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

Bayesian Dynamic Factor Models: Latent Threshold Approach
Jouchi Nakajima
Department of Statistical Science, Duke University, Durham, NC 27708-0251
jouchi.nakajima@stat.duke.edu
March 24, 2011
Abstract
We develop the dynamic sparse factor models using latent threshold approach. The time-
varying factor loadings are thresholded to induce zero values adaptively to address the uncer-
tainty of factor structure. We discuss methodologies of Bayesian model estimation and pre-
diction with application to daily exchange rate returns. The empirical results suggest that the
incorporation of data-driven dynamic sparsity in the loadings substantially reduces estimation
uncertainties and leads to improve predictive performance.
KEY WORDS: Dynamic factor models; Portfolio allocation; Latent threshold dynamic models;
Sparse time-varying loadings; Time-varying variable selection.
1

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

2 Dynamic factor models
2.1 Standard model
Suppose that the time series of m × 1 vector responses y t, (t = 1, 2, . . .), follows the model
y t = c + Bf t + εt, εt ∼ N(εt|0, Σ), (1)
f t ∼ N(f t|0, W ), (2)
where c is the m × 1 mean vector, f t is the k × 1 vector of factors with k < m, B is the m × k factor
loadings matrix, εt and f s are mutually independent for all t and s, Σ = diag(σ 2 1 , . . . σ2 m) is the
diagonal matrix of series-specific variances, and W = diag(w2 1 , . . . , w2 k ) is the diagonal matrix of
factor variances. Given the latent factors f t that extract common structure in yt = (y1t, . . . , ymt) ′ ,
the model (1)–(2) implies that yit and yjt are conditionally uncorrelated for all i and j; specifically,
V (y t|f t) = Σ, and V (y t) = BW B ′ + Σ.
Under a large m circumstance (i.e., the high-dimensional responses), in practice, the relatively
small number of latent factors isolate the idiosyncratic variation of each series, from the high-
dimensional covariance structure of the whole responses. In particular, this point is critical when
we consider the time-varying structures that follow below. The factor modelling allows for the
incorporation of flexible time-varying relations between responses and stochastic volatility not only
in each series but also in common movements that may yield substantial improvements in short-
term forecasting and interpretation, especially in application with a large data sets such as hundreds
of financial return time series.
For identification problem, the factor model requires some constraint to define the unique
model, because the model (1)–(2) is invariant under location shifts of the loading matrix. Al-
though there are several ways to ensure this identification, we assume here that B is the block
lower triangular matrix with diagonal elements equal to one, as suggested by Aguilar and West
(2000), Lopes and Carvalho (2007) (see further discussions in Lopes and West (2004)).
2.2 Time-varying parameters, stochastic volatility and autoregressive factor process
We consider a generalization of the standard factor model with three substantial compositions
that capture important features of financial time series, often discussed in literature. First, we allow
each free element in the loading matrix and the mean to change over time (Lopes and Carvalho
(2007)). Second, we assume that the variances of the errors in eqn. (1) and the factors are time-
varying in the form of stochastic volatility (Pitt and Shephard (1999), Aguilar and West (2000),
Lopes and Carvalho (2007)). Third, we suppose that the latent factors follow an autoregressive pro-
cess. These generalizations provide reasonably flexible and adaptive structure to address changes
3

