Empirical Vector Autoregressive Modeling

The main subject of this book is empirical application of multivariate linear time series model on quarterly or month- ly economic data to discoverand describe important dynamic relationships between the variables of interest. The book stresses "real-life" application and the selection of data analytic tools. Simple numerical examples and some more al- gebraicexercises are used to illustrate major points. Rele- vant old and recent results from over 400 authors and refe- rences from econometrics, mathematical statistics, time se- ries analysis, economics and descriptve statistics are dis- cussed. Appropriate use of multivariate time series models requires an intimate knowledge of relevant characteristics of thedata.One can obtain this using a method that combines influence analysis (which data points contain the major part of the information? ) and diagnostic checking (does the model describe the interesting part of the information well enough? ). For economic time series these issuses are (the type of) nonstationarity of the trend and seasonal compo- nent, be it of the (fractional) "unit root" type or of the changing parameter type (structural breaks), both in a unva- riate and a multivariate context. The book introduces new graphical and statistical methodes to improve the understan- ding of seasonality, outliers, structural breaks, pushing trends and pulling equilibria in aparticular data set.

1 Introduction.- 1.1 Integrating results.- 1.2 Goal of the study.- 1.3 Data and measurement model.- 1.4 Baseline model and methodology.- 1.5 Outline of the study.- 1.6 What is new?.- 2 The Unrestricted VAR and its components.- 2.1 Introduction.- 2.2 The model.- 2.3 Univariate processes and unit roots.- 2.4 Integrated processes.- 2.4.1 Definitions and notation.- 2.4.2 MA representation, autocorrelation and pseudo spectrum.- 2.5 Alternative models for nonstationarity, long memory and persistence.- 2.5.1 Nonstationarity.- 2.5.2 Long memory, the variance time function and adjusted range analysis.- 2.5.3 Persistence.- Appendix A2.1 MA representation integrated process.- A2.1.1 MA representations.- A2.1.2 Pseudo autocorrelation functions.- Appendix A2.2 Univariate testing for unit root nonstationarity.- A2.2.1 The pure unit root case without deterministic terms.- A2.2.1.1 Notation and model.- A2.2.1.2 Discussion.- A2.2.2 Deterministic terms and unknown residual autocorrelation.- A2.2.2.1 Generalization of the test regression.- A2.2.2.2 Interesting null hypotheses, alternatives and tests.- A2.2.2.3 The parameters ?i and ?i in (A2.2.11) and (A2.2.12).- A2.2.2.4 Test statistics and distributions.- A2.2.2.5 Evaluation of methods.- A2.2.2.6 Other approaches and some extensions.- 3 Data Analysis by Vector Autoregression.- 3.1 Introduction.- 3.2 Data-oriented measures of influence.- 3.2.1 Goal of the influence analysis.- 3.2.2 Influence measures in regression.- 3.2.3 Influence measures for dynamic and multiple equation models.- 3.2.4 Other influence measures from multivariate analysis.- 3.3 Diagnostic checking.- 3.3.1 Choosing test statistics.- 3.3.2 Theoretical consideration for choosing tests.- 3.3.3 Practical considerations for choosing tests.- 3.3.4 Dynamic specification of the mean.- 3.3.5 Distribution of the disturbances.- 3.3.6 Parameter constancy of dynamic and covariance parameters.- 3.3.7 An alternative test for parameter stability.- 3.3.8 Multivariate diagnostics.- 3.3.9 A diagnostic for multivariate unit roots.- 3.3.10 Consequences of “rejection” of the model.- Appendix A3.1 Influence measures for the normal linear model.- A3.1.1 Global influence measures.- A3.1.2 Local influence measures.- Appendix A3.2 Influence measures for the multivariate general linear model.- Appendix A3.3 Influence measures in principal component analysis.- 4 Seasonality.- 4.1 Introduction.- 4.2 Application of the idea of unobserved components.