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# Analysis of Financial Time Series, 2nd Edition. Tsay

(Parte **1** de 5)

Analysis of Financial Time Series Second Edition

University of Chicago Graduate School of Business

Analysis of Financial Time Series Analysis of Financial Time Series

WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors: David J. Balding, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, Geert Molenberghs, Louise M. Ryan, David W. Scott, Adrian F. M. Smith, Jozef L. Teugels Editors Emeriti: Vic Barnett, J. Stuart Hunter, David G. Kendall

A complete list of the titles in this series appears at the end of this volume.

Analysis of Financial Time Series Second Edition

University of Chicago Graduate School of Business

Copyright 2005 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Tsay, Ruey S., 1951–

Analysis of ﬁnancial time series/Ruey S. Tsay.—2nd ed. p. cm. “Wiley-Interscience.” Includes bibliographical references and index. ISBN-13 978-0-471-69074-0 ISBN-10 0-471-69074-0 (cloth) 1. Time-series analysis. 2. Econometrics. 3. Risk management. I. Title.

Printed in the United States of America. 1098 765 4321

To my parents and Teresa To my parents and Teresa

Contents

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, 2 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, 3 2.4.2 Identifying AR Models in Practice, 40 2.4.3 Goodness of Fit, 46 2.4.4 Forecasting, 47 vii viii CONTENTS

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 2.1 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, 9 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

CONTENTS ix

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 3.1 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 x CONTENTS

Appendix A: Some RATS Programs for Nonlinear Volatility Models, 199

Appendix B: S-Plus Commands for Neural Network, 200 Exercises, 200 References, 202

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

CONTENTS xi

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 xii CONTENTS

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

CONTENTS xiii

Appendix A: Review of Vectors and Matrices, 395 Appendix B: Multivariate Normal Distributions, 399 Appendix C: Some SCA Commands, 400 Exercises, 401 References, 402

9. Principal Component Analysis and Factor Models 405

9.1 A Factor Model, 406 9.2 Macroeconometric Factor Models, 407 9.2.1 A Single-Factor Model, 408 9.2.2 Multifactor Models, 412 9.3 Fundamental Factor Models, 414 9.3.1 BARRA Factor Model, 414 9.3.2 Fama–French Approach, 420 9.4 Principal Component Analysis, 421 9.4.1 Theory of PCA, 421 9.4.2 Empirical PCA, 422 9.5 Statistical Factor Analysis, 426 9.5.1 Estimation, 428 9.5.2 Factor Rotation, 429 9.5.3 Applications, 430 9.6 Asymptotic Principal Component Analysis, 436 9.6.1 Selecting the Number of Factors, 437 9.6.2 An Example, 437

Exercises, 440 References, 441

10. Multivariate Volatility Models and Their Applications 443

10.1 Exponentially Weighted Estimate, 4 10.2 Some Multivariate GARCH Models, 447 10.2.1 Diagonal VEC Model, 447 10.2.2 BEKK Model, 451 10.3 Reparameterization, 454 10.3.1 Use of Correlations, 454 10.3.2 Cholesky Decomposition, 455 10.4 GARCH Models for Bivariate Returns, 459 10.4.1 Constant-Correlation Models, 459 10.4.2 Time-Varying Correlation Models, 464 xiv CONTENTS

10.4.3 Some Recent Developments, 470 10.5 Higher Dimensional Volatility Models, 471 10.6 Factor–Volatility Models, 477 10.7 Application, 480 10.8 Multivariate t Distribution, 482 Appendix: Some Remarks on Estimation, 483 Exercises, 488 References, 489

1. State-Space Models and Kalman Filter 490

1.1 Local Trend Model, 490 1.1.1 Statistical Inference, 493 1.1.2 Kalman Filter, 495 1.1.3 Properties of Forecast Error, 496 1.1.4 State Smoothing, 498 1.1.5 Missing Values, 501 1.1.6 Effect of Initialization, 503 1.1.7 Estimation, 504 1.1.8 S-Plus Commands Used, 505 1.2 Linear State-Space Models, 508 1.3 Model Transformation, 509 1.3.1 CAPM with Time-Varying Coefﬁcients, 510 1.3.2 ARMA Models, 512 1.3.3 Linear Regression Model, 518 1.3.4 Linear Regression Models with ARMA Errors, 519 1.3.5 Scalar Unobserved Component Model, 521 1.4 Kalman Filter and Smoothing, 523 1.4.1 Kalman Filter, 523 1.4.2 State Estimation Error and Forecast Error, 525 1.4.3 State Smoothing, 526 1.4.4 Disturbance Smoothing, 528 1.5 Missing Values, 531 1.6 Forecasting, 532 1.7 Application, 533 Exercises, 540 References, 541

CONTENTS xv

12. Markov Chain Monte Carlo Methods with Applications 543

12.1 Markov Chain Simulation, 544 12.2 Gibbs Sampling, 545 12.3 Bayesian Inference, 547 12.3.1 Posterior Distributions, 547 12.3.2 Conjugate Prior Distributions, 548 12.4 Alternative Algorithms, 551 12.4.1 Metropolis Algorithm, 551 12.4.2 Metropolis–Hasting Algorithm, 552 12.4.3 Griddy Gibbs, 552 12.5 Linear Regression with Time Series Errors, 553 12.6 Missing Values and Outliers, 558 12.6.1 Missing Values, 559 12.6.2 Outlier Detection, 561 12.7 Stochastic Volatility Models, 565 12.7.1 Estimation of Univariate Models, 566 12.7.2 Multivariate Stochastic Volatility Models, 571 12.8 A New Approach to SV Estimation, 578 12.9 Markov Switching Models, 588 12.10 Forecasting, 594 12.1 Other Applications, 597 Exercises, 597 References, 598

Index 601

Preface

The subject of ﬁnancial time series analysis has attracted substantial attention in recent years, especially with the 2003 Nobel awards to Professors Robert Engle and Clive Granger. At the same time, the ﬁeld of ﬁnancial econometrics has undergone various new developments, especially in high-frequency ﬁnance, stochastic volatility, and software availability. There is a need to make the material more complete and accessible for advanced undergraduate and graduate students, practitioners, and researchers. The main goals in preparing this second edition have been to bring the book up to date both in new developments and empirical analysis, and to enlarge the core material of the book by including consistent covariance estimation under heteroscedasticity and serial correlation, alternative approaches to volatility modeling, ﬁnancial factor models, state-space models, Kalman ﬁltering, and estimation of stochastic diffusion models.

The book therefore has been extended to 10 chapters and substantially revised to include S-Plus commands and illustrations. Many empirical demonstrations and exercises are updated so that they include the most recent data.

The two new chapters are Chapter 9, Principal Component Analysis and Factor

Models, and Chapter 1, State-Space Models and Kalman Filter. The factor models discussed include macroeconomic, fundamental, and statistical factor models. They are simple and powerful tools for analyzing high-dimensional ﬁnancial data such as portfolio returns. Empirical examples are used to demonstrate the applications. The state-space model and Kalman ﬁlter are added to demonstrate their applicability in ﬁnance and ease in computation. They are used in Chapter 12 to estimate stochastic volatility models under the general Markov chain Monte Carlo (MCMC) framework. The estimation also uses the technique of forward ﬁltering and backward sampling to gain computational efﬁciency. A brief summary of the added material in the second edition is:

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