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
4

series because process of their latent common factor typically tends to exhibit serious time-varying
variance behaviors.
The final feature to mention here is that the factors are modelled to follow the stationary VAR(1)
process with individual autoregressive parameters, Ψ = diag(ψ1, . . . , ψk) with |ψj| < 1. That ψj ≡ 0
for all j implies the basic independent (over time) factors as in (2). These autoregressive factors as
well as the local means, time-varying loadings and stochastic volatility described so far are intended
not only to address underlying time-varying structure in relation to the variance and covariance
structure on the factor models, but also to elaborate the structured models with greater short-term
forecastabilty. Those specifications are considerably flexible and adaptive date-dependent tracking
structure in means and variances for financial time series.
3 Latent threshold dynamic factor models
3.1 Framework
The free time-varying parameters in the loading matrix grow quickly in dimensions with the
number of time series response variables. This often leads to increased uncertainty in estimation.
To address this problem, we exploit the latent threshold modelling (Nakajima and West (2010)) in
the latent process of the time-varying loadings.
Apart from the eqn. (5), suppose that we have the latent threshold model (LTM) structure for
bt = (b1t, . . . , bpt) ′ ; that is,
bit = βitsit, and sit = I(|βit| ≥ di), i = 1, . . . , p, (9)
where I(·) denotes the indicator function, d = (d1, . . . , dp) is the latent threshold vector with di ≥ 0,
for i = 1, . . . , p, and the latent time-varying parameters β t = (β1t, . . . , βpt) ′ follow a stationary
VAR(1) model
β t+1 = µ β + Φβ(β t − µ β) + η βt, η βt ∼ N(0, V β), (10)
where Φβ and V β are diagonal, and |φβi| < 1. The key idea of the LTM structure is that the
value of the time-varying loading is shrunk to zero when its absolute value falls below the latent
threshold. The relevance of the factor to the response is time-varying; each factor plays a role
in predicting the response only when the corresponding βit is large enough. For each time series
response variable, the factor may have non-zero loading in some time periods but zero in others,
depending on the data and context. This mechanism leads to the dynamic model uncertainty in
the factor loading matrix by neatly embodying sparsity/shrinkage and parameter reduction at each
time point. We refer to the model of eqns. (3), (4), and (6-10) as a Latent Threshold Dynamic
5

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)
6

process for βit, with µi = µβi and v 2 i = v2 βi /(1 − φ2 βi ). This prior covers the range of βit as reflected
in its marginal distribution, considering relevant values of the threshold. Their work suggests
K = 3, also taken in the current paper. The generation of the posterior distribution is performed
by a direct MH approach with candidate drawn from the prior.
4 Application to exchange rate returns
For illustration, we apply the LTDFM described above to multivariate series of foreign exchange
(FX) rate returns. Our focus here is particular interest in LTM-based shrinkage to zero of time-
varying factor loadings and its implications for the dynamic relations underlying the FX series.
There are some connections with previous work on FX time series using the DFMs (e.g. Aguilar and
West (2000), Lopes and West (2004), Lopes and Carvalho (2007)).
4.1 Data and priors
The data are m = 6 daily international currency exchange rates relative to US dollar over a
time period of 980 business days beginning in January 2006 and ending in December 2009, The
returns are computed as yit = 100(Pit/Pi,t−1 − 1), where Pit denotes the daily closing spot rate.
The series are listed in Table 1; the exchange rates and returns are both plotted in Figure 1. The
most striking feature in these sample periods is a quite volatile and turbulent movement triggered
by the financial crisis around 2008; there are some superficial correlated flows and shifts among
the series. Initial exploratory analysis using the univariate stochastic volatility fit to each series
separately is summarized in Figure 2. There exist marked region-specific patterns. The peaks of
stochastic volatility seem to coincide in 2008, but there exist several spikes common only to two or
three currencies. From these evidences and some computational experiments, we determined the
order of the currencies in the y t vectors as listed in Table 1.
1 GBP British Pound Sterling
2 EUR Euro
3 JPY Japanese Yen
4 CAD Canadian Dollar
5 AUD Australian Dollar
6 CHF Swiss Franc
Table 1: International currencies relative to the US dollar in exchange rate data.
Our analysis uses the LTDFM, namely the dynamic factor model with (i) local mean, (ii) stochas-
tic volatility for both individual series and factors, (iii) autoregressive factor process, and (iv) LTM-
based time-varying loading matrix. We take the following priors throughout: v −2
ci
v −2
βi
∼ G(40, 0.005),
∼ G(40, 0.02), v−2
hi ∼ G(20, 0.001), and (φγi + 1)/2 ∼ B(20, 1.5), where γ = (c, β, h, λ), and
(ψi + 1)/2 ∼ B(1, 1). For µ·i’s, we use µγi ∼ N(0, 1) for γ = (c, β), and exp(−µ˜γi) ∼ G(3, 0.03)
7

