Mike G. Tsionas, Panayotis G. Michaelides Neglected chaos...

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

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http://users.ntua.gr/pmichael/

https://www.journals.elsevier.com/physica-a-statistical-mechanics-and-its-applications/

https://www.journals.elsevier.com/physica-a-statistical-mechanics-and-its-applications/

http://dx.doi.org/10.1016/j.physa.2017.04.060

http://eprints.lse.ac.uk/80749/

{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)

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


−5 0 5 10 15x 10−3

0

200

400

600

λ

dens

ity

Switzwerland


−5 0 5 10 15 20x 10−3

0

500

1000

1500

λ

dens

ity

Netherlands


−0.01 0 0.01 0.02 0.030

50

100

150

λ

dens

ity

France


−0.01 0 0.01 0.020

200

400

600

800

λ

dens

ity

Germany


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


0 2 4 6 8 100

5

10

15

20

L

norm

aliz

ed m

argi

nal l

ikel

ihoo

d


0 200 400 600 800 10000

5

10

15

20

n

norm

aliz

ed m

argi

nal l

ikel

ihoo

d


0 5 10 15 200

2

4

6

8

embedding dimension, m

norm

aliz

ed m

argi

nal l

ikel

ihoo

d


λ

−0.05 0 0.05 0.1 0.150

20

40

60

λ

dens

ity

US


−0.05 0 0.05 0.1 0.150

20

40

60

λ

dens

ity

UK


−0.04 −0.02 0 0.02 0.040

50

100

150

200

λ

dens

ity

Switzerland


−0.01 0 0.01 0.020

50

100

150

200

λ

dens

ity

Netherlands


−5 0 5 10x 10−3

0

100

200

300

400

λ

dens

ity

France


−0.03 −0.02 −0.01 0 0.010

50

100

150

λ

dens

ity

Germany


λ

−0.04 −0.03 −0.02 −0.01 0 0.010

50

100

λ

dens

ity

US


−0.03 −0.02 −0.01 0 0.010

100

200

300

400

λ

dens

ity

UK


−10 −5 0 5x 10−3

0

500

1000

1500

λ

dens

ity

Switzerland


−0.03 −0.02 −0.01 0 0.01 0.020

20

40

60

80

λ

dens

ity

Netherlands


−6 −4 −2 0 2x 10−3

0

500

1000

1500

2000

λ

dens

ity

France


−0.03 −0.02 −0.01 0 0.010

50

100

150

200

λ

dens

ity

Germany


Mike G. Tsionas, Panayotis G. Michaelides Neglected chaos...

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Transcript of Mike G. Tsionas, Panayotis G. Michaelides Neglected chaos...