Analysis of Financial Time Series

Contents
Preface xvii
Preface to the Second Edition xix
Preface to the First Edition xxi
1 Financial Time Series and Their Characteristics 1
1.1 Asset Returns, 2
1.2 Distributional Properties of Returns, 7
1.2.1 Review of Statistical Distributions and Their Moments, 7
1.2.2 Distributions of Returns, 14
1.2.3 Multivariate Returns, 18
1.2.4 Likelihood Function of Returns, 19
1.2.5 Empirical Properties of Returns, 19
1.3 Processes Considered, 22
Appendix: R Packages, 24
Exercises, 25
References, 27
2 Linear Time Series Analysis and Its Applications 29
2.1 Stationarity, 30
2.2 Correlation and Autocorrelation Function, 30
2.3 White Noise and Linear Time Series, 36
2.4 Simple AR Models, 37
2.4.1 Properties of AR Models, 38
2.4.2 Identifying AR Models in Practice, 46
2.4.3 Goodness of Fit, 53
2.4.4 Forecasting, 54
2.5 Simple MA Models, 57
2.5.1 Properties of MA Models, 59
2.5.2 Identifying MA Order, 60
2.5.3 Estimation, 61
2.5.4 Forecasting Using MA Models, 62
2.6 Simple ARMA Models, 64
2.6.1 Properties of ARMA(1,1) Models, 64
2.6.2 General ARMA Models, 66
2.6.3 Identifying ARMA Models, 66
2.6.4 Forecasting Using an ARMA Model, 68
2.6.5 Three Model Representations for an ARMA Model, 69
2.7 Unit-Root Nonstationarity, 71
2.7.1 Random Walk, 72
2.7.2 Random Walk with Drift, 73
2.7.3 Trend-Stationary Time Series, 75
2.7.4 General Unit-Root Nonstationary Models, 75
2.7.5 Unit-Root Test, 76
2.8 Seasonal Models, 81
2.8.1 Seasonal Differencing, 82
2.8.2 Multiplicative Seasonal Models, 84
2.9 Regression Models with Time Series Errors, 90
2.10 Consistent Covariance Matrix Estimation, 97
2.11 Long-Memory Models, 101
Appendix: Some SCA Commands, 103
Exercises, 104
References, 107
3 Conditional Heteroscedastic Models 109
3.1 Characteristics of Volatility, 110
3.2 Structure of a Model, 111
3.3 Model Building, 113
3.3.1 Testing for ARCH Effect, 114
3.4 The ARCH Model, 115
3.4.1 Properties of ARCH Models, 117
3.4.2 Weaknesses of ARCH Models, 119
3.4.3 Building an ARCH Model, 119
3.4.4 Some Examples, 123
3.5 The GARCH Model, 131
3.5.1 An Illustrative Example, 134
3.5.2 Forecasting Evaluation, 139
3.5.3 A Two-Pass Estimation Method, 140
3.6 The Integrated GARCH Model, 140
3.7 The GARCH-M Model, 142
3.8 The Exponential GARCH Model, 143
3.8.1 Alternative Model Form, 144
3.8.2 Illustrative Example, 145
3.8.3 Second Example, 145
3.8.4 Forecasting Using an EGARCH Model, 147
3.9 The Threshold GARCH Model, 149
3.10 The CHARMA Model, 150
3.10.1 Effects of Explanatory Variables, 152
3.11 Random Coefficient Autoregressive Models, 152
3.12 Stochastic Volatility Model, 153
3.13 Long-Memory Stochastic Volatility Model, 154
3.14 Application, 155
3.15 Alternative Approaches, 159
3.15.1 Use of High-Frequency Data, 159
3.15.2 Use of Daily Open, High, Low, and Close Prices, 162
3.16 Kurtosis of GARCH Models, 165
Appendix: Some RATS Programs for Estimating Volatility Models, 167
Exercises, 168
References, 171
4 Nonlinear Models and Their Applications 175
4.1 Nonlinear Models, 177
4.1.1 Bilinear Model, 177
4.1.2 Threshold Autoregressive (TAR) Model, 179
4.1.3 Smooth Transition AR (STAR) Model, 184
4.1.4 Markov Switching Model, 186
4.1.5 Nonparametric Methods, 189
4.1.6 Functional Coefficient AR Model, 198
4.1.7 Nonlinear Additive AR Model, 198
4.1.8 Nonlinear State-Space Model, 199
4.1.9 Neural Networks, 199
4.2 Nonlinearity Tests, 205
4.2.1 Nonparametric Tests, 206
4.2.2 Parametric Tests, 209
4.2.3 Applications, 213
4.3 Modeling, 214
4.4 Forecasting, 215
4.4.1 Parametric Bootstrap, 215
4.4.2 Forecasting Evaluation, 215
4.5 Application, 218
Appendix A: Some RATS Programs for Nonlinear Volatility Models, 222
Appendix B: R and S-Plus Commands for Neural Network, 223
Exercises, 224
References, 226
5 High-Frequency Data Analysis and Market Microstructure 231
5.1 Nonsynchronous Trading, 232
5.2 Bid–Ask Spread, 235
5.3 Empirical Characteristics of Transactions Data, 237
5.4 Models for Price Changes, 244
5.4.1 Ordered Probit Model, 245
5.4.2 Decomposition Model, 248
5.5 Duration Models, 253
5.5.1 The ACD Model, 255
5.5.2 Simulation, 257
5.5.3 Estimation, 260
5.6 Nonlinear Duration Models, 264
5.7 Bivariate Models for Price Change and Duration, 265
5.8 Application, 270
Appendix A: Review of Some Probability Distributions, 276
Appendix B: Hazard Function, 279
