2.1. Model Specification

For each point in time t = 1, …, T, let ${\mathit{}\mathbf{y}}_t={(y_{1t},\dots ,y_{mt})}^{{}^{\text{'}}}$ be a zero-mean vector of m observed returns and let ${\mathit{}\mathbf{f}}_t={(f_{1t},\dots ,f_{rt})}^{{}^{\text{'}}}$ be a vector of r unobserved latent factors. In analogy to the static factor model, the observations are assumed to be driven by the latent factors and the idiosyncratic innovations. In the case of the factor stochastic volatility model, however, both the idiosyncratic innovations as well as the latent factors are allowed to have time-varying variances, depending on m + r latent volatilities h_t = (h^U_t, h^V_t), where ${\mathit{h}}_t^U={(h_{1t},\dots ,h_{mt})}^{{}^{\text{'}}}$ and ${\mathit{h}}_t^V={(h_{m+1,t},\dots ,h_{m+r,t})}^{{}^{\text{'}}}$. In short, we have (1) $$\begin{array}{c}\hfill {\mathit{}\mathbf{y}}_t=\mathit{}\mathbf{\Lambda }{\mathit{}\mathbf{f}}_t+{\mathit{}\mathbf{U}}_t{\left({\mathit{h}}_t^U\right)}^{1/2}{\mathit{}\mathbf{ϵ}}_t,{\mathit{}\mathbf{f}}_t={\mathit{}\mathbf{V}}_t{\left({\mathit{h}}_t^V\right)}^{1/2}{\mathit{}\mathbf{\zeta }}_t,\end{array}$$(1) where Λ is an unknown m × r factor loadings matrix, ${\mathit{}\mathbf{U}}_t\left({\mathit{h}}_t^U\right)=(exp\left(h_{1t}\right),\dots ,exp\left(h_{mt}\right))$ is a diagonal m × m matrix containing the idiosyncratic (series-specific) variances, and ${\mathit{}\mathbf{V}}_t\left({\mathit{h}}_t^V\right)=(exp\left(h_{m+1,t}\right),\dots ,exp\left(h_{m+r,t}\right))$ is a diagonal r × r matrix containing the factor variances. These variances are themselves modeled as latent variables whose logarithms follow independent autoregressive processes of order one, that is, for i = 1, …, m + r: (2) $$\begin{array}{c}\hfill h_{it}={\mu }_i+{\varphi }_i(h_{i,t-1}-{\mu }_i)+{\sigma }_i{\eta }_{it},\end{array}$$(2) with unknown initial value h_i0.

All innovations are assumed to follow independent standard normal distributions, that is, ${\mathit{}\mathbf{ϵ}}_t\sim {\mathcal{N}}_m(\mathit{}\mathbf{0},{\mathit{I}}_m)$, ${\mathit{}\mathbf{\zeta }}_t\sim {\mathcal{N}}_r(\mathit{}\mathbf{0},{\mathit{I}}_r)$, and ${\mathit{}\mathbf{\eta }}_t\sim {\mathcal{N}}_{m+r}(\mathit{}\mathbf{0},{\mathit{I}}_{m+r})$, where η_t = (η_1t, …, η_{m + r, t})′. This implies the following structure: (3) $$\begin{array}{c}\hfill {\mathit{}\mathbf{y}}_t=\mathit{}\mathbf{\Lambda }{\mathit{}\mathbf{f}}_t+{\mathit{}\epsilon }_t,{\mathit{}\mathbf{f}}_t|{\mathit{h}}_t\sim {\mathcal{N}}_r\left(\mathit{}\mathbf{0},{\mathit{}\mathbf{V}}_t\left({\mathit{h}}_t^V\right)\right),\end{array}$$(3) with ${\mathit{}\epsilon }_t|{\mathit{h}}_t\sim {\mathcal{N}}_m(\mathit{}\mathbf{0},{\mathit{}\mathbf{U}}_t\left({\mathit{h}}_t\right))$. One of the main reasons for estimating a factor SV model is to reliably estimate the potentially time-varying conditional covariance matrix of y_t which, for the model at hand, is given by $\text{cov}\left({\mathit{}\mathbf{y}}_t\right|{\mathit{h}}_t)={\mathit{}\mathbf{\Sigma }}_t\left({\mathit{h}}_t\right)=\mathit{}\mathbf{\Lambda }{\mathit{}\mathbf{V}}_t\left({\mathit{h}}_t^V\right){\mathit{}\mathbf{\Lambda }}^{\text{'}}+{\mathit{}\mathbf{U}}_t\left({\mathit{h}}_t^U\right)$. Note that because U_t(h^U_t) is diagonal, all covariances between the component series are governed by the latent factors. Marginally with respect to h_t, y_t is a process with a non-Gaussian stationary distribution.

