stick-breaking bayesian change-point var model with ......stick-breaking bayesian change-point var...

Stick-Breaking Bayesian Change-Point VAR Model with Stochastic Search Variable Selection

Stick-Breaking Bayesian Change-Point VAR Modelwith Stochastic Search Variable Selection

Stanley Iat-Meng Ko

The Chinese University of Hong Kong

3, August 2013

Stanley Iat-Meng Ko Stick-Breaking Bayesian Change-Point VAR Model with Stochastic Search Variable Selection


Outline

1 Introduction

2 The Bayesian Change-Point VAR Model

3 The MCMC

4 Monte Carlo Simulation

5 Application to Daily Hedge Fund Return

6 Conclusion



Introduction

Bayesian change-point models

Chernoff and Zacks (1964): a constant probability of changeat each point in time.

Smith (1975): single change-point model under differentassumptions of model parameters; e.g. single mean-shift withknown and unknown variance.

Carlin et al. (1992): the structural parameters areindependent of the change points; Markov chain Monte Carlomethod.



Introduction

Bayesian change-point models

Stephens (1994): multiple changes; MCMC.

Chib (1998): multiple changes with change probability todepend on the regimes.

Recent work: Wang and Zivot (2000), Giordani and Kohn(2008), Maheu and Gordon (2008) and Geweke and Yu(2011).



Introduction

Nonparametric Bayesian

Nonparametric Bayesian literature has grown rapidly in recentyears due to the advance of computing technology.

Dirichlet Process Prior: Ferguson (1973).

Blackwell and MacQueen (1973): DP can be represented bythe Polya urn model.

Sethuraman (1994): the constructive stick-breaking definition.



Introduction

Nonparametric Bayesian

Parametric Bayesian:θ ∼ p(θ).

Nonparametric Bayesin (Dirichlet Process):

θ ∼ G (θ),

G ∼ DP(G0, α).

Nonparametric Bayesian (Stick-breaking):

θ ∼ G , G =∞∑i=1

piδθ∗i , θ∗i ∼ G0,

pi = vi

i−1∏j=1

(1− vj), vi ∼ Beta(1, α).



Introduction

This paper...

Nonparametric Bayesian estimation method on change-pointmodel.

Mutivariate model: VAR.

Stochastic search variable selection (SSVS) method,over-parameterization problem, see George and McCulloch(1993).



The Bayesian Change-Point VAR Model


Let y1, y2, . . . , yT be the observed series where yt be a (k × 1)vector (y1t , y2t , . . . , ykt)

′.

yt = c + A1yt−1 + · · ·+ Apyt−p + εt , εt ∼ N(0,Σ), (1)

where εt is the k-dimensional error term, c is a k-dimensionalconstant vector and Ai is a k × k coefficient matrices. The error εtis assumed to be normally distributed with mean zero andcovariance matrix Σ.





Define x′t = (1, y′t−1, . . . , y′t−p). We can rewrite (1) in compact

matrix formY = XΦ + E,

where

Y =

y′1...

y′T

, X =

x′1...

x′T

, Φ =

c′

A′1...

A′p

, E =

ε′1...ε′T

.

Here Y and E are T × k matrices, X is a T × [1 + pk] matrix, andΦ is a [1 + pk]× k matrix.





Suppose there are structural changes in the model such that

yt = cj + Aj1yt−1 + · · ·+ Ajpyt−p + εt , εt ∼ N(0,Σj),

for τj−1 < t ≤ τj ,

where the parameters are subject to m changes at unknown timeperiod 1 < τ1 < · · · < τm < T and remain constant within eachregime. We denote correspondingly the stacked parameter matrixΦj with respect to each regimes j = 1, . . . ,m + 1.





Let s = (s1, s2, . . . , sT ) be the discrete indicating variablestaking values 1, . . . ,m + 1 such that τj−1 < t ≤ τj if and onlyif st = j .

The jth change point occurs at τj if sτj = j and sτj+1 = j + 1.

Chib (1998) proposes hidden Markov model (HMM) approach

P =

p11 p12 0 · · · 00 p22 p23 · · · 0...