Appendix C: Some RATS Programs for Duration Models, 280
Exercises, 282
References, 284
6 Continuous-Time Models and Their Applications 287
6.1 Options, 288
6.2 Some Continuous-Time Stochastic Processes, 288
6.2.1 Wiener Process, 289
6.2.2 Generalized Wiener Process, 291
6.2.3 Ito Process, 292
6.3 Ito’s Lemma, 292
6.3.1 Review of Differentiation, 292
6.3.2 Stochastic Differentiation, 293
6.3.3 An Application, 294
Estimation of μ and σ , 295
6.3.4
6.4 Distributions of Stock Prices and Log Returns, 297
6.5 Derivation of Black–Scholes Differential Equation, 298
6.6 Black–Scholes Pricing Formulas, 300
6.6.1 Risk-Neutral World, 300
6.6.2 Formulas, 300
6.6.3 Lower Bounds of European Options, 304
6.6.4 Discussion, 305
6.7 Extension of Ito’s Lemma, 309
6.8 Stochastic Integral, 310
6.9 Jump Diffusion Models, 311
6.9.1 Option Pricing under Jump Diffusion, 315
6.10 Estimation of Continuous-Time Models, 318
Appendix A: Integration of Black–Scholes Formula, 319
Appendix B: Approximation to Standard Normal Probability, 320
Exercises, 321
References, 322
7 Extreme Values, Quantiles, and Value at Risk 325
7.1 Value at Risk, 326
7.2 RiskMetrics, 328
7.2.1 Discussion, 331
7.2.2 Multiple Positions, 332
7.2.3 Expected Shortfall, 332
7.3 Econometric Approach to VaR Calculation, 333
7.3.1 Multiple Periods, 336
7.3.2 Expected Shortfall under Conditional Normality, 338
7.4 Quantile Estimation, 338
7.4.1 Quantile and Order Statistics, 338
7.4.2 Quantile Regression, 341
7.5 Extreme Value Theory, 342
7.5.1 Review of Extreme Value Theory, 342
7.5.2 Empirical Estimation, 345
7.5.3 Application to Stock Returns, 348
7.6 Extreme Value Approach to VaR, 353
7.6.1 Discussion, 356
7.6.2 Multiperiod VaR, 357
7.6.3 Return Level, 358
7.7 New Approach Based on the Extreme Value Theory, 359
7.7.1 Statistical Theory, 360
7.7.2 Mean Excess Function, 361
7.7.3 New Approach to Modeling Extreme Values, 363
7.7.4 VaR Calculation Based on the New Approach, 365
7.7.5 Alternative Parameterization, 367
7.7.6 Use of Explanatory Variables, 371
7.7.7 Model Checking, 372
7.7.8 An Illustration, 373
7.8 The Extremal Index, 377
The D(un ) Condition, 378
7.8.1
7.8.2 Estimation of the Extremal Index, 381
7.8.3 Value at Risk for a Stationary Time Series, 384
Exercises, 384
References, 387
8 Multivariate Time Series Analysis and Its Applications 389
8.1 Weak Stationarity and Cross-Correlation Matrices, 390
8.1.1 Cross-Correlation Matrices, 390
8.1.2 Linear Dependence, 392
8.1.3 Sample Cross-Correlation Matrices, 392
8.1.4 Multivariate Portmanteau Tests, 397
8.2 Vector Autoregressive Models, 399
8.2.1 Reduced and Structural Forms, 399
8.2.2 Stationarity Condition and Moments of a VAR(1)
Model, 401
8.2.3 Vector AR(p) Models, 403
8.2.4 Building a VAR(p) Model, 405
8.2.5 Impulse Response Function, 413
8.3 Vector Moving-Average Models, 417
8.4 Vector ARMA Models, 422
8.4.1 Marginal Models of Components, 427
8.5 Unit-Root Nonstationarity and Cointegration, 428
8.5.1 An Error Correction Form, 431
8.6 Cointegrated VAR Models, 432
8.6.1 Specification of the Deterministic Function, 434
8.6.2 Maximum-Likelihood Estimation, 435
8.6.3 Cointegration Test, 436
8.6.4 Forecasting of Cointegrated VAR Models, 437
8.6.5 An Example, 438
8.7 Threshold Cointegration and Arbitrage, 442
8.7.1 Multivariate Threshold Model, 444
8.7.2 The Data, 445
8.7.3 Estimation, 445
8.8 Pairs Trading, 446
8.8.1 Theoretical Framework, 446
8.8.2 Trading Strategy, 448
8.8.3 Simple Illustration, 449
Appendix A: Review of Vectors and Matrices, 456
Appendix B: Multivariate Normal Distributions, 460
Appendix C: Some SCA Commands, 461
Exercises, 462
References, 464
9 Principal Component Analysis and Factor Models 467
9.1 A Factor Model, 468
9.2 Macroeconometric Factor Models, 470
9.2.1 Single-Factor Model, 470
9.2.2 Multifactor Models, 474
9.3 Fundamental Factor Models, 476
9.3.1 BARRA Factor Model, 477
9.3.2 Fama–French Approach, 482
9.4 Principal Component Analysis, 483
9.4.1 Theory of PCA, 483
9.4.2 Empirical PCA, 485
9.5 Statistical Factor Analysis, 489
9.5.1 Estimation, 490
9.5.2 Factor Rotation, 492
9.5.3 Applications, 492
9.6 Asymptotic Principal Component Analysis, 498
9.6.1 Selecting the Number of Factors, 499
9.6.2 An Example, 500
Exercises, 501
References, 503


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