- 4.3 Application of linear filters to estimate unobserved components.- 4.3.1 Optimal extraction in multivariate series.- 4.3.2 Optimal extraction in nonstationary series.- 4.3.3 Specification of low dimensional univariate models.- 4.3.4 Optimal extraction in a finite sample.- 4.3.5 Optimal extraction in the presence of outliers.- 4.4 Data analysis of the seasonal component.- 4.5 Application of the Census X-11 filter in a VAR.- Appendix 4.1 Trigonometric seasonal processes in regression.- A4.1.1 Notation and underlying model.- A4.1.2 Zero correlation between seasonal patterns.- A4.1.3 Circularity: Unit correlation between seasonal patterns.- Appendix 4.2 Backforecasts and deterministic changes in mean.- A4.2.1 Introduction.- A4.2.2 Backforecasting and deterministic changes in mean with linear trends.- A4.2.3 Backforecasting and deterministic changes in mean with seasonal dummies.- A4.2.4 Changes in mean in multivariate model with unit roots.- 5 Outliers.- 5.1 Introduction.- 5.2 The outlier model.- 5.3 Some effects of outliers on VAR estimates.- 5.3.1 Effect of outliers on unit root tests.- 5.3.2 Effect of outliers on estimates of ?.- 5.4 Derivation of the LM-statistics.- 5.4.1 Case of known parameters and timing.- 5.4.2 Case of estimated parameters and unknown timing.- 5.4.3 Distinguishing between outlier types.- 5.4.4 Distinguishing between outliers in different equations.- 5.5 An artificial example.- 5.6 Application to macroeconomic series.- 5.7 Two simple ways to study the influence of outliers.- Appendix 5.1 Some proofs concerning outlier test statistics.- A5.1.1 Derivation simultaneous test.- A5.1.2 Finite sample alternatives for I test procedure.- Appendix 5.2 Subsample analysis outlier influence.- Appendix 5.3 Robust estimation by extraction of additive outliers.- 6 Restrictions on the VAR.- 6.1 Introduction.- 6.2 Cointegration, the number of unit roots, and common trends.- 6.2.3 Vector error correction.- 6.2.4 Other parameterizations.- 6.3 Straightforward transformation formulae.- 6.3.1 From Campbell-Shiller to vector error correction.- 6.3.2 From vector error correction to Campbell-Shiller, mean growth.- 6.3.3 From vector error correction to common trends.- 6.3.4 Examples.- 6.3.5 Conditions for VECM, I(2)-ness, and explosive systems.- 6.4 Trend stationary processes and quadratic trends.- 6.5 Estimating pushing trends and pulling equilibria.- 6.5.1 Deterministic trends.- 6.5.2 Estimating the stochastic part of the trend.- 6.5.3 Estimating pulling equilibria.- 6.6 Multivariate tests for unit roots.- 6.6.1 Models with p = 1 and zero mean.- 6.6.2 Deterministic terms and serial correlation in AR(1) residuals.- Appendix 6.1 Computation and distribution multivariate unit root test statistics.- A6.1.1 Computation.- A6.1.2 Distribution.- 7 Applied VAR Analysis for Aggregate Investment.- 7.1 Introduction.- 7.2 The variable of interest and some of its supposed relationships.- 7.2.1 Theoretical relationships.- 7.2.2 Empirical models.- 7.3 Measurement model.- 7.3.1 Investment in the national accounts.- 7.3.2 Definition of investment.- 7.3.3 Other macroeconomic price indexes.- 7.4 Univariate analysis.- 7.4.1 The variables.- 7.4.2 Graphs and influence analysis.- 7.4.3 Representations of the autocorrelation function.- 7.4.4 Adjusted range techniques.- 7.4.6 Application.- 7.4.7 Results.- 7.4.7.1 Outliers.- 7.4.7.2 Autocorrelations.- 7.4.7.3 Long memory analysis.- 7.4.7.4 Data analysis seasonal components.- 7.4.7.5 Variance time functions.- 7.4.7.6 Statistical unit root analysis.- 7.4.7.7 Parameter stability.- 7.4.7.8 Summary of univariate results.- 7.5 Multivariate analysis.- 7.5.1 Predictions and seasonality in the unrestricted VAR.- 7.5.2 Unit root analysis.- 7.5.3 Detecting a structural break.- 7.5.4 The final model.- Appendix 7.1 Data sources and construction.- Appendix 7.2 Results of final VECM model.- Appendix 7.3 Open economy stochastic dynamic general equilibrium models.- Summary.- References.- Name index.