Identifying Speculative Bubbles with an Infinite Hidden Markov Model

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where n ji is the number of {τ | s τ = i, s τ−1 = j}.Sampling Θ | S, Y uses the results of regular linear models. The prior is:(ϕ i , σ −2i) ∼ NG(ϕ, H, χ 2 , ν 2 ).By conjugacy, the posterior is:with(ϕ i , σi−2 ) | S, Y ∼ NG(ϕ i , H i , χ i2 , ν i2 )ϕ i = H −1i (Hϕ + X ′ iY i )H i = H + X ′ iX iχ i = χ + Y ′i Y i + ϕ ′ Hϕ − ϕ ′ Hϕν i = ν + n iwhere Y i is the collection of y t in regime i. x t = (1, y t−1 , · · · , y t−q ). X i and n i are the collectionof x t and the number of observations in regime i, respectively.B.3 Sample (ϕ, H, χ) | S, Θ, νThe conditional posterior is:ϕ, H | {ϕ i , σ i } K i=1 ∼ NW(m 1 , τ 1 , A 1 , a 1 )30
where K is the number of active regimes, with which at least one data point is associated. ϕ iand σ i are the parameters of regime i.m 1 =τ 1 =A 1 =1τ0 −1 ∑+ K σi−2i=11τ −10 + K ∑(i=1A −10 +a 1 = a 0 + K.σ −2iK∑i=1(τ −10 m 0 +K∑i=1)σi −2 ϕ iσi−2 ϕ i ϕ ′ i + τ0 −1 m 0m ′ 0 − τ1 −1 m 1m ′ 1) −1The conditional posterior of χ is:χ | ν, {σ i } K i=1 ∼ G(d 1 /2, c 1 /2)with d 1 = d 0 + K ∑i=1σ −2iand c 1 = c 0 + Kν.B.4 Sample ν | χ, S, ΘThe conditional posterior of ν has no regular density form:p(ν | χ, {σ i } K i=1) ∝() K ((χ/2) ν/2 K) ν/2∏σi−2 exp{− ν }.Γ(ν/2)ρi=1νThe Metropolis-Hastings method is applied to sample ν.distribution:{with acceptance probability minν | ν ′ ∼ G( ζ νν ′ , ζ ν)Draw a new ν from a proposal}, where ν ′ is the value from the pre-1, p(ν|χ,{σ i} K i=1 )f G(ν ′ ; ζ νν ,ζ ν )p(ν ′ |χ,{σ i } K i=1 )f G(ν; ζ νν ′ ,ζ ν )vious sweep. ζ ν is fine tuned to produce a reasonable acceptance rate around 0.5, as suggestedby Roberts et al. (1997) and Müller (1991).31
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Page 6 and 7: 2 Infinite Hidden Markov ModelThe i
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Page 12: 3.2 Dating Algorithm of BubblesThe
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Identifying Speculative Bubbles with an Infinite Hidden Markov Model

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