Bayesian Statistics &
Econometrics
Law 243
PP 303C
Bayes Rule Derivation
Bayes Rule
Bayes Rule Example
Bayes Factor
Cases Where Beliefs Do Not
Change
Bayes Factor = 1
Evidence equivocal
Dogmatic Prior
Prior probability is 1 for one possible outcome
Value
Flexible Models: Nonparametric
and Semiparametric Methods
Koop, Chapter 10
The Idea
Always have a likelihood function that is in
some sense parametric
But can relax various features e.g.
NLRM:
Relax linear relationship
Relax normal errors assumption
N
Modeling Covariance Matrices in
Terms of Standard Deviations
and Correlations
Barnard, McCulloch & Meng (2000)
Rousseeuw & Molenberghs (1994)
The Central Idea
A Particular Matrix Decomposition
Example
Prior Elicitation
Sometimes useful to impose prior be
Bayesian Statistics & Econometrics
Fall 2009
Assignment #1 Due 10/2/09
Problems. Please prepare written answers to the following problems:
(1) (12 points) Find an example in any field of a published article that makes strong claims
based on two-sided p-va
Bayesian Statistics & Econometrics
Fall 2009
Assignment #2 Due 10/14/09
Problems. Please prepare written answers and (where appropriate) MATLAB output for
whichever of the following problems are included in the assignment for your option. The final
page b
Bayesian Statistics & Econometrics
Fall 2009
Assignment #3 Due 10/21/09
Problems. Please prepare written answers and (where appropriate) MATLAB output for
whichever of the following problems are included in the assignment under your option.
(1) (3 points)
Bayesian Statistics & Econometrics
Fall 2009
Assignment #4 Due 10/30/09
Problems. Please prepare written answers and (where appropriate) MATLAB output for the
following problems:
(1) (6 points) Metropolis-Hastings Algorithms. See next page.
(2) (6 points)
Bayesian Statistics & Econometrics
Fall 2009
Assignment #5 Due 11/11/09
Problems. Please prepare written answers and (where appropriate) MATLAB output for the
following problems:
(1) (8 points) Heteroskedasticity of an Unknown Form.
(2) (4 points) Autocor
Qualitative and Limited
Dependent Variable Models
Koop, Chapter 9
Strategy
Qualitative or limited dependent variable
makes NLRM not directly applicable
Normality not a reasonable assumption for
dependent variable
Have normally distributed latent variab
Introduction to Time Series
Koop, Chapter 8
Strategy and Background
Already covered AR(p) models in chapter 6
Develop state space models here
Is one way of doing time series
Bayesian approaches to other ways developed
elsewhere
Has natural expression
Linear Regression Model with
Panel Data
Koop, Chapter 7
Panel Data
(also called Longitudinal Data)
Multiple observations for each unit
E.g., T observations on each of N units
Types of Models
Pooled model
Assume coefficients same for all units
Individ
A Bayesian Perspective
on p-Values
Frequentist Approach
Available data is sample from a larger real or
imagined population
Focus is on estimators
Criteria: unbiased, efficient, etc.
Many results only apply asymptotically
Small sample problem
Strict
Linear Regression Model with
a Single Explanatory Variable
Koop, chapter 2
Regression Set Up
Likelihood Function
Rewrite Sum in Likelihood
New Likelihood Expression
Normal-Gamma Form
Gamma Distribution
Special Cases of Gamma
Distribution
Conjugate and Nat
Linear Regression Model with
Many Variables
Koop, chapter 3
Regression Set Up
Error Distribution
Likelihood Function
Rewrite Sum in Likelihood
Derivation of Rewritten Sum
Error in Book - Likelihood
Gamma Distribution
Probability Density Functions for
Prio
Posterior Simulation I
Geweke, Chapter 4
Start Through Section 4.2
The Paradigm
Key Questions
for a Posterior Simulator
Does it converge?
Need a Theorem
If so, is it efficient?
Versus alternatives
Overview
Types of Posterior Simulation
Direct samplin
Nonlinear Regression Model
Koop, Chapter 5
Metropolis-Hastings algorithm
Can be used as a step within Gibbs Sampler
Gelfand-Dey
Method to compute marginal likelihoods
Posterior predictive p-value
Examples (CES, Cobb-Douglas)
Intrinsically Nonlinear v
Posterior Simulation II
Geweke, Chapter 4
Section 4.3 to End
The Paradigm
Key Questions
for a Posterior Simulator
Does it converge?
Need a Theorem
If so, is it efficient?
Versus alternatives
Gibbs Sampler for Normal Linear
Regression Model
Invariant D
Bayesian Model Averaging
Bayesian Model Comparison
Koop, Chapter 11
Fernandez, Ley & Steel (2001)
George & Foster (2000)
OHagan & Forster (2004)
Strnad (2007)
BMA Idea
Implementation Overview
Potentially difficult if number of models
very large and/or mu
Linear Regression Model with
General Error Covariance Matrix
Koop, Chapter 6
New Assumptions
General Error Covariance Matrix
Transformed Model
Transformation: Math
Independent Normal Gamma
Set Up
Likelihood Function
Regular and Transformed
GLS Form
Note E
Bayesian Statistics & Econometrics
Fall 2009
Assignment #5 Due 11/30/09
Problems. Please prepare written answers and (where appropriate) MATLAB output for the
following problems:
(1) (4 points) Probit Model.
(2) (4 points) Tobit Model.
(3) (4 points) Time