2.2. Identification Issues

Whenever certain combinations of parameter values result in (almost) identical maxima in the likelihood function, estimation of the corresponding parameter values from data can become impossible. Consequently, observationally equivalent parameter constellations must be ruled out for reliable statistical inference and a large body of literature dealing with this issue has arisen. In particular, Frühwirth-Schnatter and Lopes (2017) gave an overview of recent advances in the context of static Bayesian factor models, and Sentana and Fiorentini (2001) specifically discussed identification for models where the factors exhibit conditional heteroscedasticity (but innovations are assumed to be homoscedastic). If not dealt with properly, usually through certain restrictions on the parameter space, sensible interpretation of the posterior distribution is not possible (“nonidentifiability”). In less severe cases (“near-nonidentifiability”), MCMC algorithms and other estimation procedures often lack convergence and thus provide unreliable results. For the model at hand, we face several issues related to this problem.

First, to prevent factor rotation and column switching, one option is to follow the usual convention and set the upper triangular part of Λ to zero and $\left(\mathit{}\mathbf{\Lambda }\right)$ nonzero (e.g., Geweke and Zhou 1996). Doing so, however, imposes an—often unwanted—order dependence. We therefore also discuss the possibility to leave the factor loadings matrix unrestricted and deal with column switching through post-processing of the MCMC draws.

Second, without identifying the scaling of either the jth column of Λ or the variance of f_jt, the model is not identified. The usual remedy (e.g., Aguilar and West 2000; Chib, Nardari, and Shephard 2006; Han 2006; Lopes and Carvalho 2007; Nakajima and West 2013; Zhou, Nakajima, and West 2014) is that the diagonal loading elements in model (3) are fixed to one, that is, Λ_jj = 1, for j = 1, …, r, while the level μ_{m + j} of the factor volatilities h_{m + j, t} in model (2) (which corresponds to the scaling of f_jt) is modeled to be unknown. This approach implies that the first r variables are leading the factors and thus makes variable ordering an even more important modeling decision. To alleviate this issue, we leave the diagonal elements Λ_jj in model (3) unrestricted, an intuitive interpretation being that “leadership” of a factor can be shared by several series. Instead, we fix the level μ_{m + j} of the factor volatilities h_{m + j, t} at zero: (4) $$\begin{array}{ccc}\hfill h_{it}& =& (1-{\varphi }_i){\mu }_i+{\varphi }_i{h}_{i,t-1}+{\sigma }_i{\eta }_{it},i=1,\dots ,m,\hfill \\ \hfill h_{m+j,t}& =& {\varphi }_{m+j}{h}_{m+j,t-1}+{\sigma }_{m+j}{\eta }_{m+j,t},j=1,\dots ,r.\hfill \end{array}$$(4) This assumption, alongside the prior distribution on the loadings introduced in Section 3.1, identifies the factor variance.

Finally, each column of Λ is only identified up to a possible sign switch. We deal with this (lightweight) identification issue a posteriori, meaning that we run our MCMC sampler in the unrestricted model and identify signs afterward, see Section A.4 in Appendix A for details.

