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Mike G. Tsionas, Panayotis G. Michaelides
Neglected chaos in international stock markets: Bayesian analysis of the joint return–volatility dynamical system Article (Accepted version) (Refereed)
Original citation: Tsionas, Mike G. and Michaelides, Panayotis G. (2017) Neglected chaos in
international stock markets: Bayesian analysis of the joint return–volatility dynamical system. Physica A: Statistical Mechanics and Its Applications, 482 . pp. 95-107. ISSN 0378-4371 DOI: 10.1016/j.physa.2017.04.060 Reuse of this item is permitted through licensing under the Creative Commons: © 2017 Elsevier B.V CC BY-NC-ND
This version available at: http://eprints.lse.ac.uk/80749/ Available in LSE Research Online: June 2017
LSE has developed LSE Research Online so that users may access research output of the School. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may freely distribute the URL (http://eprints.lse.ac.uk) of the LSE Research Online website.
{xt; t = 1, ..., T }
yt = f(yt−L, yt−2L, ..., yt−mL) + ut, ut|σt ∼ N(0,σ2t ), t = 1, ..., T,
σ2t m L
F :
⎡
⎢
⎢
⎢
⎢
⎣
yt−L
yt−2L
yt−mL
⎤
⎥
⎥
⎥
⎥
⎦
→
⎡
⎢
⎢
⎢
⎢
⎣
yt = f(yt−L, yt−2L, ..., yt−mL) + εt
yt−L
yt−(m−1)L
⎤
⎥
⎥
⎥
⎥
⎦
.
y0 △y0 t △y(y0, t)
λ = limτ→∞
τ−1 ln|△y(y0, τ)|
|∆y0|,
J χ
J t(x) =df t(χ)
dχ
Jt =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
∂f∂yt−L
∂f∂yt−2L
· · · ∂f∂yt−(m−1)L
∂f∂yt−mL
1 0 · · · 0 0
0 1 · · · 0 0
0 0 0 1 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
yt = f(xt) + ut,
xt = [xt−L, ..., xt−mL] ∈ ℜd.
λ = limM→∞
1
2Mlog ν∗,
ν∗ T ′MTM
TM =M−1∏
t=1
JM−1,
M ≤ T J
f M = T 2/3
yt =G∑
g=1
αgϕ(x′tβg) + ut,
ϕ(z) = 11+exp(−z) , z ∈ ℜ βg ∈ ℜL, ∀g = 1, ..., G
α1 ≤ ... ≤ αG.
T ′MTM
log σ2t = µ+ ρ log σ2
t−1 + εt, εt ∼ iidN(
0,ω2)
.
yt =∑G1
g=1 αgϕ(x′tβg +
∑l2l=1 γglσ
2t−lL2
) + ut, ut|σt ∼ N(0,σ2t ), t = 1, ..., T,
log σ2t =
∑Gg=1 δgϕ(x
′tζg +
∑l2l=1 ψglσ2
t−lL2) + εt, εt ∼ N(0,ω2), t = 1, ..., T,
L2
α1 ≤ ... ≤ αG δ1 ≤ ... ≤ δG.
yt =∑G1
g=1 αgϕ(x′tβg +
∑l2l=1 γglσ
2t−lL2
) + σtξt1, ξt1 ∼ N(0, 1), t = 1, ..., T,
log σ2t =
∑Gg=1 δgϕ(x
′tζg +
∑l2l=1 ψglσ2
t−lL2) + ωξt2, ξt2 ∼ N(0, 1), t = 1, ..., T.
σt
ξt
ξ∗ = [ξ∗1 , ξ∗2 ]
′ ∼ iidN(0, 1) ξt1 = ξ∗1 ξt2 = ξ∗2 .
ξ∗2 = 0
λ(ξ∗) = limn→∞
n−1n−1∑
i=0
log ||∂Φ(χi)/∂χi||,
χ =[
yt−1, ....yt−m1L1 ,σ2t−1, ...,σ
2t−m2L2
]
Φ(χ) ||.||
λ(ξ∗) = limM→∞
1
2Mlog ν∗,
ν∗ T ′MTM
TM =M−1∏
t=1
JM−1,
M ≤ T M = T 2/3 Ji = ∂Φ(χi)/∂χi
Ξ∗
ξ∗
λ = minξ∗∈Ξ∗
λ(ξ∗).
