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ISSN 1431-875X ISSN 2197-4136 (electronic)
ISBN 978-1-4614-8787-6 ISBN 978-1-4614-8788-3 (eBook)
DOI 10.1007/978-1-4614-8788-3
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2013951152
© Springer Science+Business Media New York 2014
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Contents
Part I DATA EXPLORATION, ESTIMATION
AND SIMULATION 1
1 UNIVARIATE DATA DISTRIBUTIONS 3
1.1 Probability Distributions and Their Parameters . . . . . . . . . . . . . . . . . . 3
1.2 Observations and Nonparametric Density Estimation . . . . . . . . . . . . . 31
1.3 Monte Carlo Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2 HEAVY TAIL DISTRIBUTIONS 69
2.2 GEV & GPD Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.3 Semi Parametric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3 DEPENDENCE & MULTIVARIATE DATA EXPLORATION 121
3.1 Multivariate Data and First Measure of Dependence . . . . . . . . . . . . . 121
3.2 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.3 Marginals and More Measures of Dependence . . . . . . . . . . . . . . . . . . 135
3.4 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3.5 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Part II REGRESSION 197
4 PARAMETRIC REGRESSION 199
4.1 Simple Linear Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.1.6 Regression as a Minimization Problem . . . . . . . . . . . . . . . . . . 209
4.2 Regression for Prediction & Sensitivities . . . . . . . . . . . . . . . . . . . . . . . 211
4.3 Smoothing Versus Distribution Theory . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.4 Multiple Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.5 Matrix Formulation and Linear Models . . . . . . . . . . . . . . . . . . . . . . . . 228
4.6 Polynomial Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
4.7 Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
4.8 Term Structure of Interest Rates: A Crash Course . . . . . . . . . . . . . . . . 252
4.9 Parametric Yield Curve Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
5 LOCAL AND NONPARAMETRIC REGRESSION 277
5.1 Review of the Regression Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
5.2 Basis Expansion Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
5.3 Nonparametric Scatterplot Smoothers. . . . . . . . . . . . . . . . . . . . . . . . . . 283
5.4 More Yield Curve Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
5.5 Multivariate Kernel Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.6 Projection Pursuit Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
5.7 Nonparametric Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Part III TIME SERIES & STATE SPACE MODELS 343
6 TIME SERIES MODELS: AR, MA, ARMA, & ALL THAT 345
6.1 Notation and First Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
6.2 Time Dependent Statistics and Stationarity . . . . . . . . . . . . . . . . . . . . . 355
6.3 First Examples of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
6.5 Putting a Price on Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
7 MULTIVARIATE TIME SERIES, LINEAR SYSTEMSAND KALMAN FILTERING 423
7.1 Multivariate Time Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
7.2 State Space Models: Mathematical Set Up . . . . . . . . . . . . . . . . . . . . . . 435
7.3 Factor Models as Hidden Markov Processes . . . . . . . . . . . . . . . . . . . . 437
7.4 Kalman Filtering of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
7.5 Applications to Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
7.6 State Space Representation of Time Series. . . . . . . . . . . . . . . . . . . . . . 453
7.7 Example: Prediction of Quarterly Earnings . . . . . . . . . . . . . . . . . . . . . 457
8 NONLINEAR TIME SERIES: MODELS AND SIMULATION 473
8.2 More Nonlinear Models: ARCH, GARCH & All That . . . . . . . . . . . . 478
8.3 Stochastic Volatility Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
8.4 Discretization of Stochastic Differential Equations . . . . . . . . . . . . . . . 500
8.5 Random Simulation and Scenario Generation . . . . . . . . . . . . . . . . . . . 505
9 APPENDICES 537
References 559
Notation Index 565
Data Set Index 569
R Index 571
Author Index 575
Subject Index 579
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