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From the Back Cover

This book is aimed at the reader who wishes to gain a working knowledge of time series and forecasting methods as applied to economics, engineering and the natural and social sciences. It assumes knowledge only of basic calculus, matrix algebra and elementary statistics. This third edition contains detailed instructions for the use of the professional version of the Windows-based computer package ITSM2000, now available as a free download from the Springer Extras website. The logic and tools of time series model-building are developed in detail. Numerous exercises are included and the software can be used to analyze and forecast data sets of the user's own choosing. The book can also be used in conjunction with other time series packages such as those included in R. The programs in ITSM2000 however are menu-driven and can be used with minimal investment of time in the computational details.

The core of the book covers stationary processes, ARMA and ARIMA processes, multivariate time series and state-space models, with an optional chapter on spectral analysis. Many additional special topics are also covered.

New to this edition:

A chapter devoted to Financial Time Series

Introductions to Brownian motion, Lévy processes and Itô calculus

An expanded section on continuous-time ARMA processes

Peter J. Brockwell and Richard A. Davis are Fellows of the American Statistical Association and the Institute of Mathematical Statistics and elected members of the International Statistics institute. Richard A. Davis is the current President of the Institute of Mathematical Statistics and, with W.T.M. Dunsmuir, winner of the Koopmans Prize. Professors Brockwell and Davis are coauthors of the widely used advanced text, Time Series: Theory and Methods, Second Edition (Springer-Verlag, 1991).