. . .. . .

. . ....

......

. . . pmm pm,m+1

0 0 · · · 0 1

,

where pij = Pr(st = j |st−1 = i) is the probability of moving toregime j at time t given that the regime at time t − 1 is i .





y1 y2 y3 y4 y5 yT

s1 s2 s3 s4 s5 sT

θ1 θ2 θk+1

p(θ|γ)

p(γ)

Figure: Hidden Markov process change-point model.





Limitation of Chib (1998)

Model misspecification: number of states has to be specifiedin advance.

Model comparison according to Bayes factors; e.g. onechange point v.s. multiple change points.





Different from Chib (1998), we consider the transition dynamicwith left-to-right restriction, that is, a particular state st−1 = i willeither stay at the current state i or transit to a state j > i .

P =

p11 p12 p13 · · · p1,m+1

0 p22 p23 · · · p2,m+1

0 0 p33 · · · p3,m+1...

......

. . ....

0 0 · · · 0 1

,

where the summation of each row equals 1.

st | st−1 ∼ Multinomial(pst−1,st−1 , pst−1,st−1+1, . . . , pst−1,m+1), t ≥ 2,

s1 ∼ Multinomial(π1, π2, . . . , πm+1).




The Stick-breaking Process Prior

We proposed nonparametric HMM: the stick-breaking process priorto each row of the transition matrix (Paisley and Carin (2009)).

Beal et al. (2002): nonparametric Bayesian method to infiniteHMM.

Teh et al. (2006): extend Beal et al. (2002) to hierarchicalDirichlet process HMM (HDP-HMM).

Fox et al. (2011): sicky HDP-HMM.

Song (2011) proposes similar univariate change-point modeluses Fox et al. (2011).

Paisley and Carin (2009): stick-breaking HMM (SB-HMM),fully conjugate prior for an infinite-state HMM, facilitate largedata set.





For an infinite probability profile p = (p1, p2, . . . ), recallstick-breaking prior:

pi = vi

i−1∏j=1

(1− vj), vi ∼ Beta(1, α).

Now, hierarchical stick-breaking (Paisley and Carin (2009)):

pi = vi

i−1∏j=1

(1− vj) ,

vi | αi ∼ Beta(1, αi ),

αii .i .d .∼ Gamma(c, d).





When p = (p1, . . . , pd , pd+1) and pd+1 = 1−∑d

i=1 pi , p reducesto a draw from the generalized Dirichlet distribution (GDD), i.e.the truncated stick-breaking prior. See Connor and Mosimann(1969), Wong (1998) and Paisley and Carin (2009).





Our model is thus

Gi | pij ,θj =∞∑j=i

pijδθj,

pij | Vij = Vij

j−1∏k=1

(1− Vik) ,

Vij | αij ∼ Beta(1, αij),

θji .i .d .∼ G0,

αiji .i .d .∼ Gamma(c, d),

where i ≤ j and G0 is the base parameter distribution.




Stochastic Search Variable Selection (SSVS)

Recall that given regime j , i.e. t ∈ (τj−1, τj ],

yt = cj + Aj1yt−1 + · · ·+ Ajpyt−p + εt , εt ∼ N(0,Σj),

for τj−1 < t ≤ τj ,

Denote Φj = [c′j ,A′j1, . . . ,A

′jp]′ and

φj = vec(Φj) = (φj1, φj2, . . . , φ

jn)′ where n = k(1 + pk).

We assume θj ≡ (φj ,Σj) ∼ G0.




Stochastic Search Variable Selection (SSVS): Priors onφj = vec(Φj)

We assume that r elements of φj are subject to restriction and therest n − r elements are always included in the model. The prior forthe elements that are included in the model is

φincludej ∼ N(0, σ2I ), p(σ2) ∝ 1/σ2, (2)

where I is the (n − r)× (n − r) identity matrix and thehyperparameter σ2 has the noninformative prior.