Factor model (3) together with the m + r SV models (4) defines our baseline parameterization, however alternative parameterizations will be exploited in Section 3.3 in the context of efficient MCMC estimation of the factor SV model.

3.1. Prior Distributions

Independently for each i ∈ {1, …, m + r}, priors for the univariate SV processes are chosen as in Kastner and Frühwirth-Schnatter (2014): p(μ_i, φ_i, σ_i) = p(μ_i)p(φ_i)p(σ_i), where the level ${\mu }_i\in \mathbb{R}$ is equipped with the usual normal prior ${\mu }_i\sim \mathcal{N}(b_{\mu },B_{\mu })$, the persistence parameter φ_i ∈ ( − 1, 1) is chosen according to $({\varphi }_i+1)/2\sim \mathcal{B}(a_0,b_0)$ as in Kim, Shephard, and Chib (1998), and the volatility of log variance ${\sigma }_i\in {\mathbb{R}}^{+}$ is implied by ${\sigma }_i^2\sim B_{\sigma }\times {\chi }_1^2=\mathcal{G}(\frac{1}{2},\frac{1}{2B_{\sigma }})$. The initial state h_i0 is distributed according to the stationary distribution of the AR(1) process (2), that is, $h_{i0}|{\mu }_i,{\varphi }_i,{\sigma }_i\sim \mathcal{N}({\mu }_i,{\sigma }_i^2/(1-{\varphi }_i^2))$. For every unrestricted element of the factor loadings matrix, we choose independent zero-mean Gaussian distributions, that is, ${\Lambda }_{ij}\sim \mathcal{N}(0,B_{\Lambda })$.

3.2. Full Conditional MCMC Estimation

Bayesian inference operates directly in the latent variable model (3) and (4) and relies on data augmentation by introducing the latent volatilities h = {h_{i, •}}, i = 1, …, m + r, where h_{i, •} = (h_i0, h_i1, …, h_iT)′, and the latent factors f = {f_{j, •}}, j = 1, …, r, where f_{j, •} = (f_j1, …, f_jT)′, as latent data. This allows to set up a simple scheme for full conditional MCMC sampling which is outlined in Algorithm 1 and discussed in detail thereafter.

Algorithm 1. Choose appropriate starting values for μ_i, i ∈ {1, …, m}, φ_i and σ_i, i ∈ {1, …, m + r}, as well as Λ, h, and f and repeat the following steps:

For Step (a), observe that conditional on knowing the latent factors f and the loadings Λ, we are dealing with m + r independent, univariate SV models where the latent state equations in (4) are combined with following observation equations: (5) $$\begin{array}{c}\hfill log{(y_{it}-{\mathit{}\mathbf{\Lambda }}_{i,•}{\mathit{}\mathbf{f}}_t)}^2=h_{it}+log{ϵ}_{it}^2,i=1,\dots ,m,\end{array}$$(5) (6) $$\begin{array}{c}\hfill log{f}_{jt}^2=h_{m+j,t}+log{\zeta }_{jt}^2,j=1,\dots ,r.\end{array}$$(6) Hence, sampling the latent volatilities h_{i, •} as well as the parameters (μ_i, φ_i, σ_i) for i = 1, …, m + r (with μ_i = 0 for i > m) in Step (a) amounts to m + r univariate SV updates. Consequently, the substantial amount of research on this matter which has emerged in the last two decades can directly be applied. In particular, we follow recent findings in Kastner and Frühwirth-Schnatter (2014), where an efficient sampling scheme is proposed and evaluated, and simply use the implementation in the R package stochvol (Kastner 2016a) as a “plug-in” for Step (a) of the factor SV sampler presented in Algorithm 1; see Appendix A.1 for more details and additional references on MCMC estimation for univariate SV models.

