Index

ACD, see Autoregressive conditional durationActivation function, see Neural networkAirline model, 75Akaike information criterion (AIC), 41, 356Arbitrage, 391Autoregressive conditional hetereoscedastic

(ARCH) effect, 101Autoregressive conditional hetereoscedastic

(ARCH) model, 102estimation, 107

t distribution, 108GED innovation, 108normal, 107

Arranged autoregression, 189Augmented DickeyFuller test, 69Autocorrelation function (ACF), 26Autoregressive conditional duration (ACD)

model, 227exponential, 228generalized Gamma, 229threshold, 236Weibull, 228

Autoregressive integrated moving-average(ARIMA) model, 67

Autoregressive model, 32estimation, 43forecasting, 47order, 41stationarity, 40

Autoregressive moving-average(ARMA) model, 56

forecasting, 61

Back propagation (BP) neural network, 180Back-shift operator, 36Bartletts formula, 27

Analysis of Financial Time Series, Second Edition By Ruey S. TsayCopyright 2005 John Wiley & Sons, Inc.

Bayesian information criterion(BIC), 42

Bidask bounce, 211Bidask spread, 211Bilinear model, 156BlackScholes

differential equation, 263BlackScholes formula

European call option, 97, 265European put option, 265

Brownian motion, 255geometric, 258standard, 253

Business cycle, 37

Canonical correlation analysis, 385Characteristic equation, 40Characteristic root, 36, 40Cholesky decomposition, 350, 397, 455Co-integration, 82, 376Co-integration test

maximum eigenvalue, 385trace, 385

Common factor, 477Common trend, 378Companion matrix, 354Compounding, 4Conditional distribution, 8Conditional forecast, 48Conditional heteroscedasticity, 86

HAC covariance estimator, 86Conditional heteroscedasticity ARMA

(CHARMA) model, 131Conditional likelihood method, 53Conjugate prior, see Distribution

601

602 INDEX

Correlationcoefcient, 25constant, 459time-varying, 464

Cost-of-carry model, 390Covariance matrix, 340Cross-correlation matrix, 340, 341Cross-validation, 170

Data3M stock return, 19, 59, 66, 164Cisco stock return, 260, 472, 480Citi-Group stock return, 19Civilian employment number, 412consumer price index, 412equal-weighted index, 19, 52, 54, 91, 157, 190GE stock return, 591Hewlett-Packard stock return, 423Hong Kong market index, 445IBM 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, 240Intel stock return, 19, 100, 109, 299, 423, 472,

480Japan market index, 445Johnson and Johnsons earning, 72Mark/Dollar exchange rate, 104Merrill Lynch stock return, 423Microsoft stock return, 19Morgan Stanley Dean Witter stock return, 423SP 500 excess return, 116, 132SP 500 index futures, 390, 392SP 500 index return, 135, 138, 141, 343, 462,

472, 478, 569, 573, 586SP 500 spot price, 392U.S. 3-month treasury bill rate, 173U.S. government bond, 21, 345, 431U.S. interest rate, 21, 80, 556, 564U.S. monthly unemployment rate, 159U.S. real GNP, 38, 165U.S. unemployment rate, 194value-weighted index, 19, 28, 41, 91, 126, 190

Data augmentation, 544Decomposition model, 221Descriptive statistics, 19Diagonal VEC model, 447DickeyFuller test, 68Differencing, 68

seasonal, 74Distribution

beta, 549conjugate prior, 548double exponential, 276

Frechet family, 303gamma, 243, 549generalized error, 108generalized extreme value, 302generalized Gamma, 245generalized Pareto, 320, 330inverted chi-squared, 551Laplacian, 275multivariate t , 482multivariate normal, 399, 549negative binomial, 550Poisson, 550posterior, 548prior, 548Weibull, 244

Diurnal pattern, 212Donskers theorem, 254Duration

between trades, 215model, 225

DurbinWatson statistic, 85

EGARCH model, 124forecasting, 128

Eigenvalue, 396Eigenvector, 396EM algorithm, 544Error-correction model, 380Estimation, extreme value parameter, 304Exact likelihood method, 53Exceedance, 318Exceeding times, 318Excess return, 5Extended autocorrelation function, 59Extreme value theory, 301

Factor analysis, 426Factor mimicking portfolio, 420Factor model

common factor, 406estimation, 428factor loading, 406specic factor, 406

Factor rotation, varimax, 429Filtering, 493Forecast

horizon, 47origin, 47

Forecast updating formula, 513Forecasting, see Markov chain Monte Carlo

methodForward ltering and backward sampling, 583Fractional differencing, 89

INDEX 603

GARCH model, 114Cholesky decomposition, 468diagonal multivariatemultivariate, 459time-varying correlation, 466

