Introduction Bayesian estimation: the basics Priors Evaluating the posterior Bayesian inference and model comparison Bayesian estimation in Dynare Extensions Dynamic Macro Bayesian Estimation Petr Sedl´ aˇ cek Bonn University Summer 2015 1 / 114
IntroductionBayesian estimation: the basics
PriorsEvaluating the posterior
Bayesian inference and model comparisonBayesian estimation in Dynare
Extensions
Dynamic Macro
Bayesian Estimation
Petr Sedlacek
Bonn University
Summer 2015
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IntroductionBayesian estimation: the basics
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Overall plan
Motivation
Week 1: Use of computational tools, simple DSGE model X
Tools necessary to solve models and a solution method
Week 2: function approximation and numerical integration X
Week 3: theory of perturbation (1st and higher-order) X
Tools necessary for, and principles of, estimation
Week 4: Kalman filter and Maximum Likelihood estimation X
Week 5: principles of Bayesian estimation
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Plan for today
Bayesian estimation: the basic ideas
extra information over ML: priors
main challenge: evaluating the posterior
Markov Chain Monte Carlo (MCMC)
practical issues: acceptance rate, diagnostics
implementation in Dynare
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PriorsEvaluating the posterior
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Extensions
Frequentist vs. Bayesian viewsBayes’ rule
Bayesian estimation: basic concepts
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IntroductionBayesian estimation: the basics
PriorsEvaluating the posterior
Bayesian inference and model comparisonBayesian estimation in Dynare
Extensions
Frequentist vs. Bayesian viewsBayes’ rule
Frequentist vs. Bayesian views
Frequentist view:
parameters are fixed, but unknown
likelihood is a sampling distribution for the data
realizations of observables YT
just one of many possible realizations from L(YT |Ψ)
inferences about Ψ
based on probabilities of particular YT for given Ψ
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Frequentist vs. Bayesian viewsBayes’ rule
Frequentist vs. Bayesian views
Bayesian view:
observations, not parameters, are taken as given
Ψ are viewed as random
inference about Ψ
based on probabilities of Ψ conditional on data YT P(Ψ|YT )
probabilistic view of Ψ enables incorporation of prior beliefs
Sims (2007):“Bayesian inference is a way of thinking, not a basket of methods”
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Frequentist vs. Bayesian viewsBayes’ rule
Bayes’ rule
1701-1761
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Bayes’ rule
Joint density of the data and parameters is:
P(YT ,Ψ) =L(YT |Ψ)P(Ψ) or
P(YT ,Ψ) =L(Ψ|YT )P(YT )
From the above we get Bayes’ rule:
P(Ψ|YT ) =L(YT |Ψ)P(Ψ)
P(YT )
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Elements of Bayes’ rule
what we’re interested in, posterior distribution: P(Ψ|YT )
likelihood of the data: L(YT |Ψ)
our prior about the parameters: P(Ψ)
probability of the data: P(YT )
for the distribution of Ψ P(YT ) is just a constant
P(Ψ|YT ) ∝ L(YT |Ψ)P(Ψ)
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What is the challenge?
getting the posterior is typically not such a big deal
problem is that we often want to know more:
conditional expected values of a function of the posterior
like mean, variance, model etc.
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What is the challenge?
E[g(Ψ)] =
∫g(Ψ)P(Ψ|YT )dΨ∫
P(Ψ|YT )dΨ
E[g(Ψ)] is the weighted average of g(Ψ)
weights are determined by the data (likelihood) and the prior
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Frequentist vs. Bayesian viewsBayes’ rule
What is the challenge?
we need to be able to evaluate the integral!