Denote the restricted elements as φrj . Let γ j = (γj1, γ

j2, . . . , γ

jr )′ be

a vector of 0− 1 variables and let D = diag(h1, . . . , hr ) where

hi =

κ0 if γji = 0,

κ1 if γji = 1,and

κ20 ∼ Inv-Gamma(a0, b0),κ2

1 ∼ Inv-Gamma(a1, b1),

Also, γ j are assumed to be i .i .d . Bernoulli(qi ) where qi hasuniform prior on [0, 1]. Hence the prior on φr

j is

φrj | γ j ∼ N(0,D2), γ j ∼

r∏i=1

qγjii (1− qi )

1−γji , qi ∼ Uniform(0, 1),

κ20 ∼ Inv-Gamma(a0, b0), κ2

1 ∼ Inv-Gamma(a1, b1).(3)





Combining the priors for φincludej and φr

j , we denote

φj | γ j ∼ N(0,Ω(γ)j ), (4)

where the corresponding elements of Ω(γ)j are given by (2) and (3).




Stochastic Search Variable Selection (SSVS): Priors onΣj

We consider the Wishart prior such that

Σ−1j ∼Wishart(v ,Λ−1

0 ), p(Λ0) ∝ |Λ0|−(k+2)/2. (5)




The posteriors of φj and Σj

Note that the likelihood function of (Φj ,Σj) given regime (τj−1, τj ]is

f (Yj | Φj ,Σj ; Xj) ∝ |Σj |−(τj−τj−1)/2etr

−1

2(Yj − XjΦj)Σ−1

j (Yj − XjΦj)′,

(6)

where etr(A) ≡ exp(Trace(A)), Yj = (yτj−1+1, . . . , yτj )′ and

Xj = (xτj−1+1, . . . , xτj )′. Denote φj = vec(Φj), the likelihood (6)

can be rewritten as

f (Yj | φj ,Σj ; Xj )

∝ |Σj |−(τj−τj−1)/2 exp

−

1

2(φj − φj )

′[Σj ⊗ (X′jXj )−1]−1(φj − φj )−

1

2tr[SjΣ

−1j ]

φj = vec(Φj ); Φj = (X′jXj )

−1X′jYj ; Sj = (Yj − Xj Φj )′(Yj − Xj Φj ).

(7)




The posteriors of φj and Σj

Hence we have the conditional posterior of φj

φj | γ j ,Σj ,Ω(γ)j ; Yj ,Xj ∼ N(µj ,∆j),

µj =[Σ−1

j ⊗ (X′jXj) + (Ω(γ)j )−1

]−1 [(Σ−1

j ⊗ (X′jXj))· φj

],

∆j =[Σ−1

j ⊗ (X′jXj) + (Ω(γ)j )−1

]−1,

(8)and the conditional posterior of Σ−1

j

Σ−1j | φj ,γ j ,Λ0; Yj ,Xj ∼Wishart(vj ,Λ

−1j ),

Λj = Λ0 + (Yj − XjΦj)′(Yj − XjΦj),

vj = v + (τj − τj−1).

(9)




The posteriors of γji

Then under (3) and independence between elements of φrj , we

have the posterior of γji for i = 1, . . . , r

γji | φrj , κ0, κ1, qi ∼ Bernoulli(ui2/ (ui1 + ui2)) ,

ui1 =1

κ0exp

(−φrj (i)2

2κ20

)(1− qi ), ui2 =

1

κ1exp

(−φrj (i)2

2κ21

)qi ,

(10)where φr

j (i) denotes the i-th element of φrj .




The posteriors of hyperparameters

Given m change points, i.e. m + 1 regimes, the hyperparametersσ2, κ2

0, κ21, qi and Λ0 have the following posteriors

σ2∣∣ φinclude

j m+1j=1 ∼ Inv-χ2

m + 1,1

m + 1

m+1∑j=1

φincludej

′ · φincludej

;

κ20

∣∣ φrj

m+1j=1 , γ jm+1

j=1 ∼ Inv-Gamma

a0 +m + 1

2, b0 +

1

2

m+1∑j=1

φrj[γ

j

=0]′φr

j[γj

=0]

;