Preface vii
1. Introduction 1
1.1. Examples of Time Series 1
1.2. Objectives of Time Series Analysis 5
1.3. Some Simple Time Series Models 6
1.3.1. Some Zero-Mean Models 6
1.3.2. Models with Trend and Seasonality 7
1.3.3. A General Approach to Time Series Modeling 12
1.4. Stationary Models and the Autocorrelation Function 13
1.4.1. The Sample Autocorrelation Function 16
1.4.2. A Model for the Lake Huron Data 18
1.5. Estimation and Elimination of Trend and Seasonal
Components 20
1.5.1. Estimation and Elimination of Trend in the Absence
of Seasonality 21
1.5.2. Estimation and Elimination of Both Trend and
Seasonality 26
1.6. Testing the Estimated Noise Sequence 30
Problems 34
2. Stationary Processes 39
2.1. Basic Properties 39
2.2. Linear Processes 44
2.3. Introduction to ARMA Processes 47
2.4. Properties of the Sample Mean and Autocorrelation Function 50
2.4.1. Estimation of μ 50
2.4.2. Estimation of γ(·) and ρ(·) 51
2.5. Forecasting Stationary Time Series 55
2.5.1. Prediction of Second-Order Random Variables 57
2.5.2. The Prediction Operator P(·|W) 58
2.5.3. The Durbin–Levinson Algorithm 60
2.5.4. The Innovations Algorithm 62
2.5.5. Recursive Calculation of the h-Step Predictors 65
2.5.6. Prediction of a Stationary Process in Terms of
Infinitely Many Past Values 65
2.5.7. Determination of ˜PnXn+h 66
2.6. The Wold Decomposition 67
Problems 68
3. ARMA Models 73
3.1. ARMA(p, q) Processes 73
3.2. The ACF and PACF of an ARMA(p, q) Process 77
3.2.1. Calculation of the ACVF 78
3.2.2. The Autocorrelation Function 82
3.2.3. The Partial Autocorrelation Function 83
3.2.4. Examples 84
3.3. Forecasting ARMA Processes 87
3.3.1. h-Step Prediction of an ARMA(p, q) Process 91
Problems 94
4. Spectral Analysis 97
4.1. Spectral Densities 98
4.2. The Periodogram 106
4.3. Time-Invariant Linear Filters 111
4.4. The Spectral Density of an ARMA Process 115
4.4.1. Rational Spectral Density Estimation 117
Problems 117
5. Modeling and Forecasting with ARMA Processes 121
5.1. Preliminary Estimation 122
5.1.1. Yule–Walker Estimation 123
5.1.2. Burg’s Algorithm 129
5.1.3. The Innovations Algorithm 132
5.1.4. The Hannan–Rissanen Algorithm 137
5.2. Maximum Likelihood Estimation 139
5.3. Diagnostic Checking 144
5.3.1. The Graph of {ˆRt, t = 1, . . . , n} 145
5.3.2. The Sample ACF of the Residuals 146
5.3.3. Tests for Randomness of the Residuals 146
5.4. Forecasting 147
5.5. Order Selection 149
5.5.1. The FPE Criterion 149
5.5.2. The AICC Criterion 151
Problems 153
6. Nonstationary and Seasonal Time Series Models 157
6.1. ARIMA Models for Nonstationary Time Series 158
6.2. Identification Techniques 164
6.3. Unit Roots in Time Series Models 169
6.3.1. Unit Roots in Autoregressions 169
6.3.2. Unit Roots in Moving Averages 171
6.4. Forecasting ARIMA Models 173
6.4.1. The Forecast Function 175
6.5. Seasonal ARIMA Models 177
6.5.1. Forecasting SARIMA Processes 182
6.6. Regression with ARMA Errors 184
6.6.1. OLS and GLS Estimation 184
6.6.2. ML Estimation 186
Problems 191
7. Time Series Models for Financial Data 195
7.1. Historical Overview 196
7.2. GARCH Models 197
7.3. Modified GARCH Processes 204
7.3.1. EGARCH Models 205
7.3.2. FIGARCH and IGARCH Models 207
7.4. Stochastic Volatility Models 209
7.5. Continuous-Time Models 212
7.5.1. Lévy Processes 212
7.5.2. The Geometric Brownian Motion (GBM) Model for
Asset Prices 215
7.5.3. A Continuous-Time SV Model 217
7.6. An Introduction to Option Pricing 221
Problems 224
8. Multivariate Time Series 227
8.1. Examples 228
8.2. Second-Order Properties of Multivariate Time Series 232
8.2.1. Second-Order Properties in the Frequency Domain 236
8.3. Estimation of the Mean and Covariance Function 236
8.3.1. Estimation of μ 236
8.3.2. Estimation of (h) 238
8.3.3. Testing for Independence of Two Stationary Time
Series 239
8.3.4. Bartlett’s Formula 240
8.4. Multivariate ARMA Processes 243
8.4.1. The Covariance Matrix Function of a Causal ARMA
Process 245
8.5. Best Linear Predictors of Second-Order Random Vectors 246
8.6. Modeling and Forecasting with Multivariate AR Processes 247
8.6.1. Estimation for Autoregressive Processes Using
Whittle’s Algorithm 248