Special/Simple case:
we are able to draw Ψ from P(Ψ|YT )
can evaluate integral via Monte Carlo integration
you won’t be lucky enough to experience this case
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Evaluating the posterior
Our situation:
we can calculate P(Ψ|YT ), but we cannot draw from it
Solutions:
numerical integration
Markov Chain Monte Carlo (MCMC) integration
What is the standard?
although numerical integration is fast and accurate
computational burden rises exponentially with dimension
suited for low-dimension problems
→ use MCMC methods
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Priors
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Extensions
Idea of priors
summarize prior information
previous studies
data not used in estimation
pre-sample data
other countries etc.
don’t be too restrictive
more on prior selection in “extensions”
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Priors
Most commonly used distributions:
normal
beta, support ∈ [0, 1]
persistence parameters
(inverted-) gamma, support ∈ (0,∞)
volatility parameters
uniform
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Prior predictive analysis
check whether priors “make sense”
use the prior as the posterior
steady state?
impulse response functions?
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Some terminology
Jeffreys prior
non-informative prior
improper vs. proper priors
improper prior is non-integrable (integral is ∞)
important to have proper distributions for model comparison
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Some terminology
(natural) conjugate priors
family of prior distributions
after multiplication with the likelihood
produce a posterior of the same family
Minnesota (Litterman) prior
used in VARs for distribution of lags
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Importance samplingMarkov Chain Monte CarloGibbs algorithmMetropolis-Hastings algorithmPractical issues with MH algorithm
Evaluating the posterior
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Importance samplingMarkov Chain Monte CarloGibbs algorithmMetropolis-Hastings algorithmPractical issues with MH algorithm
Starting point
Aim is to be able to calculate something like
E[g(Ψ)] =
∫g(Ψ)P(Ψ|YT )dΨ∫
P(Ψ|YT )dΨ
we know how to calculate P(Ψ|YT )
but we cannot draw from it
the system is too large for numerical integration
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Principle of posterior evaluation
We cannot draw from the “target” distribution, but
1. can draw from a different, “stand-in”, distribution
2. can evaluate both stand-in and target distributions
3. comparing the two, we can re-weigh the draw “cleverly”
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Importance samplingMarkov Chain Monte CarloGibbs algorithmMetropolis-Hastings algorithmPractical issues with MH algorithm
Principle of posterior evaluation
the above procedure is the idea of “importance sampling”
MCMC methods effectively a version of importance sampling
traveling through the parameter space is more sophisticated
and or acceptance probability more sophisticated
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A few simple examples
Problem:
we want to simulate x
x comes from truncated normal with
mean µ and variance σ2
and a < x < b
Solution:
1. draw y from N(µ, σ2)
2a. if y ∈ (a, b) then keep draw (accept) and go back to 1
2b. otherwise discard draw (reject) and go back to 1
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A few simple examples
Problem:
want to draw x from F (x), but we cannot
we can sample from G (x) and f (x) ≤ cg(x) ∀xSolution:
1. sample y from G (y)
2. accept draw with probability f (y)cg(y) and go back to 1
Note:
acceptance rate higher for lower c
optimal c is c = supxf (x)g(x)
Metropolis-Hastings sampler (MCMC) is a generalization
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Importance sampling
Main idea very similar to the previous example:
cannot draw from P(Ψ|YT )
but can draw from H(Ψ)
be smart in reweighing (accepting) the draws
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Importance sampling
E[g(Ψ)] =
∫g(Ψ) P(Ψ|YT )
h(Ψ) h(Ψ)dΨ∫ P(Ψ|YT )h(Ψ) h(Ψ)dΨ
=
∫g(Ψ)ω(Ψ)h(Ψ)dΨ∫ω(Ψ)h(Ψ)dΨ
ω(Ψ) =P(Ψ|YT )
h(Ψ)
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Importance sampling
Approximate the integral using MC integration:
E[g(Ψ)] ≈∑M
m=1 ω(Ψ(m))g(Ψ(m))∑Mm=1 ω(Ψ(m))
M is the number of draws from importance function h(Ψ)
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Importance sampling
How to best choose h(.)?
we’d like h(.) to have fatter tails compared to f (.)
normal distribution has rather thin tails
→ often not a good importance function
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Before we move on
3 doors, behind one of them is a car
pick one
I will open one of the remaining two without the car
you can choose to stick with your choice or switch
who stays and who switches?