On the other hand, conditional on knowing the latent volatilities h, we are dealing in (3) with a factor model with heteroscedastic errors. Nevertheless, given h, f and Λ may be sampled conditionally on each other from the respective multivariate normal distributions in a similar manner as for a standard factor model (Lopes and West 2004). This approach is conceptually straightforward, see Appendix A.2 for details how to sample in Step (b) each row Λ_{i, •} of the factor loading matrix from Λ_{i, •}|f, y_{i, •}, h_{i, •}, where y_{i, •} = (y_i1, …, y_iT)′, and Appendix A.3 for details how to sample in Step (c) the factor f_t from f_t|Λ, y_t, h_t for t = 1, …, T.

After discarding a certain amount of initial draws (the burn-in), the standard full conditional sampler iterates steps (a)–(c) of Algorithm 1, but not (b*), and should, in principle, yield draws from the joint posterior distribution. However, when estimating factor SV models through such an MCMC scheme, slow convergence and poor mixing (i.e., high correlation of posterior draws) can become a potentially prohibitive issue. This phenomenon substantiates in enormous autocorrelation of posterior draws—even after thinning—and can render MCMC output practically useless. For certain datasets, the burn-in phase may take extremely long and a huge amount of samples has to be discarded before the draws can be considered to emerge from the posterior distribution. Additionally, even after burn-in, these draws often show extraordinarily high autocorrelation and thus only explore the target distribution painstakingly slowly. These so-called badly mixing samplers do not only prolong computation time, they also frequently lead to unreliable estimates and misleading results. The simulation study in Section 4 illustrates that this can happen for the standard full conditional sampler even with data simulated from the true model, see, for example, the top of the two panels in Figure 1. Consequently, a carefully crafted posterior simulator is of utmost importance.

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

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Gregor Kastner http://orcid.org/0000-0002-8237-8271, Sylvia Frühwirth-Schnatter & Hedibert Freitas Lopes

https://doi.org/10.1080/10618600.2017.1322091

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Figure 1. Trace plots of 10,000 draws from p(Λ₁₁|y) (left-hand side) and empirical autocorrelation functions of all 5000000 draws (right-hand side) obtained via the standard sampler (top), shallow interweaving (middle), and deep interweaving (bottom).

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To overcome this problem, Chib, Nardari, and Shephard (2006) proposed to sample the factor loading matrix Λ from the marginalized conditional posterior p(Λ|y, h), without conditioning on the factors f. This distribution, however, is not available in closed form, and to sample from it requires a rather involved Metropolis-Hastings update where the proposal distribution is based on numerically maximizing the often high-dimensional conditional likelihood function and approximating its Hessian matrix at every MCMC iteration. To avoid this potential bottleneck, we employ the simpler full conditional procedure outlined in Algorithm 1 but enhance it in Step (b^*) by employing two variants of an ancillarity-sufficiency interweaving strategy (ASIS; Yu and Meng 2011), called shallow interweaving and deep interweaving, which are explained in detail in Section 3.3.

Applications to simulated data in Section 4 as well as to exchange rate data in Section 5 illustrate how adding Step (b^*) boosts MCMC dramatically, in particular for deep interweaving; compare, for example, the top panel in Figure 1 to the remaining panels. Appendix A.5 provides comments on practical implementation of the boosted Algorithm 1 using the R package factorstochvol (Kastner 2016b).

3.3. Boosting Full Conditional MCMC Through Interweaving

As discussed in Section 3.2, the standard full conditional sampler outlined in Algorithm 1 is based on data augmentation in the parameterization (3) and (4) of the factor SV model and suffers from slow convergence like so many other MCMC schemes which alternate between sampling from the full conditionals of the latent states and the model parameters. A large literature has emerged discussing various techniques to improve such algorithms, in particular reparameterization (Papaspiliopoulos, Roberts, and Sköld 2007), marginal data augmentation (van Dyk and Meng 2001), and interweaving strategies (Yu and Meng 2011).

