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

A Tractable State-Space Model for Symmetric Positive-Definite Matrices Jesse Windle 1 Carlos Carvalho 2 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 System’s parameters, θ x t − 1 x t +1 x t Latent States: y t − 1 y t y t +1 Observations: 3

State-Space Models System’s parameters, θ x t − 1 x t x t +1 Latent States: y t − 1 y t +1 y t Observations: � T �� T � � � p ( y t | x t , θ ) p ( x t | x t − 1 , θ ) p ( x 1 | θ ) i =1 i =2 3

State-Space Models System’s parameters, θ x t − 1 x t +1 x t Latent States: y t − 1 y t y t +1 Observations: Filter: p ( x t | y 1: t ) . 3

State-Space Models System’s parameters, θ x t − 1 x t +1 x t Latent States: y t − 1 y t y t +1 Observations: Smooth: p ( x 1: T | y 1: T ) . 3

State-Space Models System’s parameters, θ x t − 1 x t +1 x t Latent States: y t − 1 y t y t +1 Observations: Infer: p ( θ | y 1: T ) . 3

State-Space Models in Finance R t ∼ N (0 , V t ) , V t ∼ P ( V t − 1 ) . 4

State-Space Models in Finance R t ∼ N (0 , V t ) , V t ∼ P ( V t − 1 ) . 4

State-Space Models in Finance R t ∼ N (0 , V t ) , V t ∼ P ( V t − 1 ) . 5

State-Space Models in Finance R t , i ∼ N (0 , V t / k ) , i = 1 , . . . , k , V t ∼ P ( V t − 1 ) . 5

State-Space Models in Finance k � R t , i R ′ Y t ∼ W m ( k , V t / k ) , Y t = t , i i =1 V t ∼ P ( V t − 1 ) . 5

State-Space Models in Finance k � Y t ∼ W m ( k , X − 1 R t , i R ′ / k ) , Y t = t t , i i =1 X t ∼ P ( X t − 1 ) . 5

Our hands are now tied � T � T � � � � p ( Y t | X t , θ ) p ( X t | X t − 1 , θ ) p ( X 1 | θ ) i =1 i =2 � �� � Wishart Problem: Moving around the state-space. x t = Lower( X t ) ∼ GP ? � � d 1 c c d 2 6

Pick a new set of coordinates? Matrix logarithm [Bauer and Vorkink, 2011]: X t = U t exp( D t ) U ′ t , log X t = U t D t U ′ t , Z t = Lower(log X t ) . � T � T � � � � p ( Y t | X t , θ ) p ( X t | X t − 1 , θ ) p ( X 0 | θ ) i =1 i =1 � �� � Wishart p ( X 1: T | Y 1: T , θ ) → Gibbs + Metropolis-Hastings. p ( X t | X − t , Y 1: T , θ ) . 7

Pick a new set of coordinates? LDL decomposition [Chiriac and Voev, 2010, Loddo et al., 2011]: X t = L t exp( D t ) L ′ t , Z t = ( StrictLower( L t ) , Diag( D t ) ) . � T � T � � � � p ( Y t | X t , θ ) p ( X t | X t − 1 , θ ) p ( X 0 | θ ) i =1 i =1 � �� � Wishart p ( X 1: T | Y 1: T , θ ) → Gibbs + Metropolis-Hastings. p ( X t | X − t , Y 1: T , θ ) . 7

Use the original coordinates? X t = S t Ψ t S ′ t , S t S ′ t = f ( X t − 1 ) Source f ( X t − 1 ) Ψ t p ( X 1: T | Y 1: T , θ ) λ − 1 X t − 1 (1) W m ( ρ, I m /ρ ) � � λ − 1 X t − 1 n 2 , 1 (2) β m 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? X t = S t Ψ t S ′ t , S t S ′ t = f ( X t − 1 ) Source f ( X t − 1 ) Ψ t p ( X 1: T | Y 1: T , θ ) λ − 1 X t − 1 (1) W m ( ρ, I m /ρ ) Gibbs + MH � � λ − 1 X t − 1 n 2 , 1 (2) β m p ( X t | Y 1: t , θ ) 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? X t = S t Ψ t S ′ t , S t S ′ t = f ( X t − 1 ) Source f ( X t − 1 ) Ψ t p ( X 1: T | Y 1: T , θ ) λ − 1 X t − 1 (1) W m ( ρ, I m /ρ ) Gibbs + MH � � λ − 1 X t − 1 n 2 , 1 (2) β m p ( X t | Y 1: t , θ ) 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

Uhlig Extension t = λ − 1 X t − 1 X t = S t Ψ t S ′ t , S t S ′ � n � 2 , k Ψ t ∼ β m , k ∈ N ; 2 Easy to compute: ◮ p ( X t | Y 1: t , θ ) Wishart ◮ p ( X t | Y 1: t , X t +1 , θ ) Shifted Wishart ◮ p ( X 1: T | Y 1: T , θ ) 9