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Some preliminaries for MCMC
Markov property:
if for all k ≥ 1 and all tP(xt+1|xt , xt−1, ..., xt−k ) = P(xt+1|xt)
Transition kernel:
K(x , y) = P(xt+1 = y |xt = x) for x , y ∈ X
X is the sample space
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Main idea behind MCMC methods
as before, we’d like to sample from P(Ψ|YT ), but we cannot
MCMC methods provide a way to
create a Markov chain transition kernel (K) for Ψ
that has an invariant density P(Ψ|YT )
given K simulate the Markov chain P ′ = KP
starting with some initial values P(Ψ0)
(eventually) distribution of Markov chain → P(Ψ|YT )
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Main idea behind MCMC methods
a principle of constructing such kernels
→ Metropolis (-Hastings) algorithm (MH)
the Gibbs sampler is a special case
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Importance samplingMarkov Chain Monte CarloGibbs algorithmMetropolis-Hastings algorithmPractical issues with MH algorithm
Gibbs algorithm
special case of the MH algorithm
applies when can sample from each conditional distribution
again, this will rarely be applicable in our case
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Gibbs algorithm
instead of draws of Ψ from P(Ψ|YT )
portion Ψ into k blocks
sample each from P(Ψj |YT ,Ψ−j ) for j = 1, ..., k
iterate until convergence
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Gibbs sampling
Iterations (k = 2):
initiate sample with Ψ0
then iterate according to:
Ψ1i+1 ∼ P(Ψ1|YT ,Ψ2
i )
Ψ2i+1 ∼ P(Ψ2|YT ,Ψ1
i )
can prove that the above converges to P(Ψ|YT )
discard first B number of draws to eliminate influence of Ψ0
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Gibbs sampling
once Markov chain has converged
proceed as if we could sample directly:
E[g(Ψ)] =1
m
m∑i=1
g(Ψi )
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Gibbs sampling
however, draws are serially correlated
standard errors are higher
σ (E[g(Ψ)]) =
[1
m
(σ2
0 + 2m−1∑l=1
γlm − 1
m
)]1/2
σ20 variance of g(Ψ)
γl lth-order autocovariance of g(Ψ)
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Importance samplingMarkov Chain Monte CarloGibbs algorithmMetropolis-Hastings algorithmPractical issues with MH algorithm
Metropolis-Hastings algorithm
Main idea same as with importance sampling:
1. draw from a stand-in distribution h(Ψ; θ)
θ explicitly shows parameters of stand-in distribution
e.g. mean (µh ) and variance (σ2h)
2. accept/reject based on probability q(Ψi+1|Ψi )
3. go back to 1
3a. stand-in density does not change (indpendent MH)
3b. mean of stand-in adjusts (random walk MH)
can show convergence to target distribution
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Acceptance probability
“Metropolis”
q(Ψi+1|Ψi ) = min
[1,
P(Ψ∗i+1|YT )
P(Ψi |YT )
]
Ψ∗i+1 is the new candidate draw from stand-in distribution
if P(Ψ∗i+1|YT ) high relative to P(Ψi |YT )
→ probability of Ψ∗i+1 relatively high and should accept
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Importance samplingMarkov Chain Monte CarloGibbs algorithmMetropolis-Hastings algorithmPractical issues with MH algorithm
Acceptance probability
“Metropolis-Hastings”
q(Ψi+1|Ψi ) = min
[1,
P(Ψ∗i+1|YT )
P(Ψi |YT )
h(Ψi ; θ)
h(Ψ∗i+1; θ)
]
scale down by relative likelihood in stand-in density
a more “common” draw from the stand-in gets less “weight”
→ q(Ψi+1|Ψi ) is lowered
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Acceptance probability
“Metropolis-Hastings”
q(Ψi+1|Ψi ) = min
[1,
