

8.8 Bibliographic Notes . 193

8.9 References . 195

8.10 Problems . 195

8.11 R Lab . 195

8.11.1 Simulating Copulas . 195

8.11.2 Fitting Copulas to Returns Data . 197

8.12 Exercises . 200

9 Time Series Models: Basics . : 201

9.1 Time Series Data . 201

9.2 Stationary Processes . 201

9.2.1 White Noise . 205

9.2.2 Predicting White Noise . 205

9.3 Estimating Parameters of a Stationary Process . 206

9.3.1 ACF Plots and the Ljung{Box Test . 206

9.4 AR(1) Processes . 208

9.4.1 Properties of a stationary AR(1) Process . 209

9.4.2 Convergence to the Stationary Distribution . 211

9.4.3 Nonstationary AR(1) Processes . 211

9.5 Estimation of AR(1) Processes . 212

9.5.1 Residuals and Model Checking . 213

9.5.2 Maximum Likelihood and Conditional Least-Squares . 217

9.6 AR(p) Models . 218

9.7 Moving Average (MA) Processes . 222

9.7.1 MA(1) Processes . 223

9.7.2 General MA Processes . 223

9.8 ARMA Processes . 225

9.8.1 The Backwards Operator . 225

9.8.2 The ARMA Model . 225

9.8.3 ARMA(1,1) Processes . 226

9.8.4 Estimation of ARMA Parameters . 227

9.8.5 The Di?erencing Operator . 227

9.9 ARIMA Processes . 228

9.9.1 Drifts in ARIMA Processes . 232

9.10 Unit Root Tests . 233

9.10.1 How Do Unit Root Tests Work? . 235

9.11 Automatic Selection of an ARIMA Model . 236

9.12 Forecasting . 237

9.12.1 Forecast Errors and Prediction Intervals . 239

9.12.2 Computing Forecast Limits by Simulation . 241

9.13 Partial Autocorrelation Coe±cients . 245

9.14 Bibliographic Notes . 247

9.15 References . 248

9.16 R Lab . 248

9.16.1 T-bill Rates . 248

9.16.2 Forecasting . 251

9.17 Exercises . 251

10 Time Series Models: Further Topics . 257

10.1 Seasonal ARIMA Models . 257

10.1.1 Seasonal and nonseasonal di?erencing . 258

10.1.2 Multiplicative ARIMA Models . 259

10.2 Box{Cox Transformation for Time Series . 262

10.3 Multivariate Time Series . 264

10.3.1 The cross-correlation function . 264

10.3.2 Multivariate White Noise . 265

10.3.3 Multivariate ARMA processes . 266

10.3.4 Prediction Using Multivariate AR Models . 268

10.4 Long-Memory Processes . 270

10.4.1 The Need for Long-Memory Stationary Models . 270

10.4.2 Fractional Di?erencing . 270

10.4.3 FARIMA Processes . 272

10.5 Bootstrapping Time Series . 276

10.6 Bibliographic Notes . 277

10.7 References . 277

10.8 R Lab . 277

10.8.1 Seasonal ARIMA Models . 277

10.8.2 VAR Models . 278

10.8.3 Long-Memory Processes . 279

10.8.4 Model-Based Bootstrapping of an ARIMA Process . 280

10.9 Exercises . 282

11 Portfolio Theory. 285

11.1 Trading O? Expected Return and Risk . 285

11.2 One Risky Asset and One Risk-Free Asset . 285

11.2.1 Estimating E(R) and ?R . 287

11.3 Two Risky Assets . 287

11.3.1 Risk Versus Expected Return . 287

11.4 Combining Two Risky Assets with a Risk-Free Asset . 289

11.4.1 Tangency Portfolio with Two Risky Assets . 289

11.4.2 Combining the Tangency Portfolio with the Risk-Free

Asset . 291

11.4.3 E?ect of ?12 . 292

11.5 Selling Short . 293

11.6 Risk-E±cient Portfolios with N Risky Assets . 294

11.7 Resampling and E±cient Portfolios . 299

11.8 Bibliographic Notes . 305

11.9 References . 305

11.10 R Lab . 306

11.10.1 E±cient Equity Portfolios . 306

