A Tractable State-Space Model for Symmetric Positive-Definite - - PowerPoint PPT Presentation

A Tractable State-Space Model for Symmetric Positive-Definite Matrices

Jesse Windle1 Carlos Carvalho2 August 9, 2015

1 Hi Fidelity Genetics 2 The University of Texas at Austin 1

The Basic Story

• 1. The Bayesian analysis of covariance-matrix-valued state-space

models can be difficult.

• 2. The subsequent model is computationally tractable, but it

comes at a cost.

2

State-Space Models

Latent States: xt−1 xt xt+1 Observations: yt−1 yt yt+1 System’s parameters, θ

3

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

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

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

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

State-Space Models in Finance

Rt ∼ N(0, Vt), Vt ∼ P(Vt−1).

4

State-Space Models in Finance

Rt ∼ N(0, Vt), Vt ∼ P(Vt−1).

4

State-Space Models in Finance

Rt ∼ N(0, Vt), Vt ∼ P(Vt−1).

5

State-Space Models in Finance

Rt,i ∼ N(0, Vt/k), i = 1, . . . , k, Vt ∼ P(Vt−1).

5

State-Space Models in Finance

Yt ∼ Wm(k, Vt/k), Yt =

k

• i=1

Rt,iR′

t,i

Vt ∼ P(Vt−1).

5

State-Space Models in Finance

Yt ∼ Wm(k, X −1

t

/k), Yt =

k

• i=1

Rt,iR′

t,i

Xt ∼ P(Xt−1).

5

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

c c d2

• 6

Pick a new set of coordinates?

Matrix logarithm [Bauer and Vorkink, 2011]: Xt = Ut exp(Dt)U′

t,

log Xt = UtDtU′

t,

Zt = Lower(log Xt). 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

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

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

• n

2, 1 2

• (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

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

• n

2, 1 2

• 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

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

• n

2, 1 2

• 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

Uhlig Extension

Xt = StΨtS′

t, StS′ t = λ−1 Xt−1

Ψt ∼ βm n 2, k 2

• , k ∈ N;

Easy to compute:

◮ p(Xt|Y1:t, θ) Wishart ◮ p(Xt|Y1:t, Xt+1, θ) Shifted Wishart ◮ p(X1:T|Y1:T, θ) 9

Uhlig Extension

Xt = StΨtS′

t, StS′ t = λ−1 Xt−1

Ψt ∼ βm n 2, k 2

• , k ∈ N;

Easy to compute:

◮ p(Xt|Y1:t, θ) Wishart ◮ p(Xt|Y1:t, Xt+1, θ) Shifted Wishart ◮ p(X1:T|Y1:T, θ) ◮ p(Yt|Yt−1, θ) Multivariate compound gamma

= ⇒ p(Y1:T|θ).

9

Uhlig Extension

Xt = StΨtS′

t, StS′ t = λ−1 Xt−1

Ψt ∼ βm n 2, k 2

• , k ∈ N;

Easy to compute:

◮ p(Xt|Y1:t, θ) Wishart ◮ p(Xt|Y1:t, Xt+1, θ) Shifted Wishart ◮ p(X1:T|Y1:T, θ) ◮ p(Yt|Yt−1, θ) Multivariate compound gamma

= ⇒ p(Y1:T|θ). Only need to record: Σt = λΣt−1 + Yt.

9

How does this work? Key Transformation

Muirhead [1982], Uhlig [1997], D´ ıaz-Garc´ ıa and J´ aimez [1997]: Wishart

• Mult. Beta

Xt−1 Ψt ⊥ Wishart Wishart λXt Zt ⊥ g

10

How does this work? Key Transformation

Muirhead [1982], Uhlig [1997], D´ ıaz-Garc´ ıa and J´ aimez [1997]: Wishart

• Mult. Beta

Xt−1 Ψt ⊥ Wishart Wishart λXt Zt ⊥ g Density of rank-deficient Wishart π−(mk−k2)/2|L|(k−m−1)/2 2mk/2Γk k

2

• |V |k/2

exp

• tr − 1

2V −1Y

• (dY ) = 2−k

k

• i=1

lm−k

i k

• i<j

(li − lj)(H′

1d H1) ∧ k

• i=1

dli. (Introductory text: Mikusi´ nski and Taylor [2002])

10

Example

◮ 30 stocks from DJIA as of Oct. 2010. ◮ Feb. 27, 2007 to Oct. 29, 2010. ◮ Yt: Realized kernels (e.g. Barndorff-Nielsen et al. [2009]) 11

Prediction Exercise

• Predictive portfolios:

π∗

t = argmin π′1=1

π′ ˆ Vtπ ˆ Vt = E[Vt|Y1:t−1].

• Performance:

portfolio variation = var(π∗

t ′rt).

root mean variation FSV Extension 0.00977 Uhlig Extension 0.00936.

12

Prediction Exercise

13

Drawbacks

Discussion:

◮ Roberto Casarin ◮ Catherine Scipione Forbes ◮ Enrique ter Horst, German Molina 14

Drawback: Xt is not stationary (realism)

15

Drawback: Xt is not stationary (predictions)

16

Drawback: Xt is not stationary (predictions)

16

Drawback: Xt is not stationary (predictions)

Predictions of future variance: Mh = E[X −1

t+h | X −1 t

], h > 0. Konno [1988]: Mh = n + k − m − 1 n − m − 1 λ Mh−1 where M0 = X −1

t

.

17

What does this work at all?

18

What does this work at all?

18

Volatility models: think in terms of forecasts

◮ Uhlig extension :

E[X −1

t+1|Y1:t, θ] =

λk n − m − 1 t−1

• i=0

λiYt−i + λtΣ0

• .

19

Volatility models: think in terms of forecasts

◮ Uhlig extension (EWMA):

E[X −1

t+1|Y1:t, θ] =

λk n − m − 1 t−1

• i=0

λiYt−i + λtΣ0

• .

n + k − m − 1 n − m − 1 λ = 1 = ⇒ E[X −1

t+1|Y1:t, θ] = (1 − λ)

t−1

• i=0

λiYt−i + λtΣ0

• .

19

Volatility models: think in terms of forecasts (continued)

◮ “GARCH” (EWMA-MR):

E[X −1

t+1|Y1:t, θ] ≃ (1 − γ)C + γ (1 − λ)

• t
• i=0

λiYt−i

• .

◮ Univariate stochastic volatility:

EWMA-MR of the log squared returns

◮ Leverage effects:

asymmetrically weight past observations depending on market movements.

20

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

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

Recapitulation

• 1. Given our specific observation distribution, it isn’t easy to

construct tractable matrix-valued state-space models.

• 2. Uhlig essentially provides a way to do this, but it comes with

a cost. Slides with references: http://www.jessewindle.com/

22

Thank you for your attention. http://www.jessewindle.com/

23

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Journal of Econometrics, 150:182–192, 2009.

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