λ > 0 λ
ξ∗
S = ξ∗1 ξ∗2S2
λ(ξ∗) ξ∗1 ξ∗2ξ∗1 ξ∗2
L(θ;Y) = (2πω2)−T/2∫
ℜT+
∏Tt=1(2π)
−1/2(σ2t )
−1 exp
{
−[yt−
∑G1g=1 αgϕ(x
′
tβg+∑l2
l=1 γglσ2t−lm2
)]2
2σ2t
}
·,
exp
{
−(logσ2
t−∑G
g=1 δgϕ(x′
tζg+∑l2
l=1 ψglσ2t−lm2
))2
2ω2
}
dσ2,
θ =[
α′,β′,γ′, δ′,ψ′,ω]′⊆ ℜp σ2 = [σ2
1 , ...,σ2T ] Y = [y1, ..., yT ] λ ≥ 0
p(θ) ∝ ω−1.
L, L2 l1, l2
m1,m2 G1, G2 L1 = L2 = L l1 = l2 = l m1 = m2 = m
G1 = G2 = G
p(θ|Y) ∝ L(θ;Y) · p(θ).
L G
M(Y) =
∫
ℜp
L(θ;Y) · p(θ)dθ.
λ
G L n
M(Y) G = 1 L = 1 n
G L m n
λ σt = σ
λ
yT ∼ p(yt|xt), st ∼ p(st|st−1),
st
Yt p(st|Yt){
s(i)t , i = 1, ..., .N} {
w(i)t , i = 1, ..., N
}
∑Ni=1 w
(i)t = 1
p(st+1|Yt) =
∫
p(st+1|st)p(st|Yt)dst ≃N∑
i=1
p(st+1|s(i)t )w(i)
t ,
p(st+1|Yt) ∝ p(yt+1|st+1)p(st+1|Yt) ≃ p(yt+1|st+1)N∑
i=1
p(st+1|s(i)t )w(i)
t .
{
s(i)t , w(i)t , i = 1, . . . , N
}
{
s(i)t+1, w(i)t+1, i = 1, . . . , N
}
θ ∈ Θ ∈ ℜk
p(θ|Yt) ≃N∑
i=1
w(i)t N(θ|aθ(i)t + (1− a)θt, b
2Vt),
θt =∑N
i=1 w(i)t θ(i)t Vt =
∑Ni=1 w
(i)t (θ(i)t − θt)(θ
(i)t − θt)′ a b
δ ∈ (0, 1) a = (1 − b2)1/2 b2 = 1− [(3δ − 1)/2δ]2.
pN (Y |θ)
p(Y |θ)
θ(j) Lj = pN (Y |θ(j))
θc q(θc|θ(j)) Lc = pN (Y |θc) θ(j+1) = θc
A = min
{
1,p(θc)Lc
p(θ(j)Lj
q(θ(j)|θc
q(θc|θ(j))
}
,
{
θ(j+1), Lj+1}
={
θ(j), Lj}
p(Y |θ)
p(st|st−1)
st = g(st−1, ut)
p(yt|st−1) =
∫
p(yt|st)p(st|st−1)dst,
p(ut|st−1; yt) = p(yt|st−1, ut)p(ut|st−1)/p(yt|st−1).
p(yt|st−1) p(ut|st−1; yt)
g(yt|st−1)
p(yt|st−1) p(yt|st−1) = N (E(yt|st−1),V(yt|st−1))
p(ut|yt, st−1) ∝ p(yt|st−1, ut)p(ut) p(ut|yt, st−1)
M umt m = 1, . . . ,M
l(ut) = log [p(yt|st−1, ut)p(ut)]
l(ut) ≃ l(umt ) + 1
2 (ut − umt )′∇2l(um
t )(ut − umt ).
g(ut|xt, st−1) =M∑
m=1
λm(2π)−d/2|Σm|−1/2 exp{
12 (ut − um
t )′Σ−1m (ut − um
t
}
,
Σm = −[
∇2l(umt )]−1
λm ∝ exp {l(umt )}
∑Mm=1 = 1 sit.