Reparameterization relies on data augmentation in a different parameterization of the model with alternative latent variables. In particular, so-called noncentered parameterizations, where unknown model parameters are moved from the latent state equation to the observation equation, proved to be useful, see, for example, Frühwirth-Schnatter and Wagner (2010) in the context of state-space modeling of time series. However, MCMC estimation based on different data augmentation schemes will often be efficient in separate regions of the parameter space, as demonstrated, for example, by Kastner and Frühwirth-Schnatter (2014) in the context of univariate SV models. This suggests to combine different data augmentation schemes to obtain an improved sampler.

Marginal data augmentation employs a randomly sampled “working parameter” to transform the baseline parameterization to an expanded, unidentified latent variable model in which the model parameters are updated conditional on the (randomly) transformed latent variables. This technique has been applied to the basic factor model, using the undefined scaling of the factors as a working parameter (Ghosh and Dunson 2009; Frühwirth-Schnatter and Lopes 2017), however, it is not easily extended to factor SV models, in particular if the latent volatilities should be part of the acceleration scheme.

The ancillarity-sufficiency interweaving strategy (ASIS), introduced by Yu and Meng (2011), provides another principled way to interweave two different data augmentation schemes by resampling certain parameters conditional on the latent variables in an alternative parameterization of the model, thereby combining “best of different worlds.” ASIS has been successfully employed in a variety of contexts such as univariate SV models (Kastner and Frühwirth-Schnatter 2014) and dynamic linear state-space models (Simpson 2015; Simpson, Niemi, and Roy 2017). To boost Algorithm 1, we apply ASIS to the factor SV model in the present article. Two interweaving strategies—called shallow interweaving and deep interweaving—are derived in Section 3.3.1, where the diagonal elements Λ₁₁, …, Λ_rr of the factor loadings matrix are resampled in Step (b*) in two alternative parameterizations of the model.

As will become clear in the following sections, deep interweaving typically yields the highest sampling efficiency gains and is thus the generally recommended strategy. However, also shallow interweaving has its merits. First, being conditionally conjugate, it is somewhat easier to implement. Second, it can be applied also to static factor models which are by construction not suited for deep interweaving (see also Bitto and Frühwirth-Schnatter 2016).

5.1. Model Specification

For selecting the number of factors in this application, it is important to keep in mind the primary purpose of the analysis. Different sampling strategies are applied, depending on whether identification of Λ is of no concern (e.g., for covariance matrix prediction only), or whether identification is instrumental for understanding the unobserved, underlying factors.

For the first case, we experimented with fitting unrestricted models to the exchange rates data. This implies that the method is completely invariant to series ordering and there are no model-implied “leading factors” as is usually the case. With respect to selecting the number of factors, we found that higher-order models without any restriction on the factor loadings matrix yield higher marginal likelihoods and are thus recommended. Figure 4 illustrates this via log predictive Bayes factors.

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

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Gregor Kastner http://orcid.org/0000-0002-8237-8271, Sylvia Frühwirth-Schnatter & Hedibert Freitas Lopes

https://doi.org/10.1080/10618600.2017.1322091

Published online:

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Figure 4. Log predictive Bayes factors in favor of the r-factor model over the no-factor model. The first 1000 returns in the dataset are treated as prior information and log one-day-ahead predictive likelihoods are accumulated over the following 1000 days. Circles connected with a solid line indicate values obtained with completely unrestricted loadings matrices; triangles connected with dashed lines indicate values, where the loadings matrix is restricted to be lower triangular. The component series are ordered alphabetically.

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If identification is warranted, an important step is the appropriate ordering of the variables, before the usual lower triangular structure is imposed on the factor loadings matrix to guarantee mathematical identifiability, as outlined in Section 2.2. This, however, makes inference on the factor loadings matrix dependent on the appropriate ordering of the variables. We exemplify this by the predictive Bayes factors for models, where the component series are ordered alphabetically and the lower triangular structure is imposed on the first three series appearing in Table 3, namely AUD, CAD, and CHF. As shown in Figure 4, for a given number of factors r, the log predictive Bayes factors of the constrained models are consistently smaller than for the unrestricted models, indicating that a purely mathematical identifiability constraint may be in conflict with the data.

