A Course in Time Series Analysis

1. Introduction 1
D. Pena and G. C. Tiao

1.1. Examples of time series problems, 1

1.1.1. Stationary series, 2

1.1.2. Nonstationary series, 3

1.1.3. Seasonal series, 5

1.1.4. Level shifts and outliers in time series, 7

1.1.5. Variance changes, 7

1.1.6. Asymmetric time series, 7

1.1.7. Unidirectional-feedback relation between series, 9

1.1.8. Comovement and cointegration, 10

1.2. Overview of the book, 10

1.3. Further reading, 19

PART I BASIC CONCEPTS IN UNIVARIATE TIME SERIES

2. Univariate Time Series: Autocorrelation, Linear Prediction, Spectrum, and State-Space Model 25
G. T. Wilson

2.1. Linear time series models, 25

2.2. The autocorrelation function, 28

2.3. Lagged prediction and the partial autocorrelation function, 33

2.4. Transformations to stationarity, 35

2.5. Cycles and the periodogram, 37

2.6. The spectrum, 42

2.7. Further interpretation of time series acf, pacf, and spectrum, 46

2.8. State-space models and the Kalman Filter, 48

3. Univariate Autoregressive Moving-Average Models 53
G. C. Tiao

3.1. Introduction, 53

3.1.1. Univariate ARMA models, 54

3.1.2. Outline of the chapter, 55

3.2. Some basic properties of univariate ARMA models, 55

3.2.1. The ø and TT weights, 56

3.2.2. Stationarity condition and autocovariance structure o f z „ 58

3.2.3. The autocorrelation function, 59

3.2.4. The partial autocorrelation function, 60

3.2.5. The extended autocorrelaton function, 61

3.3. Model specification strategy, 63

3.3.1. Tentative specification, 63

3.3.2. Tentative model specification via SEACF, 67

3.4. Examples, 68

4. Model Fitting and Checking, and the Kalman Filter 86
G. T. Wilson

4.1. Prediction error and the estimation criterion, 86

4.2. The likelihood of ARMA models, 90

4.3. Likelihoods calculated using orthogonal errors, 94

4.4. Properties of estimates and problems in estimation, 98

4.5. Checking the fitted model, 101

4.6. Estimation by fitting to the sample spectrum, 104

4.7. Estimation of structural models by the Kalman filter, 105

5. Prediction and Model Selection 111
D. Pefia

5.1. Introduction, 111

5.2. Properties of minimum mean-square error prediction, 112

5.2.1. Prediction by the conditional expectation, 112

5.2.2. Linear predictions, 113

5.3. The computation of ARIMA forecasts, 114

5.4. Interpreting the forecasts from ARIMA models, 116

5.4.1. Nonseasonal models, 116

5.4.2. Seasonal models, 120

5.5. Prediction confidence intervals, 123

5.5.1. Known parameter values, 123

5.5.2. Unknown parameter values, 124

5.6. Forecast updating, 125

5.6.1. Computing updated forecasts, 125

5.6.2. Testing model stability, 125

5.7. The combination of forecasts, 129

5.8. Model selection criteria, 131

5.8.1. The FPE and AIC criteria, 131

5.8.2. The Schwarz criterion, 133

5.9. Conclusions, 133

6. Outliers, Influential Observations, and Missing Data 136
D. Pena

6.1. Introduction, 136

6.2. Types of outliers in time series, 138

6.2.1. Additive outliers, 138

6.2.2. Innovative outliers, 141

6.2.3. Level shifts, 143

6.2.4. Outliers and intervention analysis, 146

6.3. Procedures for outlier identification and estimation, 147

6.3.1. Estimation of outlier effects, 148

6.3.2. Testing for outliers, 149

6.4. Influential observations, 152

6.4.1. Influence on time series, 152

6.4.2. Influential observations and outliers, 153

6.5. Multiple outliers, 154

6.5.1. Masking effects, 154

6.5.2. Procedures for multiple outlier identification, 156

6.6. Missing-value estimation, 160

6.6.1. Optimal interpolation and inverse autocorrelation function, 160

6.6.2. Estimation of missing values, 162

6.7. Forecasting with outliers, 164

6.8. Other approaches, 166

6.9. Appendix, 166

7. Automatic Modeling Methods for Univariate Series 171
V. Gomez and A. Maravall

7.1. Classical model identification methods, 171

