The Econometrics of Uncertainty Shocks
Jesús Fernández-Villaverde
University of Pennsylvania
March 7, 2016
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March 7, 2016
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Several challenges
Several challenges
How do we document the presence of time-varying uncertainty?
How do we distinguish time-variation in the data and in expectations?
Forecaster disagreement?
How much of the uncertainty is exogenous or endogenous?
How do we take DSGE models with time-varying uncertainty to the
data?
1
Likelihood function.
2
Method of moments.
Because of time limitations, I will focus on the …rst and last
challenges.
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Documenting time-varying uncertainty.
Interest rates
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Documenting time-varying uncertainty.
A real life example
Remember our decomposition of interest rates:
rt = |{z}
r +
mean
εtb,t and εr ,t follow:
+
εtb,t
|{z}
T-Bill shocks
εr ,t
|{z}
Spread shocks
εtb,t = ρtb εtb,t
1
+ e σtb,t utb,t , utb,t
εr ,t = ρr εr ,t
1
+ e σr ,t ur ,t , ur ,t
N (0, 1)
N (0, 1)
σtb,t and σr ,t follow:
σtb,t = 1
ρσtb σtb + ρσtb σtb,t
σr ,t = 1
Jesús Fernández-Villaverde (PENN)
ρσr σr + ρσr σr ,t
1
1
Econometrics
+ η tb uσtb ,t , uσtb ,t
+ η r uσr ,t , uσr ,t
N (0, 1)
N (0, 1)
March 7, 2016
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Documenting time-varying uncertainty.
Stochastic volatility
Remember, as well, that we postulated a general process for
stochastic volatility:
xt = ρxt
1
+ σ t εt , εt
N (0, 1).
and
log σt = (1
ρσ ) log σ + ρσ log σt
1
+ 1
ρ2σ
1
2
ηut , ut
N (0, 1).
We discussed this was a concrete example of a richer class of
speci…cations.
Main point: non-linear structure.
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Documenting time-varying uncertainty.
State space representation I
De…ne:
xt 1
σt
St =
Then, we have a transition equation:
xt
σ t +1
xt
σt
=f
,
εt
ut + 1
;γ
where the …rst row of f ( ) is:
xt = ρxt
1
+ σ t εt
and the second is
log σt +1 = (1
ρσ ) log σ + ρσ log σt + 1
ρ2σ
1
2
ηut +1
The vector of parameters:
γ = (ρ, ρσ , log σ, η )
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Documenting time-varying uncertainty.
State space representation II
In more compact notation:
St = f ( St
1 , Wt ; γ )
We also have a trivial measurement equation:
Yt =
1
0
xt
σ t +1
In more general notation:
Yt = g (St , Vt ; γ)
Note Markov structure.
Note also how we can easily accommodate more general cases.
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Documenting time-varying uncertainty.
Shocks
fWt g and fVt g are independent of each other.
fWt g is known as process noise and fVt g as measurement noise.
Wt and Vt have zero mean.
No assumptions on the distribution beyond that.
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Documenting time-varying uncertainty.
Conditional densities
From St = f (St
1 , Wt ; γ ) ,
we can compute p (St jSt
1 ; γ ).
From Yt = g (St , Vt ; γ), we can compute p (Yt jSt ; γ) .
From St = f (St
1 , Wt ; γ )
and Yt = g (St , Vt ; γ), we have:
Yt = g ( f ( St
1 , Wt ; γ ) , Vt ; γ )
and hence we can compute p (Yt jSt
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1 ; γ ).
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Filtering
Filtering, smoothing, and forecasting
Filtering: we are concerned with what we have learned up to current
observation.
Smoothing: we are concerned with what we learn with the full sample.
Forecasting: we are concerned with future realizations.
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Filtering
Goal of …ltering I
Compute conditional densities: p St jy t
1; γ
and p (St jy t ; γ) .
Why?
1
It allows probability statements regarding the situation of the system.
2
Compute conditional moments: mean, st jt and st jt 1 , and variances
Pt jt and Pt jt 1 .
