Analysis of financial time series

Analysis of Financial Time Series Second Edition RUEY S TSAY University of Chicago Graduate School of Business A JOHN WILEY & SONS, INC., PUBLICATION 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 RUEY S TSAY University of Chicago Graduate School of Business A JOHN WILEY & SONS, INC., PUBLICATION Copyright  2005 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in 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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) Time-series analysis Econometrics Risk management I Title HA30.3T76 2005 332 01 51955—dc22 2005047030 Printed in the United States of America 10 To my parents and Teresa Contents Preface xvii Preface to First Edition xix Financial Time Series and Their Characteristics 1.1 1.2 Asset Returns, Distributional Properties of Returns, 1.2.1 Review of Statistical Distributions and Their Moments, 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 Linear Time Series Analysis and Its Applications 2.1 2.2 2.3 2.4 24 Stationarity, 25 Correlation and Autocorrelation Function, 25 White Noise and Linear Time Series, 31 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 REFERENCES 599 Gelfand, A E and Smith, A F M (1990) Sampling-based approaches to calculating marginal densities Journal of the American Statistical Association 85: 398–409 Gelfand, A E., Hills, S E., Racine-Poon, A., and Smith, A F M (1990) Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American Statistical Association 85: 972–985 Gelman, A., Carlin, J B., Stern, H S., and Rubin, D B (2003) Bayesian Data Analysis, 2nd edition Chapman and Hall/CRC Press, London Geman, S and Geman, D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images IEEE Transactions on Pattern Analysis and Machine Intelligence 6: 721–741 Hasting, W K (1970) Monte Carlo sampling methods using Markov chains and their applications Biometrika 57: 97–109 Jacquier, E., Polson, N G., and Rossi, P E (1994) Bayesian analysis of stochastic volatility models (with discussion) Journal of Business & Economic Statistics 12: 371–417 Jacquier, E., Polson, N G., and Rossi, P E (2004) Bayesian analysis of stochastic volatility models with fat-tails and correlated errors Journal of Econometrics 122: 185–212 Jones, R H.(1980) Maximum likelihood fitting of ARMA models to time series with missing observations Technometrics 22: 389–395 Justel, A., Pe˜na, D., and Tsay, R S (2001) Detection of outlier patches in autoregressive time series Statistica Sinica 11: 651–673 Kim, S., Shephard, N., and Chib, S (1998) Stochastic volatility: Likelihood inference and comparison with ARCH models Review of Economic Studies 65: 361–393 Liu, J., Wong, W H., and Kong, A (1994) Correlation structure and convergence rate of the Gibbs samplers I: Applications to the comparison of estimators and augmentation schemes Biometrika 81: 27–40 McCulloch, R E and Tsay, R S (1994a), Bayesian analysis of autoregressive time series via the Gibbs sampler Journal of Time Series Analysis 15: 235–250 McCulloch, R E and Tsay, R S (1994b) Statistical analysis of economic time series via Markov switching models Journal of Time Series Analysis 15: 523–539 McCulloch, R E and Tsay, R S (2001) Nonlinearity in high-frequency financial data and hierarchical models Studies in Nonlinear Dynamics and Econometrics 5: 1–17 Metropolis, N and Ulam, S (1949) The Monte Carlo method Journal of the American Statistical Association 44: 335–341 Metropolis, N., Rosenbluth, A W., Rosenbluth, M N., Teller, A H., and Teller, E (1953) Equation of state calculations by fast computing machines Journal of Chemical Physics 21: 1087–1092 Tanner, M A (1996) Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd edition Springer-Verlag, New York Tanner, M A and Wong, W H (1987) The calculation of posterior distributions by data augmentation (with discussion) Journal of the American Statistical Association 82: 528–550 Tierney, L (1994) Markov chains for exploring posterior distributions (with discussion) Annals of Statistics 22: 1701–1762 Tsay, R S (1988) Outliers, level shifts, and variance changes in time series Journal of Forecasting 7: 1–20 600 MARKOV CHAIN MONTE CARLO METHODS WITH APPLICATIONS Tsay, R S., Pe˜na, D., and Pankratz, A (2000) Outliers in multivariate time series Biometrika 87: 789–804 Zhang, M Y., Russell, J R., and Tsay, R S (2000) Determinants of bid and ask quotes and implications for the cost of