Preface xvii
Preface to First Edition xix
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, 13
1.2.3 Multivariate Returns, 16
1.2.4 Likelihood Function of Returns, 17
1.2.5 Empirical Properties of Returns, 17
1.3 Processes Considered, 20
Exercises, 22
References, 23
2. Linear Time Series Analysis and Its Applications 24
2.1 Stationarity, 25
2.2 Correlation and Autocorrelation Function, 25
2.3 White Noise and Linear Time Series, 31
2.4 Simple Autoregressive Models, 32
2.4.1 Properties of AR Models, 33
2.4.2 Identifying AR Models in Practice, 40
2.4.3 Goodness of Fit, 46
2.4.4 Forecasting, 47
vii

2.5 Simple Moving-Average Models, 50
2.5.1 Properties of MA Models, 51
2.5.2 Identifying MA Order, 52
2.5.3 Estimation, 53
2.5.4 Forecasting Using MA Models, 54
2.6 Simple ARMA Models, 56
2.6.1 Properties of ARMA(1,1) Models, 57
2.6.2 General ARMA Models, 58
2.6.3 Identifying ARMA Models, 59
2.6.4 Forecasting Using an ARMA Model, 61
2.6.5 Three Model Representations for an ARMA Model, 62
2.7 Unit-Root Nonstationarity, 64
2.7.1 Random Walk, 64
2.7.2 Random Walk with Drift, 65
2.7.3 Trend-Stationary Time Series, 67
2.7.4 General Unit-Root Nonstationary Models, 67
2.7.5 Unit-Root Test, 68
2.8 Seasonal Models, 72
2.8.1 Seasonal Differencing, 73
2.8.2 Multiplicative Seasonal Models, 75
2.9 Regression Models with Time Series Errors, 80
2.10 Consistent Covariance Matrix Estimation, 86
2.11 Long-Memory Models, 89
Appendix: Some SCA Commands, 91
Exercises, 93
References, 96

3. Conditional Heteroscedastic Models 97
3.1 Characteristics of Volatility, 98
3.2 Structure of a Model, 99
3.3 Model Building, 101
3.3.1 Testing for ARCH Effect, 101
3.4 The ARCH Model, 102
3.4.1 Properties of ARCH Models, 104
3.4.2 Weaknesses of ARCH Models, 106
3.4.3 Building an ARCH Model, 106
3.4.4 Some Examples, 109
3.5 The GARCH Model, 113
3.5.1 An Illustrative Example, 116

3.5.2 Forecasting Evaluation, 121
3.5.3 A Two-Pass Estimation Method, 121
3.6 The Integrated GARCH Model, 122
3.7 The GARCH-M Model, 123
3.8 The Exponential GARCH Model, 124
3.8.1 An Alternative Model Form, 125
3.8.2 An Illustrative Example, 126
3.8.3 Second Example, 126
3.8.4 Forecasting Using an EGARCH Model, 128
3.9 The Threshold GARCH Model, 130
3.10 The CHARMA Model, 131
3.10.1 Effects of Explanatory Variables, 133
3.11 Random Coefficient Autoregressive Models, 133
3.12 The Stochastic Volatility Model, 134
3.13 The Long-Memory Stochastic Volatility Model, 134
3.14 Application, 136
3.15 Alternative Approaches, 140
3.15.1 Use of High-Frequency Data, 140
3.15.2 Use of Daily Open, High, Low, and Close Prices, 143
3.16 Kurtosis of GARCH Models, 145
Appendix: Some RATS Programs for Estimating Volatility Models, 147
Exercises, 148
References, 151

4. Nonlinear Models and Their Applications 154
4.1 Nonlinear Models, 156
4.1.1 Bilinear Model, 156
4.1.2 Threshold Autoregressive (TAR) Model, 157
4.1.3 Smooth Transition AR (STAR) Model, 163
4.1.4 Markov Switching Model, 164
4.1.5 Nonparametric Methods, 167
4.1.6 Functional Coefficient AR Model, 175
4.1.7 Nonlinear Additive AR Model, 176
4.1.8 Nonlinear State-Space Model, 176
4.1.9 Neural Networks, 177
4.2 Nonlinearity Tests, 183
4.2.1 Nonparametric Tests, 183
4.2.2 Parametric Tests, 186
4.2.3 Applications, 190