2.0
1.5
GBP
2006 2007 2008 2009
0.012 JPY
0.010
0.008
2006 2007 2008 2009
1.0 AUD
0.8
2006 2007 2008 2009
3
0
−3
2006 2007 2008 2009
3
0
−3
GBP
2006 2007 2008 2009
5
0
−5
JPY
AUD
2006 2007 2008 2009
(i) Exchange rates
1.6 EUR
1.4
1.2
1.0
0.8
2006 2007 2008 2009
CAD
2006 2007 2008 2009
1.0 CHF
0.9
0.8
(ii) Returns
2006 2007 2008 2009
0
−3
3 EUR
2006 2007 2008 2009
6 CAD
3
0
−3
2006 2007 2008 2009
3
0
CHF
−3
2006 2007 2008 2009
Figure 1: Daily exchange rates and returns
8

3
2
1
GBP
EUR
JPY
CAD
AUD
CHF
2006 2007 2008 2009
Figure 2: Estimated trajectories of univariate stochastic volatility (σit = exp(hit/2)) fit to each
series separately.
for ˜γ = (h, λ). The conditional prior for thresholds is U(di|0, |µi| + Kivi) with Ki = 3. MCMC
used J = 100,000 iterations after discarding a burn-in period of 20,000 samples. Computations are
obtained using Ox (Doornik 2006); code is available to interested readers.
A determination of the number of factors has been a challenging issue in literature (see e.g.
Lopes and West (2004) in Bayesian context). For the analysis here, we fit the LTDFM to the FX
data and repeated the estimation across models with different number of factors k = 1, 2, 3, or
4. We evaluated each of the resulting suite of models by forecasting over the final five business
days using first 930 observations, and repeated this five-step-ahead forecasting based on the first
930+5j observations for j = 1, . . . , 9, to obtain the 50-day forecasts. Based on the root mean
squared errors (RMSE) for these out-of-sample forecasts, the LTDFM performs best when k = 3 is
assumed, which is taken in the following analysis.
4.2 Summaries of posterior inference
Figure 3 displays the posterior estimates of the time-varying factor loadings, as well as the
posterior probabilities of sit = 0 for the LTDFM model. The estimated latent process exhibits a
considerable time variation for several loadings, which can not be captured by a constant-loading
factor model. In particular, JPY-Factor1 and AUD-Factor3 are estimated positive in 2006, but turns
to be negative after 2007, reflecting a shift of correlated dynamics among the FX series. The
9
.

