Introduction to Multiple Time Series Analysis

Inhaltsverzeichnis

1. Introduction.- 1.1 Objectives of Analyzing Multiple Time Series.- 1.2 Some Basics.- 1.3 Vector Autoregressive Processes.- 1.4 Outline of the Following Chapters.- I. Finite Order Vector Autoregressive Processes.- 2. Stable Vector Autoregressive Processes.- 2.1 Basic Assumptions and Properties of VAR Processes.- 2.1.1 Stable VAR(p) Processes.- 2.1.2 The Moving Average Representation of a VAR Process.- 2.1.3 Stationary Processes.- 2.1.4 Computation of Autocovariances and Autocorrelations of Stable VAR Processes.- 2.1.4a Autocovariances of a VAR(1) Process.- 2.1.4b Autocovariances of a Stable VAR(p) Process.- 2.1.4c Autocorrelations of a Stable VAR(p) Process.- 2.2 Forecasting.- 2.2.1 The Loss Function.- 2.2.2 Point Forecasts.- 2.2.2a Conditional Expectation.- 2.2.2b Linear Minimum MSE Predictor.- 2.2.3 Interval Forecasts and Forecast Regions.- 2.3 Structural Analysis with VAR Models.- 2.3.1 Granger-Causality and Instantaneous Causality.- 2.3.1a Definitions of Causality.- 2.3.1b Characterization of Granger-Causality.- 2.3.1c Characterization of Instantaneous Causality.- 2.3.1d Interpretation and Critique of Instantaneous and Granger-Causality.- 2.3.2 Impulse Response Analysis.- 2.3.2a Responses to Forecast Errors.- 2.3.2b Responses to Orthogonal Impulses.- 2.3.2c Critique of Impulse Response Analysis.- 2.3.3 Forecast Error Variance Decomposition.- 2.3.4 Remarks on the Interpretation of VAR Models.- 2.4 Exercises.- 3. Estimation of Vector Autoregressive Processes.- 3.1 Introduction.- 3.2 Multivariate Least Squares Estimation.- 3.2.1 The Estimator.- 3.2.2 Asymptotic Properties of the Least Squares Estimator.- 3.2.3 An Example.- 3.2.4 Small Sample Properties of the LS Estimator.- 3.3 Least Squares Estimation with Mean-Adjusted Data and Yule-Walker Estimation.- 3.3.1 Estimation when the Process Mean Is Known.- 3.3.2 Estimation of the Process Mean.- 3.3.3 Estimation with Unknown Process Mean.- 3.3.4 The Yule-Walker Estimator.- 3.3.5 An Example.- 3.4 Maximum Likelihood Estimation.- 3.4.1 The Likelihood Function.- 3.4.2 The ML Estimators.- 3.4.3 Properties of the ML Estimators.- 3.5 Forecasting with Estimated Processes.- 3.5.1 General Assumptions and Results.- 3.5.2 The Approximate MSE Matrix.- 3.5.3 An Example.- 3.5.4 A Small Sample Investigation.- 3.6 Testing for Granger-Causality and Instantaneous Causality.- 3.6.1 A Wald Test for Granger-Causality.- 3.6.2 An Example.- 3.6.3 Testing for Instantaneous Causality.- 3.7 The Asymptotic Distributions of Impulse Responses and Forecast Error Variance Decompositions.- 3.7.1 The Main Results.- 3.7.2 Proof of Proposition 3.6.- 3.7.3 An Example.- 3.7.4 Investigating the Distributions of the Impulse Responses by Simulation Techniques.- 3.8 Exercises.- 3.8.1 Algebraic Problems.- 3.8.2 Numerical Problems.- 4. VAR Order Selection and Checking the Model Adequacy.- 4.1 Introduction.- 4.2 A Sequence of Tests for Determining the VAR Order.- 4.2.1 The Impact of the Fitted VAR Order on the Forecast MSE.- 4.2.2 The Likelihood Ratio Test Statistic.- 4.2.3 A Testing Scheme for VAR Order Determination.- 4.2.4 An Example.- 4.3 Criteria for VAR Order Selection.- 4.3.1 Minimizing the Forecast MSE.- 4.3.2 Consistent Order Selection.- 4.3.3 Comparison of Order Selection Criteria.- 4.3.4 Some Small Sample Simulation Results.- 4.4 Checking the Whiteness of the Residuals.- 4.4.1 The Asymptotic Distributions of the Autocovariances and Autocorrelations of a White Noise Process.- 4.4.2 The Asymptotic Distributions of the Residual Autocovariances and Autocorrelations of an Estimated VAR Process.- 4.4.2a Theoretical Results.- 4.4.2b An Illustrative Example.- 4.4.3 Portmanteau Tests.- 4.5 Testing for Nonnormality.- 4.5.1 Tests for Nonnormality of a Vector White Noise Process.- 4.5.2 Tests for Nonnormality of a VAR Process.- 4.6 Tests for Structural Change.- 4.6.1 A Test Statistic Based on one Forecast Period.- 4.6.2 A Test Based on Several Forecast Periods.- 4.6.3 An Example.- 4.7 Exercises.- 4.7.1 Algebraic Problems