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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Particle rolling MCMC
Yasuhiro Omori
Faculty of Economics, University of Tokyo
with Naoki Awaya
August 28, 2018
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Outline
Introduction
Motivation
State space model
Rolling parameter and state estimation
Contribution of the paper
Particle rolling MCMC
Simple particle rolling MCMC
Particle rolling MCMC with double block sampling
Theoretical justification of PRMCMC
Forward block sampling
Backward block sampling
Numerical examples
Example 1 (Linear Gaussian SSM)
Example 2 (Realized stochastic volatility model)2 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
MotivationState space modelRolling parameter and state estimationContribution of the paper
1. Introduction
3 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
MotivationState space modelRolling parameter and state estimationContribution of the paper
Motivation
We consider modelling economic time series during the long
sample period where we
▶ estimate model parameters,
▶ forecast on the time series
▶ for the risk management and policy simulation ...
but, in the economic time series, it is well-known that
▶ there have been major structural changes (due to such as the
financial crisis in 2008)
▶ parameters are not necessary constant and subject to changes
in the long sample period
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
MotivationState space modelRolling parameter and state estimationContribution of the paper
Motivation
To deal with the structural change in economic times series, for
example,
▶ divide the sample periods into several subsample periods→1. How many structural changes?
2. When does the structural changes occur?
▶ fix the length of the sample period and implement the rolling
estimation→1. How to decide the length of the sample period?
2. It takes time to estimate parameters and forecast using
MCMC.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
MotivationState space modelRolling parameter and state estimationContribution of the paper
State space model
This paper considers the state space model
p(yt | y1:t−1, α1:t , θ) = p(yt | αt , θ) ≡ gθ(yt | αt),
p(αt | α1:t−1, θ) = p(αt | αt−1, θ) ≡ fθ(αt | αt−1),
p(α1 | θ) ≡ µθ(α1),
where
yt : an observation vector at time t,
αt : an unobserved state vector at time t,
θ: a static parameter vector,
ys:t = (ys , . . . , yt) and αs:t = (αs , . . . , αt). The correlation
between yt and αt+1 is also taken into account (e.g. leverage in
stochastic volatility models).6 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
MotivationState space modelRolling parameter and state estimationContribution of the paper
Rolling parameter and state estimation
We estimate the posterior distribution of θ and αs:t given the
observations ys:t with t = s + L for s = 1, 2 . . .:
π(θ, αs:t | ys:t)
∝ p(θ)µθ(αs)gθ(ys | αs)
t∏
j=s+1
fθ(αj | αj−1, yj−1)gθ(yj | αj)
∝ p(θ)µθ(αs)
t∏
j=s+1
fθ(αj | αj−1)gθ(yj−1 | αj , αj−1)
gθ(yt | αt),
where p(θ): the prior probability density function of θ.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
MotivationState space modelRolling parameter and state estimationContribution of the paper
Contribution of the paper
This paper considers the state space model for economic times
series and
▶ proposes the efficient rolling estimation method for both
parameter θ and states αs:t .
▶ gives the theoretical justification of the algorithm by proving
the marginal density of parameters and states.
▶ the algorithm is based on Particle Gibbs
▶ proposes, as a special case, the sequential Bayesian estimation
method (which is alternative to SMC2 based on Particle MH).
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
2. Particle rolling MCMC (PRMCMC)
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Simple particle rolling MCMC
Suppose, at time t − 1, we have
▶ (θn, αns−1:t−1): a collection of particles for n = 1, . . . ,N
▶ W ns−1:t−1: the importance weight which is a discrete
approximation of π(θ, αs−1:t−1 | ys−1:t−1).
We update these particles in two steps:
1. Step 1. Add a new observation yt .
(θn, αns−1:t−1) → (θn, αn
s−1:t), Wns−1:t−1 → W n
s−1:t .
2. Step 2. Discard the old observation ys−1.
(θn, αns−1:t) → (θn, αn
s:t), Wns−1:t → W n
s:t .
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Simple particle rolling MCMC
-ss − 1
ss
st − 1
st
-� Sample period
Step 1. Generate αnt given (θn, αn
s−1:t−1) and ys−1:t
Update W ns−1:t−1 → W n
s−1:t
time
where
W ns−1:t ∝
fθn(αnt | αn
t−1, yt−1)gθn(yt | αnt )
qt,θn(αnt | αn
t−1, yt)W n
s−1:t−1,
qt,θn(· | αnt−1, yt): proposal density to generate αn
t .
