On the contribution of international shocks in Australian business cycle fluctuations

Abstract

What proportion of Australian business cycle fluctuations are caused by international shocks? We address this question by estimating a panel VAR model that has time-varying parameters and a common stochastic volatility factor. The time-varying parameters capture the inter-temporal nature of Australia’s various bilateral trade relationships, while the common stochastic volatility factor captures various episodes of volatility clustering among macroeconomic shocks, e.g., the 1997/98 Asian Financial Crisis and the 2007/08 Global Financial Crisis. Our main result is that international shocks from Australia’s five largest trading partners: China, Japan, the EU, the USA and the Republic of Korea, have caused around half of all Australian business cycle fluctuations over the past two decades. We also find important changes in the relative importance of each country’s economic impact. For instance, China’s positive contribution increased throughout the mining boom of the 2000s, while the overall US influence has almost halved since the 1990s.

Appendix A. Appendices

1.1 Appendix A.1 Markov Chain Monte Carlo algorithm

In appendix, we present the Markov chain Monte Carlo (MCMC) algorithm used to estimate the TVP-PVAR model. All alternative models considered in this paper are nested versions of this model and can therefore be estimated by straightforward modifications of the MCMC procedure. To detail the MCMC procedure, let $\varvec{Y}=(\varvec{Y}_{1},\ldots ,\varvec{Y}_{T})'$, $\varvec{\theta }=(\varvec{\theta }_{1},\ldots ,\varvec{\theta }_{T})'$, $\varvec{h}=(\varvec{h}_{1},\ldots ,\varvec{h}_{T})'$ and $\varvec{\lambda }=(\lambda _{1},\ldots ,\lambda _{T})'$. The posterior draws are obtained through a six-block Metropolis-within-Gibbs sampler that sequentially samples each variable from their respective full conditional distribution:

• 1. Draw from $p\left( \varvec{\theta }\mid {\mathbf {Y}},{\mathbf {h}},\varvec{\Sigma }_{u},\varvec{\Omega },\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

• 2. Draw from $p\left( \varvec{\Sigma }_{u}\mid {\mathbf {Y}},\varvec{\theta },{\mathbf {h}},\varvec{\Omega },\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

• 3. Draw from $p\left( \varvec{\Omega }\mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},{\mathbf {h}},\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

• 4. Draw from $p\left( {\mathbf {h}}\mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},\varvec{\Omega },\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

• 5. Draw from $p\left( \rho \mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},{\mathbf {h}},\varvec{\Omega },\sigma _{h}^{2},\varvec{\kappa }\right)$

• 6. Draw from $p\left( \sigma _{h}^{2}\mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},{\mathbf {h}},\varvec{\Omega },\rho ,\varvec{\kappa }\right)$

• 7. Draw from $p\left( \varvec{\kappa }\mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},{\mathbf {h}},\varvec{\Omega },\rho ,\sigma _{h}^{2}\right)$

In our analysis, we use 35,000 posterior draws, discarding the first 15,000 to allow for convergence of the Markov chain to its stationary distribution. Under the previously defined conjugate priors, Steps 2, 3 and 6 can be directly sampled from their resulting posterior distributions. Next, following Canova et al. (2007, 2012), the latent states in Step 1 can be sampled using standard Kalman filter algorithms as in Chib and Greenberg (1995). In this paper, we instead follow Poon (2018b) and make use of an efficient precision sampling algorithm. The increased efficiency of the algorithm comes from exploiting the fact that the precision matrices of the latent states are block banded and sparse. This means that computational savings can be made in necessary operations when solving linear systems—such as taking a Cholesky decomposition, matrix multiplication and forward–backward substitution. Next, Step 4 involves a nonlinear non-Gaussian measurement equation, and the standard linear Kalman filter cannot be applied. To overcome this issue, we follow Poon (2018a) and make use of the auxiliary mixture sampler developed by Kim et al. (1998) along with an efficient sampling algorithm in Chan and Hsiao (2014). The auxiliary mixture sampler uses a seven-Gaussian mixture to convert the nonlinear measurement equation in the stochastic volatility model, into a log-linear equation that is conditionally Gaussian. Chan and Hsiao (2014) then show how to adapt to sample the log-volatilities using an efficient precision sampling algorithm. Finally, the full conditional distribution in Steps 5 results in nonstandard distribution. Sampling is therefore achieved through an independence chain Metropolis–Hastings Algorithm adapted from the univariate models in Chan and Hsiao (2014). For completeness, we now discuss the derivation of the conditional posterior distribution of each block in the Gibbs sampler. In each case, we use the fact that the posterior distribution for a parameter of interest can be obtained by working with the kernel of the resulting product from the prior and likelihood functions.