8.6.2. Forecasting Multivariate Autoregressive Processes 250
8.7. Cointegration 254
Problems 255
9. State-Space Models 259
9.1. State-Space Representations 260
9.1.1. State-Space Models with t ∈ {0,±1, . . .} 262
9.2. The Basic Structural Model 263
9.3. State-Space Representation of ARIMA Models 266
9.4. The Kalman Recursions 270
9.4.1. h-Step Prediction of {Yt} Using the Kalman
Recursions 272
9.5. Estimation for State-Space Models 275
9.5.1. Application to Structural Models 276
9.6. State-Space Models with Missing Observations 280
9.6.1. The Gaussian Likelihood of {Yi1, . . . ,Yir
},
1 ≤ i1 < i2 < · · · < ir ≤ n 281
9.6.2. Estimation of Missing Values for State-Space Models 283
9.7. The EM Algorithm 285
9.7.1. Missing Data 286
9.8. Generalized State-Space Models 287
9.8.1. Parameter-Driven Models 288
9.8.2. Observation-Driven Models 294
9.8.3. Exponential Family Models 296
Problems 303
10.Forecasting Techniques 309
10.1.The ARAR Algorithm 310
10.1.1.Memory Shortening 310
10.1.2.Fitting a Subset Autoregression 311
10.1.3.Forecasting 311
10.1.4.Application of the ARAR Algorithm 312
10.2.The Holt–Winters Algorithm 314
10.2.1.The Algorithm 314
10.2.2.Holt–Winters and ARIMA Forecasting 316
10.3.The Holt–Winters Seasonal Algorithm 317
10.3.1.The Algorithm 317
10.3.2.Holt–Winters Seasonal and ARIMA Forecasting 318
10.4.Choosing a Forecasting Algorithm 318
Problems 320
11.Further Topics 323
11.1.Transfer Function Models 323
11.1.1.Prediction Based on a Transfer Function Model 327
11.2.Intervention Analysis 331
11.3.Nonlinear Models 334
11.3.1.Deviations from Linearity 335
11.3.2.Chaotic Deterministic Sequences 335
11.3.3.Distinguishing Between White Noise and iid
Sequences 337
11.3.4.Three Useful Classes of Nonlinear Models 338
11.4.Long-Memory Models 338
11.5.Continuous-Time ARMA Processes 342
11.5.1.The Gaussian CAR(1) Process, {Y(t), t ≥ 0} 343
11.5.2.The Gaussian CARMA(p, q) Process, {Y(t), t ∈ R} 345
11.5.3.Lévy-driven CARMA Processes, {Y(t), t ∈ R} 347
Problems 350
A. Random Variables and Probability Distributions 353
A.1. Distribution Functions and Expectation 353
A.1.1. Examples of Continuous Distributions 354
A.1.2. Examples of Discrete Distributions 355
A.1.3. Expectation, Mean, and Variance 356
A.2. Random Vectors 357
A.2.1. Means and Covariances 359
A.3. The Multivariate Normal Distribution 360
Problems 363
B. Statistical Complements 365
B.1. Least Squares Estimation 365
B.1.1. The Gauss–Markov Theorem 367
B.1.2. Generalized Least Squares 367
B.2. Maximum Likelihood Estimation 368
B.2.1. Properties of Maximum Likelihood Estimators 369
B.3. Confidence Intervals 369
B.3.1. Large-Sample Confidence Regions 370
B.4. Hypothesis Testing 370
B.4.1. Error Probabilities 371
B.4.2. Large-Sample Tests Based on Confidence Regions 371
C. Mean Square Convergence 373
C.1. The Cauchy Criterion 373
D. Lévy Processes, Brownian Motion and Itô Calculus 375
D.1. Lévy Processes 375
D.2. Brownian Motion and the Itô Integral 377
D.3. Itô Processes and Itô’s Formula 381
D.4. Itô Stochastic Differential Equations 383
E. An ITSM Tutorial 387
E.1. Getting Started 388
E.1.1. Running ITSM 388
E.2. Preparing Your Data for Modeling 388
E.2.1. Entering Data 389
E.2.2. Information 389
E.2.3. Filing Data 389
E.2.4. Plotting Data 390
E.2.5. Transforming Data 390
E.3. Finding a Model for Your Data 394
E.3.1. Autofit 394
E.3.2. The Sample ACF and PACF 394
E.3.3. Entering a Model 396
E.3.4. Preliminary Estimation 397
E.3.5. The AICC Statistic 398
E.3.6. Changing Your Model 399
E.3.7. Maximum Likelihood Estimation 399
E.3.8. Optimization Results 400
E.4. Testing Your Model 401
E.4.1. Plotting the Residuals 401
E.4.2. ACF/PACF of the Residuals 402
E.4.3. Testing for Randomness of the Residuals 403
E.5. Prediction 404
E.5.1. Forecast Criteria 404
E.5.2. Forecast Results 405
E.6. Model Properties 405
E.6.1. ARMA Models 406
E.6.2. Model ACF, PACF 406
E.6.3. Model Representations 408
E.6.4. Generating Realizations of a Random Series 409
E.6.5. Spectral Properties 409
E.7. Multivariate Time Series 409
References 411
Index 419
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