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Simple particle rolling MCMC
-ss − 1
ss
st − 1
st
-� Sample period
Step 2. Discard αns−1 and construct (θn, αn
s:t) given ys:t
Update W ns−1:t → W n
s:t for n = 1, . . . ,N
time
Ws:t ∝ gθn(ys−1 | αns−1, α
ns )
−1Ws−1:t ,
gθ(ys−1 | αs−1, αs) =µθ(αs−1)fθ(αs | αs−1)gθ(ys−1 | αs−1, αs)
p(αs−1 | αs , θ)µθ(αs)
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Simple particle rolling MCMC
Remarks.
▶ If some degeneracy criterion is fulfilled (such as ESS < 0.5N),
resample all the particles by implementing MCMC algorithm.
▶ We often use a prior density fθ(αt |αt−1, yt−1) as a proposal
density qt,θ to generate αt in Step 1. It is known to cause the
weight degeneracy problem since the prior density does not
take account of the new observation yt .
▶ Even if we are able to use a fully adapted proposal density,
qt,θ(αt | αt−1, ys−1:t) = p(αt | αt−1, yt , θ), we still have the
weight degeneracy phenomenon.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Simple particle rolling MCMC
Example 1.
yt = αt + ϵt , ϵt ∼ N (0, σ2), t = 1, . . . , 2000
αt+1 = µ+ 0.25(αt − µ) + ηt , ηt ∼ N (0, 2σ2), t = 1, . . . , 2000,
α1 = µ+η0√
1− 0.252, η0 ∼ N (0, 2σ2),
where θ = (µ, σ2)′, µ | σ2 ∼ N (0, 10σ2), σ2 ∼ IG(5/2, 0.05/2).The window size = 1000. In order to evaluate the weight
degeneracy in Steps 1a and 2a, we define two ratios:
R1t =ESSs−1:t
ESSs−1:t−1, R2t =
ESSs:tESSs−1:t
, ESSs:t =1∑N
n=1(Wns:t)
2.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Simple particle rolling MCMC
1000 1200 1400 1600 1800 2000
0.2
0.4
0.6
0.8
1.0R1t
R2t
1000 1200 1400 1600 1800 2000
0.2
0.4
0.6
0.8
1.0 R2t
0.00 0.25 0.50 0.75 1.00
100
200
300
400R1t
0.00 0.25 0.50 0.75 1.00
100
200
300
400
Figure: Rolling window estimation of p(θ, αt−999:t | yt−999:t),
t = 1001, . . . , 2000 in the linear Gaussian state space model. The fully
adapted proposal density is used for qt,θ.15 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Particle rolling MCMC with double block sampling
We update these particles in two steps:
Step 1. Add a new obs yt (forward block sampling)
1. Generate αnt−K :t given (θn, αn
s−1:t−K−1) and ys−1:t
2. Construct a collection of particles (θn, αns−1:t) with the
importance weight W ns−1:t
3. Implement the particle simulation smoother to improve the
mixing property.
Step 2. Remove the old obs ys−1 (backward block sampling)
1. Generate αns−1:s+K−1 given (θn, αn
s+K :t) and ys:t .
2. Construct a collection of particles (θn, αns:t) with the
importance weight W ns:t
3. Discard αns−1 and implement the particle simulation smoother.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Particle rolling MCMC with double block sampling
-ss − 1
ss
st − K
st − 1
st
-s� s
Forward block sampling
Particle simulation smoother
Step 1. Generate αnt−K :t given (θn, αn
s−1:t−K−1) and ys−1:t
Update W ns−1:t−1 → W n
s−1:t
time
where
W ns−1:t ∝ p(yt | ys−1:t−1, α
nt−K−1, θ
n)×W ns−1:t−1.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Particle rolling MCMC with double block sampling
-ss − 1
ss
ss + K − 1
st − 1
st
-s� sBackward block sampling
Particle simulation smoother
Step 2. Generate αns−1:s+K−1 given (θn, αn
s+K :t) and ys:t
Discard αns−1 and update W n
s−1:t → W ns:t
time
where
W ns:t ∝
1
p(ys−1 | ys:t , αns+K , θ
n)W n
s−1:t
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Particle rolling MCMC with double block sampling
Forward block sampling
1. ‘Move’ the particles αnt−K :t−1 using the conditional SMC
update for each n = 1, . . . ,N based on particle Gibbs(Andrieu et al. (2010)).