3
Other functions of the states. Examples of interest.
Theoretical point: do the conditional densities exist?
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Filtering
Goals of …ltering II
Evaluate the likelihood function of the observables y T at parameter
values γ:
p yT ; γ
Given the Markov structure of our state space representation:
T
p y T ; γ = p (y1 jγ) ∏ p yt jy t
1
;γ
t =2
Then:
T
p y ;γ =
Z
T
p (y1 js1 ; γ) dS1 ∏
t =2
Z
p (yt jSt ; γ) p St jy t
1
; γ dSt
T
Hence, knowledge of p St jy t 1 ; γ t =1 and p (S1 ; γ) allow the
evaluation of the likelihood of the model.
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Filtering
Two fundamental tools
1
Chapman-Kolmogorov equation:
p St j y t
2
1
;γ =
Z
p ( St j St
1 ; γ) p
St
1 jy
t 1
; γ dSt
1
Bayes’theorem:
p St j y t ; γ =
p (yt jSt ; γ) p St jy t
p (yt jy t 1 ; γ)
1; γ
Z
p (yt jSt ; γ) p St jy t
1
where:
p yt jy t
Jesús Fernández-Villaverde (PENN)
1
;γ =
Econometrics
; γ dSt
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Filtering
Interpretation
All …ltering problems have two steps: prediction and update.
1
Chapman-Kolmogorov equation is one-step ahead predictor.
2
Bayes’theorem updates the conditional density of states given the new
observation.
We can think of those two equations as operators that map measures
into measures.
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Filtering
Recursion for conditional distribution
Combining the Chapman-Kolmogorov and the Bayes’theorem:
R R
R
p St j y t ; γ =
p (St jSt 1 ; γ) p St 1 jy t 1 ; γ dSt 1
p (yt jSt ; γ)
p (St jSt 1 ; γ) p (St 1 jy t 1 ; γ) dSt 1 p (yt jSt ; γ) dSt
To initiate that recursion, we only need a value for s0 or p (S0 ; γ).
Applying the Chapman-Kolmogorov equation once more, we get
T
p St jy t 1 ; γ t =1 to evaluate the likelihood function.
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Filtering
Initial conditions
From previous discussion, we know that we need a value for s1 or
p ( S1 ; γ ) .
Stationary models: ergodic distribution.
Non-stationary models: more complicated. Importance of
transformations.
Forgetting conditions.
Non-contraction properties of the Bayes operator.
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Filtering
Smoothing
We are interested on the distribution of the state conditional on all
the observations, on p St jy T ; γ and p yt jy T ; γ .
We compute:
p St j y T ; γ = p St j y t ; γ
Z
p St + 1 j y T ; γ p ( St + 1 j St ; γ )
dSt +1
p ( St + 1 j y t ; γ )
a backward recursion that we initialize with p ST jy T ; γ ,
fp (St jy t ; γ)gTt=1 and p St jy t
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1; γ
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T
t =1
we obtained from …ltering.
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Filtering
Forecasting
We apply the Chapman-Kolmogorov equation recursively, we can get
p (St +j jy t ; γ) , j 1.
Integrating recursively:
p yl +1 jy l ; γ =
Z
p (yl +1 jSl +1 ; γ) p Sl +1 jy l ; γ dSl +1
from t + 1 to t + j, we get p yt +j jy T ; γ .
Clearly smoothing and forecasting require to solve the …ltering
problem …rst!
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Filtering
Problem of …ltering
We have the recursion
R R
R
p St j y t ; γ =
p (St jSt 1 ; γ) p St 1 jy t 1 ; γ dSt 1
p (yt jSt ; γ)
p (St jSt 1 ; γ) p (St 1 jy t 1 ; γ) dSt 1 p (yt jSt ; γ) dSt
A lot of complicated and high dimensional integrals (plus the one
involved in the likelihood).
In general, we do not have closed form solution for them.