Also for the constrained models, the log predictive Bayes factors are ever increasing for the exchange rate data. However, it stands out that the relative gain per additional factor are highest for few factors, flattening out quickly. Furthermore, draws of the factor loading matrix in these higher-order models are difficult to identify, in particular, if the lower triangular constraint is in conflict with the data and spurious factors, that is, factors which are significantly loaded on by only few series, are present (see Frühwirth-Schnatter and Lopes 2017, for a detailed discussion of this issue in the static factor context). Thus, to keep presentation feasible and to avoid spurious factors, we restrict ourselves to a model with r = 4 factors for the following in-depth discussion.

One way to find constraints that are not in conflict with the data is to post-process the MCMC draws of an unrestricted sampler with r = 4 factors, a method that has been applied in Conti et al. (2014) and Aßmann, Boysen-Hogrefe, and Pape (2016). Note that rather than reordering the variables before imposing a lower triangular constraint, we can choose three (out of the 26) currencies, and impose the r(r − 1)/2 = 6 zero restrictions on the corresponding factor loadings, see, for example, Dunn (1973). While the choice of these currencies is not unique, inference is robust to specific choices, as long as the corresponding currencies serve as “leaders” for specific factors. This is exemplified by Figure C.3 which displays the posterior median of the MCMC draws of the factor loadings obtained from an unrestricted sampler with r = 4. To solve column switching, the columns of Λ are rearranged by the size of their maximum median loading. According to Figure C.3 in Appendix C, USD is a definite candidate to lead factor one, PLN leads a second factor, and AUD leads a third factor. Alternatively, identification could be based on any other currency strongly loading on factor 1 (such as HKD or CNY), in combination with HUF (instead of PLN) and NZD (instead of AUD).

Prior hyperparameters are the same as for the simulation study in Section 4. A sensitivity analysis shows that none of the hyperparameter choices turn out to be very influential in this particular application with the exception of the prior factor loadings variances $B_{\Lambda }$. These, however, are only important for the absolute scaling of the factors and do not notably influence the relative loadings sizes or predictive Bayes factors. We run each sampler for 550,000 iterations, discard the first 50,000 draws as burn-in, leaving 500,000 which we use for posterior inference. Even after this substantial amount of iterations, it is not clear that the sampler without interweaving has properly converged; we therefore omit its presentation. IFs from the interwoven samplers are presented in Table C.1 in Appendix C.

Finally, we identify the signs of the loadings in the post-processing phase by investigating the MCMC draws. For each factor, the series whose posterior absolute loadings distribution is furthest away from zero is assigned a positive sign, the other loadings are aligned thereafter, see also Section A.4 in Appendix A.

5.2. Posterior Factor Volatilities and Their Loadings

We begin by discussing the log-variances of the latent factors, visualized in Figure 5, alongside the corresponding factor loadings whose posterior distribution is depicted in Figure 6 and whose posterior means are listed in Table 4.

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

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Gregor Kastner http://orcid.org/0000-0002-8237-8271, Sylvia Frühwirth-Schnatter & Hedibert Freitas Lopes

https://doi.org/10.1080/10618600.2017.1322091

Published online:

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Figure 5. Marginal posteriors of the factor log-variances h_{m + j, t}, j = 1, …, 4 ($\text{mean}\pm 2\times \text{sd}$).

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Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

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Gregor Kastner http://orcid.org/0000-0002-8237-8271, Sylvia Frühwirth-Schnatter & Hedibert Freitas Lopes

https://doi.org/10.1080/10618600.2017.1322091

Published online:

11 October 2017

Figure 6. Marginal posterior distribution of the factor loadings, visualized through arbitrarily colored scatterplots of MCMC draws.