11.11 Exercises . 307

12 Regression: Basics . 309

12.1 Introduction . 309

12.2 Straight-Line Regression . 310

12.2.1 Least-Squares Estimation . 310

12.2.2 Variance of bˉ1 . 314

12.3 Multiple Linear Regression . 315

12.3.1 Standard Errors, t-Values, and p-Values . 317

12.4 Analysis of Variance, Sums of Squares, and R2 . 318

12.4.1 AOV Table . 318

12.4.2 Degrees of Freedom (DF) . 320

12.4.3 Mean Sums of Squares (MS) and F-Tests . 321

12.4.4 Adjusted R2 . 323

12.5 Model Selection . 323

12.6 Collinearity and Variance In°ation . 325

12.7 Partial Residual Plots . 332

12.8 Centering the Predictors . 334

12.9 Orthogonal Polynomials . 334

12.10 Bibliographic Notes . 335

12.11 References . 335

12.12 R Lab . 335

12.12.1 U.S. Macroeconomic Variables . 335

12.13 Exercises . 338

13 Regression: Troubleshooting . 341

13.1 Regression Diagnostics . 341

13.1.1 Leverages . 343

13.1.2 Residuals . 344

13.1.3 Cook's D . 346

13.2 Checking Model Assumptions . 348

13.2.1 Nonnormality . 349

13.2.2 Nonconstant Variance . 351

13.2.3 Nonlinearity . 351

13.2.4 Residual Correlation and Spurious Regressions . 354

13.3 Bibliographic Notes . 361

13.4 References . 361

13.5 R Lab . 361

13.5.1 Current Population Survey Data . 361

13.6 Exercises . 364

14 Regression: Advanced Topics . : 369

14.1 Linear Regression with ARMA Errors . 369

14.2 The Theory Behind Linear Regression . 373

14.2.1 The E?ect of Correlated Noise and Heteroskedasticity . 374

14.2.2 Maximum Likelihood Estimation for Regression . 374

14.3 Nonlinear Regression . 376

14.4 Estimating Forward Rates from Zero-Coupon Bond Prices . 381

14.5 Transform-Both-Sides Regression . 386

14.5.1 How TBS Works . 388

14.6 Transforming Only the Response . 389

14.7 Binary Regression . 390

14.8 Linearizing a Nonlinear Model . 396

14.9 Robust Regression . 397

14.10 Regression and Best Linear Prediction . 401

14.10.1 Best Linear Prediction . 401

14.10.2 Prediction Error in Best Linear Prediction . 402

14.10.3 Regression Is Empirical Best Linear Prediction . 402

14.10.4 Multivariate Linear Prediction . 403

14.11 Regression Hedging . 403

14.12 Bibliographic Notes . 405

14.13 References . 405

14.14 R Lab . 406

14.14.1 Regression with ARMA Noise . 406

14.14.2 Nonlinear Regression . 406

14.14.3 Response Transformations . 409

14.14.4 Binary Regression: Who Owns an Air Conditioner? . 410

14.15 Exercises . 410

15 Cointegration. : 413

15.1 Introduction . 413

15.2 Vector Error Correction Models . 415

15.3 Trading Strategies . 419

15.4 Bibliographic Notes . 419

15.5 References . 419

15.6 R Lab . 420

15.6.1 Cointegration Analysis of Midcap Prices . 420

15.6.2 Cointegration Analysis of Yields . 421

15.6.3 Simulation . 421

15.7 Exercises . 422

16 The Capital Asset Pricing Model . : 423

16.1 Introduction to the CAPM . 423

16.2 The Capital Market Line (CML) . 424

16.3 Betas and the Security Market Line . 426

16.3.1 Examples of Betas . 428

16.3.2 Comparison of the CML with the SML . 428

16.4 The Security Characteristic Line . 429

16.4.1 Reducing Unique Risk by Diversiˉcation . 430

16.4.2 Are the Assumptions Sensible? . 432

16.5 Some More Portfolio Theory . 432

16.5.1 Contributions to the Market Portfolio's Risk . 432