Step 1: Sample from $p\left( \varvec{\theta }\mid {\mathbf {Y}},{\mathbf {h}},\varvec{\Sigma }_{u},\varvec{\Omega },\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

To sample $\varvec{\theta }$, first note that (4) can be rewritten as:

$$\begin{aligned} {\mathbf {Y}}={\mathbf {Z}}\varvec{\theta }+{\mathbf {u}},\,{\mathbf {u}}\sim N({\mathbf {0}},{\varvec{\Sigma }}), \end{aligned}$$

(A.1)

where $\varvec{Z}=\text {diag}\left( \varvec{Z}_{1},\dots ,\varvec{Z}_{T}\right)$, $\varvec{u}=\left[ \begin{array}{cccc} \varvec{u}_{1}^{'}&\varvec{u}_{2}^{'}&\dots&\varvec{u}_{T}^{'}\end{array}\right] '$and $\varvec{\Sigma }=\text {diag}\left( e^{h_{1}}\varvec{\Sigma }_{u},\dots ,e^{h_{T}}\varvec{\Sigma }_{u}\right)$. By a change of variable:

$$\begin{aligned} {\mathbf {Y}}\sim N\left( {\mathbf {Z}}\varvec{\theta },\varvec{\Sigma }\right) . \end{aligned}$$

(A.2)

Next, rewrite (5) as:

$$\begin{aligned} {\mathbf {H}}_{\theta }\varvec{\theta }=\tilde{\varvec{\alpha }}_{\theta }+\varvec{\eta },\,\varvec{\eta }\sim N\left( \varvec{0},\varvec{S}_{\theta }\right) , \end{aligned}$$

(A.3)

where $\tilde{\varvec{\alpha }}_{\theta }=\left[ \begin{array}{cccc} \varvec{\theta }_{0}^{'}&{\mathbf {0}}&\dots&{\mathbf {0}}\end{array}\right]$, ${\varvec{\eta }}=({\varvec{\eta }}_{1},\ldots ,{\varvec{\eta }}_{T})'$, ${\mathbf {S}}_{\theta }=\text {diag}({\mathbf {V}}_{\theta },\varvec{\Omega },\dots ,\varvec{\Omega })$ and ${\mathbf {H}}_{\theta }$ is a $Tm\times Tm$ block diagonal matrix, where $m=N_{1}+N+G$, with ${\mathbf {I}}_{m}$ on the main diagonal, $-{\mathbf {I}}_{m}$ on the lower diagonal and ${\mathbf {0}}_{m}$ elsewhere. Since ${\mathbf {H}}_{\theta }$ is a lower triangular matrix with ones along the main diagonal, $\left| {\mathbf {H}}_{\theta }\right| =1$, implying that it is invertible. Using this result, (A.3) can be rewritten as:

$$\begin{aligned} \varvec{\theta }=\varvec{\alpha }_{\theta }+{\mathbf {H}}^{-1}\eta , \end{aligned}$$

(A.4)

where $\varvec{\alpha }_{\theta }={\mathbf {H}}_{\theta }^{-1}\tilde{\varvec{\theta }}_{0}$. By a change of variable:

$$\begin{aligned} \varvec{\theta }|\varvec{\Omega }\sim N\left( \varvec{\alpha }_{\theta },\left( {{\mathbf {H}}_{\theta }^{'}}{\mathbf {S}}_{\theta }^{-1}{{\mathbf {H}}_{\theta }}\right) ^{-1}\right) . \end{aligned}$$

