VIII. Model selection
A. Marginal likelihood
Suppose were trying to choose
among a series of models:
Model 1: p y| 1
Model M: p y|
M
where m are possibly of
different dimension
The Bayesian might think in
terms of an unobserved
random variable:
s 1 if Mod
VII. Time-varying variances
A.
B.
C.
D.
Introduction to ARCH models
Extensions
Markov-switching GARCH
Stochastic volatility
GARCH family:
yt
ut
vt
ht
xt
ut
ht v t
i.i.d. 0, 1 (e.g. N 0, 1 )
h u t 1, ut 2, . . .
Implication:
the difference between
the real
VII. Time-varying variances
A. Introduction to ARCH models
return on a stock in period t
population mean return
yt
ut
Observation: u t is almost impossible
to predit
0
E u t |u t 1 , u t 2 , . . .
However: u 2 does seem to be
t
quite forecastable
yt
Quest
VI. Spatiotemporal models
A. Introduction
1
s location s 1, 2, . . . , N
t date t 1, 2, . . . , T
yt s
variable of interest
Example 1 (economics):
s1
Alabama
sN
W yoming
t1
1954:I
tT
2004:IV
yt s
unemployment rate in Colorado
in 1972:II
Example 2 (Wikle,
V. Estimation of continuous-time
models
asset price or interest
rate at instant t
dy t
a y t , t; dt b y t , t; dW t
Wt
standard Brownian motion
Wt Ws
N 0, t s
yt
yt
1
yt
t1
st
t1
st
a y s , s ; ds
b y s , s ; dW s
Presumptions:
(1) y . is only observed a
IV. Markov-switching models
A. Introduction
B. Bayesian analysis of Markov-switching
models
C. State-space models with Markov
switching
D. Panel models with Markov switching
y nt
y nt
s nt
s nt
growth of employment
in state n for quarter t
n
n s nt
nt
1 w
III. Linear state-space models
A. State-space representation of a dynamic
system
B. Kalman filter
C. Using the Kalman filter
D. Bayesian analysis of linear state-space
models
E. Solutions to linear rational expectations
models
F. Estimating DSGE models
in
III. Linear state-space models
A. State-space representation of a dynamic
system
B. Kalman filter
C. Using the Kalman filter
D. Bayesian analysis of linear state-space
models
E. Solutions to linear rational expectations
models
1. Problem statement
AE t y
III. Linear state-space models
A. State-space representation of a dynamic
system
Consider following model
State equation:
F
t1
r1
rr
t
r1
vt
1
r1
Observation equation:
yt
A xt
n kk 1
n1
H
nr
t
r1
wt
n1
Observed variables: y t , x t
Unobserved variables:
t
II. Vector autoregressions
A.
B.
C.
D.
Introduction
Normal-Wishart priors for VARs
Bayesian analysis of structural VARs
Identification using inequality constraints
Structural model:
B 0 yt b0 B 1 y t 1 B 2 y t
B p y t p ut
E u t ut
2
I n for m
m
0
0 for m
II. Vector autoregressions
A. Introduction
1. VARs as forecasting models
Forecasting:
Let y 1 t be the first elem ent of a vector y t
y 1 t |t
1
ft y t 1 , y t 2, . . . , y 1
Convenient assumptions:
1) time invariant
2) linear
3) finite parameter
y 1t|t
1
I. Bayesian econometrics
A. Introduction
B. Bayesian inference in the univariate
regression model
C. Statistical decision theory
D. Large sample results
E. Diffuse priors
F. Numerical Bayesian methods
1. Importance sampling
Generic Bayesian problem:
p Y|
I. Bayesian econometrics
A. Introduction
B. Bayesian inference in the univariate
regression model
C. Statistical decision theory
1. Example: portfolio allocation problem
aj
quantity of asset j purchased
j 1, . . . , J
y income
budget constraint:
J
aj
y
j1
Econ 226: Bayesian and Numerical
Methods
Course requirements: two exams (based
primarily on lectures)
Slides: sometimes, not always, check web
page night before
Office hours: Tuesdays 9:30-10:30 a.m.
Theme: Bayesian econometrics and
applications of nu