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.

This was originally written for the VAR e-course, offered in the fall of 2009. The second edition expands and updates the original offering. In particular,

- Chapter 3 on "Error Bands" adds a discussion of Kilian (1998)?s bias-corrected bootstrap, and the newer @MCPROCESSIRF procedure, and updates the descriptions to use features added to RATS with version 9.
- Chapter 5 on "Structural VARs" has been greatly expanded to include detailed discussion of methods of computing error bands for impulse responses in structural models.
- Chapter 6 on "Semi-Structural VARs" describes procedures for incorporating "medium-run" constraints in addition to impact (short-run) and long-run constraints.
- Chapter 7 on "Sign Restrictions" has been expanded to include coverage of zero constraints (in addition to the sign constraints) and calculation of the Fry-Pagan median target responses.

(171 pages, 24 examples)

1.2 Log Likelihood Function

1.3 Choosing Lag Length

1.4 SYSTEM definition and ESTIMATE

1.5 Variables and Residuals

1.6 Alternative Estimation Methods

2.2 Computing Impulse Responses

2.3 Orthogonalization

2.4 Variance Decomposition

2.5 RATS Tips and Tricks

3.2 Bootstrapping

3.2.1 Kilian bootstrap-after-bootstrap

3.3 Monte Carlo Integration

3.3.1 Creating %%RESPONSES

3.3.2 The @MCPROCESSIRF procedure

3.4 RATS Tips and Tricks

3.4.1 The %MODEL function family

3.4.2 Block Sampling

4.2 Counterfactual Simulations

4.3 Error Bands

5.2 Generalized Impulse Responses

5.3 Parametric Structural VAR?s

5.4 Identification

5.5 Estimation

5.6 Structural Residuals

5.7 Error Bands

5.7.1 Bootstrapping

5.7.2 Monte Carlo: Deriving the Posterior Distribution

5.7.3 Monte Carlo with Importance Sampling

5.7.4 Monte Carlo with Random Walk Metropolis

5.7.5 Monte Carlo with the Waggoner-Zha Sampler

5.8 Tips and Tricks

5.8.1 Using PARMSETS

5.8.2 Waggoner-Zha sampler

6.2 Short- and Long-Run Restrictions

6.3 Multi-step Restrictions

6.4 Error Bands

7.2 Penalty functions

7.3 Multiple Shocks

7.3.1 FORCEDFACTOR and the QR Decomposition

7.4 Zero Constraints

7.5 Fry-Pagan Critique

7.6 Historical Decomposition

7.7 Convenience functions for sign restrictions

A.2 Wishart Distribution

A.3 Gamma Distribution

A.4 Inverse Gamma Distribution