(A.5)

Combining (A.2) and (A.5) gives the conditional posterior distribution:

$$\begin{aligned}&p\left( \varvec{\theta }|\varvec{Y},\varvec{h},\varvec{\Sigma }_{u},\varvec{\Omega },\rho ,\sigma _{h}^{2}\right) \\&\quad \propto p\left( \varvec{Y}|\varvec{\theta },\varvec{h},\varvec{\Sigma }_{u},\varvec{\Omega },\rho ,\sigma _{h}^{2}\right) p\left( \varvec{\theta }|\varvec{\Omega }\right) \\&\quad \propto \exp \left\{ -\frac{1}{2}\left[ \left( \varvec{Y}-\varvec{Z}\varvec{\theta }\right) \varvec{\varvec{\Sigma }}^{-1}\left( \varvec{Y}-\varvec{Z}\varvec{\theta }\right) '+\left( \varvec{\theta }-\varvec{\alpha }_{\varvec{\theta }}\right) \left( \varvec{H}_{\varvec{\theta }}^{'}\varvec{S}_{\varvec{\theta }}^{-1}\varvec{H}_{\varvec{\theta }}\right) \left( \varvec{\theta }-\varvec{\alpha }_{\varvec{\theta }}\right) '\right] \right\} \\&\quad \propto \exp \left\{ -\frac{1}{2}\left[ \varvec{\theta }\left( \varvec{Z}'\varvec{\varvec{\Sigma }}^{-1}\varvec{Z}\right) \theta '-2\varvec{\theta }'\left( \varvec{Z}'\varvec{\varvec{\Sigma }}^{-1}\varvec{Y}+\varvec{H}_{\varvec{\theta }}^{'}\varvec{S}_{\varvec{\theta }}^{-1}\varvec{H}_{\varvec{\theta }}\varvec{\alpha }_{\varvec{\theta }}\right) \right] \right\} \end{aligned}$$

Thus, by standard linear regression results (Kroese et al. 2014, pp. 237–240):

$$\begin{aligned} \left( \varvec{\theta }\mid {\mathbf {Y}},{\mathbf {h}},\varvec{\Sigma }_{u},\varvec{\Omega },\rho ,\sigma _{h}^{2},\varvec{\kappa }\right) \sim N\left( \hat{\varvec{\varvec{\theta }}},{\mathbf {D}}_{\beta }^{^{-1}}\right) , \end{aligned}$$

(A.6)

where $\hat{\varvec{\theta }}={\mathbf {D}}_{\beta }^{^{-1}}\left( {\mathbf {Z}}'{\varvec{\Sigma }}^{-1}{\mathbf {Y}}+{{\mathbf {H}}_{\theta }^{'}}\varvec{S}_{\theta }^{-1}{{\mathbf {H}}_{\theta }\varvec{\alpha }_{\theta }}\right)$ and ${\mathbf {D}}_{\beta }={\mathbf {Z}}'\varvec{\varvec{\Sigma }}^{-1}{\mathbf {Z}}+{{\mathbf {H}}_{\theta }^{'}}{\mathbf {S}}_{\theta }^{-1}{{\mathbf {H}}_{\theta }}$. Following Poon (2018b), sampling from this distribution is conducted with the precision sampling algorithm in Chan and Jeliazkov (2009).

Step 2: Sample from $p\left( \varvec{\Sigma }_{u}\mid {\mathbf {Y}},\varvec{\theta },{\mathbf {h}},\varvec{\Omega },\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

To sample, $\varvec{\Sigma }_{u}$, combine the inverse-Wishart prior distribution with the Gaussian likelihood function. Since this is a conjugate distribution, it is easy to show that:

$$\begin{aligned}&\left( {\varvec{\Sigma }}_{u}|{\mathbf {Y}},\varvec{\theta },{\mathbf {h}},\sigma _{h}^{2},\varvec{\Omega },\rho ,\varvec{\kappa }\right) \nonumber \\&\quad \sim {\mathrm{IW}}\left( \nu _{2}+T,\sum _{t=1}^{T}\frac{({\mathbf {Y}}_{t}-{\mathbf {Z}}_{t}\theta _{t})({\mathbf {Y}}_{t}-{\mathbf {Z}}_{t}\theta _{t})'}{e^{h_{t}}}+\nu _{2}k_{2}^{2}{\mathbf {V}}_{2}\right) . \end{aligned}$$