trading Working paper, Statistics Research Center, Graduate School of Business, University of Chicago Index ACD, see Autoregressive conditional duration Activation function, see Neural network Airline model, 75 Akaike information criterion (AIC), 41, 356 Arbitrage, 391 Autoregressive conditional hetereoscedastic (ARCH) effect, 101 Autoregressive conditional hetereoscedastic (ARCH) model, 102 estimation, 107 t distribution, 108 GED innovation, 108 normal, 107 Arranged autoregression, 189 Augmented Dickey−Fuller test, 69 Autocorrelation function (ACF), 26 Autoregressive conditional duration (ACD) model, 227 exponential, 228 generalized Gamma, 229 threshold, 236 Weibull, 228 Autoregressive integrated moving-average (ARIMA) model, 67 Autoregressive model, 32 estimation, 43 forecasting, 47 order, 41 stationarity, 40 Autoregressive moving-average (ARMA) model, 56 forecasting, 61 Back propagation (BP) neural network, 180 Back-shift operator, 36 Bartlett’s formula, 27 Analysis of Financial Time Series, Second Edition Copyright  2005 John Wiley & Sons, Inc Bayesian information criterion (BIC), 42 Bid–ask bounce, 211 Bid–ask spread, 211 Bilinear model, 156 Black−Scholes differential equation, 263 Black−Scholes formula European call option, 97, 265 European put option, 265 Brownian motion, 255 geometric, 258 standard, 253 Business cycle, 37 Canonical correlation analysis, 385 Characteristic equation, 40 Characteristic root, 36, 40 Cholesky decomposition, 350, 397, 455 Co-integration, 82, 376 Co-integration test maximum eigenvalue, 385 trace, 385 Common factor, 477 Common trend, 378 Companion matrix, 354 Compounding, Conditional distribution, Conditional forecast, 48 Conditional heteroscedasticity, 86 HAC covariance estimator, 86 Conditional heteroscedasticity ARMA (CHARMA) model, 131 Conditional likelihood method, 53 Conjugate prior, see Distribution By Ruey S Tsay 601 602 Correlation coefficient, 25 constant, 459 time-varying, 464 Cost-of-carry model, 390 Covariance matrix, 340 Cross-correlation matrix, 340, 341 Cross-validation, 170 Data 3M stock return, 19, 59, 66, 164 Cisco stock return, 260, 472, 480 Citi-Group stock return, 19 Civilian employment number, 412 consumer price index, 412 equal-weighted index, 19, 52, 54, 91, 157, 190 GE stock return, 591 Hewlett-Packard stock return, 423 Hong Kong market index, 445 IBM stock return, 19, 28, 126, 135, 136, 161, 180, 190, 259, 292, 295, 298, 300, 307, 313, 323, 343, 423, 462, 478, 573 IBM transactions, 213, 215, 219, 223, 234, 240 Intel stock return, 19, 100, 109, 299, 423, 472, 480 Japan market index, 445 Johnson and Johnson’s earning, 72 Mark/Dollar exchange rate, 104 Merrill Lynch stock return, 423 Microsoft stock return, 19 Morgan Stanley Dean Witter stock return, 423 SP 500 excess return, 116, 132 SP 500 index futures, 390, 392 SP 500 index return, 135, 138, 141, 343, 462, 472, 478, 569, 573, 586 SP 500 spot price, 392 U.S 3-month treasury bill rate, 173 U.S government bond, 21, 345, 431 U.S interest rate, 21, 80, 556, 564 U.S monthly unemployment rate, 159 U.S real GNP, 38, 165 U.S unemployment rate, 194 value-weighted index, 19, 28, 41, 91, 126, 190 Data augmentation, 544 Decomposition model, 221 Descriptive statistics, 19 Diagonal VEC model, 447 Dickey−Fuller test, 68 Differencing, 68 seasonal, 74 Distribution beta, 549 conjugate prior, 548 double exponential, 276 INDEX Frechet family, 303 gamma, 243, 549 generalized error, 108 generalized extreme value, 302 generalized Gamma, 245 generalized Pareto, 320, 330 inverted chi-squared, 551 Laplacian, 275 multivariate t, 482 multivariate normal, 399, 549 negative binomial, 550 Poisson, 550 posterior, 548 prior, 548 Weibull, 244 Diurnal pattern, 212 Donsker’s theorem, 254 Duration between trades, 215 model, 225 Durbin−Watson statistic, 85 EGARCH model, 124 forecasting, 128 Eigenvalue, 396 Eigenvector, 396 EM algorithm, 544 Error-correction model, 380 Estimation, extreme value parameter, 304 Exact likelihood method, 53 Exceedance, 318 Exceeding times, 318 Excess return, Extended autocorrelation function, 59 Extreme value theory, 301 Factor analysis, 426 Factor mimicking portfolio, 420 Factor model common factor, 406 estimation, 428 factor loading, 406 specific factor, 406 Factor rotation, varimax, 429 Filtering, 493 Forecast horizon, 47 origin, 47 Forecast updating formula, 513 Forecasting, see Markov chain Monte Carlo method Forward filtering and backward sampling, 583 Fractional