A Course in Time Series Analysis

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1 A Course in Time Series Analysis Edited by DANIEL PENA Universidad Carlos III de Madrid GEORGE C. TIAO University of Chicago RUEY S. TSAY University of Chicago A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

2 Contents Preface About ECAS Contributors xv xvi xvii 1. Introduction 1 D. Pena and G. C. Tiao 1.1. Examples of time series problems, Stationary series, Nonstationary series, Seasonal series, Level shifts and outliers in time series, Variance changes, Asymmetrie time series, Unidirectional-feedback relation between series, Comovement and cointegration, Overview of the book, Further reading, 19 PART I BASIC CONCEPTS IN UNIVARIATE TIME SERIES 2. Univanate Time Series: Autocorrelation, Linear Prediction, Spectrum, and State-Space Model 25 G. T. Wilson 2.1. Linear time series modeis, The autocorrelation function, Lagged prediction and the parüal autocorrelation function, 33

3 CONTENTS 2.4. Transformations to stationarity, Cycles and the periodogram, Thespectrum, Further Interpretation of time series acf, pacf, andspectrum, State-space modeis and the Kaiman Filter, 48 Univariate Autoregressive Moving-Average Models 53 G. C. Tiao 3.1. Introduction, Univariate ARMA modeis, Outline of the chapter, Some basic properties of univariate ARMA modeis, The I J and ir weights, Stationarity condition and autocovariance structure ofz The autocorrelation function, The partial autocorrelation function, The extended autocorrelaton function, Model specification strategy, Tentative specification, Tentative model specification via SEACF, Examples, 68 Model Fitting and Checking, and the Kaiman Filter 86 G. T. Wilson 4.1. Prediction error and the estimation criterion, The likelihoodof ARMA modeis, Likelihoods calculated using orthogonal errors, Properties of estimates and problems in estimation, Checking thefittedmodel, Estimation byfittingto the sample spectrum, Estimation of structural modeis by the Kaiman filter, 105 Prediction and Model Selection 111 D. Pena 5.1. Introduction, Properties of minimum mean-square error prediction, Prediction by the conditional expectation, Linear predictions, 113

4 CONTENTS 5.3. The computation of ARIMA forecasts, Interpreting the forecasts from ARIMA modeis, Nonseasonal modeis, Seasonal modeis, Prediction confidence intervals, Known parameter values, Unknown parameter values, Forecast updating, Computing updated forecasts, Testing model stability, The combination of forecasts, Model selection criteria, The FPE and AIC criteria, The Schwarz criterion, Conclusions, Outliers, Influential Observations, and Missing Data D.Pena 6.1. Introduction, Types of outliers in time series, Additive outliers, Innovative outliers, Level shifts, Outliers and Intervention analysis, Procedures for outlier identificatiqn and estimation, Estimation of outlier effects, Testing for outliers, Influential observations, Influence ontimeseries, Influential observations and outliers, Multiple outliers, Masking effects, Procedures for multiple outlier identification, Missing-value estimation, Optimal interpolation and inverse autocorrelation function, Estimation of missing values, Forecasting with outliers, Other approaches, Appendix, 166

5 CONTENTS Automatic Modeling Methods for Univariate Series 171 V. Gömez anda. Maravall 7.1. Classical model identification methods, Subjectivity of the classical methods, The difficulties with mixed ARMA modeis, Automatic model identification methods, Unit root testing, Penalty function methods, Pattern identification methods, Uniqueness of the Solution and the purpose of modeling, Tools for automatic model identification, Test for the log-level specification, Regression techniques for estimating unit roots, The Hannan-Rissanen method, Liu's filtering method, Automatic modeling methods in the presence of outliers, Algorithms for automatic outlier detection and correction, Estimation and filtering techniques to speed up the algorithms, The need to robustify automatic modeling methods, An algorithm for automatic model identification in the presence of outliers, An automatic procedure for the general regression-arima model in the presence of outlierw, special effects, and, possibly, missing observations, Missing observations, Tradingday and Easter effects, Intervention and regression effects, Examples, Tabular summary, 196 Seasonal Adjustment and Signal Extraction Time Series 202 V. Gömez anda. Maravall 8.1. Introduction, Some remarks on the evolution of seasonal adjustment methods, 204

