A Tractable State-Space Model for Symmetric Positive...
Transcript of A Tractable State-Space Model for Symmetric Positive...
![Page 1: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/1.jpg)
A Tractable State-Space Model for SymmetricPositive-Definite Matrices
Jesse Windle1
Carlos Carvalho2
August 9, 2015
1 Hi Fidelity Genetics2 The University of Texas at Austin
1
![Page 2: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/2.jpg)
The Basic Story
1. The Bayesian analysis of covariance-matrix-valued state-spacemodels can be difficult.
2. The subsequent model is computationally tractable, but itcomes at a cost.
2
![Page 3: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/3.jpg)
State-Space Models
Latent States: xt−1 xt xt+1
Observations: yt−1 yt yt+1
System’s parameters, θ
3
![Page 4: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/4.jpg)
State-Space Models
Latent States: xt−1 xt xt+1
Observations: yt−1 yt yt+1
System’s parameters, θ
[ T∏i=1
p(yt |xt , θ)][ T∏
i=2
p(xt |xt−1, θ)]p(x1|θ)
3
![Page 5: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/5.jpg)
State-Space Models
Latent States: xt−1 xt xt+1
Observations: yt−1 yt yt+1
System’s parameters, θ
Filter: p(xt |y1:t).
3
![Page 6: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/6.jpg)
State-Space Models
Latent States: xt−1 xt xt+1
Observations: yt−1 yt yt+1
System’s parameters, θ
Smooth: p(x1:T |y1:T ).
3
![Page 7: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/7.jpg)
State-Space Models
Latent States: xt−1 xt xt+1
Observations: yt−1 yt yt+1
System’s parameters, θ
Infer: p(θ|y1:T ).
3
![Page 8: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/8.jpg)
State-Space Models in Finance
Rt ∼ N(0,Vt),
Vt ∼ P(Vt−1).
4
![Page 9: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/9.jpg)
State-Space Models in Finance
Rt ∼ N(0,Vt),
Vt ∼ P(Vt−1).
4
![Page 10: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/10.jpg)
State-Space Models in Finance
Rt ∼ N(0,Vt),
Vt ∼ P(Vt−1).
5
![Page 11: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/11.jpg)
State-Space Models in Finance
Rt,i ∼ N(0,Vt/k), i = 1, . . . , k,
Vt ∼ P(Vt−1).
5
![Page 12: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/12.jpg)
State-Space Models in Finance
Yt ∼Wm(k ,Vt/k), Yt =k∑
i=1
Rt,iR′t,i
Vt ∼ P(Vt−1).
5
![Page 13: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/13.jpg)
State-Space Models in Finance
Yt ∼Wm(k ,X−1t /k), Yt =
k∑i=1
Rt,iR′t,i
Xt ∼ P(Xt−1).
5
![Page 14: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/14.jpg)
Our hands are now tied
[ T∏i=1
p(Yt |Xt , θ)]
︸ ︷︷ ︸Wishart
[ T∏i=2
p(Xt |Xt−1, θ)]p(X1|θ)
Problem: Moving around the state-space.
xt = Lower(Xt) ∼ GP ?
[d1 cc d2
]
6
![Page 15: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/15.jpg)
Pick a new set of coordinates?
Matrix logarithm [Bauer and Vorkink, 2011]:
Xt = Ut exp(Dt)U′t ,
logXt = UtDtU′t ,
Zt = Lower(logXt).
[ T∏i=1
p(Yt |Xt , θ)]
︸ ︷︷ ︸Wishart
[ T∏i=1
p(Xt |Xt−1, θ)]p(X0|θ)
p(X1:T |Y1:T , θ)→ Gibbs + Metropolis-Hastings.
p(Xt |X−t ,Y1:T , θ).
7
![Page 16: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/16.jpg)
Pick a new set of coordinates?
LDL decomposition [Chiriac and Voev, 2010, Loddo et al., 2011]:
Xt = Lt exp(Dt)L′t ,
Zt = ( StrictLower(Lt),Diag(Dt) ).
[ T∏i=1
p(Yt |Xt , θ)]
︸ ︷︷ ︸Wishart
[ T∏i=1
p(Xt |Xt−1, θ)]p(X0|θ)
p(X1:T |Y1:T , θ)→ Gibbs + Metropolis-Hastings.
p(Xt |X−t ,Y1:T , θ).