differencing, 89 603 INDEX GARCH model, 114 Cholesky decomposition, 468 diagonal multivariate multivariate, 459 time-varying correlation, 466 GARCH-M model, 123, 588 Generalized least squares, 415 Generalized Pareto distribution, 320 Geometric ergodicity, 158 Gibbs sampling, 545 Global minimum variance portfolio, 411 Griddy Gibbs, 553 Hazard function, 245 Hh function, 281 Hill estimator, 306 Hyperparameter, 554 Identifiability, 371 IGARCH model, 122, 290 Implied volatility, 98 Impulse response function, 63, 362 Innovation, 31 Inverted yield curve, 82 Invertibility, 52, 379 Invertible ARMA model, 62 Ito’s lemma, 258 multivariate, 273 Ito process, 256 Joint distribution function, Jump diffusion, 275 Kalman filter, 496, 524 Kalman gain, 495, 524 Kernel, 168 bandwidth, 169 Epanechnikov, 169 Gaussian, 169 Kernel regression, 168 Kurtosis, excess, Lag operator, 36 Lead-lag relationship, 341 Leptokurtic, Leverage effect, 99, 125, 579 Likelihood function, 17 Linear time series, 31 Liquidity, 210 Ljung−Box statistic, 27, 101 multivariate, 346 Local linear regression, 173 Local trend model, 491 Log return, Logit model, 239 Long position, Long-memory stochastic volatility, 135 time series, 89 Marginal distribution, Market model, 408 Markov chain Monte Carlo method (MCMC), 177, 594 Markov process, 543 Markov property, 32 Markov switching model, 164, 588 Martingale difference, 114 Maximum likelihood estimate (MLE), exact, 368 Mean equation, 101 Mean excess function, 321 Mean excess plot, 321 Mean reversion, 49, 63 half-life, 49 Metropolis algorithm, 551 Metropolis−Hasting algorithm, 552 Missing value, 531, 558 Model checking, 44 Moment of a random variable, Moving-average model, 50 Nadaraya−Watson estimator, 169 Neural network, 177 activation function, 178 feed-forward, 177 skip layer, 179 Neuron, see Neural network Node, see Neural network Nonlinearity test, 183 Brock-Dechert-Scheinkman (BDS), 185 bispectral, 184 F-test, 188 Keenan, 187 RESET, 186 Tar-F, 190 Nonstationarity, unit-root, 64 Nonsynchronous trading, 207 Nuisance parameter, 188 Options American, 252 at-the-money, 252 European call, 97 604 in-the-money, 252 out-of-the-money, 252 stock, 252 strike price, 97, 252 Order statistics, 299 Ordered probit model, 218 Orthogonal factor model, 427 Outlier additive, 558 detection, 561 Parametric bootstrap, 192 Partial autoregressive function (PACF), 40 Peaks over thresholds, 318 π -weight, 62 Pickands estimator, 306 Platykurtic, Poisson process, 275 inhomogeneous, 329 intensity function, 322 Portmanteau test, 27 See also Ljung–Box statistic Positive definite matrix, 396 Prediction, 493 Present value, Price change and duration (PCD) model, 238 Principal component analysis, 421, 478 ψ-weight, 31 Put-call parity, 266 Quantile, definition, 289 R-square, 46 adjusted, 47 Random coefficient (RCA) model, 133 Random walk, 64 with drift, 65 Realized volatility, 141, 492 Reduced form model, 349 Regression, with time series errors, 80 Return level, 317 stress period, 317 RiskMetrics, 290 Sample autocorrelation, 26 Scree plot, 425 Seasonal adjustment, 72 Seasonal model, 72 multiplicative, 75 INDEX Shape parameter of a distribution, 302 Shock, 31, 48, 101 Short position, Simple return, Skewness, Smoothed disturbance, 528 Smoothing, 167, 493 Square root of time rule, 291 Standard Brownian motion, 69 State space model, 509 nonlinear, 176 Stationarity, 25 weak, 340 Steady state, 525 Stochastic diffusion equation, 256 Stochastic volatility model, 134, 565 multivariate, 571 Structural equation, 350 Structural form, 350 Structural time series model (STSM), 491, 521 Student-t distribution standardized, 108 Survival function, 322 Tail index, 302 TGARCH model, 130 general form, 161 Threshold, 159 Threshold autoregressive model multivariate, 392 self-exciting, 159 smooth, 163 Threshold co-integration, 392 Time plot, 17 Transactions data, 212 Trend stationary model, 67 Unit-root test, 68 Unit-root time series, 64 Unobserved component model, 521 Value at Risk (VaR), 287, 480 econometric approach, 294 homogeneous Poisson process, 324 inhomogeneous Poisson process, 328 RiskMetrics, 290 of a short position, 316 traditional extreme value, 312 Vector autoregressive (VAR) model, 349 Vector ARMA model, 371 marginal models, 375 605 INDEX Vector moving average model, 365 VIX Volatility Index, 98 Volatility, 97 Volatility equation, 101 Volatility model factor, 477 Volatility smile, 274 Weighted least squares, 415 White noise, 31 Wiener process, 253 generalized, 255 Yule–Walker equation multivariate, 354 [...]