(1) Generate a number of candidates αn,mt−K :t−1 (m = 1, . . . ,M)
where the current ‘lineage’ αnt−K :t−1 is fixed as one of
candidates.
(2) The αn,mt is also generated conditional on αn,m
t−K :t−1.
2. Determine which lineage is appropriate as αnt−k:t and compute
its importance weight for each n = 1, . . . ,N.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Particle rolling MCMC with double block sampling
Figure: (θn, αns−1:t−1) is fixed and shown in the rectangle. Other state
variables in the circle are generated.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Particle rolling MCMC with double block sampling
Backward block sampling
1. Generate a cloud of particles of αn,ms+K−1, α
n,ms+K−2, . . . , α
n,ms−1
(m = 1, . . . ,M) given (αns+K :t , θ
n) and ys:t targeting
π(θ, αs−1:t | ys:t)2. Stochastically determine the new values of αn
s:t (discarding
αns−1) with the updated importance weight.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Particle rolling MCMC with double block sampling
Figure: (θn, αns−1:t−1) is fixed and shown in the rectangle. Other state
variables in the circle are generated.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Initialization of the rolling MCMC
1. In order to sample from the initial posterior distribution, it is
straightforward to use MCMC methods as in the warm-up
period for the practical filtering in Polson et al. (2008).
2. SMC-based methods are preferred when we need to compute
the marginal likelihood p(y1:s−1).
3. Thus we implement only forward block sampling (with particle
simulation smoother) to sample α1:L+1 and θ where L = t − s.
# If some degeneracy criterion is fulfilled (such as ESS < 0.5N),
resample all the particles by implementing MCMC algorithm.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Estimation of the marginal likelihood
We compute
p(ys:t) =
∫p(ys:t | αs:t , θ)p(αs:t | θ)p(θ)dαs:tdθ
=p(yt | ys−1:t−1)
p(ys−1 | ys:t)p(ys−1:t−1),
using the estimates
p(yt | ys−1:t−1) =N∑
n=1
W ns−1:t−1p(yt | ys−1:t−1, α
nt−K−1, θ
n),
p(ys−1 | ys:t) =N∑
n=1
W ns−1:t p(ys−1 | ys:t , αn
s+K , θn).
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Simple particle rolling MCMCParticle rolling MCMC with double block sampling
Estimation of the marginal likelihood
The initial estimate is given by
p(y1:L+1) = p(y1)L+1∏j=2
p(yj | y1:j−1),
where
p(y1) =N∑
n=1
p(y1 | θn),
p(yj | y1:j−1) =N∑
n=1
W nj−1p(yj | y1:j−1, α
nj−K−1, θ
n),
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Forward block samplingBackward block sampling
3. Theoretical justification of PRMCMC
26 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Forward block samplingBackward block sampling
Forward block sampling
� �Proposition 4.1
The marginal density of (θ, αs−1:t−K−1, αk∗t−K
t−K , . . . , αk∗t
t ) for the
artificial target density
π(θ, αs−1:t−K−1, α1:Mt−K :t , a
1:Mt−K :t−1, k
∗t | ys−1:t)
is π(θ, αs−1:t−K−1, αk∗t−K
t−K , . . . , αk∗t
t | ys−1:t) for the forward
block sampling where aj : ‘parent’ index variable and k∗j : (ran-
dom) index (j = t − K , . . . , t).� �27 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Forward block samplingBackward block sampling
Forward block sampling
� �Proposition 4.2
E [p(yt | ys−1:t−1, αt−K−1, θ)|ys−1:t ]
= E [p(yt | ys−1:t−1, αt−K−1, θ)|ys−1:t , αt−K−1, θ]
= p(yt | ys−1:t−1).� �
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Forward block samplingBackward block sampling
Particle simulation smoother
� �Proposition 4.3 The joint conditional density of (k∗t−K , . . . , k
∗t )
is given by
π(k∗t−K , . . . , k∗t |θ, αs−1:t−K−1, α
1:Mt−K :t , a
1:Mt−K :t−1, ys−1:t)
= π(k∗t |θ, αs−1:t−K−1, α1:Mt−K :t , a
1:Mt−K :t−1, ys−1:t)
×t−K∏
t0=t−1
π(k∗t0 |θ, αs−1:t−K−1, α1:Mt−K :t0 , a
1:Mt−K :t0−1, α
k∗t0+1
t0+1 , . . . ,
αk∗t
t , k∗t0+1:t , ys−1:t),� �as in Whitely et al. (2010) and the discussion of Whitely following
Andrieu et al. (2010).