κ21

∣∣ φrj

m+1j=1 , γ jm+1

j=1 ∼ Inv-Gamma

a1 +m + 1

2, b1 +

1

2

m+1∑j=1

φrj[γ

j

=1]′φr

j[γj

=1]

;

qi∣∣ γ jm+1

j=1 ∼ Beta

m+1∑j=1

γji + 1,m+1∑j=1

(1− γji ) + 1

, i = 1, . . . , r ;

Λ0

∣∣ Σjm+1j=1 ∼Wishart

(m + 1)v ,

m+1∑j=1

Σ−1j

−1 ;

(11)

where φrj[γ j=0] and φr

j[γ j=1] are the subvectors of φrj that contain

the elements correspond to γ j = 0 and γ j = 1 respectively.



The MCMC

MCMC Inference for SB Change-Point Model: A GibbsSampler

Denote D as the total number of used states (regimes) followingan iteration. Note that the number of change points m = D − 1.To sample from the full posterior of our model, we sample eachparameter from its full conditional posterior as follows:

Step 1. Sample αij from its Gamma posterior for i ≤ j

αij

∣∣ Vij , c , d ∼ Gamma(c + 1, d − ln(1− Vij)). (12)

Draw new values of the corresponding α from the priorGamma(c , d) if new states are generated from the previousiteration.



The MCMC


Step 2. Construct the transition probabilities of P from thestick-breaking conditional posteriors

Vij

∣∣ s, αijDj=i ∼ Beta

1 +

T−1∑t=1

1(st = i , st+1 = j), αij +D∑

k=j+1

T−1∑t=1

1(st = i , st+1 = k)

,

pii = Vii , pij = Vij

j−1∏k=1

(1− Vik ),

(13)

for i = 1, . . . ,D and i ≤ j . Set pi ,D+1 = 1−∑D

j=i pij . And we setthe new D + 1 row of P as (0, . . . , 0, 1).



The MCMC


Step 3. Sample the innovation parameters, απj , for π from theGamma posteriors

απj∣∣ V π

j , c , d ∼ Gamma(c + 1, d − ln(1− V πj )). (14)

Step 4. Sample π from its stick-breaking conditional posterior

V πj

∣∣ s1, απj Dj=1 ∼ Beta(1 + 1(s1 = j), απj

),

π1 = V π1 , πj = V π

j

j−1∏k=1

(1− V πk ).

(15)

Set πD+1 = 1−∑D

j=1 πj .



The MCMC


Step 5. Conditional on s, sample the parameters θj from (8) and(9) for j = 1, . . . ,D. Update the hyperparameters σ2, κ2

0, κ21, qi

and Λ0 using (11). Draw new parameter θD+1 from the priors (4)and (5).



The MCMC


Step 6. Sample the indicating variables s from the conditionalposteriors

p(s1

∣∣ s2, π,P, φjD+1j=1 , ΣjD+1

j=1 ; y1, x1

)∝ p(s1)f (y1|Φs

1

,Σs1

; x1)p(s2|s1),

p(st∣∣ st−1, st+1,P, φD+1

j=1 , ΣjD+1j=1 ; yt , xt

)∝ p(st |st−1)f (yt |Φs

t

,Σst

; xt)p(st+1|st),

2 ≤ t ≤ T − 1,

p(sT∣∣ sT−1,P, φD+1

j=1 , ΣjD+1j=1 ; yT , xT

)∝ f (yT |Φs

T

,ΣsT

; xT )p(sT |sT−1).

(16)

Step 7. Set D equals to the number of unique states drawn in theprevious step and prune away any unused states. The updatednumber of change points is m = D − 1.



Monte Carlo Simulation


We simulate 1000 observations from a hypothetical DGP thatsubjects to abrupt changes.

y1t = ϕj0 + ϕj1y1,t−1 + ϕj2y2,t−1 + ϕj3y1,t−2 + ϕj4y2,t−1 + ε1t ,

y2t = ψj0 + ψj1y1,t−1 + ψj2y2,t−1 + ψj3y1,t−2 + ψj4y2,t−1 + ε2t ,

where (ε1t , ε2t)′ | st = j ∼ N(0,Σj ).

φincludej = (ϕj0, ϕj1, ψj0, ψj2)′.

