## Bayesian Econometrics Course

### Workbook Preface

This workbook is based upon the content of the RATS e-course on Bayesian
Econometrics, offered in spring 2009. It covers most of the most important
methods now used in Bayesian analysis in econometrics, including Gibbs sampling,
Metropolis-Hastings and importance sampling. The applications are to
a broad range of topics, include time series, cross-section and panel data. It
assumes that the user is comfortable with such basic instructions as COMPUTE,
DISPLAY, GRAPH, SCATTER and LINREG, and can use simple programming
techniques such as DO loops. In each chapter, there is a Tips and Tricks section
which covers in greater detail any functions or instructions that might be
unfamiliar.

The presentation is based largely on Gary Koop's *Bayesian Econometrics* (Koop
(2003)). We've added to that in several areas, with a chapter on vector autoregressions,
and examples from the literature for panel, cross-sectional data
and state-space models. In most cases, we've included much of the statistical
derivations from the book, presented in a way to highlight the calculations as
they are done with RATS, so even those without the book can benefit.

### Workbook Contents

(194 pages, 30 examples)

#### Preface

#### 1 Introduction

1.1 Bayesian Statistics: An Overview

1.2 Single Parameter--Brute Force

1.3 RATS Tips and Tricks

**Example 1.1** Brute Force: Analyzing on a Grid

#### 2 Linear Regression Model with Conjugate Prior

2.1 LRM with a Single Variable

2.2 Normal Linear Model: Theory

2.3 Using Cross Product Matrices

2.4 Calculations

2.5 Simulations

2.6 RATS Tips and Tricks

**Example 2.1** Linear Model: Single Variable

**Example 2.2** Multiple Regression: Conjugate Prior

**Example 2.3** Multiple Regression with Conjugate Prior: Simulations

#### 3 Normal Linear Model with Independent Prior

3.1 Theory

3.2 Calculations

3.3 Diagnostics

3.4 The Bayesian Approach to Hypothesis Testing

3.5 Hypothesis Testing with the Linear Model

3.6 RATS Tips and Tricks

**Example 3.1** Linear Model with Independent Prior

**Example 3.2** Linear Regression: Conjugate Prior with Restrictions

#### 4 Nonlinear Regression: Introduction to Metropolis-Hastings

4.1 Theory

4.2 Calculations

4.3 RATS Tips and Tricks

**Example 4.1** Non-linear Regression: Random Walk MH

**Example 4.2** Non-linear Regression: Independence MH

#### 5 Linear Regression with Non-Spherical Errors

5.1 Heteroscedasticity of Known Form

5.2 Heteroscedasticity of Unknown Form

5.3 Serially Correlated Errors

5.4 Seemingly Unrelated Regressions

5.5 RATS Tips and Tricks

**Example 5.1** Heteroscedastic errors with a known form

**Example 5.2** Heteroscedastic errors with a unknown functional form

**Example 5.3** Linear regression with AR(1) errors

**Example 5.4** Seemingly unrelated regression

#### 6 Vector Autoregressions

6.1 Flat Prior

6.2 Antithetic Acceleration

6.3 An Application: Blanchard-Quah Model

6.4 Structural VAR's

6.5 Informative Prior for Univariate Autoregressions

6.6 VAR with Informative Prior (BVAR)

6.7 RATS Tips and Tricks

**Example 6.1** Antithetic Acceleration

**Example 6.2** VAR with flat prior

**Example 6.3** Structural VAR: Importance sampling

**Example 6.4** Structural VAR: Random Walk MH

**Example 6.5** Structural VAR: Independence Chain MH

**Example 6.6** Univariate autoregression with prior

**Example 6.7** Univariate Autoregression: Out-of-sample forecast performance

**Example 6.8** Bayesian VAR: Gibbs sampling

#### 7 Cross Section and Panel Data

7.1 Panel Data

7.2 Probit and Tobit Models

7.3 RATS Tips and Tricks

**Example 7.1** Panel data: LSDV

**Example 7.2** Panel data: Fixed Effects

**Example 7.3** Panel data: Random Effects (hierarchical prior)

**Example 7.4** Panel data: Random coefficients model

**Example 7.5** Tobit model

**Example 7.6** Probit model

#### 8 State Space Models

**Example 8.1** State-space model: Independence MH

**Example 8.2** State Space Model: Gibbs sampling

**Example 8.3** State space model: Time-varying coefficients

#### A General Information on RATS Instructions

#### B Probability Distributions

B.1 Uniform

B.2 Univariate Normal

B.3 Univariate Student (t)

B.4 Beta distribution

B.5 Gamma Distribution

B.6 Inverse Gamma Distribution

B.7 Bernoulli Distribution

B.8 Multivariate Normal

B.9 Multivariate Student (t)

B.10 Wishart Distribution

#### C Properties of Multivariate Normals

#### D Non-Standard Matrix Calculations

#### Bibliography

#### Index