16.5.2 Derivation of the SML . 433

16.6 Estimation of Beta and Testing the CAPM . 434

16.6.1 Estimation Using Regression . 434

16.6.2 Testing the CAPM . 436

16.6.3 Interpretation of Alpha . 437

16.7 Using the CAPM in Portfolio Analysis . 437

16.8 Bibliographic Notes . 437

16.9 References . 438

16.10 R Lab . 438

16.11 Exercises . 440

17 Factor Models and Principal Components . 443

17.1 Dimension Reduction . 443

17.2 Principal Components Analysis . 443

17.3 Factor Models . 453

17.4 Fitting Factor Models by Time Series Regression . 454

17.4.1 Fama and French Three-Factor Model . 455

17.4.2 Estimating Expectations and Covariances of Asset

Returns . 460

17.5 Cross-Sectional Factor Models . 463

17.6 Statistical Factor Models . 466

17.6.1 Varimax Rotation of the Factors . 469

17.7 Bibliographic Notes . 470

17.8 References . 470

17.9 R Lab . 471

17.9.1 PCA . 471

17.9.2 Fitting Factor Models by Time Series Regression . 473

17.9.3 Statistical Factor Models . 475

17.10 Exercises . 475

18 GARCH Models . 477

18.1 Introduction . 477

18.2 Estimating Conditional Means and Variances . 478

18.3 ARCH(1) Processes . 479

18.4 The AR(1)/ARCH(1) Model . 481

18.5 ARCH(p) Models . 482

18.6 ARIMA(pA; d; qA)/GARCH(pG; qG) Models . 483

18.6.1 Residuals for ARIMA(pA; d; qA)/GARCH(pG; qG)

Models . 484

18.7 GARCH Processes Have Heavy Tails . 484

18.8 Fitting ARMA/GARCH Models . 484

18.9 GARCH Models as ARMA Models . 488

18.10 GARCH(1,1) Processes . 489

18.11 APARCH Models . 491

18.12 Regression with ARMA/GARCH Errors . 494

18.13 Forecasting ARMA/GARCH Processes . 497

18.14 Bibliographic Notes . 498

18.15 References . 499

18.16 R Lab . 500

18.16.1 Fitting GARCH Models . 500

18.17 Exercises . 501

19 Risk Management . 505

19.1 The Need for Risk Management . 505

19.2 Estimating VaR and ES with One Asset . 506

19.2.1 Nonparametric Estimation of VaR and ES . 507

19.2.2 Parametric Estimation of VaR and ES . 508

19.3 Conˉdence Intervals for VaR and ES Using the Bootstrap . 511

19.4 Estimating VaR and ES Using ARMA/GARCH Models . 512

19.5 Estimating VaR and ES for a Portfolio of Assets . 514

19.6 Estimation of VaR Assuming Polynomial Tails . 516

19.6.1 Estimating the Tail Index . 518

19.7 Pareto Distributions . 522

19.8 Choosing the Horizon and Conˉdence Level . 523

19.9 VaR and Diversiˉcation . 524

19.10 Bibliographic Notes . 526

19.11 References . 526

19.12 R Lab . 527

19.12.1 VaR Using a Multivariate-t Model . 527

19.13 Exercies . 528

20 Bayesian Data Analysis and MCMC. 531

20.1 Introduction . 531

20.2 Bayes's Theorem . 532

20.3 Prior and Posterior Distributions . 534

20.4 Conjugate Priors . 536

20.5 Central Limit Theorem for the Posterior . 543

20.6 Posterior Intervals . 543

20.7 Markov Chain Monte Carlo . 545

20.7.1 Gibbs Sampling . 546

20.7.2 Other Monte Carlo Samplers . 547

20.7.3 Analysis of MCMC Output . 548

20.7.4 WinBUGS . 549

20.7.5 Monitoring MCMC Convergence and Mixing . 551

20.7.6 DIC and pD for Model Comparisons . 556

20.8 Hierarchical Priors . 558

20.9 Bayesian Estimation of a Covariance Matrix . 562

20.9.1 Estimating a Multivariate Gaussian Covariance Matrix 562