Translate, spread, and deform (TSD) the conditional densities in ways
that impossibilities to …t them within any known parametric family.
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Filtering
Exception
There is one exception: linear and Gaussian case.
Why? Because if the system is linear and Gaussian, all the conditional
probabilities are also Gaussian.
Linear and Gaussian state spaces models translate and spread the
conditional distributions, but they do not deform them.
For Gaussian distributions, we only need to track mean and variance
(su¢ cient statistics).
Kalman …lter accomplishes this goal e¢ ciently.
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Nonlinear Filtering
Nonlinear …ltering
Di¤erent approaches.
Deterministic …ltering:
1
Kalman family.
2
Grid-based …ltering.
Simulation …ltering:
1
McMc.
2
Particle …ltering.
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Nonlinear Filtering
Particle …ltering I
Remember,
1
Transition equation:
St = f (St 1 , Wt ; γ)
2
Measurement equation:
Yt = g (St , Vt ; γ)
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Nonlinear Filtering
Particle …ltering II
Some Assumptions:
1
We can partition fWt g into two independent sequences fW1,t g and
fW2,t g, s.t. Wt = (W1,t , W2,t ) and
dim (W2,t ) + dim (Vt ) dim (Yt ).
2
We can always evaluate the conditional densities
p yt jW1t , y t 1 , S0 ; γ .
3
The model assigns positive probability to the data.
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Nonlinear Filtering
Rewriting the likelihood function
Evaluate the likelihood function of the a sequence of realizations of
the observable y T at a particular parameter value γ:
p yT ; γ
We factorize it as (careful with initial condition!):
p yT ; γ =
T
∏p
t =1
T
=
∏
t =1
Z Z
p yt jW1t , y t
Jesús Fernández-Villaverde (PENN)
1
yt jy t
1
;γ
, S0 ; γ p W1t , S0 jy t
Econometrics
1
; γ dW1t dS0
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Nonlinear Filtering
A law of large numbers
If
n
t jt 1,i
s0
t jt 1,i
, w1
p W1t , S0 jy t
1; γ
oN
T
N i.i.d. draws from
i =1
T
,
t =1
then:
T
N
p yT ; γ '
t =1
1
∏N
t =1
∑p
i =1
t jt 1,i
yt jw1
, yt
1
t jt 1,i
, s0
;γ
The problem of evaluating the likelihood is equivalent to the problem of
drawing from
p W1t , S0 jy t
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1
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;γ
T
t =1
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Nonlinear Filtering
Introducing particles I
n
s0t
1,i
, w1t
1,i
oN
i =1
Each s0t 1,i , w1t
particles.
n
t jt 1,i
s0
1,i
t jt 1,i
, w1
Jesús Fernández-Villaverde (PENN)
N i.i.d. draws from p W1t
is a particle and
oN
i =1
n
s0t
1,i
, w1t
1
, S0 j y t
1,i
oN
i =1
N i.i.d. draws from p W1t , S0 jy t
Econometrics
1; γ
.
a swarm of
1; γ
.
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Nonlinear Filtering
Introducing particles II
t jt 1,i
t jt 1,i
Each s0
, w1
is a proposed particle and
n
o
t jt 1,i
t jt 1,i N
s0
, w1
a swarm of proposed particles.
i =1
Weights:
qti =
Jesús Fernández-Villaverde (PENN)
t jt 1,i
p yt jw1
, yt
t jt 1,i
∑N
i =1 p yt jw1
Econometrics
1 , s t jt 1,i ; γ
0
, yt
1 , s t jt 1,i ; γ
0
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Nonlinear Filtering
A proposition
Theorem
Let e
s0i , w
e1i
N
i =1
be a draw with replacement from
and probabilities
qti .