... out of an MBA course in analysis of financial time series that I have been teaching at the University of Chicago since 1999 It also covers materials of Ph.D courses in time series analysis that I taught over the years It is an introductory book intended to provide a comprehensive and systematic account of financial econometric models and their application to modeling and prediction of financial time series. .. University of Chicago Chicago, Illinois CHAPTER 1 Financial Time Series and Their Characteristics Financial time series analysis is concerned with the theory and practice of asset valuation over time It is a highly empirical discipline, but like other scientific fields theory forms the foundation for making inference There is, however, a key feature that distinguishes financial time series analysis from other time. .. from other time series analysis Both financial theory and its empirical time series contain an element of uncertainty For example, there are various definitions of asset volatility, and for a stock return series, the volatility is not directly observable As a result of the added uncertainty, statistical theory and methods play an important role in financial time series analysis The objective of this book... nonlinear models The chapter also introduces nonparametric Analysis of Financial Time Series, Second Edition Copyright  2005 John Wiley & Sons, Inc By Ruey S Tsay 1 2 FINANCIAL TIME SERIES AND THEIR CHARACTERISTICS estimation methods and neural networks and shows various applications of nonlinear models in finance Chapter 5 is concerned with analysis of high-frequency financial data and its application to... to provide some knowledge of financial time series, introduce some statistical tools useful for analyzing these series, and gain experience in financial applications of various econometric methods We begin with the basic concepts of asset returns and a brief introduction to the processes to be discussed throughout the book Chapter 2 reviews basic concepts of linear time series analysis such as stationarity... Bayesian analysis Some data sets and programs are accessible from the World Wide Web at http://www.gsb.uchicago.edu/fac/ruey.tsay/teaching/fts The book begins with some basic characteristics of financial time series data in Chapter 1 The other chapters are divided into three parts The first part, consisting of Chapters 2 to 7, focuses on analysis and application of univariate financial time series The... application to modeling and prediction of financial time series data The goals are to learn basic characteristics of financial data, understand the application of financial econometric models, and gain experience in analyzing financial time series The book will be useful as a text of time series analysis for MBA students with finance concentration or senior undergraduate and graduate students in business,... developed recently to describe the evolution of volatility of an asset return over time The chapter also discusses alternative methods to volatility modeling, including use of high-frequency transactions data and daily high and low prices of an asset In Chapter 4, we address nonlinearity in financial time series, introduce test statistics that can discriminate nonlinear series from linear ones, and discuss... 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... empirical characteristics of financial time series are used to motivate the development of econometric models Computer programs and commands used in data analysis are provided when needed In some cases, the programs are given in an appendix Many real data sets are also used in the exercises of each chapter 1.1 ASSET RETURNS Most financial studies involve returns, instead of prices, of assets Campbell, Lo, .. .Analysis of Financial Time Series Second Edition RUEY S TSAY University of Chicago Graduate School of Business A JOHN WILEY & SONS, INC., PUBLICATION Analysis of Financial Time Series WILEY SERIES. .. complete list of the titles in this series appears at the end of this volume Analysis of Financial Time Series Second Edition RUEY S TSAY University of Chicago Graduate School of Business A... grew out of an MBA course in analysis of financial time series that I have been teaching at the University of Chicago since 1999 It also covers materials of Ph.D courses in time series analysis