An Efficient Bayesian Algorithm for Markov-Switching ARMA Models ...

An Efficient Bayesian Algorithm for Markov-SwitchingARMA Models : An Application to US real GDPJune. 06. 2012Chang-Jin KimUniversity of Washington and Korea UniversityandJaeho KimUniversity of Washington

Consider the following Markov-Switching ARMA (p,q):py t = μ St + ∑ φ i,St ( y t−i − μ St−i )i=1t = 1,2, … , T.q+ e t + ∑ θ j,St e t−jj=1, e t ~ i. i. d N(0, σ St 2 ) Model parameters are dependent upon hidden state, S twhere S t = 1,2, . . , k. Pr[ S t | S t−1 , S t−2 , … , S 1 ] = Pr[ S t | S t−1 ]

Consider the following Markov-Switching ARMA (1,1):y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1 ,t = 1,2, … , T. e t ~ i. i. d N(0, σ 2 ) The mean, μ St , is dependent upon hidden state, S t = 1,2. Pr[ S t | S t−1 , S t−2 , … , S 1 ] = Pr[ S t | S t−1 ] Pr[ S t = 1| S t−1 = 1] = p 11 , Pr[ S t = 2| S t−1 = 2] = p 22

Consider the following Markov-Switching ARMA (1,1):y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1 ,t = 1,2, … , T. Ψ = [ μ 1 , μ 2 , φ , θ, σ 2 ] P = [ p 11 , p 22 ] S̃ = [ S 1 , S 2 , … , S T ], S̃t = [ S 1 , S 2 , … , S t ] Ỹ = [ y 1 , y 2 , … , y T ], Ỹt = [ y 1 , y 2 , … , y t ]

Consider the following Markov-Switching ARMA (1,1):y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1 ,t = 1,2, … , T. Infinite order AR process:∞y t = μ St + ∑ ω i ( y t−i − μ St−i )i=1t = 1,2, … , T.+ e t ,

Joint Posterior Density:Pr[ ψ , P, S̃ | Ỹ]. Markov Chain Monte Carlo AlgorithmStep1. Draw S̃ from Pr[ S̃ | P, ψ, Ỹ].(Billio et al. (1999), Yoo (2010), Henneke et al. (2011))Step2. Draw ψ from Pr[ ψ | P, S̃ , Ỹ].(Chib and Greenberg (1994), Nakatuma (2000))Step3. Draw P from Pr[ P | ψ, S̃ , Ỹ] = Pr[ P | S̃ ].(Beta Distribution)

Target Density: Pr⁡[S̃|ψ⁡, P, Ỹ]Single-move sampler:Pr⁡[S t |S̃≠t , ψ⁡, P, Ỹ],⁡⁡⁡⁡t = 1,2, … , T.⁡

Target Density: Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |Ỹ]Single-move sampler:Pr⁡[S 1 |S̃≠1 , Ỹ],⁡⁡⁡⁡t = 1,2,3,4,5.⁡S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5

Target Density: Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |Ỹ]Single-move sampler:Pr⁡[S 2 |S̃≠2 , Ỹ],⁡⁡⁡⁡t = 1,2,3,4,5.⁡S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5

Target Density: Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |Ỹ]Single-move sampler:Pr⁡[S 3 |S̃≠3 , Ỹ],⁡⁡⁡⁡t = 1,2,3,4,5.⁡S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5

Target Density: Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |Ỹ]Single-move sampler:Pr⁡[S 4 |S̃≠4 , Ỹ],⁡⁡⁡⁡t = 1,2,3,4,5.⁡S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5

Target Density: Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |Ỹ]Single-move sampler:Pr⁡[S 5 |S̃≠5 , Ỹ],⁡⁡⁡⁡t = 1,2,3,4,5.⁡S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5 Problems1. Huge Computation2. High Autocorrelations across MCMC samples

Target Density: Pr⁡[⁡S̃⁡|⁡ψ⁡, P, Ỹ]Multi-move sampler: Pr⁡[⁡S̃⁡|⁡ψ⁡, P, Ỹ]

Target Density: Pr⁡[⁡S 1 , S 2 , S 3 , S 4 , S 5 ⁡|⁡Ỹ]Multi-move sampler:Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |⁡Ỹ]S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5

Multi-move sampler:Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |⁡Ỹ]⁡= Pr⁡[S 5 |⁡Ỹ]⁡Pr⁡[S 4 |⁡Ỹ, S 5 ]⁡… ⁡Pr⁡[S 1 |⁡Ỹ, S 2 , S 3 , S 4 , S 5 ⁡]S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5