(A.7)

Step 3: Sample from $p\left( \varvec{\Omega }\mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},{\mathbf {h}},\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

To sample $\varvec{\Omega }$, we combine the inverse-Wishart prior distribution with the Gaussian likelihood function to get

$$\begin{aligned} \left( \varvec{\Omega }|{\mathbf {Y}},\varvec{\theta },{\mathbf {h}},\sigma _{h}^{2},\varvec{{\Sigma }}_{u},\rho ,\varvec{\kappa }\right) \sim {\mathrm{IW}}\left( \nu _{1}+T-1,{\mathbf {H}}_{\theta }\varvec{\theta }\varvec{\theta }'{\mathbf {H}}_{\theta }'+\nu _{1}{\mathbf {k}}_{1}{\mathbf {V}}_{1}{\mathbf {k}}_{1}\right) . \end{aligned}$$

(A.8)

Step 4: Sample from $p\left( {\mathbf {h}}\mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},\varvec{\Omega },\rho ,\sigma _{h}^{2},\varvec{\kappa }\right)$

To sample ${\mathbf {h}}$, we follow Poon (2018b) and apply the auxiliary mixture sampler from Kim et al. (1998) along with the precision sampler from Chan and Hsiao (2014). To this end, note that the measurement equation in (4) can be written as:

$$\begin{aligned} \varvec{P}^{-1}\left( {\mathbf {Y}}_{t}-{\mathbf {Z}}_{t}\varvec{\theta }_{t}\right) =e^{\frac{1}{2}h_{t}}\varvec{\epsilon }_{t},\varvec{\epsilon }_{t}\sim N\left( 0,{\mathbf {I}}_{n}\right) , \end{aligned}$$

(A.9)

where ${\varvec{P}}$ is the (lower) Cholesky factor of $\varvec{\Sigma }_{u}$. Squaring both sides of (A.9) and taking the (natural) logarithm give the log-linear equation:

$$\begin{aligned} \varvec{y}_{t}^{*}=\varvec{\iota }_{NG}h_{t}+\varvec{\epsilon }_{t}^{*}, \end{aligned}$$

(A.10)

where $\varvec{y}_{t}^{*}=\ln \left( \left( {\varvec{P}}^{-1}\left( {\mathbf {Y}}_{t}-{\mathbf {X}}_{t}\Xi \varvec{\theta _{t}}\right) \right) ^{2}\right)$, $\varvec{\iota }_{NG}$ is a $NG\times 1$ unit vector and $\varvec{\epsilon }_{t}^{*}=\left[ \begin{array}{ccc} \ln \left( \varvec{\epsilon }_{1,t}^{2}\right)&\dots&\ln \left( \varvec{\epsilon }_{n,t}^{2}\right) \end{array}\right] '$. In practice, it is common to set some small constant, c,  to $\varvec{y}_{t}^{*}$ to avoid numerical problems when $\varvec{y}_{t}^{*}$ is close to zero—in this paper, we set $c=0.0001$. Note that the resulting disturbance term in (A.10) is no longer Gaussian distributed but instead follows a $\log \chi _{1}^{2}$ distribution. This means that despite being linear in the log-volatility term, standard linear Gaussian state–space algorithms cannot be directly applied. To overcome this difficulty, Kim et al. (1998) show that the moments of the $\log \chi _{1}^{2}$ distribution can be well approximated through a seven-Gaussian mixture, in which an auxiliary random variable, denoted $s_{t}$, serves as the mixture component indicator—hence the name of the algorithm, that is,

$$\begin{aligned} f\left( \epsilon _{t}^{*}\right) \approx \sum _{j=1}^{7}p_{j}f_{N}\left( \epsilon _{t}^{*}|\mu _{j}-1.2704,\sigma _{j}^{2}\right) , \end{aligned}$$

(A.11)

where $p_{j}=P\left( s_{t}=j\right)$, $j=1,\dots ,7$, $f_{N}\left( \cdot |\mu ,\sigma ^{2}\right)$ is a Gaussian density with mean $\mu$ and variance $\sigma ^{2}$ and the values of the probabilities and moments associated with each Gaussian distribution are given in Table 3.