7
![Page 17: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/17.jpg)
Use the original coordinates?
Xt = StΨtS′t , StS
′t = f (Xt−1)
Source f (Xt−1) Ψt p(X1:T |Y1:T , θ)
(1) λ−1Xt−1 Wm(ρ, Im/ρ)
(2) λ−1Xt−1 βm
(n2 ,
12
)(1) Philipov and Glickman [2006], Asai and McAleer [2009](2) Uhlig [1997], Rank m=1 Case Only
Other relevant work: Gourieroux et al. [2009], Fox and West [2011];Prado and West [2010], Jin and Maheu [2013], Shirota et al. [2015].GARCH literature... Bauwens et al. [2006].
8
![Page 18: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/18.jpg)
Use the original coordinates?
Xt = StΨtS′t , StS
′t = f (Xt−1)
Source f (Xt−1) Ψt p(X1:T |Y1:T , θ)
(1) λ−1Xt−1 Wm(ρ, Im/ρ) Gibbs + MH
(2) λ−1Xt−1 βm
(n2 ,
12
)p(Xt |Y1:t , θ)
(1) Philipov and Glickman [2006], Asai and McAleer [2009](2) Uhlig [1997], Rank m=1 Case Only
Other relevant work: Gourieroux et al. [2009], Fox and West [2011];Prado and West [2010], Jin and Maheu [2013], Shirota et al. [2015].GARCH literature... Bauwens et al. [2006].
8
![Page 19: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/19.jpg)
Use the original coordinates?
Xt = StΨtS′t , StS
′t = f (Xt−1)
Source f (Xt−1) Ψt p(X1:T |Y1:T , θ)
(1) λ−1Xt−1 Wm(ρ, Im/ρ) Gibbs + MH
(2) λ−1Xt−1 βm
(n2 ,
12
)p(Xt |Y1:t , θ)
(1) Philipov and Glickman [2006], Asai and McAleer [2009](2) Uhlig [1997], Rank m=1 Case Only
Other relevant work: Gourieroux et al. [2009], Fox and West [2011];Prado and West [2010], Jin and Maheu [2013], Shirota et al. [2015].GARCH literature... Bauwens et al. [2006].
8
![Page 20: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/20.jpg)
Uhlig Extension
Xt = StΨtS′t , StS
′t = λ−1 Xt−1
Ψt ∼ βm(n
2,k
2
), k ∈ N;
Easy to compute:
I p(Xt |Y1:t , θ) Wishart
I p(Xt |Y1:t ,Xt+1, θ) Shifted Wishart
I p(X1:T |Y1:T , θ)
I p(Yt |Yt−1, θ) Multivariate compound gamma=⇒ p(Y1:T |θ).
Only need to record:
Σt = λΣt−1 + Yt .
9
![Page 21: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/21.jpg)
Uhlig Extension
Xt = StΨtS′t , StS
′t = λ−1 Xt−1
Ψt ∼ βm(n
2,k
2
), k ∈ N;
Easy to compute:
I p(Xt |Y1:t , θ) Wishart
I p(Xt |Y1:t ,Xt+1, θ) Shifted Wishart
I p(X1:T |Y1:T , θ)
I p(Yt |Yt−1, θ) Multivariate compound gamma=⇒ p(Y1:T |θ).
Only need to record:
Σt = λΣt−1 + Yt .
9
![Page 22: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/22.jpg)
Uhlig Extension
Xt = StΨtS′t , StS
′t = λ−1 Xt−1
Ψt ∼ βm(n
2,k
2
), k ∈ N;
Easy to compute:
I p(Xt |Y1:t , θ) Wishart
I p(Xt |Y1:t ,Xt+1, θ) Shifted Wishart
I p(X1:T |Y1:T , θ)
I p(Yt |Yt−1, θ) Multivariate compound gamma=⇒ p(Y1:T |θ).
Only need to record:
Σt = λΣt−1 + Yt .