7.1.1. Subjectivity of the classical methods, 172

7.1.2. The difficulties with mixed ARMA models, 173

7.2. Automatic model identification methods, 173

7.2.1. Unit root testing, 174

7.2.2. Penalty function methods, 174

7.2.3. Pattern identification methods, 175

7.2.4. Uniqueness of the solution and the purpose of modeling, 176

7.3. Tools for automatic model identification, 177

7.3.1. Test for the log-level specification, 177

7.3.2. Regression techniques for estimating unit roots, 178

7.3.3. The Hannan-Rissanen method, 181

7.3.4. Liu's filtering method, 185

7.4. Automatic modeling methods in the presence of outliers, 186

7.4.1. Algorithms for automatic outlier detection and correction, 186

7.4.2. Estimation and filtering techniques to speed up the algorithms, 190

7.4.3. The need to robustify automatic modeling methods, 191

7.4.4. An algorithm for automatic model identification in the presence of outliers, 191

7.5. An automatic procedure for the general regression-ARIMA model in the presence of outlierw, special effects, and, possibly, missing observations, 192

7.5.1. Missing observations, 192

7.5.2. Trading day and Easter effects, 193

7.5.3. Intervention and regression effects, 194

7.6. Examples, 194

7.7. Tabular summary, 196

8. Seasonal Adjustment and Signal Extraction Time Series 202
V. Gomez and A. Maravall

8.1. Introduction, 202

8.2. Some remarks on the evolution of seasonal adjustment methods, 204

8.2.1. Evolution of the methodologic approach, 204

8.2.2. The situation at present, 207

8.3. The need for preadjustment, 209

8.4. Model specification, 210

8.5. Estimation of the components, 213

8.5.1. Stationary case, 215

8.5.2. Nonstationary series, 217

8.6 Historical or final estimator, 218

8.6.1. Properties of final estimator, 218

8.6.2. Component versus estimator, 219

8.6.3. Covariance between estimators, 221

8.7. Estimators for recent periods, 221

8.8. Revisions in the estimator, 223

8.8.1. Structure of the revision, 223

8.8.2. Optimality of the revisions, 224

8.9. Inference, 225

8.9.1. Optical Forecasts of the Components, 225

8.9.2. Estimation error, 225

8.9.3. Growth rate precision, 226

8.9.4. The gain from concurrent adjustment, 227

8.9.5. Innovations in the components (pseudoinnovations), 228

8.10. An example, 228

8.11. Relation with fixed filters, 235

8.12. Short-versus long-term trends; measuring economic cycles, 236

PART II ADVANCED TOPICS IN UNIVARIATE TIME SERIES

9. Heteroscedastic Models
R. S. Tsay

9.1. The ARCH model, 250

9.1.1. Some simple properties of ARCH models, 252

9.1.2. Weaknesses of ARCH models, 254

9.1.3. Building ARCH models, 254

9.1.4. An illustrative example, 255

9.2. The GARCH Model, 256

9.2.1. An illustrative example, 257

9.2.2. Remarks, 259

9.3. The exponential GARCH model, 260

9.3.1. An illustrative example, 261

9.4. The CHARMA model, 262

9.5. Random coefficient autoregressive (RCA) model, 263

9.6. Stochastic volatility model, 264

9.7. Long-memory stochastic volatility model, 265

10. Nonlinear Time Series Models: Testing and Applications 267
R. S. Tsay

10.1. Introduction, 267

10.2. Nonlinearity tests, 268

10.2.1. The test, 268

10.2.2. Comparison and application, 270

10.3. The Tar model, 274

10.3.1. U.S. real GNP, 275

10.3.2. Postsample forecasts and discussion, 279

10.4. Concluding remarks, 282

11. Bayesian Time Series Analysis 286
R. S. Tsay

11.1. Introduction, 286

11.2. A general univariate time series model, 288

11.3. Estimation, 289

11.3.1. Gibbs sampling, 291

11.3.2. Griddy Gibbs, 292

11.3.3. An illustrative example, 292

11.4. Model discrimination, 294

11.4.1. A mixed model with switching, 295

11.4.2. Implementation, 296

11.5. Examples, 297

12 Nonparametric Time Series Analysis: Nonparametric Regression, Locally Weighted Regression, Autoregression, and Quantile Regression 308
S. Heiler