Table 3 A seven-component Gaussian mixture for approximation the $\log -\chi _{1}^{2}$ distribution.

In summary, given the vector of mixture component indicators, $\varvec{s}=\left[ \begin{array}{ccc} s_{1}&\dots&s_{T}\end{array}\right] '$, and parameter values in Table 3, the state–space model in (A.10) and (9) is linear and conditionally Gaussian; thus, standard sampling methods can be used. Instead of adopting traditional Kalman filter-based algorithms, we follow Poon (2018b) and implement the efficient precision sampling-based algorithm in Chan and Jeliazkov (2009) and Chan and Hsiao (2014).

To summarize, define $\varvec{y}^{*}=\left( \varvec{y}_{1}^{*},\dots ,\varvec{y}_{T}^{*}\right) '$, ${\mathbf {h}}=(h_{1},...,h_{T})'$ is a $T\times 1$ and $\varvec{\epsilon }^{*}=(\varvec{\epsilon }_{1}^{*},\ldots \varvec{\epsilon }_{t}^{*})'$. Stacking the measurement equation in (A.10) over dates $t=1,\dots ,T$ gives:

$$\begin{aligned} {\mathbf {y}}^{*}={\mathbf {X}}_{h}{\mathbf {h}}+\varvec{\epsilon }^{*}, \end{aligned}$$

(A.12)

where ${\mathbf {X}}_{h}={\mathbf {I}}_{T}\otimes \varvec{\iota }_{NG}$ and $\varvec{\epsilon }^{*}\sim N({\mathbf {d}}_{s},\varvec{\Sigma }_{\varvec{y}^{*}})$ where ${\mathbf {d}}_{s}=(\varvec{\mu }_{s_{1}},\dots ,\varvec{\mu }_{s_{T}})'$ and $\varvec{\Sigma }_{\varvec{y}^{*}}=\text {diag}(\varvec{\sigma }_{s_{1}},\dots ,\varvec{\sigma }_{s_{T}})$ in which $\varvec{\mu }_{s_{t}}=\left( {\mu _{s_{t}^{1}}-1.2704,\dots ,\mu _{s_{t}^{n}}-1.2704}\right)$ and $\varvec{\sigma }_{s_{t}}=\left( {\sigma _{s_{t}^{1}}^{2},\dots ,\sigma _{s_{t}^{n}}^{2}}\right)$. Thus, by a change of variable:

$$\begin{aligned} \left( \varvec{y}_{i}^{*}|\varvec{s}_{i},\varvec{h}_{i}\right) \sim N\left( \varvec{h}_{i}+\varvec{d}_{i},\varvec{\Sigma }_{{\mathbf {y}}_{i}^{*}}\right) . \end{aligned}$$

(A.13)

To complete the state–space representation, stack the state equation for the log stochastic volatility factor in (9) over all dates $t=1,\dots ,T$ to get:

$$\begin{aligned} \begin{array}{cc} {\mathbf {h}}={\varvec{\alpha }}_{h}+{\mathbf {H}}_{h}^{-1}\xi ,&\xi \sim N(0,\Phi ),\end{array} \end{aligned}$$

(A.14)

where $\varvec{\alpha }_{h}={\mathbf {H}}_{h}^{-1}\left( \begin{array}{cccc} h_{0}&0&\dots&0\end{array}\right) '$, $\varvec{\Phi }=\text {diag}\left( \frac{\sigma _{h}^{2}}{(1-\rho ^{2})},\sigma _{h}^{2},\ldots ,\sigma _{h}^{2}\right)$ and

$$\begin{aligned} {\mathbf {H}}_{h}=\left[ \begin{array}{ccccc} 1 &{} 0 &{} 0 &{} \cdots &{} 0\\ -\rho &{} 1 &{} 0 &{} \cdots &{} \vdots \\ 0 &{} -\rho &{} 1 &{} \ddots &{} 0\\ \vdots &{} &{} \ddots &{} \ddots &{} 0\\ 0 &{} \cdots &{} 0 &{} -\rho &{} 1 \end{array}\right] . \end{aligned}$$