trajectories of posterior mean of several loadings also evidently change over time. Time-varying
sparsity is observed for the loadings of JPY-Factor1, AUD-Factor3 and CHF-Factor3, with estimated
values shrinking to zero for some time periods but not others, whereas other loadings such as
CAD-Factor2 and -Factor3 are totally selected out over the entire time-frame.
The interesting observation of the shrinkage pattern of JPY-Factor1 in 2008 implies that the JPY
returns turned to be negatively correlated with Factor1. The loadings to Factor1 are all positive in
the currencies except JPY. In 2008, the financial crisis led investors to reduce their size of US dollar
short position and the currencies in our analysis except JPY depreciated against US dollar, whereas
the JPY temporally appreciated.
Figure 4 shows the proportion of variation of the time-series explained by each factor; the time
variation of each factor’s contribution to the time-series is evident. By construction, Factor1 is a
GBP-leading factor, although its explanatory ratio decreases from 2006 to 2009, and there are a
few downward spots in the crisis period, which indicates an idiosyncratic variation of the GBP.
Factor2 is recognized as an European factor that mainly explains the variation of EUR and CHF.
The variation of JPY is of interest; the Factor1 primarily explains its fluctuation in 2006, but later
its role is shifted to Factor3, namely a JPY-leading factor. A clear shrinkage of the contribution in
JPY-Factor1, CAD-Factor2, and CAD-Factor3 is detected by the LTM structure as we observed in
Figure 3. It would eliminate unnecessary fluctuation of the time-varying loadings and lead to an
efficient distinction of the factors.
Figure 5 plots the posterior estimates of the factors and their stochastic volatility. The trajec-
tories and peak of the stochastic volatility remarkably differ among the factors. The stochastic
volatility of Factor1 has the peak around September 2008 corresponding to the major crush of the
market, whereas that of Factor2 has its peak around December 2008, when EUR and CHF returns
exhibit relatively higher volatility in reaction to the US Senate’s rejection of the financial bailout
for the automotive industry. The stochastic volatility of Factor1 exhibits a steeper hike towards the
peak and a more moderate diminishing afterwards compared to that of Factor2.
Figure 6 graphs posterior means of series-specific stochastic volatility, σit = exp(hit/2), that is,
volatility elements excluding the common factors, exhibiting quite a different picture from Figure
2. GBP has a unique hike of volatility in the second half of 2008 and CHF has several spikes in 2008
and 2009, which are not explained by the factors. Because most of the fluctuation in EUR returns
are explained by Factor1 and Factor2 as can be seen in Figure 4, the remaining series-specific shocks
are relatively smaller than the others as shown in Figure 6.
We determined the number of factors as k = 3 based on the forecasting performance, but also
report experimental results from k = 4 factors case. As can be seen in Figure 7, the estimated
trajectory of Factor4 is clearly shrunk with significantly smaller stochastic volatility than the other
10

1.0
0.5
β 2,1t (EUR−Factor)
2006
1.0
0.5
2007 2008 2009
1
0
β 3,1t (JPY−Factor1)
2006
1.0
0.5
2007 2008 2009
2
1
β 4,1t (CAD−Factor1)
2006
1.0
0.5
2007 2008 2009
2
1
β 5,1t (AUD−Factor1)
2006
1.0
0.5
2007 2008 2009
1.5
1.0
0.5
β6,1t (CHF−Factor1)
2006
1.0
0.5
2007 2008 2009
1 0 0
1
β 3,2t (JPY−Factor2)
0
2006
1.0
0.5
2007 2008 2009
1.0
0.5
0.0
β4,2t (CAD−Factor2)
2006
1.0
0.5
2007 2008 2009
1.0
0.5
0.0
β5,2t (AUD−Factor2)
2006
1.0
0.5
2007 2008 2009
1.5
1.0
0.5
β6,2t (CHF−Factor2)
2006
1.0
0.5
2007 2008 2009
1 0
0.5
0.0
1
β 4,3t (CAD−Factor3)
2006
1.0
0.5
2007 2008 2009
1
0
−1
β5,3t (AUD−Factor3)
2006
1.0
0.5
2007 2008 2009
1.0
0.5
0.0
β 6,3t (CHF−Factor3)
2006
1.0
0.5
2007 2008 2009
Figure 3: Posterior means (solid) and 95% credible intervals (dotted) of time-varying factor loadings.
Posterior probability of sit = 0 are plotted below each trajectory.
11