P(Ψ∗i+1|YT )
P(Ψi |YT )
h(Ψi ; θ)
h(Ψ∗i+1; θ)
]
P(Ψ∗i+1|YT )/h(Ψ∗i+1; θ) high
→ high probability of Ψ∗i+1 in target distribution
→ should accept → higher q(Ψi+1|Ψi )
P(Ψi |YT )/h(Ψi ; θ) high → lower q(Ψi+1|Ψi )
last draw was already in a ”likely” part of the parameter space
force the algorithm to explore less likely areas
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Updating the stand-in density
“Independence chain variant”
stand-in distribution does not change
it is independent across Monte Carlo replications
this is also the case in importance-sampling
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Updating the stand-in density
“Random walk variant”
candidate draws are obtained according to Ψ∗i+1 = Ψi + εi+1
εi from a symmetric density around 0 and variance σ2h
as if the mean of the stand-in density adjusts with eachaccepted draw
in θ, µh = Ψi
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Summary of MCMC with MH algorithm
1. maximize log-posterior logP(YT |Ψ) + logP(Ψ)
this yields the posterior mode Ψ
2. draw from a stand-in distribution h(Ψ; θ)
should have fatter tails than posterior
3. accept/reject based on probability q(Ψi+1|Ψi )
Metropolis vs. Metropolis-Hastings specification
4. go back to 2
adjust (random walk variant) stand-in distribution
do not adjust (independence variant) stand-in distribution
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Summary of MCMC with MH algorithm
evaluation of the likelihood (step 1 and 3) requires
computation of the steady state
solution of the model
constructing the likelihood function (via the Kalman filter)
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Choice of stand-in density
stand-in should have fatter tails
variance parameter important for acceptance rate
optimal acceptance rates:
around 0.44 for estimation of 1 parameter
around 0.23 for estimation of more than 5 parameters
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Choice of stand-in density
often, stand-in is N(Ψ, c2ΣΨ)
Ψ is the posterior mode
ΣΨ is the inverse (negative) Hessian at the mode
tip: start with c = 2.4/√d
d is number of estimated parameters
increase (decrease) c if acceptance rate is too high (low)
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Convergence statistics
theory says that distribution will converge to target
when does this happen?
→ diagnostic tests
sequence of draws should be from the invariant distribution
moments should not change within/between sequences
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Brooks and Gelman statistics
I draws and J sequences
W =1
J
J∑j=1
1
I − 1
I∑i=1
(Ψi ,j −Ψj
)2
B =I
J
J∑j=1
(Ψj −Ψ
)2
B/I : estimate of the variance of the mean across sequences
W : estimate of average variance within sequences
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Brooks and Gelman statistics
Combine the two measures of variance:
V =I − 1
IW +
B
I
as the length of the simulation increases
want these statistics to “settle down”
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Geweke statistic
partition a sequence into 3 subsets s = I , II , III
compute mean (Ψs) and standard errors (σs
Ψ)
s.e.’s must be corrected for serial correlation
then, under convergence CD is distributed N(0, 1)
CD =Ψ
I −ΨIII
σIΨ + σIII
Ψ
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Bayesian inference and model comparison
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Bayesian vs. frequentist inference
Bayesian inference cannot use frequentist principles
t-test, F-test, LR-test etc.