20.9.2 Estimating a multivariate-t Scale Matrix . 564

20.9.3 Non-conjugate Priors for the Covariate Matrix . 566

20.10 Sampling a Stationary Process . 566

20.11 Bibliographic Notes . 567

20.12 References . 569

20.13 R Lab . 570

20.13.1 Fitting a t-Distribution by MCMC. 570

20.13.2 AR Models . 574

20.13.3 MA Models . 575

20.13.4 ARMA Models . 577

20.14 Exercises . 577

21 Nonparametric Regression and Splines . : 579

21.1 Introduction . 579

21.2 Local Polynomial Regression . 581

21.2.1 Lowess and Loess . 584

21.3 Linear Smoothers . 584

21.3.1 The Smoother Matrix and the E?ective Degrees of

Freedom . 585

21.3.2 AIC and GCV . 585

21.4 Polynomial Splines . 586

21.4.1 Linear Splines with One Knot . 586

21.4.2 Linear Splines with Many Knots . 587

21.4.3 Quadratic Splines . 588

21.4.4 pth Degree Splines . 589

21.4.5 Other Spline Bases . 589

21.5 Penalized Splines . 589

21.5.1 Selecting the Amount of Penalization . 591

21.6 Bibliographic Notes . 593

21.7 References . 593

21.8 R Lab . 594

21.8.1 Additive Model for Wages, Education, and Experience 594

21.8.2 An Extended CKLS model for the Short Rate . 595

21.9 Exercises . 596

A Facts from Probability, Statistics, and Algebra . : 597

A.1 Introduction . 597

A.2 Probability Distributions . 597

A.2.1 Cumulative Distribution Functions . 597

A.2.2 Quantiles and Percentiles . 597

A.2.3 Symmetry and Modes . 598

A.2.4 Support of a Distribution . 598

A.3 When Do Expected Values and Variances Exist? . 598

A.4 Monotonic Functions . 599

A.5 The Minimum, Maximum, Inˉnum, and Supremum of a Set . 599

A.6 Functions of Random Variables . 600

A.7 Random Samples . 601

A.8 The Binomial Distribution . 601

A.9 Some Common Continuous Distributions . 602

A.9.1 Uniform Distributions . 602

A.9.2 Transformation by the CDF and Inverse CDF . 602

A.9.3 Normal Distributions . 603

A.9.4 The Lognormal Distribution . 603

A.9.5 Exponential and Double-Exponential Distributions . 604

A.9.6 Gamma and Inverse-Gamma Distributions . 605

A.9.7 Beta Distributions . 606

A.9.8 Pareto Distributions . 606

A.10 Sampling a Normal Distribution . 607

A.10.1 Chi-Squared Distributions . 607

A.10.2 F-distributions . 607

A.11 Law of Large Numbers and the Central Limit Theorem for

the Sample Mean . 608

A.12 Bivariate Distributions . 608

A.13 Correlation and Covariance . 609

A.13.1 Normal Distributions: Conditional Expectations and

Variance . 612

A.14 Multivariate Distributions . 613

A.14.1 Conditional Densities . 613

A.15 Stochastic Processes . 614

A.16 Estimation . 614

A.16.1 Introduction . 614

A.16.2 Standard Errors . 615

A.17 Conˉdence Intervals . 615

A.17.1 Conˉdence Interval for the Mean . 615

A.17.2 Conˉdence Intervals for the Variance and Standard

Deviation . 616

A.17.3 Conˉdence Intervals Based on Standard Errors . 617

A.18 Hypothesis Testing . 617

A.18.1 Hypotheses, Types of Errors, and Rejection Regions . 617

A.18.2 p-Values . 618

A.18.3 Two-Sample t-Tests . 618

A.18.4 Statistical Versus Practical Signiˉcance . 620

A.19 Prediction . 620

A.20 Facts About Vectors and Matrices . 621

A.21 Roots of Polynomials and Complex Numbers . 621

A.22 Bibliographic Notes . 622

A.23 References . 622

Index 623