12.1 Introduction, 308

12.2 Nonparametric regression, 309

12.3 Kernel estimation in time series, 314

12.4 Problems of simple kernel estimation and restricted approaches, 319

12.5 Locally weighted regression, 321

12.6 Applications of locally weighted regression to time series, 329

12.7 Parameter selection, 330

12.8 Time series decomposition with locally weighted regression, 336

13. Neural Network Models 348
K. Hornik and F. Leisch

13.1. Introduction, 348

13.2. The multilayer perceptron, 349

13.3. Autoregressive neural network models, 354

13.3.1. Example: Sunspot series, 355

13.4. The recurrent perceptron, 356

13.4.1. Examples of recurrent neural network models, 357

13.4.2. A unifying view, 359

PART III MULTIVARIATE TIME SERIES

14. Vector ARMA Models 365
G. C. Tiao

14.1. Introduction, 365

14.2. Transfer function or unidirectional models, 366

14.3. The vector ARMA model, 368

14.3.1. Some simple examples, 368

14.3.2. Relationship to transfer function model, 371

14.3.3. Cross-covariance and correlation matrices, 371

14.3.4. The partial autoregression matrices, 372

14.4. Model building strategy for multiple time series, 373

14.4.1. Tentative specification, 373

14.4.2. Estimation, 378

14.4.3. Diagnostic checking, 379

14.5. Analyses of three examples, 380

14.5.1. The SCC data, 380

14.5.2. The gas furnace data, 383

14.5.3. The census housing data, 387

14.6. Structural analysis of multivariate time series, 392

14.6.1. A canonical analysis of multiple time series, 395

14.7. Scalar component models in multiple time series, 396

14.7.1. Scalar component models, 398

14.7.2. Exchangeable models and overparameterization, 400

14.7.3. Model specification via canonical correlation analysis, 402

14.7.4. An illustrative example, 403

14.7.5. Some further remarks, 404

15. Cointegration in the VAR Model 408
5. Johansen

15.1. Introduction, 408

15.1.1. Basic definitions, 409

15.2. Solving autoregressive equations, 412

15.2.1. Some examples, 412

15.2.2. An inversion theorem for matrix polynomials, 414

15.2.3. Granger's representation, 417

15.2.4. Prediction, 419

15.3. The statistical model for / ( l ) variables, 420

15.3.1. Hypotheses on cointegrating relations, 421

15.3.2. Estimation of cointegrating vectors and calculation of test statistics, 422

15.3.3. Estimation of â under restrictions, 426

15.4. Asymptotic theory, 426

15.4.1. Asymptotic results, 427

15.4.2. Test for cointegrating rank, 427

15.4.3. Asymptotic distribution of â and test for restrictions on â, 429

15.5. Various applications of the cointegration model, 432

15.5.1. Rational expectations, 432

15.5.2. Arbitrage pricing theory, 433

15.5.3. Seasonal cointegration, 433

16. Identification of Linear Dynamic Multiinput/Multioutput Systems 436
M. Deistler

16.1. Introduction and problem statement, 436

16.2. Representations of linear systems, 438

16.2.1. Input/output representations, 438

16.2.2. Solutions of linear vector difference equations (VDEs), 440

16.2.3. ARMA and state-space representations, 441

16.3. The structure of state-space systems, 443

16.4. The structure of ARMA systems, 444

16.5. The realization of state-space systems, 445

16.5.1. General structure, 445

16.5.2. Echelon forms, 447

16.6. The realization of ARMA systems, 448

16.7. Parametrization, 449

16.8. Estimation of real-valued parameters, 452

16.9. Dynamic specification, 454

INDEX 457