6 CONTENTS Evolution of the methodologic approach, The Situation at present, The need for preadjustment, Model specification, Estimationof the components, Stationary case, Nonstationary series, Historical or final estimator, Properties of final estimator, Component versus estimator, Covariance between estimators, Estimators for recent periods, Revisions in the estimator, Structureoftherevision, Optimality of the revisions, Inference, Optical Forecasts of the Components, Estimation error, Growth rate precision, The gain from concurrent adjustment, Innovations in the components (pseudoinnovations), Anexample, Relation with fixed Alters, Short-versus long-term trends; measunng economic cycles, 236 PART H ADVANCED TOPICS IN UNTVARIATE TIME SERIES 9. Heteroscedastic Models R. S. Tsay 9.1. The ARCH model, Some simple properties of ARCH modeis, Weaknesses of ARCH modeis, Building ARCH modeis, An illustrative example, The GARCH Model, An illustrative example, Remarks, 259

7 X CONTENTS 9.3. The exponential GARCH model, An illustrative example, The CHARMA model, Random coefficient autoregressive (RCA) model, Stochastic volatility model, Long-memory stochastic volatility model, Nonlinear Time Series Models: Testing and Applications 267 R. S. Tsay Introduction, Nonünearity tests, Thetest, Comparison and application, The Tar model, U.S. real GNP, Postsample forecasts and discussion, Concluding remarks, Bayesian Time Series Analysis 286 R. S. Tsay Introduction, A general univariate time series model, Estimation, Gibbs sampling, Griddy Gibbs, An illustrative example, Model discrimination, A mixed model with switching, Implementation, Examples, Nonparametric Time Series Analysis: Nonparametric Regression, Locally Weighted Regression, Autoregression, and Quantile Regression 308 S. Heiler 12.1 Introduction, Nonparametric regression, Kernel estimation in time series, Problems of simple kernel estimation and restricted approaches, 319

8 v / CONTENTS xi 12.5 Locally weighted regression, Applications of locally weighted regression to time series, Parameter selection, Time series decomposition with locally weighted regression, Neural Network Models 348 K. Hornik and F. Leisch Introduction, The multilayer perceptron, Autoregressive neural network modeis, Example: Sunspot series, The recurrent perceptron, Examples of recurrent neural network modeis, A unifying view, 359 PART III MULTIVARIATE TIME SERIES 14. Vector ARMA Models 365 G. C. Tiao Introduction, Transfer function or unidirectional modeis, The vector ARMA model, Some simple examples, Relationship to transfer function model, Cross-covariance and correlation matrices, The partial autoregression matrices, Model building strategy for multiple time series, Tentative specification, Estimation, Diagnostic checking, Analysesof three examples, The SCC data, The gas furnace data, The census housing data, Structural analysis of multivariate time series, A canonical analysis of multiple time series, 395

9 xii CONTENTS Scalar component modeis in multiple time series, Scalar component modeis, Exchangeable modeis and overparameterization, Model specification via canonical correlation analysis, An illustrative example, Some further remarks, Cointegration in the VAR Model 408 S. Johansen Introduction, Basic definitions, Solving autoregressive equations, Some examples, An inversion theorem for matrix polynomials, Granger's representation, Prediction, The Statistical model for 7(1) variables, Hypotheses on cointegrating relations, Estimation of cointegrating vectors and calculation of test statistics, Estimation of ß under restrictions, Asymptotic theory, Asymptotic results, Test for cointegrating rank, Asymptotic distribution of $ and test for restrictions onß, Various applications of the cointegration model, Rational expectations, Arbitrage pricing theory, Seasonal cointegration, Identification of Linear Dynamic Multünput/Multioutput Systems 436 M. Deistler Introduction and problem Statement, Representations of linear Systems, Input/output representations, 438

10 CONTENTS xüi Solutions of linear vector difference equations (VDEs), ARMA and state-space representations, The structure of state-space Systems, The structure of ARMA Systems, The realization of state-space Systems, General structure, Echelon forms, The realization of ARMA Systems, Parametrization, Estimation of real-valued parameters, Dynamic specification, 454 INDEX 457

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