具体描述
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
作者简介
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).
目录信息
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
· · · · · · (收起)
读后感
用户评价
这本书的实用性远远超出了我对一本理论教材的预期。我个人更倾向于那些能提供大量实际案例和代码实现的著作，而这本书在这方面做得非常平衡。它在讲解每一个核心模型（比如GARCH族模型来处理波动率）时，都会紧接着提供一个清晰的例子，并且深入探讨了软件实现上的细节和陷阱。我特别欣赏它对模型诊断的强调。很多新手容易犯的错误是，拟合出一个看起来不错的模型参数就万事大吉了，但这本书反复提醒我们，模型的好坏最终取决于残差是否满足白噪声假设，以及我们是否遗漏了重要的信息。它对异方差性和序列相关性的处理描述得极其细致，让你在实际操作中，不会因为一个微小的诊断步骤的缺失而得出完全错误的结论。对我来说，这本书更像是一本“实战手册”与“理论基石”的完美结合体。它不只是告诉你“是什么”，更重要的是告诉你“怎么做”以及“为什么这么做”。
总的来说，这是一本需要时间和精力的投资，但回报率极高的参考书。我将它放在我书架上“常备不懈”的位置，因为它不仅适用于初学者打基础，也适用于有经验的研究者在回顾和深化理解时查阅。它没有追逐最新的、那些仅存在于前沿论文中的花哨模型，而是聚焦于那些经过时间检验、在工业界和学术界被广泛接受的核心工具。这种务实的选择，使得这本书的生命力经久不衰。每次当我需要重新审视一个经典模型背后的数学直觉，或者想确认一个特定的模型假设是否被当前数据满足时，我总能从中找到清晰、权威的解答。这本书建立的是一种思维框架，让你在面对任何新的时间序列问题时，都能有一个结构化的分析路径可循，而不是盲目地尝试各种算法。它提供的知识体系是坚固而可靠的基座。
我必须承认，阅读过程中我偶尔会感到一丝挑战，尤其是在某些涉及概率分布和推导的深入部分。但这种“挑战感”是积极的，它促使我不得不停下来，拿起笔进行演算，而不是仅仅被动地接受信息。这本书的文字风格非常精确，用词考究，几乎没有模糊不清的地方，这在处理时间序列这种高度依赖精确定义的学科时显得尤为重要。我特别喜欢它对“预测区间”和“置信区间”之间细微差别的解释，这往往是区分业余和专业分析师的关键点。它不仅给出了计算方法，还从统计学意义上解释了当我们说“未来值有95%的概率落在这个范围内”时，我们到底在对什么做出保证。这种对统计哲学层面的探讨，提升了整本书的深度。它不满足于停留在“如何预测”，更深入地探讨了“我们对预测的确定程度究竟了解多少”，这对于建立科学的决策流程至关重要。
说实话，当我翻到关于状态空间模型和卡尔曼滤波的那几章时，我几乎就要打退堂鼓了。那部分内容在很多教材里都是一笔带过，或者直接扔一堆矩阵代数公式，让人望而生畏。但是，这本书的处理方式简直是教科书级别的“化繁为简”。它没有直接跳到复杂的矩阵运算，而是先用一个非常生活化的例子，比如追踪一个移动中的物体，来阐述什么是“状态”以及我们如何通过不完全的观测来“估计”这个状态。这种叙事方式极大地降低了理解门槛。等我真正理解了背后的逻辑后，再去看那些数学表达时，它们就不再是冷冰冰的符号，而是服务于那个清晰逻辑的工具。而且，这本书对不同模型的适用场景的讨论也极为到位。它不会武断地说某个模型是“最优的”，而是会列出各种假设条件，并引导读者思考：“如果我的数据有季节性但没有明显的趋势，我应该优先考虑哪个方向？”这种启发式的引导，培养的不是机械套用公式的能力，而是独立思考和解决实际问题的能力，这对于任何希望在量化领域有所建树的人来说，都是无价之宝。
这本书的封面设计就透露着一股严谨与经典的气息，那种沉稳的蓝色调和工整的字体，让人一看就知道这不是一本轻飘飘的“速成秘籍”，而是脚踏实地的学术著作。我一开始接触时间序列分析时，确实被各种复杂的数学公式和抽象的概念搞得晕头转向，总觉得缺少一个能把理论和实践完美串联起来的桥梁。这本书的厉害之处就在于，它没有一股脑地把所有高级模型都堆砌在前面，而是非常耐心地从最基础的平稳性、自相关性讲起，用非常直观的方式解释了为什么我们需要这些工具。特别是对于ARIMA模型的介绍，作者似乎非常清楚初学者在哪个步骤容易卡住，总能提前给出“预警”和详细的步骤拆解。我记得有一次我尝试用它教的方法去拟合一个金融数据，那些关于残差检验和模型选择的细微差别，以前我都是囫囵吞枣地跳过去，但读完这一章后，突然间就清晰了。这种循序渐进的教学法，让原本枯燥的统计学知识变得有血有肉，仿佛你不是在阅读一本教材，而是在跟随一位经验丰富的导师进行一对一辅导。它给我的感觉是，作者不仅精通理论，更重要的是，他真的懂得如何将这些知识有效地传授给下一代学习者。