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The first factor can clearly be interpreted as the USD-driven one, as the pegged triplet USD, CNY, and HKD loads very highly on this factor, alongside many other currencies. Its volatility is generally very smooth, rising in the aftermath of the 2008 financial crisis and going down again after 2009; a second increase can be seen in the second half of 2014, possibly in connection with the Greek government-debt crisis. Factor 2’s log-variance appears slightly less persistent and more volatile, it is driven by ZAR, the only African currency in the sample, alongside Eastern Europe’s/Southwestern Asia’s HUF, PLN and TRY. Interestingly, JPY loads negatively on this factor.

The third factor shows a similar overall pattern as the first. The highest loading series for this factor are AUD and NZD, emphasizing the Trans–Tasman relations. Other commodity currencies such as ZAR and CAD also load highly on this factor.

Factor 4 is clearly driven by the currencies of the Tiger Cub economies such as MYR, KRW, PHP, and SGD.

5.3. Posterior Volatilities and Correlations

In order not to overload the graphical displays used to visualize the results of the analysis, we display the results for a two-year period only for the rest of this section. More specifically, we look at the years 2008 and 2009, covering the most volatile span during the financial crisis. Irrespectively of that, the full dataset has been used for estimation and other time spans could be displayed analogously.

We start out by visualizing the marginal posterior means of univariate volatilities for all 26 currencies in Figure 7 from the last day of 2007 until the last day of 2009. Series such as DKK or HRK are (very) closely pegged to EUR and unsurprisingly show very low volatility throughout the crisis. Other European currencies (CHF¹, RON, SEK, CZK) follow suit. Tiger Cub economies such as PHP, HKD, THB, and MYR align very closely with USD and CNY. The most volatile currencies during this period are KRW, ZAR, IDR, JPY, and also TRY, followed by NZD, AUD, and CAD. Overall, it stands out that even though some series-specific ups and downs can be spotted, a common trend is clearly visible.

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

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Gregor Kastner http://orcid.org/0000-0002-8237-8271, Sylvia Frühwirth-Schnatter & Hedibert Freitas Lopes

https://doi.org/10.1080/10618600.2017.1322091

Published online:

11 October 2017

Figure 7. Posterior volatilities of exchange rate log returns with respect to EUR, from the last day of 2007 until the last day of 2009 ($\text{mean}\pm 2\times \text{sd}$).

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Next, implied correlation matrices are displayed in Figure 8, exemplified for the last day of 2007, 2008, and 2009. Additionally to displaying the mean posterior pairwise correlations (at the given dates) via color and shading, these plots visualize posterior uncertainty; the outer and inner circles’ sizes correspond to $\text{posterior}\text{mean}\pm 2\text{standard}\text{deviations}$, respectively. The images were generated using the R package corrplot (Wei 2016); its option hclust (hierarchical clustering) was used for ordering the series to emphasize the blocks of currencies.

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

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Gregor Kastner http://orcid.org/0000-0002-8237-8271, Sylvia Frühwirth-Schnatter & Hedibert Freitas Lopes

https://doi.org/10.1080/10618600.2017.1322091

Published online:

11 October 2017

Figure 8. Posterior correlation matrices on the last trading days of 2006, 2007, and 2008. For each element, the size of the outer/inner circle is determined by the posterior mean plus/minus two posterior standard deviations, thereby indicating posterior uncertainty. Color and opacity are determined by the posterior mean. The remaining days are visualized in a video to be found online at https://vimeo.com/212887492.

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To further illustrate variability over time, we determine the posterior means of the pairwise time-varying correlations of USD against the other currencies which are plotted in Figure C.2 in Appendix C. As was to be expected, correlations of CNY and HKD with USD are almost always very close to one; IDR, THB, SGD, MYR, and PHP show rather high correlation throughout. The correlation between USD and RUB on the other hand falls from around 0.9 in early 2008 to around 0.4 in late 2009, whereas THB moves in the opposite direction; its correlation with USD is around 0.5 at the beginning of the time window and increases quickly to around 0.9. Eastern European non-euro currencies, in particular PLN and HUF, appear to be slightly negatively correlation with USD throughout the entire period.