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

in variance and covariance structure as well as underlying time-varying common movements in
multivariate time series.
In the general setting, eqns. (1)–(2) are replaced by
y t = ct + Btf t + εt, εt ∼ N(εt|0, Σt), (3)
f t ∼ N(f t|Ψf t−1, W t) (4)
where εt and es = f s − Ψf s−1 are mutually independent for all t and s. Bt is the time-varying
loadings matrix; any form of model may be defined for the time-varying structure, although one
of the simplest, and easily most widely useful in practice, is the vector autoregressive (VAR) model
taken here. For each time t, define the p [= mk − k(k + 1)/2] × 1 vector bt by stacking the free
parameters in Bt. Then, we assume that
bt = µ b + Φb(bt−1 − µ b) + η bt, η bt ∼ N(0, V b), (5)
a VAR(1) model with individual AR parameters φbi, in the p × p matrix, Φb = diag(φb1, . . . , φbp).
We assume that Vb is a diagonal matrix and |φi| < 1, which implies that each factor loading follows
a stationary AR(1) model assumed here. We also define the local mean (or time-varying intercept),
ct, following the stationary VAR(1) model:
where Φc = diag(φc1, . . . , φcm), with |φci| < 1.
ct = µ c + Φc(ct−1 − µ c) + η ct, η ct ∼ N(0, V c), (6)
The time-varying variance matrices, Σt = diag(σ 2 1t , . . . , σ2 mt), and W t = diag(w 2 1t , . . . , w2 kt )
are governed by a standard stochastic volatility process with hit = log σ2 it , and λjt = log w2 jt , for
i = 1, . . . , m, j = 1 . . . , k. That is, with ht = (h1t, . . . , hmt) ′ , and λt = (λ1t, . . . , λkt) ′ ,
ht = µ h + Φh(ht−1 − µ h) + η ht, (7)
λt = µ λ + Φλ(λt−1 − µ λ) + η λt, (8)
where (ε ′ t, η ′ bt , η′ ct, η ′ ht , η′ λt ) ∼ N[0, diag(Σ, V b, V c, V h, V λ)], with each of the matrices (Φh, Φλ,
V c, V h, V λ) diagonal. This process of log variances defines traditional univariate stochastic volatil-
ity models widely used both alone and as components of more complex models in financial liter-
ature (Jacquier et al. 1994; Kim et al. 1998; Aguilar and West 2000; Omori et al. 2007; Prado
and West 2010, chap. 7). Note that here not only the series-specific variances but also the factor
variances change over time, which is a quite reasonable assumption for financial multivariate time
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