they have a frequentist justification of repeated sampling
instead, there are two common Bayesian principles:
Highest Posterior Density (HPD) interval
Bayes factors (posterior odds)
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Highest posterior density intervals
A 100(1− α)% posterior interval for Ψ is given by
P(b < Ψ < b) =
∫ b
bP(Ψ|YT )dΨ = 1− α
there exists many such intervals
the HPD interval is the smallest one of them
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HPD “tests”
the HPD test amounts to checking whether Ψi ∈ HPD1−α
this is an informal way of comparing nested models
i.e. different parameter values
Bayesians can also compare non-nested models
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Bayes factors
B =P(YT |Ψ1)P(Ψ1)
P(YT |Ψ2)P(Ψ2)
where Ψ1 and Ψ2 are two different sets of parameter values
if B > 1 → Ψ1 is a posteriori more likely than Ψ2
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Model comparison
posterior densities can be used to evaluate
conditional probabilities of particular parameter values
conditional probabilities of different model specifications
use Bayes factors (posterior odds ratio) to compare models
advantage is that all models are treated symmetrically
there is no “null” model compared to an alternative
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Model comparison
BA|B =PA(YT |ΨA)PA(ΨA)
PB(YT |ΨB)PB(ΨB)
it is also possible to assign priors on models
the posterior odds ratio is then
POA|B =P(A|YT )
P(B|YT )= BA|B
P(A)
P(B)
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Model comparison
Bayes factor is related to Bayesian information criterion (BIC)
BA|B ≈PA(YT |ΨA)
PB(YT |ΨB)T
kB−kA2
the RHS is the BIC where
Ψi denote ML estimates of parameters
ki denote the number of parameters
important to use proper priors
if not, always prefer model with less parameters
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How much information in Bayes factor?
Kass and Raftery (1995), if the value of BA|B is
between 1 and 3 → barely worth mentioning
between 3 and 20 → positive evidence
between 20 and 150 → strong evidence
over 150 → very strong evidence
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Bayesian estimation in Dynare
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Extensions
PreliminariesPriors and steady stateEstimation commandDecompositionOutputExample
Preliminaries
setup is the same as with ML estimation
always a good idea to solve model first
some parameter values are likely to remain calibrated
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Extensions
PreliminariesPriors and steady stateEstimation commandDecompositionOutputExample
Bayesian estimation in Dynare: initialization
initialize as usual
var c, k, z, y;
varexo e;
parameters beta, rho, alpha, nu, delta, sigma;
set parameter values that are not estimated
alpha = 0.36;
rho = 0.95;
beta = 0.99;
nu = 1;
delta = 0.025;
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Extensions
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Bayesian estimation in Dynare: setting it up
after model part, and specification of steady state
tell Dynare which parameters he should estimate
estimated params;
stderr e, inv gamma pdf, 0.01, inf;
end;
the above tells Dynare to
estimate σ, the st. error of the productivity disturbance
the prior distribution is an inverted gamma
the prior mean is 0.01 and the prior st. error is ∞
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Extensions
PreliminariesPriors and steady stateEstimation commandDecompositionOutputExample
Bayesian estimation in Dynare: steady state
steady state calculated for many different values of Ψ!
solve for the steady state yourself (linearizing makes it easier)
give the exact steady state to Dynare for the initial values
option to provide own function that calculates steady state!