Then
Importance:
1
It shows how a draw
can be used to draw
2
n
N
e
s0i , w
e1i i =1
n
n
t jt 1,i
s0
s0t,i , w1t,i
oN
t jt 1,i
s0
is a draw from
t jt 1,i
, w1
oN
s0t,i , w1t,i
n
i =1
oN
i =1
t jt 1,i
, w1
oN
i =1
t
t
p (W1 , S0 jy ; γ).
from p W1t , S0 jy t 1 ; γ
from p (W1t , S0 jy t ; γ).
from p (W1t , S0 jy t ; γ) we can use
n
o
t +1 jt,i
t +1 jt,i N
p (W1,t +1 ; γ) to get a draw s0
, w1
and iterate the
i =1
procedure.
With a draw
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i =1
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Nonlinear Filtering
Algorithm
Step 0, Initialization: Set t
1 and set
p W1t 1 , S0 jy t 1 ; γ = p (S0 ; γ).
n
t jt
Step 1, Prediction: Sample N values s0
the density p W1t , S0 jy t
1; γ
1,i
t jt 1,i
, w1
oN
from
i =1
= p (W1,t ; γ) p W1t 1 , S0 jy t 1 ; γ
t jt 1,i
t jt 1,i
each draw s0
, w1
the
.
Step 2, Weighting: Assign to
weight qti .
n
oN
Step 3, Sampling: Draw s0t,i , w1t,i
with rep. from
i =1
n
oN
N
t jt 1,i
t jt 1,i
s0
, w1
with probabilities qti i =1 . If t < T set
i =1
t
t + 1 and go to step 1. Otherwise go to step 4.
T
n
o
t jt 1,i
t jt 1,i N
s0
Step 4, Likelihood: Use
, w1
to compute:
i =1
p yT ; γ '
Jesús Fernández-Villaverde (PENN)
T
1
N
∏N ∑p
t =1
i =1
t jt 1,i
yt jw1
Econometrics
, yt
t =1
1
t jt 1,i
, s0
;γ
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Nonlinear Filtering
Evaluating a Particle …lter
We just saw a plain vanilla particle …lter.
How well does it work in real life?
Is it feasible to implement in large models?
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Nonlinear Filtering
Why did we resample?
We could have not resampled and just used the weights as you would
have done in importance sampling (this is known as sequential
importance sampling ).
Most weights go to zero.
But resampling impoverish the swarm.
Eventually, this becomes a problem.
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Nonlinear Filtering
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Nonlinear Filtering
Why did we resample?
E¤ective Sample Size:
ESSt = h
1
t jt 1,i
∑N
i =1 p yt jw1
Alternatives:
, yt
1 , s t jt 1,i ; γ
0
i2
1
Strati…ed resampling (Kitagawa, 1996): optimal in terms of variance.
2
Adaptive resampling.
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Nonlinear Filtering
Simpler notation
To simplify notation:
1
Let me write the conditional distributions in terms of the current state
(instead of the innovations and the initial state).
2
Let me forget about the special notation required for period 1
(y t 1 = ?).
Then, the evaluation of the likelihood is just:
p yT ; γ =
T
∏
t =1
Thus, we are looking for
p St j y t
1; γ
Jesús Fernández-Villaverde (PENN)
T
t =1
.
Z Z
n
p yt jS t ; γ p St jy t
s t jt
1,i
oN
Econometrics
i =1
1
; γ dSt
T
N i.i.d. draws from
t =1
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Nonlinear Filtering
Improving our Particle …lter I
Remember what we did:
1
2
We draw from
s t jt 1,i
p St js t 1,i ; γ
We resample them with:
qti =
p yt js t jt 1,i , y t 1 ; γ
t jt 1,i , y t 1 ; γ
∑N
i = 1 p yt j s
But, what if I can draw instead from s t jt
1,i
q St j s t
1,i , y
t; γ
?
Intuition.
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Nonlinear Filtering
Improving our Particle …lter II
New weights:
p yt js t jt
qti =
1,i , y t 1 ; γ
t jt
∑N
i =1 p yt js
p (S t js t 1,i ;γ)
q (S t js t 1,i ,yt ;γ)
1,i , y t 1 ; γ
p (S t js t 1,i ;γ)
q (S t js t 1,i ,yt ;γ)
Clearly, if
q St j s t
1,i
, y t ; γ = p St j s t
1,i
;γ
we get back our basic Particle Filter.