1.0 GBP−Factor1
0.5
0.0
2006 2008
1.0 EUR−Factor1
0.5
0.0
2006 2008
1.0
JPY−Factor1
0.5
0.0
2006 2008
1.0 CAD−Factor1
0.5
0.0
2006 2008
1.0 AUD−Factor1
0.5
0.0
2006 2008
1.0 CHF−Factor1
0.5
0.0
2006 2008
1.0 EUR−Factor2
0.5
0.0
2006 2008
1.0 JPY−Factor2
0.5
0.0
2006 2008
1.0 CAD−Factor2
0.5
0.0
2006 2008
1.0 AUD−Factor2
0.5
0.0
2006 2008
1.0 CHF−Factor2
0.5
0.0
2006 2008
1.0 JPY−Factor3
0.5
0.0
2006 2008
1.0 CAD−Factor3
0.5
0.0
2006 2008
1.0 AUD−Factor3
0.5
0.0
2006 2008
1.0 CHF−Factor3
0.5
0.0
2006 2008
Figure 4: Time-series trajectories of proportion of variances explained by factors.
12

3
0
Factor1 (f 1t )
−3
2006 2007 2008 2009
−3
3 Factor2 (f 2t )
0
2006 2007 2008 2009
3 Factor3 (f 3t )
0
−3
2006 2007 2008 2009
1.5 (w 1t )
1.0
0.5
1.0
0.5
2006 2007 2008 2009
(w 2t )
2006 2007 2008 2009
2
1
(w 3t )
2006 2007 2008 2009
Figure 5: Left panels: posterior means of factors fit. Right panels: posterior means (solid) and
95% credible intervals (dotted) of factor stochastic volatility wit = exp(λit/2).
2.0
1.5
1.0
0.5
GBP
EUR
JPY
CAD
AUD
CHF
2006 2007 2008 2009
Figure 6: Posterior means of series-specific stochastic volatility (σit = exp(hit/2)).
13

three relevant factors. Figure 8 shows results of time-varying loadings and sparsity, indicating that
the loadings of Factor4 are mostly shrunk in the entire periods. This finding implies that the LTM
structure can facilitate the data-driven model search in the factor models via shrinkage of time-
varying loadings.
Our experiences with various ordering of FX series in y t vectors are that the ordering re-
ported here yields more reasonable factor estimates with plausible implications than other ordering.
Therefore, we fix the ordering and the number of factors in the following portfolio analysis.
3 Factor1 (f 1t )
0
−3
2006 2007 2008 2009
3 Factor2 (f 2t )
0
−3
2006 2007 2008 2009
3 Factor3 (f 3t )
0
−3
2006 2007 2008 2009
3 Factor4 (f 4t )
0
−3
2006 2007 2008 2009
2 (w 1t )
1
2006 2007 2008 2009
2 (w 2t )
1
2006 2007 2008 2009
2 (w 3t )
1
2006 2007 2008 2009
2 (w 4t )
1
2006 2007 2008 2009
Figure 7: Results of 4 factors. Left panels: posterior means of factors fit. Right panels: posterior
means (solid) and 95% credible intervals (dotted) of factor stochastic volatility wit = exp(λit/2).
5 Portfolio allocation analysis
This section explores the forecasting performance of the LTDFMs in the context of dynamic
portfolio allocations. This portfolio analysis is related to some existing literature (Quintana (1992),
Putnum and Quintana (1994), Quintana and Putnum (1996), Aguilar and West (2000), Carvalho
and West (2007), Carvalho et al. (2011)). Our main focus here is the utility of the LTM structure
on the DFMs in forecast accuracy, considering how the dynamic sparsity/shrinkage in time-varying
factor loadings plays a relevant role in investment experiment. We compare six competing mod-
14