9
![Page 23: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/23.jpg)
How does this work? Key Transformation
Muirhead [1982],Uhlig [1997],Dıaz-Garcıa andJaimez [1997]:
Wishart Mult. BetaXt−1 Ψt
⊥
Wishart Wishart
λXt Zt⊥
g
Density of rank-deficient Wishart
π−(mk−k2)/2|L|(k−m−1)/2
2mk/2Γk
(k2
)|V |k/2
exp(
tr − 1
2V−1Y
)
(dY ) = 2−kk∏
i=1
lm−ki
k∏i<j
(li − lj)(H ′1d H1) ∧k∧
i=1
dli .
(Introductory text: Mikusinski and Taylor [2002])
10
![Page 24: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/24.jpg)
How does this work? Key Transformation
Muirhead [1982],Uhlig [1997],Dıaz-Garcıa andJaimez [1997]:
Wishart Mult. BetaXt−1 Ψt
⊥
Wishart Wishart
λXt Zt⊥
g
Density of rank-deficient Wishart
π−(mk−k2)/2|L|(k−m−1)/2
2mk/2Γk
(k2
)|V |k/2
exp(
tr − 1
2V−1Y
)
(dY ) = 2−kk∏
i=1
lm−ki
k∏i<j
(li − lj)(H ′1d H1) ∧k∧
i=1
dli .
(Introductory text: Mikusinski and Taylor [2002])
10
![Page 25: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/25.jpg)
Example
I 30 stocks from DJIA as of Oct. 2010.
I Feb. 27, 2007 to Oct. 29, 2010.
I Yt : Realized kernels (e.g. Barndorff-Nielsen et al. [2009])
11
![Page 26: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/26.jpg)
Prediction Exercise
• Predictive portfolios:
π∗t = argminπ′1=1
π′Vtπ
Vt = E[Vt |Y1:t−1].
• Performance:
portfolio variation = var(π∗t′rt).
root meanvariation
FSV Extension 0.00977Uhlig Extension 0.00936.
12
![Page 27: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/27.jpg)
Prediction Exercise
13
![Page 28: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/28.jpg)
Drawbacks
Discussion:
I Roberto Casarin
I Catherine Scipione Forbes
I Enrique ter Horst, German Molina
14
![Page 29: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/29.jpg)
Drawback: Xt is not stationary (realism)
15
![Page 30: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/30.jpg)
Drawback: Xt is not stationary (predictions)
16
![Page 31: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/31.jpg)
Drawback: Xt is not stationary (predictions)
16
![Page 32: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/32.jpg)
Drawback: Xt is not stationary (predictions)
Predictions of future variance:
Mh = E[X−1t+h | X
−1t ], h > 0.
Konno [1988]:
Mh =n + k −m − 1
n −m − 1λ Mh−1
where M0 = X−1t .
17
![Page 33: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/33.jpg)
What does this work at all?
18
![Page 34: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/34.jpg)
What does this work at all?
18
![Page 35: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/35.jpg)
Volatility models: think in terms of forecasts
I Uhlig extension :
E[X−1t+1|Y1:t , θ] =
λk
n −m − 1
( t−1∑i=0
λiYt−i + λtΣ0
).
n + k −m − 1
n −m − 1λ = 1 =⇒
E[X−1t+1|Y1:t , θ] = (1− λ)
( t−1∑i=0
λiYt−i + λtΣ0
).
19
![Page 36: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/36.jpg)
Volatility models: think in terms of forecasts
I Uhlig extension (EWMA):
E[X−1t+1|Y1:t , θ] =
λk
n −m − 1
( t−1∑i=0
λiYt−i + λtΣ0
).
n + k −m − 1
n −m − 1λ = 1 =⇒
E[X−1t+1|Y1:t , θ] = (1− λ)
( t−1∑i=0
λiYt−i + λtΣ0
).
19
![Page 37: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/37.jpg)
Volatility models: think in terms of forecasts (continued)
I “GARCH” (EWMA-MR):
E[X−1t+1|Y1:t , θ] ' (1− γ)C + γ (1− λ)
( t∑i=0
λiYt−i
).
I Univariate stochastic volatility:
EWMA-MR of the log squared returns
I Leverage effects:
asymmetrically weight past observationsdepending on market movements.
20
![Page 38: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/38.jpg)
Estimating θ = (n, k , λ,Σ0)
The model:
Yt = Wm(k , (kXt)−1),
Xt = StΨtS′t , StS
′t = λ−1 Xt−1,
Ψt ∼ βm(n
2,k
2
), k ∈ N.