How do we create the proposal q St js t
1
Linearized model.
2
Unscented Kalman …lter.
3
Information from the problem.
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1,i , y ; γ
t
?
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Nonlinear Filtering
Improving our Particle …lter III
Auxiliary Particle Filter: Pitt and Shephard (1999).
Imagine we can compute either
p yt +1 js t,i
or
p
e yt +1 js t,i
Then:
qti =
p
e yt +1 js t,i p yt js t jt
1,i , y t 1 ; γ
e (yt +1 js t,i ) p yt js t jt
∑N
i =1 p
p (S t js t 1,i ;γ)
q (S t js t 1,i ,yt ;γ)
1,i , y t 1 ; γ
p (S t js t 1,i ;γ)
q (S t js t 1,i ,yt ;γ)
Auxiliary Particle Filter tends to work well when we have fat tail...
...but it can temperamental.
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Nonlinear Filtering
Improving our Particle …lter IV
Resample-Move.
Blocking.
Many others.
A Tutorial on Particle Filtering and Smoothing: Fifteen years later, by
Doucet and Johansen (2012)
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Nonlinear Filtering
Nesting it in a McMc
Fernández-Villaverde and Rubio Ramírez (2007) and Flury and
Shepard (2008).
You nest the Particle …lter inside an otherwise standard McMc.
Two caveats:
1
Lack of di¤erentiability of the Particle …lter.
2
Random numbers constant to avoid chatter and to be able to swap
operators.
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Nonlinear Filtering
Parallel programming
Why?
Divide and conquer.
Shared and distributed memory.
Main approaches:
1
OpenMP.
2
MPI.
3
GPU programming: CUDA and OpenCL.
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Nonlinear Filtering
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Nonlinear Filtering
Tools
In Matlab: parallel toolboox.
In R: package parallel.
In Julia: built-in procedures.
In Mathematica: parallel computing tools.
GPUs: ArrayFire.
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Nonlinear Filtering
Parallel Particle …lter
Simplest strategy: generating and evaluating draws.
A temptation: multiple swarms.
How to nest with a McMc?
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Nonlinear Filtering
A problem
Basic Random Walk Metropolis Hastings is di¢ cult to parallelize.
Why? Proposal draw θ i +1 depends on θ i .
Inherently serial procedure.
Assume, instead, that we have N processors.
Possible solutions:
1
Run parallel chains.
2
Independence sampling.
3
Pre-fetching.
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Nonlinear Filtering
Multiple chains
We run N chains, one in each processor.
We merge them at the end.
It goes against the principle of one, large chain.
But it may works well when the burn-in period is small.
If the burn-in is large or the chain has subtle convergence issues, it
results in waste of time and bad performance.
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Nonlinear Filtering
Independence sampling
j
We generate N proposals e
θ i +1 from an independent distribution.
We evaluate the posterior from each proposal in a di¤erent processor.
We do N Metropolis steps with each proposal.
Advantage: extremely simple to code, nearly linear speed up.
Disadvantage: independence sampling is very ine¢ cient.
Solution)design a better proposal density.
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Nonlinear Filtering
Prefetching I
Proposed by Brockwell (2006).
Idea: we can compute the relevant posteriors several periods in
advance.
Set superindex 1 for rejection and 2 for acceptance.
Advantage: if we reject a draw, we have already evaluated the next
step.
Disadvantage: wasteful. More generally, you can show that the speed
up will converge only to log2 N.
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Nonlinear Filtering
Prefetching II
1
2
3
4
5
6
Assume we are at θ i .
n 1
2
θ i +1 = θ i , e
We draw 2 paths for iteration i + 1, e
θ i +1
We
i +2
n 11draw 41 paths12 for iteration
1
21
2
22
e
θ i +2 = e
θ i +1 , e
θ i +2 g e
θ i +1 , e
θ i +2 = e
θ i +1 , e
θ i +2
o
g (θ i ) .
