1# yizhengchina
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

2# yizhengchina
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