Multi-move sampler:Pr⁡[S 1 , S 2 , S 3 , S 4 , S 5 |⁡Ỹ]⁡= Pr⁡[S 5 |⁡Ỹ]⁡Pr⁡[S 4 |⁡Ỹ, S 5 ]⁡… ⁡Pr⁡[S 1 |⁡Ỹ, S 2 , S 3 , S 4 , S 5 ⁡]S 1 S 2 S 3 S 4 S 5t : 1 2 3 4 5

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] Computational Efficiency Fast Convergence Accurate Bayesian Inference Theoretical justification:(Liu et al. (1994, 1995), and Scott (2002))

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] Approximated Filtering Algorithm by Kim (1994) Metro-Police Hastings Algorithm

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Pr⁡[⁡S̃⁡|⁡Ỹ]⁡= ⁡Pr⁡[⁡S 1 , S 2 , … , S T ⁡|⁡Ỹ]= ⁡Pr⁡[⁡S T ⁡|⁡Ỹ]⁡Pr⁡[⁡S T−1 ⁡|S T , Ỹ]⁡Pr⁡[⁡S T−2 ⁡|S T , S T−1 , Ỹ]⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡… Pr[S 2 | S 3 , S 4 , … , S T , Ỹ]⁡Pr[S 1 | S 2 , S 3 , … , S T , Ỹ]⁡

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Pr⁡[⁡S t ⁡|S t+1 , … , S T , Ỹ]⁡∝ ⁡ Pr[S t+1 |S t ] Pr[S t |Ỹt]⁡ ∏ f(y k |S̃k⁡, Ỹk−1 )Tk=t+1

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Pr⁡[⁡S t ⁡|S t+1 , … , S T , Ỹ]⁡∝ ⁡ Pr[S t+1 |S t ] Pr[S t |Ỹt]⁡ ∏ f(y k |S̃k⁡, Ỹk−1 )Tk=t+1∝ ⁡ Pr[S t+1 |S t ] Pr[S t |Ỹt] ⁡L(S t ),where L(S t ) = ⁡ ∏Tk=t+1 f(y k |S̃k⁡, Ỹk−1 ).

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Pr⁡[⁡S t = h⁡|S t+1 , … , S T , Ỹ]⁡⁡= ⁡ Pr [S t+1 |S t = h] Pr[S t = h|Ỹt] ⁡L(S t = h)∑ Pr[S t+1 |S t ] Pr[S t |Ỹt] ⁡L(S t )S t= ⁡∑S tPr[S t+1 |S t = h] Pr[S t = h|Ỹt]L(SPr[S t+1 |S t ] Pr[S t |Ỹt]⁡ t )L(S t = h)

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Pr⁡[⁡S t = h⁡|S t+1 , … , S T , Ỹ]⁡⁡= ⁡ Pr [S t+1 |S t = h] Pr[S t = h|Ỹt] ⁡L(S t = h)∑ Pr[S t+1 |S t ] Pr[S t |Ỹt] ⁡L(S t )S t≈ ⁡ Pr [S t+1 |S t = h] Pr[S t = h|Ỹt]∑ Pr[S t+1 |S t ] Pr[S t |Ỹt]⁡S t

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Pr⁡[⁡S t = h⁡|S t+1 , … , S T , Ỹ]⁡⁡≈ ⁡ Pr [S t+1 |S t = h] Pr[S t = h|Ỹt]∑ Pr[S t+1 |S t ] Pr[S t |Ỹt]⁡S t Pr[S t+1 |S t ] is known. Pr[S t |Ỹt] is obtained by Kim (1994)’s approximated filteringalgorithm.

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Pr⁡[⁡S t = h⁡|S t+1 , … , S T , Ỹ]⁡⁡≈ ⁡ Pr [S t+1 |S t = h] Pr[S t = h|Ỹt]∑ Pr[S t+1 |S t ] Pr[S t |Ỹt]⁡S t Pr[S t+1 |S t ] is known. Pr[S t |Ỹt] is obtained by Kim (1994)’s approximated filteringalgorithm. Metro-Policy Hastings Algorithm

Multi-move sampler: Pr⁡[⁡S̃⁡|⁡Ỹ] ⁡Multi-move Sampling⁡Algorithm:Kim (1994)’s filtering Algorithm: Pr[S t |Ỹt] ⁡for⁡t = 1,2, … , TStep1. Draw S T from Pr[S T |Ỹ]Step2. Draw S T−1 from Pr[S T |S T−1 ] Pr[S T−1 |ỸT−1 ]StepT. Draw S 1 from Pr[S 2 |S 1 ] Pr[S 1 |Ỹ1]Metro-police Hastings Algorithm⋮