modfilename steadystate.m or
steady state model; block
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Bayesian estimation in Dynare: estimation
then also tell Dynare which are the observable variables
varobs y;
estimation(options);
options include
specify data file for estimation: datafile=data
number of MH sequences: mh nblocks
number of MH replications: mh replic
parameter of stand-in distribution variance (c): mh jscale
variance of initial draw: mh init scale
first observation (default first): first obs
sample size (default all): nobs
many more!67 / 114
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Bayesian estimation in Dynare: decomposition
decompose endogenous variables into contribution of shocks
possible also after stoch simul
shock decomposition(options) variables;
options include e.g. parameter set
use calibrated values: =calibration
use prior/posterior mode: =prior mode/=posterior mode
variables specifies for which variables to run the decomposition
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Bayesian estimation in Dynare: output
RESULTS FROM POSTERIOR MAXIMIZATION:
most important is the mode
other stuff based on normality assumptions (typically violated)
when Dynare gets to MCMC part it shows:
in which MCMC sequence you are
which fraction has been completed
acceptance rate: adjust mh jscale appropriately
remember that low acceptance rate
→ algorithm travels through a larger part of Ψ domain
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Bayesian estimation in Dynare: plots
priors
MCMC diagnostics
prior and posterior densities
shocks implied at the mode
observables and corresponding implied values
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Extensions
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Estimating the neoclassical growth model
use neoclassical growth model as data generating process
265 observations of output
use Bayesian estimation to estimate
σ
σ, ρ, δ, α
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Extensions
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Estimating the neoclassical growth model
Easy case:
estimated params;
stderr e, inv gamma pdf, 0.01, inf;
end;
varobs y;
estimation(datafile=y,mh nblocks=1,mh replic=10000,
mh jscale=3,mh init scale=12) c, k, y;
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MCMC prior plots-easy case
0 0.01 0.02 0.03 0.04 0.05 0.060
50
100
150SE_e
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Shocks-easy case
50 100 150 200 250-0.03
-0.02
-0.01
0
0.01
0.02
0.03e
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Observables and implied values-easy case
50 100 150 200 2503.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2y
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Posterior density plots-easy case
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.0550
100
200
300
400
500
600
700
800
900
SE_e
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Printed results - easy case
Posterior mode:
0.0103 (0.0004)
Average acceptance rate:
37.7%
Diagnostic statistics (Geweke):p-values on equality of means in sub-samples
0.037 (no taper) 0.33 (4% taper) 0.38 (8% taper) etc.
Posterior mean and HPD interval:
0.0104 (0.0096 - 0.0111)
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Extensions
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What we did today
Basic concept of Bayesian estimation
priors
evaluating the posterior
Markov Chain Monte Carlo (MCMC)
practical issues
acceptance rate, diagnostics
implementation in Dynare
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What we did in the first half of course
Motivation
Week 1: Use of computational tools, simple DSGE model X
Tools necessary to solve models and a solution method
Week 2: function approximation and numerical integration X
Week 3: theory of perturbation (1st and higher-order) X
Tools necessary for, and principles of, estimation
Week 4: Kalman filter and Maximum Likelihood estimation X
Week 5: principles of Bayesian estimation X
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Extensions
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Extensions
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Trends
Problem:
methodology works for stationary environments
data has trends
not clear which trend the model represents?
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Trends
1940 1950 1960 1970 1980 1990 2000 2010 2020-15
-10
-5
0
5
10
devi
atio
ns fr
om tr
end
(%)
HP (1600)
HP(105)
linear
quadratic
BP(6,32)
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Trends
we could build in a trend within the model
e.g. productivity is trending
“stationarize” non-stationary variables within the model
i.e. inspect variables relative to productivity
however, not clear that data satisfies balanced growth
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Trends
b) Build-in a trend into the model. Detrend the data with model-based-trend. Problem: data does not seem to satisfy balanced growth.
1950:1 1962:4 1975:1 1987:2 2000:10.55
0.6
0.65Great ratios
c/y re
al
1950:1 1962:2 1975:1 1987:2 2000:10.05
0.1
0.15
0.2
i/y re
al
1950:1 1962:2 1975:1 1987:2 2000:10.5
0.6
0.7
c/y n
omina
l
1950:1 1962:2 1975:1 1987:2 2000:10.05
0.1
0.15
0.2
i/y n
omina
l
0 0.2 0.4 0.6 0.8 1−20
−10
0Log spectra
0 0.2 0.4 0.6 0.8 1−20
−10
0
0 0.2 0.4 0.6 0.8 1−20
−10
0
0 0.2 0.4 0.6 0.8 1−20
−10
0
Real and nominal Great ratios in US, 1950-2008.
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Trends
Solutions:
use differenced data
highlights high-frequency movements (measurement error)
detrend prior to estimation
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Estimation on detrended data
use e.g. quadratic trend:
yt = a0 + a1t + a2t2 + ut
each variable can have its own trend
using HP or Band Pass filter:
yobs−filteredt = B(L)yobs
t
B(L) is a 2-sided filter!