0.5
0.0
β2,1t (EUR−Factor1)
2006 2008
1.0
0.5
0
−2
2006
1.0
2008
0.5
1
0
1 0 0 0
β 3,1t (JPY−Factor1)
β 4,1t (CAD−Factor1)
2006 2008
1.0
0.5
2
0
2006 2008
1.0
0.5
1
0
2006 2008
1.0
0.5
1 0 0
β 3,2t (JPY−Factor2)
1.0
0.5
β4,2t (CAD−Factor2)
0.0
−0.5
2006 2008
1.0
0.5
β5,1t (AUD−Factor1)
1.0 β5,2t (AUD−Factor2)
β
1 6,1t (CHF−Factor1)
0
−1
2006 2008
1.0
0.5
0.5
2006 2008
1.0
0.5
1.5
1.0
β6,2t (CHF−Factor2)
2006 2008
1.0
0.5
2006 2008
1.0
0.5
1.0
0.5
1 0
β 4,3t (CAD−Factor3)
β 5,3t (AUD−Factor3)
2006 2008
1.0
0.5
1
0
β 6,3t (CHF−Factor3)
2006 2008
1.0
0.5
1
0.3
0.0
β5,4t (AUD−Factor4)
2006 2008
1.0
0.5
1
β 6,4t (CHF−Factor4)
0
2006 2008
1.0
Figure 8: Results of 4 factors. Posterior means (solid) and 95% credible intervals (dotted) of
time-varying factor loadings. Posterior probability of sit = 0 are plotted below each trajectory.
15
0.5

els by implementing sequential portfolio allocations based on forecast means and variances of FX
series.
5.1 Portfolio allocation rule
We base the following forecasting strategy and portfolio allocation rules. Suppose that we
initially have the first n = 880 observations of our FX data analyzed above. That is, in our FX data
set, we have the remaining time intervals, a total of 100 business days, as periods of investment
study. After observing the closing rates on the business day t − 1 (≥ n), one-step-ahead forecasts
for means and variances, denoted by (g t, Q t), of y t are computed via the MCMC based on draws
from one-step-ahead predictive posterior distribution using the available data, y 1:t−1. Resources
are allocated according to a vector of portfolio weights wt optimized by a specific allocation rule
described below. The realized portfolio return at time t is rt = w ′ ty t. The portfolio is reallocated
on each business day based on one-step-ahead forecasting computed via the MCMC given updated
data. We fix the total sum invested on each business day by restricting w ′ t1 = 1.
We use the traditional optimization technique for portfolio allocation originally developed by
Markowitz (1959). Given a fixed scalar return target m, we optimize the portfolio weights wt,
by minimizing the one-step ahead variance of returns among the restricted portfolios whose one-
step-ahead expectation is equal to m. Specifically, at time t, we minimize an ex-ante portfolio
variance w ′ tQtwt, subject to w ′ tgt = m, and w ′ t1 = 1. The solution of this quadratic minimization
problem is derived via Lagrange multipliers as w (m)
t = Q−1 t (atgt + bt1), where at = 1 ′ Q −1
t e, and
bt = −g ′ tQ −1
t e, where e = (1m − gt)/d, and d = (1 ′ Q −1
t 1)(g′ tQ −1
t gt) − (1 ′ Q −1
t gt) 2 . This optimal
portfolio is called as efficient frontier. In our analysis, we also consider a target-free minimum-
variance portfolio, whose solution is given by w∗ t = Q −1
t 1/(1′ Q −1
t 1). We implicitly assume that we
can freely reallocate the resources to arbitrary long or short positions across the currencies without
any transaction cost.
In addition to one-step-ahead prediction, we also examine the analysis from five-step-ahead
forecasts. Every five business days, the posterior predictive distribution of five-step horizons,
(y t, y t+1, . . . , y t+4), is computed via MCMC based on the available data y 1:t−1. This experiment
assumes a possible situation that investors allocate their resource every business day based on
weekly-updated forecasts.
5.2 Model comparisons
We consider the following three models from the class of LTDFM:
• LM-AF: Local means, and autoregressive factors.
• LM-IF: Local means, and time-independent factors (Φ = O).
16