By a change of variable:

$$\begin{aligned} \left( {\mathbf {h}}\mid \varvec{\Phi },h_{0}\right) \sim N\left( \varvec{\alpha }_{h},\left( {\mathbf {H}}'_{h}\varvec{\Phi }^{-1}{\mathbf {H}}_{h}\right) ^{-1}\right) . \end{aligned}$$

(A.15)

Finally, combining (A.13) and (A.15) gives the conditional posterior distribution:

$$\begin{aligned} \left( {\mathbf {h}}|{\mathbf {Y}},\varvec{\theta },\varvec{\Omega },\varvec{\Sigma }_{u},\sigma _{h}^{2},\rho ,\varvec{\kappa }\right) \sim N\left( \hat{{\mathbf {h}}},{\mathbf {K}}_{h}^{-1}\right) , \end{aligned}$$

(A.16)

where $\hat{{\mathbf {h}}}={\mathbf {K}}_{h}^{-1}({\mathbf {H}}_{h}^{'}\varvec{\Phi }^{-1}{\mathbf {H}}_{h}^{'}\varvec{\alpha }_{h}+{\mathbf {X}}_{h}^{'}\varvec{\Sigma }_{{\mathbf {y}}_{i}^{*}}^{-1}({\mathbf {y}}^{*}-{\mathbf {d}}_{s}))$ and ${\mathbf {K}}_{h}={\mathbf {H}}_{h}^{'}\varvec{\Phi }^{-1}{\mathbf {H}}_{h}+{\mathbf {X}}_{h}^{'}\varvec{\Sigma }_{{\mathbf {y}}_{i}^{*}}^{-1}{\mathbf {X}}_{h}$. As in Step 1, sampling from this distribution is conducted with the precision sampling algorithm in Chan and Jeliazkov (2009).

Step 5: Sample from $p\left( \rho \mid {\mathbf {Y}},\varvec{\theta },\varvec{\Sigma }_{u},{\mathbf {h}},\varvec{\Omega },\sigma _{h}^{2},\varvec{\kappa }\right)$

To sample $\rho$, first note that combining the prior with (9) gives:

$$\begin{aligned} \left( \rho |\mathbf {y,}\varvec{\theta },\varvec{\Omega },{\mathbf {h}},\sigma _{h}^{2},\varvec{\Sigma }_{u},\varvec{\kappa }\right) \propto p(\rho )g(\rho )\text{ exp }\left\{ -\frac{1}{2\sigma _{h}^{2}}\sum _{t=2}^{T}(h_{t}-\rho h_{t-1})^{2}\right\} ,\nonumber \\ \end{aligned}$$

(A.17)

where $g(\rho )=(1-\rho ^{2})^{\frac{1}{2}}\text{ exp }(-\frac{1}{2\sigma _{h}^{2}}(1-\rho ^{2})(h_{1}-h_{0})^{2})$ and $p(\rho )$ is a truncated normal. Since this conditional distribution is nonstandard, we follow Chan and Hsiao (2014) and implement an independence chain Metropolis–Hastings step with a truncated normal distribution. More precisely, $\rho \sim N(\hat{\rho },D_{\rho })1(|\rho |<1)$, where $D_{\rho }=\left( V_{\rho }+{\varvec{X}}'_{\rho }{\varvec{X}}_{\rho }/\sigma _{h}^{2}\right) ^{-1}$ and $\hat{\rho }=D_{\rho }\left( V_{\rho }^{-1}\mu _{\rho }+{\varvec{X}}'_{\rho }\varvec{{\mathbf {z}}}_{\rho }/\sigma _{h}^{2}\right)$, in which $\mathbf {\varvec{X}}_{\rho }=\left( h_{1},...,h_{T-1}\right) '$ and $\mathbf {\varvec{z}}_{\rho }=\left( h_{2},...,h_{T}\right) '$. Then, given the current draw $\rho ^{d}$, a proposal draw $\rho ^{c}$ is accepted with probability $\min \left\{ 1,\frac{g(\rho ^{c})}{g(\rho ^{d})}\right\}$; otherwise, the Markov chain stays at the current draw.

