Introduction to Time Series and Forecasting

by Brockwell, Peter J.; Davis, Richard A.

• ISBN13: 9780387953519
• ISBN10: 0387953515
• Edition: 2nd
• Format: Hardcover
• Copyright: 3/1/2002
• Publisher: Springer Verlag

Summary

This book is aimed at the reader who wishes to gain a working knowledge of time series and forecasting methods as applied in economics, engineering, and the natural and social sciences. The book assumes knowledge only of basic calculus, matrix algebra and elementary statistics. This second edition contains detailed instructions on the use of the new totally windows-based computer package ITSM2000, the student version of which is included with the text. Expanded treatments are also given of several topics treated only briefly in the first edition. These include regression with time series errors, which plays an important role in forecasting and inference, and ARCH and GARCH models, which are widely used for the modeling of financial time series. These models can be fitted using the new version of ITSM.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. Additional topics include the Burg and Hannan-Rissanen algorithms, unit roots, the EM algorithm, structural models, generalized state-space models with applications to time series of count data, exponential smoothing, the Holt-Winters and ARAR forecasting algorithms, transfer function models and intervention analysis. Brief introductions are also given to cointegration and to non-linear, continuous-time and long-memory models.

Table of Contents

Preface	vii
Introduction	1	(44)
Examples of Time Series	1	(5)
Objectives of Time Series Analysis	6	(1)
Some Simple Time Series Models	7	(8)
Some Zero-Mean Models	8	(1)
Models with Trend and Seasonality	9	(5)
A General Approach to Time Series Modeling	14	(1)
Stationary Models and the Autocorrelation Function	15	(8)
The Sample Autocorrelation Function	18	(3)
A Model for the Lake Huron Data	21	(2)
Estimation and Elimination of Trend and Seasonal Components	23	(12)
Estimation and Elimination of Trend in the Absence of Seasonality	24	(7)
Estimation and Elimination of Both Trend and Seasonality	31	(4)
Testing the Estimated Noise Sequence	35	(10)
Problems	40	(5)
Stationary Processes	45	(38)
Basic Properties	45	(6)
Linear Processes	51	(4)
Introduction to ARMA Processes	55	(2)
Properties of the Sample Mean and Autocorrelation Function	57	(6)
Estimation of μ	58	(1)
Estimation of γ(.) and ρ(.)	59	(4)
Forecasting Stationary Time Series	63	(14)
The Durbin-Levinson Algorithm	69	(2)
The Innovations Algorithm	71	(4)
Prediction of a Stationary Process in Terms of Infinitely Many Past Values	75	(2)
The Wold Decomposition	77	(6)
Problems	78	(5)
ARMA Models	83	(28)
ARMA (p, q) Processes	83	(5)
The ACF and PACF of an ARMA(p, q) Process	88	(12)
Calculation of the ACVF	88	(6)
The Autocorrelation Function	94	(1)
The Partial Autocorrelation Function	94	(2)
Examples	96	(4)
Forecasting ARMA Processes	100	(11)
Problems	108	(3)
Spectral Analysis	111	(26)
Spectral Densities	112	(9)
The Periodogram	121	(6)
Time-Invariant Linear Filters	127	(5)
The Spectral Density of an ARMA Process	132	(5)
Problems	134	(3)
Modeling and Forecasting with ARMA Processes	137	(42)
Preliminary Estimation	138	(20)
Yule-Walker Estimation	139	(8)
Burg's Algorithm	147	(3)
The Innovations Algorithm	150	(6)
The Hannan-Rissanen Algorithm	156	(2)
Maximum Likelihood Estimation	158	(6)
Diagnostic Checking	164	(3)
The Graph of {Rt, t = 1,...,n}	165	(1)
The Sample ACF of the Residuals	166	(1)
Tests for Randomness of the Residuals	166	(1)
Forecasting	167	(2)
Order Selection	169	(10)
The FPE Criterion	170	(1)
The AICC Criterion	171	(3)
Problems	174	(5)
Nonstationary and Seasonal Time Series Models	179	(44)
ARIMA Models for Nonstationary Time Series	180	(7)
Identification Techniques	187	(6)
Unit Roots in Time Series Models	193	(5)
Unit Roots in Autoregressions	194	(2)
Unit Roots in Moving Averages	196	(2)
Forecasting ARIMA Models	198	(5)
The Forecast Function	200	(3)
Seasonal ARIMA Models	203	(7)
Forecasting SARIMA Processes	208	(2)