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Forward block samplingBackward block sampling
Backward block sampling
� �Proposition 4.4
The marginal density of (θ, αk∗s
s , . . . , αk∗s+K−1
s+K−1, αs+K :t) for the
artificial target density
π(θ, α1:Ms−1:s+K−1, αs+K :t , a
1:Ms:s+K−1, ks−1, k
∗s | ys:t)
is π(θ, αk∗s
s , . . . , αk∗s+K−1
s+K−1, αs+K :t | ys−1:t) for the backward block
sampling where aj : ‘parent’ index variable and k∗j : (random)
index (j = t − K , . . . , t).� �30 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Forward block samplingBackward block sampling
Backward block sampling
� �Proposition 4.5
E [p(ys−1 | ys:t , αs+K , θ)−1]
= E [p(ys−1 | ys:t , αs+K , θ)−1 | ys:t , αs+K , θ]
= p(ys−1 | ys:t , θ)−1.� �
31 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
4. Numerical examples
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 1 (Continued)
yt = αt + ϵt , ϵt ∼ N (0, σ2), t = 1, . . . , 2000
αt+1 = µ+ 0.25(αt − µ) + ηt , ηt ∼ N (0, 2σ2), t = 1, . . . , 2000,
α1 = µ+η0√
1− 0.252, η0 ∼ N (0, 2σ2),
where θ = (µ, σ2)′, µ | σ2 ∼ N (0, 10σ2), σ2 ∼ IG(5/2, 0.05/2).▶ The window size = 1000
▶ The number of particles: N = 1000.
▶ The size of block K = 1, 2, 3, 5
▶ For each n = 1, . . . ,N, we generate M particle paths in the
forward and backward block sampling.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 1 (Continued)
Table: # resampling steps in PRMCMC with block sampling ((2))
(1) Fully (2) M (3) SimpleEstimation period K adapted 100 300 500 sampling
1 30 54 37 32 184
Initial estimation 2 10 39 21 15
(t = 1, . . . , 1000) 3 8 38 19 15
5 6 37 18 15
10 8 38 18 14
1 48 104 71 61 1027
Rolling estimation 2 8 74 33 23
(t = 1001, . . . , 2000) 3 7 74 31 22
5 6 72 32 22
10 5 69 31 23
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 1 (Continued)
0.00 0.25 0.50 0.75 1.00
100
200
300
400
500
600
700
800
R1t
0.00 0.25 0.50 0.75 1.00
100
200
300
400
500
600
700
800R2t
Simple Block (K=2)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R1t
R2t
Simple Block (K=2)
Figure: The histograms (left) and scatter plots (right) of R1t and R2t
(left) (t = 1001, . . . , 2000) for the simple sampler (dotted blue) and the
sampler with block sampling with K = 2 and M = 100 (solid red).
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 1 (Continued)
Table: Summary statistics of R1t and R2t for the simple sampling
and the block sampling (M = 100,K = 2) for t = 1001, . . . , 2000
Method Mean Median Std. dev.
R1t Simple 0.862 0.924 0.145
Block 0.975 0.988 0.057
R2t Simple 0.161 0.127 0.139
Block 0.970 0.988 0.068
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 1 (Continued)
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
-1.2
-1.0
-0.8
µ
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
0.018
0.019
0.020
0.021
σ2
Figure: True posterior means and 95% credible intervals (dotted black)
with their estimates (solid red) for µ and σ2.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 1 (Continued)
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000-110
-100
-90
-80
log p(yt−999:t)
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
log p(yt−999:t)−log p(yt−999:t)
Figure: Top: true log marginal likelihoods log p(yt−999:t) (dotted black)
and their estimates (solid red). Bottom: estimation errors
log p(yt−999:t)− log p(yt−999:t) for t = 1001, . . . , 2000.38 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (Realized stochastic volatility model)
Let y1,t : daily stock log return and y2,t : log realized volatility.
y1,t = exp(αt/2)ϵt , ϵt ∼ N (0, 1), t = 1, . . . ,T
y2,t = αt + ξ + ut , ut ∼ N (0, σ2u), t = 1, . . . ,T
αt+1 = µ+ ϕ(αt − µ) + ηt , ηt ∼ N (0, σ2η), t = 1, . . . ,T ,
α1 = µ+1√
1− ϕ2η0, η0 ∼ N (0, σ2
η),
where ϵtutηt
∼ N
0
0
0
,
1 0 ρση0 σ2
u 0
ρση 0 σ2η
.