Uhlig Extension t = λ − 1 X t − 1 X t = S t Ψ t S ′ t , S t S ′ � n � 2 , k Ψ t ∼ β m , k ∈ N ; 2 Easy to compute: ◮ p ( X t | Y 1: t , θ ) Wishart ◮ p ( X t | Y 1: t , X t +1 , θ ) Shifted Wishart ◮ p ( X 1: T | Y 1: T , θ ) ◮ p ( Y t | Y t − 1 , θ ) Multivariate compound gamma = ⇒ p ( Y 1: T | θ ). 9

Uhlig Extension t = λ − 1 X t − 1 X t = S t Ψ t S ′ t , S t S ′ � n � 2 , k Ψ t ∼ β m , k ∈ N ; 2 Easy to compute: ◮ p ( X t | Y 1: t , θ ) Wishart ◮ p ( X t | Y 1: t , X t +1 , θ ) Shifted Wishart ◮ p ( X 1: T | Y 1: T , θ ) ◮ p ( Y t | Y t − 1 , θ ) Multivariate compound gamma = ⇒ p ( Y 1: T | θ ). Only need to record: Σ t = λ Σ t − 1 + Y t . 9

How does this work? Key Transformation Wishart Mult. Beta ⊥ X t − 1 Ψ t Muirhead [1982], Uhlig [1997], g D´ ıaz-Garc´ ıa and J´ aimez [1997]: λ X t Z t ⊥ Wishart Wishart 10

How does this work? Key Transformation Wishart Mult. Beta ⊥ X t − 1 Ψ t Muirhead [1982], Uhlig [1997], g D´ ıaz-Garc´ ıa and J´ aimez [1997]: λ X t Z t ⊥ Wishart Wishart Density of rank-deficient Wishart π − ( mk − k 2 ) / 2 | L | ( k − m − 1) / 2 � tr − 1 � 2 V − 1 Y exp � k � 2 mk / 2 Γ k | V | k / 2 2 k k k � � � ( dY ) = 2 − k l m − k ( l i − l j )( H ′ 1 d H 1 ) ∧ dl i . i i =1 i < j i =1 (Introductory text: Mikusi´ nski and Taylor [2002]) 10

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

Prediction Exercise • Predictive portfolios: π ′ ˆ π ∗ t = argmin V t π π ′ 1 =1 ˆ V t = E [ V t | Y 1: t − 1 ] . • Performance: ′ r t ) . portfolio variation = var( π ∗ t 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: X t is not stationary (realism) 15

Drawback: X t is not stationary (predictions) 16

Drawback: X t is not stationary (predictions) 16

Drawback: X t is not stationary (predictions) Predictions of future variance: M h = E [ X − 1 t + h | X − 1 ] , h > 0 . t Konno [1988]: M h = n + k − m − 1 λ M h − 1 n − m − 1 where M 0 = 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 : � t − 1 λ k � � E [ X − 1 λ i Y t − i + λ t Σ 0 t +1 | Y 1: t , θ ] = . n − m − 1 i =0 19

Volatility models: think in terms of forecasts ◮ Uhlig extension (EWMA): � t − 1 λ k � � E [ X − 1 λ i Y t − i + λ t Σ 0 t +1 | Y 1: t , θ ] = . n − m − 1 i =0 n + k − m − 1 λ = 1 = ⇒ n − m − 1 � t − 1 � � E [ X − 1 λ i Y t − i + λ t Σ 0 t +1 | Y 1: t , θ ] = (1 − λ ) . i =0 19

Volatility models: think in terms of forecasts (continued) ◮ “GARCH” (EWMA-MR): t � � � E [ X − 1 λ i Y t − i t +1 | Y 1: t , θ ] ≃ (1 − γ ) C + γ (1 − λ ) . i =0 ◮ 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, Σ 0 ) The model: Y t = W m ( k , ( kX t ) − 1 ) , t = λ − 1 X t − 1 , X t = S t Ψ t S ′ t , S t S ′ � n � 2 , k Ψ t ∼ β m , k ∈ N . 2 Conjugate prior: X 1 ∼ W m ( n , ( λ k Σ 0 ) − 1 ) . Y − τ , . . . , Y 0 , Y 1 , . . . , Y T . t − 1 − τ � � λ i Y t − i + λ t Σ 0 → Σ 0 ( λ ) = λ i Y − i + 0 . Σ t = i =0 i =0 21

Estimating θ = ( n , k , λ, Σ 0 ) The model: Y t = W m ( k , ( kX t ) − 1 ) , t = λ − 1 X t − 1 , X t = S t Ψ t S ′ t , S t S ′ � n � 2 , k Ψ t ∼ β m , k ∈ N . 2 Conjugate prior: X 1 ∼ W m ( n , ( λ k Σ 0 ) − 1 ) . Y − τ , . . . , Y 0 , Y 1 , . . . , Y T . t − 1 − τ � � λ i Y t − i + λ t Σ 0 → Σ 0 ( λ ) = λ i Y − i + 0 . Σ t = i =0 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