st
p(yt|st)
p(sit|sit−1)
p(yt|sit)
t = 0, . . . , T − 1 skt ∼ p(st|Y1:t) πkt k = 1, ..., N
k = 1, . . . , N ωkt|t+1 = g(yt+1|skt )π
kt , π
kt|t+1 = ωk
t|t+1/∑N
i=1 ωit|t+1
k = 1, . . . , N skt ∼∑N
i=1 πit|t+1δ
ist(dst)
k = 1, . . . , N ukt+1 ∼ g(ut+1|skt , yt+1) skt+1 = h(skt ;u
kt+1)
k = 1, . . . , N
ωkt+1 =
p(yt+1|skt+1)p(ukt+1)
g(yt+1|skt )g(ukt+1|s
kt , yt+1)
,πkt+1 =
ωkt+1
∑Ni=1 ω
it+1
.
p(Y1:T ) =T∏
t=1
(
N∑
i=1
ωit−1|t
)(
N−1N∑
i=1
ωit
)
.
p(st|Y1:t, θ)
p(st|Y1:t, θ) ∝ g(yt|xt, θ)
∫
f(st|st−1, θ)p(st−1|y1:t−1, θ)dst−1,
p(st−1|y1:t−1, θ) t−1 t−1{
sit−1, i = 1, . . . , N}
{
wit−1, i = 1, . . . .N
}
p(st−1|y1:t−1, θ)
p(st−1|y1:t−1, θ) ∝N∑
i=1
wit−1f(st|s
it−1, θ).
sit−1 st
ωt =wi
t−1g(yt|st, θ)f(s|sit−1, θ)
ξitq(st|sit−1, yt, θ)
,
q(st|sit−1, yt, θ)
ξitq(st|sit−1, yt, θ) ≃ cg(yt|st, θ)f(st|s
it−1, θ),
c
θc = θ(s) + λz + λ2
2 ∇logp(θ(s)|Y1:T ),
z ∼ N(0, I)
λ
a(θc|θ(s)) = min
{
1,p(Y1:T |θc)q(θ(s)|θc)
p(Y1:T |θ(s))q(θc|θ(s))
}
.
∇ log p(Y1:T |θ) = E [∇ log p(s1:T , Y1:T |θ)|Y1:T , θ] ,
∇ log p(s1:T , Y1:T |θ) = ∇ log p(|s1:T−1, Y1:T−1|θ) +∇ log g(yT |sT , θ) +∇ log f(sT |s|T−1, θ),
s1:T p(s1:T |Y1:T , θ)
p(s1:T |Y1:T , θ) αit−1 = ∇ log p(si1:t−1, Y1:t−1|θ)
κi t−1
i
αit = aκi
t−1 +∇ log g(yt|sit, θ) +∇ log f(sit|s
it−1, θ).
∇ log p(Y1:t|θ)
αit−1 ∼ N(mi
t−1, Vt−1).
n
λ O(n−1/2) O(n−1/6)
αit−1 αt−1
mit−1 = δαi
t−1 + (1− δ)N∑
i=1
wit−1α
it−1,
δ ∈ (0, 1) αit
mit = δmκi
t−1 + (1− δ)N∑
i=1
wit−1m
it−1 +∇ log g(yt|s
it, θ) +∇ log f(sit|s
κi
t−1, θ),
∇ log p(Y1:t|θ) =N∑
i=1
witm
it.