ARMA(1,1) with a switching mean:y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1 ,for t = 1,2, … , T. e t ~ i. i. d N(0, σ 2 ) The mean, μ St , is dependent upon hidden state, S t = 1,2. Pr[ S t | S t−1 , S t−2 , … , S 1 ] = Pr[ S t | S t−1 ] Pr[ S t = 1| S t−1 = 1] = p 11 , Pr[ S t = 2| S t−1 = 2] = p 22

ARMA(1,1) with a switching mean:y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1 ,for t = 1,2, … , T. Two Competing AlgorithmsSingle-move Sampler by Billio et al. (1999)Multi-move Sampler by the New Algorithm

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 1 0.9(10 periods)0.95(20 periods)4 0 0.5 −0.8 1Figure1. Cumulative Averages of MCMC Samples(p 1,1 ) (p 2,2 )

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 1 0.9(10 periods)0.95(20 periods)4 0 0.5 −0.8 1Figure2. Posterior Probabilities of S t = 2(a) New AlgorithmBurn-in / Total iterations: 5,000 / 15,000(b) Billio et. al (1999)Burn-in / Total iterations: 5,000 / 15,000

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 2 0.98(50 periods)0.99(100 periods)4 0 0.5 −0.8 1Figure3. Cumulative Averages of MCMC Samples(p 1,1 ) (p 2,2 )

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 1 0.9(10 periods)0.95(20 periods)4 0 0.5 −0.8 1Figure4. Posterior Probabilities of S t = 2(a) New AlgorithmBurn-in / Total iterations: 5,000 / 15,000(b) Billio et. al (1999)Burn-in / Total iterations: 190,000 / 200,000

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 3 0.8(5 periods)0.9(10 periods)5 0 0.2 −0.7 1Figure5. Cumulative Averages of MCMC Samples(p 1,1 ) (p 2,2 )

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 3 0.8(5 periods)0.9(10 periods)5 0 0.2 −0.7 1Figure6. Posterior Probabilities of S t = 2(a) New AlgorithmBurn-in / Total iterations: 5,000 / 15,000(b) Billio et. al (1999)Burn-in / Total iterations: 20,000 / 30,000

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 4 0.8(5 periods)0.9(10 periods)1.5 0 0.2 −0.7 1Figure7. Cumulative Averages of MCMC Samples(p 1,1 ) (p 2,2 )

ARMA (1,1) :y t = μ St + φ( y t−1 − μ St−1 ) + e t + θe t−1T = 300 p 11 p 22 μ 1 μ 2 φ θ σModel 4 0.8(5 periods)0.9(10 periods)1.5 0 0.2 −0.7 1Figure8. Posterior Probabilities of S t = 2(a) New AlgorithmBurn-in / Total iterations: 5,000 / 15,000(b) Billio et. al (1999)Burn-in / Total iterations: 140,000 / 150,000

ARIMA (2,1,2) with Structural Breaks:e t = ∑2i=1∆y t = μ S1,t + e t,φ i,S2,t e t−i2+ u t + ∑ θ j,S2,t u t−jj=1u t ~ i. i. d N(0, σ 2 t ), ln(σ 2 t ) = ln(σ 2 t−1 ) + ε t ,ε t ~ i. i. d N(0, σ 2 ).,

ARIMA (2,1,2) with Structural Breaks:e t = ∑2i=1∆y t = μ S1,t + e t,φ i,S2,t e t−i + u t + ∑ θ j,S2,t u t−j ,2j=1u t ~ i. i. d N(0, σ 2 t ), ln(σ 2 t ) = ln(σ 2 t−1 ) + ε t ,ε t ~ i. i. d N(0, σ 2 ).Structural Break in Uncondional MeansStructural Break in Persistence

ARIMA (2,1,2) with Structural Breaks:e t = ∑2i=1∆y t = μ S1,t + e t,φ i,S2,t e t−i + u t + ∑ θ j,S2,t u t−j ,2j=1u t ~ i. i. d N(0, σ 2 t ), ln(σ 2 t ) = ln(σ 2 t−1 ) + ε t ,ε t ~ i. i. d N(0, σ 2 ).Cumulative Impulse Responseof State 0Cumulative Impulse Responseof State 1

ARIMA (2,1,2) with Structural Breaks:e t = ∑2i=1∆y t = μ S1,t + e t,φ i,S2,t e t−i + u t + ∑ θ j,S2,t u t−j ,2j=1u t ~ i. i. d N(0, σ 2 t ), ln(σ 2 t ) = ln(σ 2 t−1 ) + ε t ,ε t ~ i. i. d N(0, σ 2 ).Bayesian 90% Confidence Band ofStochastic Volatilities