GARCH-M model, 123, 588Generalized least squares, 415Generalized Pareto distribution, 320Geometric ergodicity, 158Gibbs sampling, 545Global minimum variance portfolio, 411Griddy Gibbs, 553

Hazard function, 245Hh function, 281Hill estimator, 306Hyperparameter, 554

Identiability, 371IGARCH model, 122, 290Implied volatility, 98Impulse response function, 63, 362Innovation, 31Inverted yield curve, 82Invertibility, 52, 379Invertible ARMA model, 62Itos lemma, 258

multivariate, 273Ito process, 256

Joint distribution function, 7Jump diffusion, 275

Kalman lter, 496, 524Kalman gain, 495, 524Kernel, 168

bandwidth, 169Epanechnikov, 169Gaussian, 169

Kernel regression, 168Kurtosis, 9

excess, 9

Lag operator, 36Lead-lag relationship, 341Leptokurtic, 9Leverage effect, 99, 125, 579Likelihood function, 17Linear time series, 31Liquidity, 210LjungBox statistic, 27, 101

multivariate, 346

Local linear regression, 173Local trend model, 491Log return, 5Logit model, 239Long position, 6Long-memory

stochastic volatility, 135time series, 89

Marginal distribution, 7Market model, 408Markov chain Monte Carlo method (MCMC),

177, 594Markov process, 543Markov property, 32Markov switching model, 164, 588Martingale difference, 114Maximum likelihood estimate (MLE),

exact, 368Mean equation, 101Mean excess function, 321Mean excess plot, 321Mean reversion, 49, 63

half-life, 49Metropolis algorithm, 551MetropolisHasting algorithm, 552Missing value, 531, 558Model checking, 44Moment of a random variable, 8Moving-average model, 50

NadarayaWatson estimator, 169Neural network, 177

activation function, 178feed-forward, 177skip layer, 179

Neuron, see Neural networkNode, see Neural networkNonlinearity test, 183

Brock-Dechert-Scheinkman (BDS), 185bispectral, 184F-test, 188Keenan, 187RESET, 186Tar-F, 190

Nonstationarity, unit-root, 64Nonsynchronous trading, 207Nuisance parameter, 188

OptionsAmerican, 252at-the-money, 252European call, 97

604 INDEX

in-the-money, 252out-of-the-money, 252stock, 252strike price, 97, 252

Order statistics, 299Ordered probit model, 218Orthogonal factor model, 427Outlier

additive, 558detection, 561

Parametric bootstrap, 192Partial autoregressive function (PACF), 40Peaks over thresholds, 318 -weight, 62Pickands estimator, 306Platykurtic, 9Poisson process, 275

inhomogeneous, 329intensity function, 322

Portmanteau test, 27. See also LjungBoxstatistic

Positive denite matrix, 396Prediction, 493Present value, 4Price change and duration (PCD) model, 238Principal component analysis, 421, 478-weight, 31Put-call parity, 266

Quantile, 8denition, 289

R-square, 46adjusted, 47

Random coefcient (RCA) model, 133Random walk, 64

with drift, 65Realized volatility, 141, 492Reduced form model, 349Regression, with time series errors, 80Return level, 317

stress period, 317RiskMetrics, 290

Sample autocorrelation, 26Scree plot, 425Seasonal adjustment, 72Seasonal model, 72

multiplicative, 75

Shape parameter of a distribution, 302Shock, 31, 48, 101Short position, 6Simple return, 3Skewness, 9Smoothed disturbance, 528Smoothing, 167, 493Square root of time rule, 291Standard Brownian motion, 69State space model, 509

nonlinear, 176Stationarity, 25

weak, 340Steady state, 525Stochastic diffusion equation, 256Stochastic volatility model, 134, 565

multivariate, 571Structural equation, 350Structural form, 350Structural time series model (STSM), 491, 521Student-t distribution

standardized, 108Survival function, 322

Tail index, 302TGARCH model, 130

general form, 161Threshold, 159Threshold autoregressive model

multivariate, 392self-exciting, 159smooth, 163

Threshold co-integration, 392Time plot, 17Transactions data, 212Trend stationary model, 67

Unit-root test, 68Unit-root time series, 64Unobserved component model, 521

Value at Risk (VaR), 287, 480econometric approach, 294homogeneous Poisson process, 324inhomogeneous Poisson process, 328RiskMetrics, 290of a short position, 316traditional extreme value, 312

Vector autoregressive (VAR) model, 349Vector ARMA model, 371

marginal models, 375

INDEX 605

Vector moving average model, 365VIX Volatility Index, 98Volatility, 97Volatility equation, 101Volatility model

factor, 477Volatility smile, 274

Weighted least squares, 415White noise, 31Wiener process, 253

generalized, 255

YuleWalker equationmultivariate, 354