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