• CM-IF: Constant means (ct ≡ c), and time-independent factors.
For each model of these three specification, our study involves the LT (latent threshold) and NT
(non-threshold) assumption; total six competing models are analyzed over 100 business days. For
CM-IF model, an additional prior is assumed: c ∼ N(0, I). The other prior specifications and
simulation size are as in the previous analysis. We use daily target returns of m = 0.02%, 0.04%,
and 0.06%, corresponding to a monthly (20 day) return of approximately 0.4%, 0.8% and 1.2%,
respectively.
Table 2 reports cumulative returns of the portfolios resulting from the sequential investment
over 100 business days. It is evident that the use of LT structure remarkably dominates the NT
models regardless of other aspects of model specification or the portfolio allocation rules. Figure
9 displays the cumulative returns across time periods from one-step-ahead forecasting. LT models
clearly outperform NT models and their difference is larger in the mean-target strategy of m =
0.06% than in the target-free minimum-variance portfolio. A sudden loss of CM-IF-NT model in
the late periods of the mean-target analysis would indicate a weak point of the constant mean,
independent factor model, not fully reacting a rapid change of market circumstance. By contrast.
it is remarkable that CM-IF-LT model avoids such a significant drop due to a reasonably flexible
shrinkage in the loading matrix that may also reduce uncertainty of the time-varying parameters,
resulting in the improvement of forecast performance.
LM-AF LM-IF CM-IF
NT LT NT LT NT LT
(i) One-step-ahead forecasts
Target return: m = 0.02 -1.08 1.25 1.18 3.11 -3.03 2.06
0.04 -2.20 1.52 -0.33 2.55 -4.85 1.61
0.06 -3.31 1.78 -1.85 1.99 -6.67 1.15
Target-free
(ii) Five-step-ahead forecasts
-1.42 2.11 -1.26 0.98 -1.12 -0.04
Target return: m = 0.02 -1.26 2.09 -1.54 0.95 -0.55 1.20
0.04 -0.96 1.88 -1.33 1.03 -3.12 0.82
0.06 -0.66 1.68 -1.12 1.11 -5.69 0.45
Target-free -0.98 0.52 -0.55 1.25 -0.53 0.50
Table 2: Cumulative returns (%) over 100 business days.
Another important insight of portfolio allocation analysis is a level of portfolio risk. Figure 10
plots an ex-ante portfolio standard deviation derived by (w ′ tQ twt) 1/2 for the optimizing weight
wt, from one-step-ahead forecasting of LM-AF models. LT models clearly yield smaller variance of
optimizing portfolio than NT models. The risk ratio, based on these portfolio standard deviation,
for NT model relative to LT model at each time point is also displayed in Figure 10 (right panels);
17

(%)
5
0
−5
5
0
(a) Mean−target (m = 0.06%)
LM−AF−NT
LM−IF−NT
CM−IF−NT
LM−AF−LT
LM−IF−LT
CM−IF−LT
0 10 20 30 40 50 60 70 80 90 100
(b) Target−free
0 10 20 30 40 50 60 70 80 90 100
Figure 9: Cumulative returns over 100 business days based on one-step-ahead forecasting for (a)
portfolio under the target returns of 0.06% and (b) target-free minimum-variance portfolio.
18

LT models provide averagely 20% less risky portfolios. Table 3 reports average risk ratio of full
analyses, indicating that the portfolio variances are likely smaller in the use of LT models, and the
ratio tends to be larger in rather more flexible specifications, LM-AF and LM-IF models, than in
CM-IF models. These evidence implies that the LTM structure plays a considerably relevant role in
reducing predicted investment risk on the portfolio allocation.
0.8
0.6
0.4
0.5
0.4
0.3
(a) Mean−target (m = 0.06%)
NT
LT
0 20 40 60 80 100
30
20
10
NT/LT
0 20 40 60 80 100 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
(b) Target−free
30
20
10
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Figure 10: Portfolio risk: Ex-ante portfolio standard deviation, (w ′ tQ twt) 1/2 based on one-stepahead
forecasting of LM-AF models for (a) the portfolio under the target return of m = 0.06%
(top), and (b) the target-free minimum-variance portfolio (bottom). Time trajectory (left panels)
and histogram of the risk ratio for NT model relative to LT model (right panels). The vertical line
in the histogram refers to the ratio of one.
A further look in the optimized portfolios is the trajectories of portfolio weights based on one-
step-ahead and five-step-ahead forecasting, displayed in Figures 11 and 12, respectively. In NT
model, the allocation is mainly led by a long position of EUR and short position of CHF. This
reflects quite a high correlation between these two currencies in the estimated forecast variance
matrix. By contrast, LT model provides milder degree of correlation between those two currencies,
yielding smaller proportion of resources allocated to the entire currencies. The lower degree of
variation in portfolio weights resulting from the LT models leads to more stable portfolios, which
would also reduce transaction costs in reality.
19