Regression with ARMA Errors	210	(13)
OLS and GLS Estimation	210	(3)
ML Estimation	213	(6)
Problems	219	(4)
Multivariate Time Series	223	(36)
Examples	224	(5)
Second-Order Properties of Multivariate Time Series	229	(5)
Estimation of the Mean and Covariance Function	234	(7)
Estimation of μ	234	(1)
Estimation of Γ(h)	235	(2)
Testing for Independence of Two Stationary Time Series	237	(1)
Bartlett's Formula	238	(3)
Multivariate ARMA Processes	241	(3)
The Covariance Matrix Function of a Causal ARMA Process	244	(1)
Best Linear Predictors of Second-Order Random Vectors	244	(2)
Modeling and Forecasting with Multivariate AR Processes	246	(8)
Estimation for Autoregressive Processes Using Whittle's Algorithm	247	(3)
Forecasting Multivariate Autoregressive Processes	250	(4)
Cointegration	254	(5)
Problems	256	(3)
State-Space Models	259	(58)
State-Space Representations	260	(3)
The Basic Structural Model	263	(4)
State-Space Representation of ARIMA Models	267	(4)
The Kalman Recursions	271	(6)
Estimation For State-Space Models	277	(6)
State-Space Models with Missing Observations	283	(6)
The EM Algorithm	289	(3)
Generalized State-Space Models	292	(25)
Parameter-Driven Models	292	(7)
Observation-Driven Models	299	(12)
Problems	311	(6)
Forecasting Techniques	317	(14)
The ARAR Algorithm	318	(4)
Memory Shortening	318	(1)
Fitting a Subset Autoregression	319	(1)
Forecasting	320	(1)
Application of the ARAR Algorithm	321	(1)
The Holt-Winters Algorithm	322	(4)
The Algorithm	322	(2)
Holt-Winters and ARIMA Forecasting	324	(2)
The Holt-winters Seasonal Algorithm	326	(2)
The Algorithm	326	(2)
Holt-Winters Seasonal and ARIMA Forecasting	328	(1)
Choosing a Forecasting Algorithm	328	(3)
Problems	330	(1)
Further Topics	331	(38)
Transfer Function Models	331	(9)
Prediction Based on a Transfer Function Model	337	(3)
Intervention Analysis	340	(3)
Nonlinear Models	343	(14)
Deviations from Linearity	344	(1)
Chaotic Deterministic Sequences	345	(2)
Distinguishing Between White Noise and iid Sequences	347	(1)
Three Useful Classes of Nonlinear Models	348	(1)
Modeling Volatility	349	(8)
Continuous-Time Models	357	(4)
Long-Memory Models	361	(8)
Problems	365	(4)
A. Random Variables and Probability Distributions	369	(14)
A.1. Distribution Functions and Expectation	369	(5)
A.2. Random Vectors	374	(3)
A.3. The Multivariate Normal Distribution	377	(6)
Problems	381	(2)
B. Statistical Complements	383	(10)
B.1. Least Squares Estimation	383	(3)
B.1.1. The Gauss-Markov Theorem	385	(1)
B.1.2. Generalized Least Squares	386	(1)
B.2. Maximum Likelihood Estimation	386	(2)
B.2.1. Properties of Maximum Likelihood Estimators	387	(1)
B.3. Confidence Intervals	388	(1)
B.3.1. Large-Sample Confidence Regions	388	(1)
B.4. Hypothesis Testing	389	(4)
B.4.1. Error Probabilities	390	(1)
B.4.2. Large-Sample Tests Based on Confidence Regions	390	(3)
C. Mean Square Convergence	393	(2)
C.1. The Cauchy Criterion	393	(2)
D. An ITSM Tutorial	395	(28)
D.1. Getting Started	396	(1)
D.1.1. Running ITSM	396	(1)
D.2. Preparing Your Data for Modeling	396	(7)
D.2.1. Entering Data	397	(1)
D.2.2. Information	397	(1)
D.2.3. Filing Data	397	(1)
D.2.4. Plotting Data	398	(1)
D.2.5. Transforming Data	398	(5)
D.3. Finding a Model for Your Data	403	(8)
D.3.1. Autofit	403	(1)
D.3.2. The Sample ACF and PACF	403	(1)
D.3.3. Entering a Model	404	(2)
D.3.4. Preliminary Estimation	406	(2)
D.3.5. The AICC Statistic	408	(1)
D.3.6. Changing Your Model	408	(1)
D.3.7. Maximum Likelihood Estimation	409	(1)
D.3.8. Optimization Results	410	(1)
D.4. Testing Your Model	411	(4)
D.4.1. Plotting the Residuals	412	(1)
D.4.2. ACF/PACF of the Residuals	412	(2)
D.4.3. Testing for Randomness of the Residuals	414	(1)
D.5. Prediction	415	(1)
D.5.1. Forecast Criteria	415	(1)
D.5.2. Forecast Results	415	(1)
D.6. Model Properties	416	(5)
D.6.1. ARMA Models	417	(1)
D.6.2. Model ACF, PACF	418	(1)
D.6.3. Model Representations	419	(1)
D.6.4. Generating Realizations of a Random Series	420	(1)
D.6.5. Spectral Properties	421	(1)
D.7. Multivariate Time Series	421	(2)
References	423	(6)
Index	429