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3.6 Single Equation Implications and Examples

Preface; Contents; 1 Introduction; References; 2 Multivariate Time Series; 2.1 Introduction; 2.2 Stationarity; 2.2.1 Strict Stationarity; 2.2.2 Strict (Joint Distribution) Stationarity; 2.2.3 Describing Covariance Non-Stationarity: Parametric Models; 2.2.4 The White Noise Process; 2.2.4.1 White Noise; 2.2.5 The Moving Average Process; 2.2.6 Wold's Representation Theorem; 2.2.7 The Autoregressive Process; 2.2.8 Lag Polynomials and Their Roots; 2.2.8.1 The Lag Operator and Lag Polynomials; 2.2.9 Non-Stationarity and the Autoregressive Process; 2.2.9.1 Stationarity of an Autoregressive Process

2.2.10 The Random Walk and the Unit Root2.2.10.1 The Random Walk Process; 2.2.10.2 Differencing and Stationarity; 2.2.10.3 The Random Walk as a Stochastic Trend; 2.2.10.4 The Random Walk with Drift; 2.2.11 The Autoregressive Moving Average Process and Operator Inversion; 2.2.11.1 Illustration of Operator Inversion; 2.2.12 Testing Stationarity in Single Series; 2.2.12.1 Reparameterizing the Autoregressive Model; 2.2.12.2 Semi-parametric Methods; 2.3 Multivariate Time Series Models; 2.3.1 The VAR and VECM Models; 2.3.2 The VMA Model; 2.3.3 Estimation; 2.3.4 The Procedure; 2.4 Persistence

2.4.1 Reparameterizing the VAR2.4.2 Long-Run Growth Models; 2.5 Impulse Responses; 2.5.1 Impulse Responses and VAR Models; 2.5.2 Orthogonality and the IRF; 2.5.3 The Choleski Decomposition; 2.5.4 IRFs in the General VAR Case; 2.5.4.1 IRFs and Time Series Identification; 2.6 Variance Decomposition; 2.6.1 Prediction Errors and Forecasts; 2.7 Conclusion; References; 3 Cointegration; 3.1 Cointegration of the VMA, VAR and VECM; 3.1.1 The Granger Representation Theorem: Systems Representation of Cointegrated Variables; 3.1.1.1 Cointegration Starting from a VMA and Deriving VAR and VECM Forms

3.1.2 VARMA Representation of CI(1,1) Variables3.2 The Smith-McMillan-Yoo Form; 3.2.1 Using the Smith Form to Reparameterize a Finite Order VMA; 3.2.1.1 Reparameterizing a VMA in Differences; 3.2.2 The SM Form in General Applied to a Rational VMA: The SMY Form; 3.2.2.1 The SMY Form and Cointegration of Order (1,1); 3.2.3 Cointegrating Vectors in the VMA and VAR Representations of CI(1,1); 3.2.3.1 A(L) as Partial Inverse of C(L) in the CI(1,1) Case; 3.2.4 Equivalence of VAR and VMA Representations in the CI(1,1) Case; 3.3 Johansen's VAR Representation of Cointegration

3.3.1 Cointegration Assuming Integration of Order 13.3.1.1 Cointegrated VARs with I(1) Processes; 3.3.2 Conditions for the VAR Process to be I(1) and Cointegrated; 3.3.2.1 Discussion; 3.3.3 The MA Representation; 3.4 Cointegration with Intercept and Trend; 3.4.1 Levels Process for the VECM with Intercept; 3.4.2 Levels Process for the VECM with Higher Order Trends and Other Deterministic Terms; 3.5 Alternative Representations of the Cointegrating VAR, VMA and VARMA; 3.5.1 The Sargan-Bézout Factorization; 3.5.2 A VAR(1) Representation of a VMA(1) Model Under Cointegration

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This book examines conventional time series in the context of stationary data prior to a discussion of cointegration, with a focus on multivariate models. The authors provide a detailed and extensive study of impulse responses and forecasting in the stationary and non-stationary context, considering small sample correction, volatility and the impact of different orders of integration. Models with expectations are considered along with alternate methods such as Singular Spectrum Analysis (SSA), the Kalman Filter and Structural Time Series, all in relation to cointegration. Using single equations methods to develop topics, and as examples of the notion of cointegration, Burke, Hunter, and Canepa provide direction and guidance to the now vast literature facing students and graduate economists