The Vector Autoregression (VAR) was introduced to the economics literature in the famous paper “Macroeconomics and Reality” (Sims (1980b)). Since then it, and its close relatives, have become the standard for analyzing multiple time series. Even when more complicated and tightly parameterized models are used, it’s the stylized facts gleaned from VAR analysis that they are expected to explain.
In this course, we will be examining techniques that use “flat priors”; that is, the techniques designed to elicit information from the data without the use of informative Bayesian priors. Strongly informative priors (such as the so-called Minnesota prior) are widely used for building forecasting models, but they tend to improve forecasts by shutting down much of the cross-variable interaction. The techniques we will examine are designed primarily to analyze precisely that type of interaction.
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Preface iv 1 Introduction 1.1 Vector Autoregressions . . . . . . . . . . . . . . . . . . . 1 1.2 Log Likelihood Function . . . . . . . . . . . . . . . . . . . 2 1.3 Choosing Lag Length . . . . . . . . . . . . . . . . . . . . . 3 1.4 SYSTEM definition and ESTIMATE . . . . . . . . . . . . . . . 6 1.5 Variables and Residuals. . . . . . . . . . . . . . . . . . . 8 1.6 Alternative Estimation Methods . . . . . . . . . . . . . . . 9 1.1 Lag Selection by AIC . . . . . . . . . . . . . . . . . . . . 11 1.2 Estimation Techniques . . . . . . . . . . . . . . . . . . . . 12 1.3 Long Lag VAR . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Impulse Response Functions 2.1 Moving Average Representation . . . . . . . . . . . . . . . . 14 2.2 Computing Impulse Responses . . . . . . . . . . . . . . . . . 16 2.3 Orthogonalization . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Variance Decomposition. . . . . . . . . . . . . . . . . . . . 24 2.5 RATS Tips and Tricks . . . . . . . . . . . . . . . . . . . . .25 2.1 IRF with input shocks . . . . . . . . . . . . . . . . . . . . 28 2.2 IRF with Choleski shocks . . . . . . . . . . . . . . . . . . .29 3 Error Bands 3.1 Delta method . . . . . . . . . . . . . . . . . . . . . . . . .30 3.2 Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Monte Carlo Integration . . . . . . . . . . . . . . . . . . . 34 3.4 RATS Tips and Tricks . . . . . . . . . . . . . . . . . . . . .37 3.1 Error Bands by Delta Method . . . . . . . . . . . . . . . . . 39 3.2 Error Bands by Bootstrapping . . . . . . . . . . . . . . . . .40 3.3 Error Bands by Monte Carlo . . . . . . . . . . . . . . . . . .41 4 Historical Decomposition and Counterfactual Simulations 4.1 Historical Decomposition . . . . . . . . . . . . . . . . . . .42 4.2 Counterfactual Simulations . . . . . . . . . . . . . . . . . .44 4.3 Error Bands . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1 Historical Decomposition . . . . . . . . . . . . . . . . . . .45 5 Structural VAR’s 5.1 Eigen Factorizations . . . . . . . . . . . . . . . . . . . . .47 5.2 Generalized Impulse Responses . . . . . . . . . . . . . . . . 48 5.3 Structural VAR’s . . . . . . . . . . . . . . . . . . . . . . .50 5.4 Identification. . . . . . . . . . . . . . . . . . . . . . . . 51 5.5 Structural Residuals . . . . . . . . . . . . . . . . . . . . .52 5.6 Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.1 Eigen factorization . . . . . . . . . . . . . . . . . . . . . 55 5.2 SVAR: A-style Model . . . . . . . . . . . . . . . . . . . . . 56 5.3 SVAR: A-B style model . . . . . . . . . . . . . . . . . . . . 57 6 Semi-Structural VAR’s 6.1 ForcedFactor Procedure . . . . . . . . . . . . . . . . . . . .60 6.2 Short and Long Run Restrictions . . . . . . . . . . . . . . . 61 6.1 Blanchard-Quah Decomposition . . . . . . . . . . . . . . . . .64 7 Sign Restrictions 7.1 Generating Impulse Vectors . . . . . . . . . . . . . . . . . .67 7.2 Penalty functions . . . . . . . . . . . . . . . . . . . . . . 72 7.3 Zero Constraints . . . . . . . . . . . . . . . . . . . . . . .75 7.4 Multiple Shocks . . . . . . . . . . . . . . . . . . . . . . . 76 7.5 Historical Decomposition . . . . . . . . . . . . . . . . . . .77 7.1 Sign Restrictions: Part I . . . . . . . . . . . . . . . . . . 79 7.2 Sign Restrictions: Part II . . . . . . . . . . . . . . . . . .81 7.3 Sign Restrictions: Part III . . . . . . . . . . . . . . . . . 84 A Probability Distributions A.1 Multivariate Normal . . . . . . . . . . . . . . . . . . . . . 88 A.2 Wishart Distribution . . . . . . . . . . . . . . . . . . . . .89 B Likelihood Function C Properties of Multivariate Normals Bibliography Index