→ creates artificial serial correlation in the filtered data
→ apply filter also to model data
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Estimation on detrended data
the above implies that the model is fitted to low(er)frequencies only
Canova (2010) points out that the above can lead to:
underestimated volatility of shocks
persistence of shocks is overestimated
less perceived noise → decisions rules imply higherpredictability
substitution and income effects may be distorted due to theabove
proposes to estimate flexible trend specifications within model
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More on selecting priors
what we’ve described is based on selecting (independent)priors about deep parameters
however, often we have priors about observables
moreover, reasonable independent priors may form ratherunreasonable properties of the model
solutions proposed in the literature:
Del Negro, Schorfheide (2008)
Andrle, Benes (2013)
Jarocinsky, Marcet (2013)
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Del Negro, Schorfheide (2008)
more guidance for eliciting priors
three main issues with (independent) priors about deepparameters:
may lead to probability mass on unrealistic properties of themodel
most exogenous shock processes are latent, i.e. difficult toform priors about
priors are often transfered to different models
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Del Negro, Schorfheide (2008)
they group parameters into three categories:
those determining the steady state
those determining exogenous shocks
those determining the endogenous propagation mechanism
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Del Negro, Schorfheide (2008)
Parameters related to steady state relationships
discount rate, depreciation, returns to scale, inflation targetetc.
let SD(Ψss) be a vector of steady state relationshipsdepending on a set of parameters Ψss
then S = SD(Ψss) + η are measurements of thoserelationships with measurement error η
S has a probabilistic interpretation and therefore
using Bayes’ rule, one can write P(Ψss |S) ∝ L(S |Ψss)P(Ψss)
allows for overidentification
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Del Negro, Schorfheide (2008)
Exogenous processes
volatility and persistence parameters
use implied moments of endogenous variables to “back out”priors
the above is given values for Ψss and Ψendo
→ valid for a particular model and should not be directlytransfered across models
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Del Negro, Schorfheide (2008)
Endogenous propagation mechanisms
price rigidity, labor supply elasticity etc.
one could use similar principle as above
authors suggest independent priors because researchers oftenhave a relatively good idea
note that the joint prior induces non-trivial non-linearrelationships between parameters
joint prior becomes
P(Ψ|S) ∝ L(S |Ψss)P(Ψss)P(Ψendo)
requires an additional step in MCMC algorithm
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Andrle, Benes (2013)
Andrle and Benes do not distinguish between groups ofparameters
their “system priors” are priors about concepts such as
impulse response functions
conditional correlations etc.
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Andrle, Benes (2013)
even sensible individual-parameter priors can lead tounintended properties of the aggregate model
independence of priors can lead to substantial mass on suchparameter regions
call for careful prior-predictive analysis:
IRFs, second moments ...
compare with posterior results
is it the data or the model driving the results?
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Andrle, Benes (2013)
Candidates for system priors:
steady states
sensible values in levels or growth rates
(un-)conditional moments
cross-correlations (conditional on shocks)
impulse response properties
peak impacts, duration, horizon of monetary policyeffectiveness etc.
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Andrle, Benes (2013)
Implementation:
use Bayes’ rule again
specify model properties you care about Z = h(Ψ)
these can be characterized by a probabilistic modelZ ∼ D(Z s)
D(Z s) is a distribution function
Z s are parameters of that function (hyper-parameters)
its likelihood function (the system prior): P(Z s |Ψ, h)
composite joint prior: P(Ψ|Z s , h) ∝ P(Z s |Ψ, h)P(Ψ)
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Andrle, Benes (2013)
The posterior becomes
P(Ψ|YT ,Z s) ∝ L(YT |Ψ)P(Z s |Ψ, h)P(Ψ)
evaluation is in principle the same as before
use of MCMC methods
additional step in evaluating the system prior
slows things down - have to run MCMC on prior (withlikelihood “switched off”) and then posterior
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Jarocinsky, Marcet (2013)
similar ideas as above, but in the context of Bayesian VARs
their point is that widely used priors about parameters
can lead to behavior of observables that is counterfactual
→ always a good to do prior-predictive analysis of you model!