0.5
0.0
0.5
0.0
GBP
CAD
0.5 GBP
0.0
0.4 CAD
0.2
0.0
NT
LT
50 100
50 100
NT
LT
(a) Mean-target (m = 0.06%)
2
0
−0.5
50 100
50 100
0.5
0.0
1
0
0.3
0.0
−0.3
EUR
AUD
50 100
50 100
(b) Target-free
EUR
AUD
50 100
50 100
0.5
0.0
0
−1
0.5
0.0
0
−1
JPY
CHF
JPY
CHF
50 100
50 100
50 100
50 100
Figure 11: Trajectories of portfolio weights based on one-step-ahead forecasts of the LM-AF models
for (a) the portfolio under the target return of m = 0.06% (upper), and (b) the target-free
minimum-variance portfolio (lower).
20

0.5 GBP
0.0
0.5
0.0
0.4
0.0
CAD
GBP
0.4 CAD
0.2
0.0
NT
LT
50 100
50 100
NT
LT
(a) Mean-target (m = 0.06%)
1
0
0.5
0.0
EUR
AUD
50 100
0.3 AUD
50 100
1.5
1.0
0.5
0.0
0.0
−0.3
0.5
0.0
JPY
50 100
1 CHF
50 100
(b) Target-free
EUR
0
−1
0.5
0.0
50 100
0.5
0.0
−0.5
−1.0
50 100
JPY
CHF
50 100
50 100
50 100
50 100
Figure 12: Trajectories of portfolio weights based on five-step-ahead forecasts of the LM-AF models
for (a) the portfolio under the target return of m = 0.06% (upper), and (b) the target-free
minimum-variance portfolio (lower).
21

LM-AF LM-IF CM-IF
(i) One-step-ahead forecasts
Target return: m = 0.02 1.033 1.007 1.185
0.04 1.149 1.187 1.062
0.06 1.232 1.407 0.969
Target-free
(ii) Five-step-ahead forecasts
1.234 1.205 1.193
Target return: m = 0.02 1.067 1.027 1.178
0.04 1.158 1.102 1.058
0.06 1.223 1.150 0.979
Target-free 1.249 1.224 1.172
Table 3: Portfolio risk ratio: ex-ante ratio of portfolio standard deviation under the NT models
relative to the LT models, averaged across 100 business days.
6 Concluding remarks
We proposed the flexible structure of latent thresholding for time-varying loading matrix in
context of dynamic factor models. Our substantive example of model fit and sequential portfolio
allocation analysis using FX return series revealed the considerable utility of the LTM structure that
plausibly dynamically selecting potential loadings in the factor analysis, eliminating unnecessary
fluctuations of time-varying parameters as well as improving forecasting performance and model
implication. There are a number of methodological and computational areas for further investiga-
tion. Among them, we note the potential of more elaborate factor models such as factor-augmented
VAR (FAVAR, Bernanke et al. (2005), Baumeister et al. (2010)) and other methodological approach
including particle filter (also known as sequential Monte Carlo) and particle learning algorithm
(Carvalho et al. (2010)) for the LTDFMs.
22

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