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

1 Introduction
Dynamic factor models have been widely developed in both methodological and practical issues,
and become a standard tool for increasingly high-dimensional modelling of time series. In particu-
lar, flexible and parsimonious innovation structures for underlying latent process such as stochastic
volatility (Aguilar and West (2000)) and time-varying loadings (Lopes and Carvalho (2007)) make
the factor models more flexible and reasonable to assess the complex multivariate time series. In
application of such factor analysis, the latent time-varying loadings grow quickly in dimension with
the number of time series response variables and the number of factors, which typically leads to
increased uncertainty in estimation. Much progress has been recently made for sparse factor analy-
sis (West 2003; Carvalho et al. 2008; Lopes et al. 2010; Yoshida and West 2010) using the sparsity
priors for variable selection (George and McCulloch 1993, 1997; Clyde and George 2004); but
these studies assume that the loadings are constant over time.
This paper develops dynamic sparse factor models by embedding the latent threshold (LT)
approach of Nakajima and West (2010); the time-varying loadings are thresholded to induce zero
values adaptively. The idea is that some factor loadings are practically relevant in some time
periods, but irrelevant or redundant in others. Nakajima and West (2010) propose latent threshold
models (LTM), a general approach to dynamic sparsity modelling in time series and state-space
models. The LTM structure allows time-varying parameters for non-zero values when supported
by the data while collapses them fully to zero when redundant or irrelevant. This data-driven
shrinkage mechanism induces dynamic sparsity in high-dimensional parameters that change over
time. In the current paper, we design the time-varying loadings with the LTM structure to focus on
relevances of factors to each time series response variable. The methodology is applied to real-data
study using daily exchange rate returns. Our objectives of this empirical analysis are model fitting,
forecasting and dynamic portfolio decisions, and reporting practical utility of our proposed models.
In Section 2, we review the dynamic factor models with time-varying loadings and stochastic
volatility. In Section 3, we generalize the factor models by embedding latent threshold modelling
approach, and develop a Bayesian inference and MCMC algorithm. In Section 4, we apply our
proposed models to six series of daily exchange rate returns. In Section 5, a portfolio allocation
study is investigated. In Section 5, concluding comments are provided.
Some notation: We use the distributional notation y ∼ N(a, A), d ∼ U(a, b), p ∼ B(a, b),
v ∼ G(a, b), for the normal, uniform, beta, and gamma distributions, respectively. We also use,
for example, N(y|a, A) to denote the actual density function f(y) when y ∼ N(a, A), in cases
where the specificity is needed; and s : t stands for indices s, s + 1, . . . , t when s < t, for succinct
subscripting such as in use of y 1:T to denote {y 1, . . . , y T }.
2