φrj = (ϕj2, ϕj3, ϕj4, ψj1, ψj3, ψj4)′.

Four structural changes or, equivalently, five states(j = 1, 2, 3, 4, 5).

Notes that the DGP has 65 unknown parameters to estimate.





Table: The structural change DGP

φincludej φr

j Σj

State ϕj0 ϕj1 ψj0 ψj2 ϕj2 ϕj3 ϕj4 ψj1 ψj3 ψj4 σ2j1 σ2

j2 σj12

st = 1, t ∈ [1, 200] 1.40 0.30 0.60 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.50 0.25 0.00st = 2, t ∈ [201, 350] 0.20 0.40 0.30 0.50 0.00 -0.50 0.00 0.00 0.00 0.00 1.00 0.50 0.70st = 3, t ∈ [351, 550] 0.80 0.50 0.50 0.20 0.00 -0.30 -0.10 0.00 0.00 -0.30 0.80 0.50 0.40st = 4, t ∈ [551, 850] 0.80 0.30 1.00 0.50 0.00 -0.20 -0.10 0.20 0.00 -0.10 0.50 0.30 0.40st = 5, t ∈ [851, 1000] 1.00 0.60 1.50 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.30 0.25 0.00





Figure: Data simulated from the structural break DGP.

0 100 200 300 400 500 600 700 800 900 1000−3

−2

−1

0

1

2

3

4

5

Time

y 1

0 100 200 300 400 500 600 700 800 900 1000−2

−1

0

1

2

3

4

Time

y 2





Figure: Posterior mean, 5-th and 95-th quantiles of the includedparameters φinclude

j compared to the true values.

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

Time

ϕ0

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time

ϕ1

0 100 200 300 400 500 600 700 800 900 10000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time

ψ0

0 100 200 300 400 500 600 700 800 900 1000−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time

ψ2





Figure: Posterior mean, 5-th and 95-th quantiles of the SSVS parametersφr

j compared to the true values.

0 200 400 600 800 1000−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Time

ϕ2

0 200 400 600 800 1000−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Time

ϕ3

0 200 400 600 800 1000−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time

ϕ4

0 200 400 600 800 1000−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time

ψ1

0 200 400 600 800 1000−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time

ψ3

0 200 400 600 800 1000−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time

ψ4





Figure: Posterior mean, 5% and 95% quantiles of the covariance matrixΣj and posterior mean of st compared to the true values.

0 100 200 300 400 500 600 700 800 900 10000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Time

σ2 1

0 100 200 300 400 500 600 700 800 900 1000

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Time

σ2 2

0 100 200 300 400 500 600 700 800 900 1000−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time

σ12

100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time

s t



Application to Daily Hedge Fund Return


We study structural changes on hedge fund data covers the periodApril 1, 2003 to June 29, 2012 with 2332 observations.

Two aggregate hedge fund market indices, HFRX EqualWeighted Strategies Index (HFREW) and HFRX GlobalHedge Fund Index (HFRGL);

Four strategy indices: HFRX Equity Hedge (EH), HFRXEvent Driven (ED), HFRX Macro and CTA (MACRO), andHFRX Relative Value (RV).





Equity factors: excess returns on the S&P 500 index(SNPMNR), the spread between small-cap and large capstock returns (SCMLC); the spread between value and growthstocks (HML).

Fixed income factors: the change in the 10-year treasuryyields (BD10RET) and the difference between the change inthe Moody’s Baa bonds yields and the change in the 10-yearT-bonds yields (BAAMTSY).





Table: Big events during the period April 1, 2003 - June 29, 2012.