(The correlation between ϵt and ηt is introduced to express the
leverage effect)39 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (Realized stochastic volatility model)
Data
▶ y1t and y2t : the daily log returns and log realized volatilities
of Standard and Poor’s (S&P) 500 index (obtained from
Oxford-Man Institute Realized Library)
▶ The initial estimation period: from January 1, 2000 (t = 1) to
December 31, 2007 (t = 1988) with L+ 1 = 1988.
▶ The rolling estimation started from sampling from the
posterior distribution using this initial sample period and
moved the window until December 30, 2016 (T = 4248).
▶ K = 10, M = 300 and N = 1000.
▶ We implement 10 MCMC iterations for the comparison below.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (RSV)
2000 2500 3000 3500 4000
0.2
0.4
0.6
0.8
1.0
1.2 R2tR1t
2000 2500 3000 3500 4000
0.2
0.4
0.6
0.8
1.0
1.2
Figure: Traceplot of R1t (left) and R2t (right), (t = 1988, . . . , 4248) in
RSV model.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (RSV)
K Mean Median Std. dev.
R1t 5 0.981 0.995 0.058
10 0.985 0.996 0.053
15 0.986 0.997 0.055
R2t 5 0.983 0.993 0.044
10 0.988 0.994 0.036
15 0.988 0.994 0.035
Table: Summary statistics for R1t and R2t (t = 1988, . . . , 4248) in RSV
model with leverage using K = 5, 10 and 15.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (RSV)
K=5 K=10 K=15
2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200
50
75
100
125
150
175
200
225
K=5 K=10 K=15
Figure: Cumulative computation times (wall time, unit time =1000
seconds) using K = 5, 10 and 15 (t = 1988, . . . , 4248) in RSV model.
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IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (RSV)
Block (1 update) Block (10 updates)
Block (5 updates) MCMC
-0.8 -0.6 -0.4 -0.2 0.0
0.5
1.0µ Block (1 update)
Block (10 updates) Block (5 updates) MCMC
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97
0.5
1.0φ
0.08 0.10 0.12 0.14 0.16
0.5
1.0σ2
η
-0.25 -0.20 -0.15 -0.10 -0.05 0.00
0.5
1.0ξ
0.20 0.22 0.24 0.26 0.28
0.5
1.0σ2
u
-0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.35
0.5
1.0ρ
Figure: The estimated posterior distribution functions of θ for
t = 2261, . . . , 4248. MCMC and Particle rolling MCMC: 1, 5 and 10
iterations.44 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (RSV)
Block (5 updates) Block (10 updates)
2000 2500 3000 3500 4000
-0.50
-0.25
0.00
0.25 µBlock (5 updates) Block (10 updates)
2000 2500 3000 3500 40000.925
0.950
0.975
φ
2000 2500 3000 3500 4000
0.050
0.075
0.100
0.125 σ2η
2000 2500 3000 3500 4000
-0.2
-0.1ξ
2000 2500 3000 3500 4000
0.20
0.25σ2
u
2000 2500 3000 3500 4000
-0.7
-0.5
ρ
Figure: Traceplot of estimated posterior means and 95% credible
intervals for parameters using S&P500 return in RSV model (from
December 31, 2007 to December 30, 2016).45 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Example 2 (RSV)
RSV with leverage RSV without leverage
2000 2500 3000 3500 4000
-4500
-4400
-4300
-4200
-4100
RSV with leverage RSV without leverage
2000 2500 3000 3500 4000
80
100
120
Figure: Left: Estimates of log p(yt−1987:t) (t = 1988, . . . , 4248) for S&P
500 index return in RSV model with leverage (solid red) and in RSV
model without leverage (dotted black). Right: Difference between two
log marginal likelihoods.
46 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Comparison with SMC2 (without rolling window)
Figure: Cumulative computation times (wall time, unit time =1000
seconds) over 10 runs with SMC2 with c = 0.3 (left), 0.5 (middle) and
with the PRMCMC-based algorithm (right).
