Empirical Vector Autoregressive Modeling

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.