2
g e
θ i +1
We iterate h steps, until we have N = 2h possible sequences.
1,...,1
2,...,2
We evaluate each of the posteriors p e
θ i +h , ..., p e
θ i +h
of the N processors.
o
.
in each
We do a MH in each step of the path using the previous posteriors
(note that any intermediate posterior is the same as the corresponding
…nal draw where all the following children are “rejections”).
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Nonlinear Filtering
A simpler prefetching algorithm I
Algorithm for N processors, where N is small.
Given θ i :
1
2
We draw N
n
o
e
θ i +1,1 , ..., e
θ i +1,N and we evaluate the posteriors
p e
θ i +1,1 , ..., p e
θ i +1,N .
We evaluate the …rst proposal, e
θ i +1,1 :
1
2
3
If accepted, we disregard θ i +1,2 , ..., θ i +1,2 .
If rejected, we make θ i +1 = θ i and e
θ i +2,1 = e
θ i +1,2 is our new proposal
for i + 2.
We continue down the list of N proposals until we accept one.
Advantage: if we reject a draw, we have already evaluated the next
step.
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Nonlinear Filtering
A simpler prefetching algorithm II
Imagine we have a PC with N = 4 processors.
Well gauged acceptance for a normal SV model
me …x it to 25% for simplicity.
20%
25%. Let
Then:
1
P (accepting 1st draw ) = 0.25. We advance 1 step.
2
P (accepting 2nd draw ) = 0.75 0.25. We advance 2 steps.
3
P (accepting 3tr draw ) = 0.752
0.25. We advance 3 steps.
4
P (accepting 4th draw ) = 0.753
0.25. We advance 4 steps.
5
P (not accepting any draw ) = 0.75 4 . We advance 4 steps.
Therefore, expected numbers of steps advanced in the chain:
1 0.25 + 2 0.75 0.25 + 3 0.75
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Econometrics
2
0.25 + 4 0.753 = 2.7344
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Taking DSGE-SV Models to the Data
First-order approximation
Remember that the …rst-order approximation of a canonical RBC
model without persistence in productivity shocks:
kbt +1 = a1 kbt + a2 εt , εt
Then:
kbt +1 = a1 a1 kbt
= a12 kbt
1
1
+ a2 ε t
+ a1 a2 ε t
Since a1 < 1 and assuming kb0 = 0
kbt +1 = a2
N (0, 1)
1
+ a2 ε t
+ a2 ε t
1
t
∑ a1j εt
j
j =0
which is a well-understood MA system.
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Taking DSGE-SV Models to the Data
Higher-order approximations
Second-order approximation:
kbt +1 = a0 + a1 kbt + a2 εt + a3 kbt2 + a4 ε2t + a5 kbt εt , εt
N (0, 1)
Then:
kbt +1 = a0 + a1 a0 + a1 kbt + a2 εt + a3 kbt2 + a4 ε2t + a5 kbt εt + a2 εt
+a3 a0 + a1 kbt + a2 εt + a3 kbt2 + a4 ε2t + a5 kbt εt
2
+ a4 ε2t
+a5 a0 + a1 kbt + a2 εt + a3 kbt2 + a4 ε2t + a5 kbt εt εt
We have terms in kbt3 and kbt4 .
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Taking DSGE-SV Models to the Data
Problem
For a large realization of εt , the terms in kbt3 and kbt4 make the system
explode.
This will happen as soon as we have a large simulation)
no unconditional moments would exist based on this approximation.
This is true even when the corresponding linear approximation is
stable.
Then:
1
How do you calibrate? (translation, spread, and deformation).
2
How do you implement GMM or SMM?
3
Asymptotics?
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Taking DSGE-SV Models to the Data
A solution
For second-order approximations, Kim et al. (2008): pruning.