Date EventsSep 15, 2008 Bankruptcy of Lehman BrothersNov 25, 2008 Quantitative Easing 1, from Dec 2008 to Mar 2010Nov 03, 2010 Quantitative Easing 2, from Nov 2010 to Jun 2011Sep 21, 2011 Operation TwistAug 02, 2011 United States debt-ceiling crisisAug 06, 2011 Standard & Pool downgraded America’s credit rating from AAA to

AA+ for the first time





We propose the following VAR model for the hedge fund dataYt

SNPMNRtSCMLCt

HMLtBD10RETtBAAMTSYt

=

[ϕj0][cj1][cj2][cj3][cj4][cj5]

+

[ϕj1] [ϕj2] [ϕj3] [ϕj4] [ϕj5] [ϕj6]ψj11 [ψj12] ψj13 ψj14 ψj15 ψj16ψj21 ψj22 [ψj23] ψj24 ψj25 ψj26ψj31 ψj32 ψj33 [ψj34] ψj35 ψj36ψj41 ψj42 ψj43 ψj44 [ψj45] ψj46ψj51 ψj52 ψj53 ψj54 ψj55 [ψj56]

·

Yt−1

SNPMNRt−1SCMLCt−1

HMLt−1BD10RETt−1BAAMTSYt−1

+

ε1tη1tη2tη3tη4tη5t

,where τj−1 < t ≤ τj , Yt takes HFREW, HFRGL, EH, ED, MACROand RV in each estimation, and (ε1t , η1t , η2t , η3t , η4t , η5t)′ ∼ N(0,Σj ). Allthe parameters in bracket [] are always included in the model.Others are to be selected by SSVS. Thus the five factors areassumed to follow the AR(1) dynamic. Note that we have 63unknown parameters in each regime.




Application to Daily Hedge Fund Return: HFREW

Figure: Posterior distribution of the break-date(s) for the HFREWindeices.

Apr 03 Oct 03 May 04 Dec 04 Jul 05 Feb 06 Sep 06 Apr 07 Nov 07 Jun 08 Jan 09 Aug 09 Mar 10 Oct 10 May 11 Nov 11 Jun 12−2

−1

0

1

2

Time

HF

RE

W

Apr 03 Oct 03 May 04 Dec 04 Jul 05 Feb 06 Sep 06 Apr 07 Nov 07 Jun 08 Jan 09 Aug 09 Mar 10 Oct 10 May 11 Nov 11 Jun 120

0.5

1

Time

Pro

ba

bili

ty


1000

1500

2000

Time

SP

50

0




Application to Daily Hedge Fund Return: HFRGL

Figure: Posterior distribution of the break-date(s) for the HFRGLindeices.


−1

0

1

2

Time

HF

RG

L


0.2

0.4

0.6

0.8

Time

Pro

babili

ty


1000

1500

2000

Time

SP

500




Application to Daily Hedge Fund Return: Equity Hedge

Figure: Posterior distribution of the break-date(s) for the equity hedgeindeices.


−2

0

2

4

Time

EH


0.5

1

Time

Pro

ba

bili

ty


1000

1500

2000

Time

SP

50

0




Application to Daily Hedge Fund Return: Event Driven

Figure: Posterior distribution of the break-date(s) for the ED indeices.


−2

0

2

4

Time

ED


0.5

1

Time

Pro

babili

ty


1000

1500

2000

Time

SP

500




Application to Daily Hedge Fund Return: Macro

Figure: Posterior distribution of the break-date(s) for the macro indeices.


−2

0

2

4

Time

MA

CR

O


0.5

1

Time

Pro

ba

bili

ty


1000

1500

2000

Time

SP

50

0




Application to Daily Hedge Fund Return: Relative Value

Figure: Posterior distribution of the break-date(s) for the relative valueindeices.


−2

0

2

4

Time

RV


0.5

1

Time

Pro

ba

bili

ty


1000

1500

2000

Time

SP

50

0



Conclusion

Conclusion

New algorithm to estimate change-point model in multivariatedimension system (VAR).

Nonparametric Bayesian stick-breaking HMM.

SSVS for the VAR to deal with possible over-parameterization.

Efficient Gibbs sampler.

Monte Carlo simulation: accurate change-point locationestimates and good inference of unknown parameters.

Detects three change-points in all hedge fund return series.

All the hedge fund returns have the same second and thirdchange-points which coincide with the turning point of themarket and the day of the U.S. debt-ceiling crisis.



Conclusion

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stick-breaking bayesian change-point var model with ......stick-breaking bayesian change-point var...

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