5. High-Frequency Data Analysis and Market Microstructure 206
5.1 Nonsynchronous Trading, 207
5.2 Bid–Ask Spread, 210
5.3 Empirical Characteristics of Transactions Data, 212
5.4 Models for Price Changes, 218
5.4.1 Ordered Probit Model, 218
5.4.2 A Decomposition Model, 221
5.5 Duration Models, 225
5.5.1 The ACD Model, 227
5.5.2 Simulation, 229
5.5.3 Estimation, 232
5.6 Nonlinear Duration Models, 236
5.7 Bivariate Models for Price Change and Duration, 237
Appendix A: Review of Some Probability Distributions, 242
Appendix B: Hazard Function, 245
Appendix C: Some RATS Programs for Duration Models, 246
Exercises, 248
References, 250
6. Continuous-Time Models and Their Applications 251
6.1 Options, 252
6.2 Some Continuous-Time Stochastic Processes, 252
6.2.1 The Wiener Process, 253
6.2.2 Generalized Wiener Processes, 255
6.2.3 Ito Processes, 256
6.3 Ito’s Lemma, 256
6.3.1 Review of Differentiation, 256
6.3.2 Stochastic Differentiation, 257

6.3.3 An Application, 258
6.3.4 Estimation of μ and σ , 259
6.4 Distributions of Stock Prices and Log Returns, 261
6.5 Derivation of Black–Scholes Differential Equation, 262
6.6 Black–Scholes Pricing Formulas, 264
6.6.1 Risk-Neutral World, 264
6.6.2 Formulas, 264
6.6.3 Lower Bounds of European Options, 267
6.6.4 Discussion, 268
6.7 An Extension of Ito’s Lemma, 272
6.8 Stochastic Integral, 273
6.9 Jump Diffusion Models, 274
6.9.1 Option Pricing Under Jump Diffusion, 279
6.10 Estimation of Continuous-Time Models, 282
Appendix A: Integration of Black–Scholes Formula, 282
Appendix B: Approximation to Standard Normal
Probability, 284
Exercises, 284
References, 285
7. Extreme Values, Quantile Estimation, and Value at Risk 287
7.1 Value at Risk, 287
7.2 RiskMetrics, 290
7.2.1 Discussion, 293
7.2.2 Multiple Positions, 293
7.3 An Econometric Approach to VaR Calculation, 294
7.3.1 Multiple Periods, 296
7.4 Quantile Estimation, 298
7.4.1 Quantile and Order Statistics, 299
7.4.2 Quantile Regression, 300
7.5 Extreme Value Theory, 301
7.5.1 Review of Extreme Value Theory, 301
7.5.2 Empirical Estimation, 304
7.5.3 Application to Stock Returns, 307
7.6 Extreme Value Approach to VaR, 311
7.6.1 Discussion, 314
7.6.2 Multiperiod VaR, 316
7.6.3 VaR for a Short Position, 316
7.6.4 Return Level, 317

7.7 A New Approach Based on the Extreme Value Theory, 318
7.7.1 Statistical Theory, 318
7.7.2 Mean Excess Function, 320
7.7.3 A New Approach to Modeling Extreme Values, 322
7.7.4 VaR Calculation Based on the New Approach, 324
7.7.5 An Alternative Parameterization, 325
7.7.6 Use of Explanatory Variables, 328
7.7.7 Model Checking, 329
7.7.8 An Illustration, 330
Exercises, 335
References, 337
8. Multivariate Time Series Analysis and Its Applications 339
8.1 Weak Stationarity and Cross-Correlation Matrices, 340
8.1.1 Cross-Correlation Matrices, 340
8.1.2 Linear Dependence, 341
8.1.3 Sample Cross-Correlation Matrices, 342
8.1.4 Multivariate Portmanteau Tests, 346
8.2 Vector Autoregressive Models, 349
8.2.1 Reduced and Structural Forms, 349
8.2.2 Stationarity Condition and Moments of a VAR(1)
Model, 351
8.2.3 Vector AR(p) Models, 353
8.2.4 Building a VAR(p) Model, 354
8.2.5 Impulse Response Function, 362
8.3 Vector Moving-Average Models, 365
8.4 Vector ARMA Models, 371
8.4.1 Marginal Models of Components, 375
8.5 Unit-Root Nonstationarity and Cointegration, 376
8.5.1 An Error-Correction Form, 379
8.6 Cointegrated VAR Models, 380
8.6.1 Specification of the Deterministic Function, 382
8.6.2 Maximum Likelihood Estimation, 383
8.6.3 A Cointegration Test, 384
8.6.4 Forecasting of Cointegrated VAR Models, 385
8.6.5 An Example, 385
8.7 Threshold Cointegration and Arbitrage, 390
8.7.1 Multivariate Threshold Model, 391
8.7.2 The Data, 392

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