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Alternatives to Bayesian estimation
Maximum likelihood
calibration
GMM
SMM & indirect inference
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Maximum likelihood
we’ve seen it yesterday
conceptually different from Bayesian estimation
tools required part of Bayesian estimation
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Calibration
wide-spread methodology at least since Kydland and Prescott(1982)
prior to this, state-of-the-art were systems of simultaneousequations
those were viewed as “true statistical” models to be estimated
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Calibration
although calibration is also an empirical exercise
it lacks the probabilistic interpretation
the constraint is that the model mimics (a priori identified)features in the data
Kydland and Prescott (1996):It is important to emphasize that the parameter values selected arenot the ones that provide the best fit in some statistical sense.
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Calibration
Parameters are pinned down by a selection of real-world features
long-run averages (labor share, hours worked)
micro studies (preference parameters)
certain business cycle properties of the data (shockparameters) etc.
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Calibration
compare different features of the data to model predictions
closely related to moment-matching (estimating models)
however, calibration lacks the statistical formality
the above is a strong source of criticism of calibration
no formal rules on selecting dimension to which model is fit
no formal rules of comparing alternatives - models arenecessarily misspecified
not that the last point does not hold for Bayesian modelcomparison
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Matching moments (GMM, SMM, II)
idea similar to calibration:
a set of moments (features) of the data used to parameterizemodel
a different set of moments used to judge the performance ofmodel
matching moments adds statistical rigor
estimation
hypothesis testing
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Matching moments (GMM, SMM, II)
as with calibration, moment matching is based on a selectionof moments
often referred to as limited-information procedures
a full range of statistical implications contained in model’slikelihood function
disadvantages of limited-information procedures
potential loss of efficiency
inference potentially sensitive to selected moments
advantages of limited-information procedures
no need to make distributional assumptions
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Generalized method of moments
attributed to Hansen (1982), generalization, asymptoticproperties
the main idea is to use “orthogonality conditions” (e.g.first-order-conditions)
E[f (xt ,Ψ)] = 0
xt is a vector of variables
Ψ are model parameters
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Generalized method of moments
pick Ψ s.t. the sample analogs of orthogonality conditionsg(X ,Ψ) = 1/T
∑t f (xt ,Ψ)
hold exactly, exactly identified case
number of parameters = number of moment conditions
are as close to zero as possible, overidentified case
are number of parameters < number of moment conditions
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Generalized method of moments
in the over-identified case
minΨ
g(X ,Ψ)′Ωg(X ,Ψ)
Ω is a weighting matrix
the optimal weighting matrix is the inverse of the var-covarmatrix of g(X ,Ψ)
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Simulated method of moments
in some cases the orthogonality conditions cannot be assessedanalytically
moment-matching estimation based on simulations retainsasymptotic properties of GMM
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Simulated method of moments
let zt be model variables corresponding to data xt
let empirical targets be summarized by h(xt)
SMM estimation is based on
E[h(xt)] = E[h(zt ,Ψ)]
→ f (xt ,Ψ) = h(xt)− h(zt ,Ψ)
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Indirect inference
based on reduced-form models
main idea is to use structural model to interpret reduced-formresults
can simulated data from a structural model replicate areduced-form estimate using real-world data?
i.e. it is a moment-matching exercise
moments are clearly defined by prior reduced-form analysis
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Indirect inference
let δ be a vector of reduced-form estimates
δ(xt) are those in the data and δ(zt ,Ψ) are those from themodel
pick Ψ s.t.δ(xt) = δ(zt ,Ψ)
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