a tractable state-space model for symmetric positive...

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

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

State-Space Models

Latent States: xt−1 xt xt+1

Observations: yt−1 yt yt+1

System’s parameters, θ

3

State-Space Models




[ T∏i=1

p(yt |xt , θ)][ T∏

i=2

p(xt |xt−1, θ)]p(x1|θ)

3

State-Space Models




Filter: p(xt |y1:t).

3

State-Space Models




Smooth: p(x1:T |y1:T ).

3

State-Space Models




Infer: p(θ|y1:T ).

3

State-Space Models in Finance

Rt ∼ N(0,Vt),

Vt ∼ P(Vt−1).

4


Rt ∼ N(0,Vt),

Vt ∼ P(Vt−1).

5


Rt,i ∼ N(0,Vt/k), i = 1, . . . , k,

Vt ∼ P(Vt−1).

5


Yt ∼Wm(k ,Vt/k), Yt =k∑

i=1

Rt,iR′t,i

Vt ∼ P(Vt−1).

5


Yt ∼Wm(k ,X−1t /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 cc d2

]

6

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(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(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

(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

Use the original coordinates?


′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

Uhlig Extension


′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

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

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

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

Prediction Exercise

13

Drawbacks

Discussion:

I Roberto Casarin

I Catherine Scipione Forbes

I 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)

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

What does this work at all?

18

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

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

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

Estimating θ = (n, k , λ,Σ0)

The model:

Yt = Wm(k , (kXt)−1),


′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 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


Thank you for your attention.


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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.

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http://econpapers.repec.org/paper/tkyfseres/2015cf979.htm

a tractable state-space model for symmetric positive...

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