Vector Autoregression (VAR) Course (2nd Edition)

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.

Workbook Contents

(171 pages, 24 examples)

Chapter 1: Introduction

1.1 Vector Autoregressions
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
Example 1.1 Lag Selection by AIC
Example 1.2 Estimation Techniques
Example 1.3 Long Lag VAR

Chapter 2: Impulse Response Functions

2.1 Moving Average Representation
2.2 Computing Impulse Responses
2.3 Orthogonalization
2.4 Variance Decomposition
2.5 RATS Tips and Tricks
Example 2.1 IRF with input shocks
Example 2.2 IRF with Choleski shocks

Chapter 3: Error Bands

3.1 Delta method
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
Example 3.1 Error Bands by Delta Method
Example 3.2 Error Bands by Bootstrapping
Example 3.3 Error Bands by Kilian bootstrap
Example 3.4 Error Bands by Monte Carlo
Example 3.5 Error Bands by Bootstrapping with Random Initial Values

Chapter 4: Historical Decomposition and Counterfactual Simulations

4.1 Historical Decomposition
4.2 Counterfactual Simulations
4.3 Error Bands
Example 4.1 Historical Decomposition

Chapter 5: Structural VAR's

5.1 Eigen Factorizations
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
Example 5.1 Eigen factorization
Example 5.2 SVAR: A-style Model
Example 5.3 SVAR: A-B style model
Example 5.4 SVAR: Importance Sampling for Error Bands
Example 5.5 SVAR: Random Walk Metropolis for Error Bands
Example 5.6 SVAR: Waggoner-Zha Sampler

Chapter 6: Semi-Structural VAR?s

6.1 ForcedFactor Procedure
6.2 Short- and Long-Run Restrictions
6.3 Multi-step Restrictions
6.4 Error Bands
Example 6.1 Blanchard-Quah Decomposition
Example 6.2 Multi-Step Restrictions
Example 6.3 Short- and Long-Run Restrictions with Error Bands

Chapter 7: Sign Restrictions

7.1 Generating Impulse Vectors
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
Example 7.1 Sign Restrictions: Part I
Example 7.2 Sign Restrictions: Part II
Example 7.3 Sign Restrictions: Part III
Example 7.4 Sign Restrictions: Median Target Method

A: Probability Distributions

A.1 Multivariate Normal
A.2 Wishart Distribution
A.3 Gamma Distribution
A.4 Inverse Gamma Distribution

B: VAR Likelihood Function

C: Properties of Multivariate Normals

D: Deriving the Schwarz criterion

E: Delta method

F: Gibbs Sampling and Markov Chain Monte Carlo