In the particle MH update steps of SMC2, the parameter θ is generated
using the random walk MH algorithm using the normal proposal
N (θ∗, cΣ) where θ∗ is a current value of the parameter θ, c is a tuning
parameter (0.3 or 0.5), and Σ is the estimated covariance matrix.47 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Comparison with SMC2 (without rolling window)
-0.5 0.0
0.5
1.0
µ (c=0.3)
0.96 0.98
0.5
1.0
φ (c=0.3)
0.03 0.04 0.05
0.5
1.0
σ2η (c=0.3)
-0.3 -0.2 -0.1
0.5
1.0
ξ (c=0.3)
0.175 0.200 0.225
0.5
1.0
σ2u (c=0.3)
-0.7 -0.5
0.5
1.0
ρ (c=0.3)
-0.5 0.0
0.5
1.0
µ (c=0.5)
0.96 0.97 0.98 0.99
0.5
1.0
φ (c=0.5)
0.03 0.04 0.05
0.5
1.0
σ2η (c=0.5)
-0.3 -0.2 -0.1
0.5
1.0
ξ (c=0.5)
0.175 0.200 0.225
0.5
1.0
σ2u (c=0.5)
-0.8 -0.7 -0.6
0.5
1.0
ρ (c=0.5)
Figure: Estimated posterior cumulative distribution functions of θ given
y1:1988 over 10 runs with SMC2 with c = 0.3, 0.5 (red and blue,
respectively) compared to the result of MCMC (gray).48 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Comparison with SMC2 (without rolling window)
-0.6 -0.4 -0.2 0.0 0.2
0.5
1.0µ
0.955 0.96 0.965 0.97 0.975 0.98 0.985
0.5
1.0φ
0.025 0.03 0.035 0.04 0.045 0.05 0.055
0.5
1.0σ2
η
-0.35 -0.30 -0.25 -0.20 -0.15 -0.10
0.5
1.0ξ
0.17 0.18 0.19 0.20 0.21 0.22 0.23
0.5
1.0σ2
u
-0.85 -0.80 -0.75 -0.70 -0.65 -0.60
0.5
1.0ρ
Figure: Estimated posterior cumulative distribution functions of θ given
y1:1988 over 10 runs with the PRMCMC-based method (red) compared to
the result of MCMC (gray).49 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Conclusion
This paper considers the state space model for economic times
series and
▶ proposes the efficient rolling estimation method for both
parameter θ and states αs:t .
▶ gives the theoretical justification of the algorithm by proving
the marginal density of parameters and states.
▶ the algorithm is based on Particle Gibbs
▶ proposes, as a special case, the sequential Bayesian estimation
method (which is alternative to SMC2 based on Particle MH).
50 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Refereneces
▶ Andrieu, C., Doucet, A., & Holenstein, R. (2010). Particle markov
chain monte carlo methods. Journal of the Royal Statistical Society:
Series B (Statistical Methodology), 72(3), 269-342.
▶ Chopin, N., Jacob, P. E., & Papaspiliopoulos, O. (2013). SMC2: an
efficient algorithm for sequential analysis of state space models.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 75(3), 397-426.
▶ Del Moral, P., Doucet, A., & Jasra, A. (2006). Sequential monte
carlo samplers. Journal of the Royal Statistical Society: Series B
(Statistical Methodology), 68(3), 411-436.
▶ Doucet, A., Briers, M., & Senecal, S. (2006). Efficient block
sampling strategies for sequential Monte Carlo methods. Journal of
Computational and Graphical Statistics, 15(3), 693-711.
51 / 52
IntroductionParticle rolling MCMC
Theoretical justification of PRMCMCNumerical examples
Example 1 (Linear Gaussian SSM)Example 2 (Realized stochastic volatility model)
Refereneces
▶ Fulop, A., & Li, J. (2013). Efficient learning via simulation: A
marginalized resample-move approach. Journal of Econometrics,
176(2), 146-161.
▶ Polson, N. G., Stroud, J. R., & Muller, P. (2008). Practical filtering
with sequential parameter learning. Journal of the Royal Statistical
Society: Series B (Statistical Methodology), 70(2), 413-428.
▶ Takahashi, M., Omori, Y., & Watanabe, T. (2009). Estimating
stochastic volatility models using daily returns and realized volatility
simultaneously. Computational Statistics & Data Analysis, 53(6),
2404-2426.
▶ Whiteley, N., Andrieu, C., & Doucet, A. (2010). Efficient Bayesian
inference for switching state-space models using discrete particle
Markov chain Monte Carlo methods. arXiv preprint
arXiv:1011.2437.
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