Idea:
kbt +1 = a0 + a1 a0 + a1 kbt + a2 εt + a3 kbt2 + a4 ε2t + a5 kbt εt + a2 εt
+a3 a0 + a1 kbt + a2 εt + a3 kbt2 + a4 ε2t + a5 kbt εt
2
+ a4 ε2t
+a5 a0 + a1 kbt + a2 εt + a3 kbt2 + a4 ε2t + a5 kbt εt εt
We omit terms raised to powers higher than 2.
Pruned approximation does not explode.
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Taking DSGE-SV Models to the Data
What do we do?
Build a pruned state-space system.
Apply pruning to an approximation of any arbitrary order.
Prove that …rst and second unconditional moments exist.
Closed-form expressions for …rst and second unconditional moments
and IRFs.
Conditions for the existence of some higher unconditional moments,
such as skewness and kurtosis.
Apply to a New Keynesian model with EZ preferences.
Software available for distribution.
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Taking DSGE-SV Models to the Data
Practical consequences
1
GMM and IRF-matching can be implemented without simulation.
2
First and second unconditional moments or IRFs can be computed in
a trivial amount of time for medium-sized DSGE models
approximated up to third-order.
3
Use the unconditional moment conditions in optimal GMM estimation
to build a limited information likelihood function for Bayesian
inference (Kim, 2002).
4
Foundation for indirect inference as in Smith (1993) and SMM as in
Du¢ e and Singleton (1993).
5
Calibration.
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State-Space Representations
Dynamic models and state-space representations
Dynamic model:
xt +1 = h (xt , σ) + σηet +1 , et +1
IID (0, I)
yt = g (xt , σ )
Comparison with our previous structure.
Again, general framework (augmented state vector).
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State-Space Representations
The state-space system I
Perturbation methods approximate h (xt , σ) and g (xt , σ) with
Taylor-series expansions around xss = σ = 0.
A …rst-order approximated state-space system replaces g (xt , σ) and
h (xt , σ) with gx xt and hx xt .
If 8 mod (eig (hx )) < 1, the approximation ‡uctuates around the
steady state (also its mean value).
Thus, easy to calibrate the model based on …rst and second moments
or to estimate it using Bayesian methods, MLE, GMM, SMM, etc.
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State-Space Representations
The state-space system II
We can replace g (xt , σ) and h (xt , σ) with their higher-order
Taylor-series expansions.
However, the approximated state-space system cannot, in general, be
shown to have any …nite moments.
Also, it often displays explosive dynamics.
This occurs even with simple versions of the New Keynesian model.
Hence, it is di¢ cult to use the approximated state-space system to
calibrate or to estimate the parameters of the model.
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State-Space Representations
The pruning method: second-order approximation I
Partition states:
h
xft
0
(xst )0
i
Original state-space representation:
1
(2 )
xt +1 = hx xft + xst + Hxx
2
(2 )
yt
xft + xst
1
(2 )
(2 )
= gx xt + Gxx xt
2
Jesús Fernández-Villaverde (PENN)
Econometrics
xft + xst
(2 )
xt
1
+ hσσ σ2 + σηet +1
2
1
+ gσσ σ2
2
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State-Space Representations
The pruning method: second-order approximation II
New state-space representation:
xft +1 = hx xft + σηet +1
1
xst +1 = hx xst + Hxx xft xft +
2
ytf = gx xft
1
yts = gx xft + xst + Gxx xft xft
2
1
hσσ σ2
2
1
+ gσσ σ2
2
All variables are second-order polynomials of the innovations.