δ = 0.95
p(Y1:T |θ) = p(y1|θ)T∏
t=2
p(yt|Y1:t−1, θ),
p(yt|Y1:t−1, θ) =
∫
g(yt|st)
∫
f(st|st−1, θ)p(st−1|Y1:T−1, θ)dst−1dst,
λ
−0.01 0 0.01 0.02 0.03 0.040
50
100
150
200
λ
dens
ity
US
before crisisafter crisis
−0.01 0 0.01 0.02 0.030
100
200
300
400
λ
dens
ity
UK
before crisisafter crisis
−5 0 5 10 15x 10−3
0
200
400
600
λ
dens
ity
Switzwerland
before crisisafter crisis
−5 0 5 10 15 20x 10−3
0
500
1000
1500
λ
dens
ity
Netherlands
before crisisafter crisis
−0.01 0 0.01 0.02 0.030
50
100
150
λ
dens
ity
France
before crisisafter crisis
−0.01 0 0.01 0.020
200
400
600
800
λ
dens
ity
Germany
before crisisafter crisis
λ(ξ∗)
0.02
0.02
0.02
0.02
0.02
0.02
0.04
0.040.04
0.04
0.06
0.06
0.06
0.06
0.080.08
0.08
0.08
ξ1
ξ 2
U.S, before crisis (returns)
−2 0 2
−3
−2
−1
0
1
2
3 0.020001
0.020001
0.020001
0.040001
0.040
001
0.040001 0.06
0.06
0.060.08
0.08
ξ1
ξ 2
U.S, after crisis (returns)
−2 0 2
−3
−2
−1
0
1
2
3
0.02
0.02
0.02
0.02
0.02
0.02
0.04
0.04
0.06
0.060.08
0.08
ξ1
ξ 2
U.S, before crisis (log volatility)
−2 0 2
−3
−2
−1
0
1
2
3 0.020002
0.020
002
0.020002
0.0400020.0
4000
2
0.040002
0.060001
0.060001
0.0600010.080001
0.080001
ξ1
ξ 2
U.S, after crisis (log volatility)
−2 0 2
−3
−2
−1
0
1
2
3
λ(ξ∗)
0.020001
0.020001
0.020001
0.020001
0.040001
0.040001
0.060001
0.060001
0.08
0.08
ξ1*
ξ 2*
U.K, before crisis (returns)
−2 0 2
−3
−2
−1
0
1
2
3
0.0200010.020001
0.020001
0.0200010.04
0.04
0.06
0.060.08
0.08
ξ1*
ξ 2*
U.K, after crisis (returns)
−2 0 2
−3
−2
−1
0
1
2
3
0.020002
0.020002
0.020002
0.040002
0.040002
0.0400020.060001
0.060001
0.080001
0.080
001
ξ1*
ξ 2*
U.K, before crisis (log volatility)
−2 0 2
−3
−2
−1
0
1
2
3
0.020002
0.02
0002
0.020002
0.0200020.040001
0.040001
0.060001
0.060001
0.08
0.08
ξ1*
ξ 2*
U.K, after crisis (log volatility)
−2 0 2
−3
−2
−1
0
1
2
3
G L n m
0 2 4 6 8 100
5
10
15
20
G
norm
aliz
ed m
argi
nal l
ikel
ihoo
d
before crisisafter crisis
0 2 4 6 8 100
5
10
15
20
L
norm
aliz
ed m
argi
nal l
ikel
ihoo
d
before crisisafter crisis
0 200 400 600 800 10000
5
10
15
20
n
norm
aliz
ed m
argi
nal l
ikel
ihoo
d
before crisisafter crisis
0 5 10 15 200
2
4
6
8
embedding dimension, m
norm
aliz
ed m
argi
nal l
ikel
ihoo
d
before crisisafter crisis
λ
−0.05 0 0.05 0.1 0.150
20
40
60
λ
dens
ity
US
before crisisafter crisis
−0.05 0 0.05 0.1 0.150
20
40
60
λ
dens
ity
UK
before crisisafter crisis
−0.04 −0.02 0 0.02 0.040
50
100
150
200
λ
dens
ity
Switzerland
before crisisafter crisis
−0.01 0 0.01 0.020
50
100
150
200
λ
dens
ity
Netherlands
before crisisafter crisis
−5 0 5 10x 10−3
0
100
200
300
400
λ
dens
ity
France
before crisisafter crisis
−0.03 −0.02 −0.01 0 0.010
50
100
150
λ
dens
ity
Germany
before crisisafter crisis
λ
−0.04 −0.03 −0.02 −0.01 0 0.010
50
100
λ
dens
ity
US
before crisisafter crisis
−0.03 −0.02 −0.01 0 0.010
100
200
300
400
λ
dens
ity
UK
before crisisafter crisis
−10 −5 0 5x 10−3
0
500
1000
1500
λ
dens
ity
Switzerland
before crisisafter crisis
−0.03 −0.02 −0.01 0 0.01 0.020
20
40
60
80
λ
dens
ity
Netherlands
before crisisafter crisis
−6 −4 −2 0 2x 10−3
0
500
1000
1500
2000
λ
dens
ity
France
before crisisafter crisis
−0.03 −0.02 −0.01 0 0.010
50
100
150
200
λ
dens
ity
Germany
before crisisafter crisis