Conjugate prior:X1 ∼Wm(n, (λ k Σ0)−1).
Y−τ , . . . ,Y0,Y1, . . . ,YT .
Σt =t−1∑i=0
λiYt−i + λtΣ0 → Σ0(λ) =−τ∑i=0
λiY−i + 0.
21
![Page 39: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/39.jpg)
Estimating θ = (n, k , λ,Σ0)
The model:
Yt = Wm(k , (kXt)−1),
Xt = StΨtS′t , StS
′t = λ−1 Xt−1,
Ψt ∼ βm(n
2,k
2
), k ∈ N.
Conjugate prior:X1 ∼Wm(n, (λ k Σ0)−1).
Y−τ , . . . ,Y0,Y1, . . . ,YT .
Σt =t−1∑i=0
λiYt−i + λtΣ0 → Σ0(λ) =−τ∑i=0
λiY−i + 0.
21
![Page 40: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/40.jpg)
Recapitulation
1. Given our specific observation distribution, it isn’t easy toconstruct tractable matrix-valued state-space models.
2. Uhlig essentially provides a way to do this, but it comes witha cost.
Slides with references:
http://www.jessewindle.com/
22
![Page 42: A Tractable State-Space Model for Symmetric Positive ...jwindle.github.io/doc/JSM-presentation.pdf · A Tractable State-Space Model for Symmetric Positive-De nite Matrices Jesse Windle1](https://reader034.fdocuments.in/reader034/viewer/2022051511/601c9ed910c98b28bd221dca/html5/thumbnails/42.jpg)
M. Asai and M. McAleer. The structure of dynamic correlations in multivariate stochastic volatility models.Journal of Econometrics, 150:182–192, 2009.
O. E. Barndorff-Nielsen, P. R. Hansen, A. Lunde, and N. Shephard. Realized kernels in practice: Trades andquotes. Econometrics Journal, 12(3):C1–C32, 2009.
G. H. Bauer and K. Vorkink. Forecasting multivariate realized stock market volatility. Journal of Econometrics,160:93–101, 2011.
L. Bauwens, S. Laurent, and J. V. K. Rombouts. Multivariate GARCH models: a survey. Journal of AppliedEconometrics, 21:7109, 2006.
R. Chiriac and V. Voev. Modelling and forecasting multivariate realized volatility. Journal of Applied Econometrics,26:922–947, 2010.
J. A. Dıaz-Garcıa and R. G. Jaimez. Proof of the conjectures of H. Uhlig on the singular multivariate beta and theJacobian of a certain matrix transformation. The Annals of Statistics, 25:2018–2023, 1997.
E. B. Fox and M. West. Autoregressive models for variance matrices: Stationary inverse Wishart processes.Technical report, Duke University, July 2011.
C. Gourieroux, J. Jasiak, and R. Sufana. The Wishart autoregressive process of multivariate stochastic volatility.Journal of Econometrics, 150:167–181, 2009.
X. Jin and J. M. Maheu. Modeling realized covariances and returns. Journal of Financial Econometrics, 11(2):335–369, 2013.
Y. Konno. Exact moments of the multivariate F and beta distributions. Journal of the Japanese Statistical Society,18:123–130, 1988.
A. Loddo, S. Ni, and D. Sun. Selection of multivariate stochastic volatility models via bayesian stochastic search.Journal of Business and Economic Statistics, 29:342–355, 2011.
P. Mikusinski and M. D. Taylor. An Introduction to Multivariate Analysis. Birkhauser, 2002.
R. J. Muirhead. Aspects of Multivariate Statistical Theory. Wiley, 1982.
A. Philipov and M. E. Glickman. Multivariate stochastic volatility via Wishart processes. Journal of Business andEconomic Statistics, 24:313–328, July 2006.
R. Prado and M. West. Time Series: Modeling, Computation, and Inference, chapter Multivariate DLMs andCovariance Models, pages 263–319. Chapman & Hall/CRC, 2010.
S. Shirota, Y. Omori, H. F. Lopes, and H. Piao. Cholesky realized stochastic volatility, July 2015. URLhttp://econpapers.repec.org/paper/tkyfseres/2015cf979.htm. Shirota attends Duke University.
H. Uhlig. Bayesian vector autoregressions with stochastic volatility. Econometrica, 65(1):59–73, Jan. 1997.
24