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State-Space Representations
The pruning method: third-order approximation I
Partition states:
h
xft
0
(xst )0
xrd
t
0
i
Original state-space representation:
(3 )
1
1
(3 )
(3 )
(3 )
(3 )
+ Hxx xt
xt
+ Hxxx xt
xt
2
6
1
3
1
(3 )
+ hσσ σ2 + hσσx σ2 xt + hσσσ σ3 + σηet +1
2
6
6
1
1
(3 )
(3 )
(3 )
(3 )
(3 )
= gx xt + Gxx xt
xt
+ Gxxx xt
xt
2
6
3
1
1
(3 )
+ gσσ σ2 + gσσx σ2 xt + gσσσ σ3
2
6
6
(3 )
xt +1 = hx xt
(3 )
yt
Jesús Fernández-Villaverde (PENN)
Econometrics
March 7, 2016
(3 )
xt
(3 )
xt
62 / 69
State-Space Representations
The pruning method: third-order approximation II
New state-space representation:
xrd
t +1
ytrd
Second-order pruned state-space representation+
1
f
= hx xrd
xst + Hxxx xft xft xf
t + Hxx xt
6
3
1
2 f
3
+ hσσx σ xt + hσσσ σ
6
6
1
= gx xft + xst + xrd
+ Gxx xft xft + 2 xft xst
t
2
1
1
3
1
f
f
f
+ Gxxx xt xt xt + gσσ σ2 + gσσx σ2 xft + gσσσ σ3
6
2
6
6
All variables are third-order polynomials of the innovations.
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State-Space Representations
Higher-order approximations
We can generalize previous steps:
1
Decompose the state variables into …rst-, second-, ... , and kth-order
e¤ects.
2
Set up laws of motions for the state variables capturing only …rst-,
second-, ... , and kth-order e¤ects.
3
Construct the expression for control variables by preserving only
e¤ects up to kth-order.
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State-Space Representations
Statistical properties: second-order approximation I
Theorem
If 8 mod (eig (hx )) < 1 and et +1 has …nite fourth moments, the pruned
state-space system has …nite …rst and second moments.
Theorem
If 8 mod (eig (hx )) < 1 and et +1 has …nite sixth and eighth moments, the
pruned state-space system has …nite third and fourth moments.
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State-Space Representations
Statistical properties: second-order approximation II
We introduce the vectors
h
(2 )
zt
xft
2
(2 )
ξ t +1
First moment:
0
6 e t +1
6
4
(xst )0
hx )
1
Jesús Fernández-Villaverde (PENN)
xft
e t +1
0
i0
3
et +1 vec (Ine ) 7
7
5
et +1 xft
f
xt et +1
h
i
h i
(2 )
E xt
= E xft + E [xst ]
| {z } | {z }
=0
E [xst ] = (I
xft
1
Hxx (I
2
6 =0
1
hx ) 1 (ση ση) vec (Ine ) + hσσ σ2
2
h
i
(2 )
E [yts ] = C(2 ) E zt
+ d(2 )
hx
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State-Space Representations
Statistical properties: second-order approximation III
Second moment:
(2 )
V zt
(2 )
= A(2 ) V zt
(2 )
A(2 )
l
(2 )
0
(2 )
+ B(2 ) V ξ t
B(2 )
0
(2 )
Cov zt +l , zt
= A(2 ) V zt
for l = 1, 2, 3, ...
h
i
(2 )
V xt
= V xft + V (xst ) + Cov xft , xst + Cov xst , xft
V [yts ] = C(2 ) V [zt ] C(2 )
(2 )
(2 )
Cov (yts , yts +l ) = C(2 ) Cov zt +l , zt
(2 )
where we solve for V zt
Lyapunov equations.
Jesús Fernández-Villaverde (PENN)
C(2 )
0
0
for l = 1, 2, 3, ...
by standard methods for discrete
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State-Space Representations
Statistical properties: second-order approximation IV
Generalized impulse response function (GIRF): Koop et al. (1996)
GIRFvar (l, ν, wt ) = E [vart +l jwt , et +1 = ν]
E [vart +l jwt ]
Importance in models with volatility shocks.
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State-Space Representations
Statistical properties: third-order approximation
Theorem
If 8 mod (eig (hx )) < 1 and et +1 has …nite sixth moments, the pruned
state-space system has …nite …rst and second moments.
Theorem
If 8 mod (eig (hx )) < 1 and et +1 has …nite ninth and twelfth moments,
the pruned state-space system has …nite third and fourth moments.
Similar (but long!!!!!) formulae for …rst and second moments and
IRFs.
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