Bayesian Dynamic Factor Models - Department of Statistical Science ...

Factor Model (LTDFM). Throughout we denote the hyper-parameters of the latent VAR(1) models
in the LTFLM by Θ = {µ γ, Φγ, V γ} γ=(c,β,h,λ).
3.2 Priors and posterior computation
Bayesian estimation of standard dynamic factor models using Markov chain Monte Carlo (MCMC)
methods are described in Aguilar and West (2000), and Lopes and West (2004). Based on their
works, we develop a MCMC algorithm for the LTDFM, exploring the full joint posterior distribution
p(β 1:T , f 1:T , c1:T , h1:T , λ1:T , Θ, Ψ, d|y 1:T ) under certain priors distributions.
First, assuming traditional priors for the VAR parameters in Θ simplifies the conditional pos-
terior distribution of Θ given the other parameters and data; we obtain the set of conditionally
independent posteriors, π(θγi|γ 1:T ), where γ ∈ {c, β, h, λ}, θγi = (µγi, φγi, v 2 γi ), and v2 γi denotes
the i-th diagonal in V γ. Similarly, the conditional posterior distribution of ψi reduces to the con-
ditionally independent posterior, π(ψi|{fit}t=1:T ). To generate the sample of these parameters,
we can use standard Metropolis-Hastings (MH) methods of sampling parameters in univariate AR
models.
Second, sampling the factors f t can be performed by a forward-filtering, backward sampling
(FFBS) algorithm (see e.g., Prado and West (2010)) to obtain the sample from the joint conditional
posterior distribution of the full sequence f 1:T , which reduces a possible high autocorrelations in
the MCMC draws. When we assume the standard independent factors by ψj ≡ 0 as in the model (2),
the conditional posterior distribution of f t reduces to a simple multivariate normal distribution.
Third, the conditional independence structure of the stochastic volatility, (h1:T , λ1:T ), allows
us to use the standard MCMC technique of the traditional univariate stochastic volatility models
(Kim et al. (1998), Omori et al. (2007). Shephard and Pitt (1997), Watanabe and Omori (2004)).
Again, these works provide the efficient algorithm to obtain the sample from the joint conditional
posterior distribution of {hit}t=1:T , and {λit}t=1:T (Prado and West (2010)).
Fourth, with regard to sampling β 1:T , Nakajima and West (2010) suggest a Metropolis-within-
Gibbs sampling strategy. To target on the conditional distribution of β t given β −t = β 1:T \β t and
other parameters, a proposal draw can be straightforwardly generated based on the non-threshold
model by fixing (s1t, . . . , spt) = (1, . . . , 1). The standard MH algorithm is completed by accept-
ing the candidate with the MH probability (See Section 2.3 of Nakajima and West (2010)). This
generation is implemented sequentially for t = 1 : T .
Finally, following Nakajima and West (2010), we assume the prior for the latent thresholds by
di ∼ U(0, |µi| + Kivi), i = 1, . . . , p,
where (µi, v2 i ) are the mean and variance of the stationary distribution of the univariate AR(1)
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