Step 6: Sample from $p\left( \sigma _{h}^{2}\mid {\mathbf {Y}},\varvec{\theta },\varvec{\varvec{\Sigma }}_{u},{\mathbf {h}},\varvec{\Omega },\rho ,\varvec{\kappa }\right)$

To sample $\sigma _{h}^{2}$, we combine the inverse-Gamma prior distribution with the Gaussian likelihood function to get

$$\begin{aligned}&(\sigma _{h}^{2}|{{\mathbf {Y}},}\varvec{\theta },\varvec{\Omega },{\mathbf {h}},\sigma _{h}^{2},\varvec{\varvec{\Sigma }}_{u},\rho ,\varvec{\kappa })\nonumber \\&\quad \sim {\mathrm{IG}}\left( k_{3}^{2}+\frac{T-1}{2},+[(1-\rho )^{2}(h_{1})^{2}+\sum _{t=2}^{T}(h_{t}-\rho h_{t-1})^{2}+k_{3}^{2}V_{3}]/2\right) .\nonumber \\ \end{aligned}$$

(A.18)

Step 7: Sample from $p\left( \varvec{\kappa }\mid {\mathbf {Y}},\varvec{\theta },\varvec{\varvec{\Sigma }}_{u},{\mathbf {h}},\varvec{\Omega },\rho ,\sigma _{h}^{2}\right)$

Finally, to sample the hyperparameters in $\varvec{\kappa }$ we follow Amir-Ahmadi et al. (2018) and employ a version of the (Gaussian) random walk Metropolis–Hastings algorithm for the proposal variance in a burn-in phase. In what follows, we describe the sampling procedure for a generic scaling factor $\varvec{\kappa }_{X}$ where $X\in \left\{ \varvec{\varvec{\Sigma }}_{u},\varvec{\Omega },\sigma _{h}^{2}\right\}$.⁹ Starting from an initial condition in which we set $\varvec{\kappa }_{X}^{0}$ to be the unit vector, the details are as follows:

• 1. At step i, take a candidate draw $\varvec{\kappa }_{X}^{c}$ from $N\left( \varvec{\kappa }_{X}^{i-1},\sigma _{\kappa _{X}}^{2}{\mathbf {I}}\right)$ where $\sigma _{\kappa _{X}}^{2}$ is a tuning parameter which is changed in the burn-in phase to achieve a target acceptance rate.¹⁰

• 2. Calculate the acceptance probability $p^{i}=\left\{ 1,\frac{p\left\langle X|\varvec{\kappa }_{X}^{c}\right\rangle p\left( \varvec{\kappa }_{X}^{c}\right) }{p\left\langle X|\varvec{\kappa }_{X}^{i-1}\right\rangle p\left( \varvec{\kappa }_{X}^{i-1}\right) }\right\}$

• 3. Accept the candidate draw by setting $\varvec{\kappa }_{X}^{i}=\varvec{\kappa }_{X}^{c}$ with probability $p^{i}$. Otherwise set $\varvec{\kappa }_{X}^{i}=\varvec{\kappa }_{X}^{i-1}$.

1.2 Appendix A.2 Model comparison

In appendix, we discuss how to compute the marginal likelihood using the one-step-ahead predictive likelihood. To this end, let $\varvec{Y}_{t}^{o}$ denote a vector of observed variables up to date t. Following Geweke and Amisano (2011), the one-step- ahead predictive likelihood for model $M_{i}$, given data up to date $t-1$, is given by:

$$\begin{aligned} p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{i}\right)= & {} \int p\left( {\mathbf {Y}}_{t},\varvec{\Theta }_{i}|{\mathbf {Y}}_{t-1}^{o},M_{i}\right) {\mathrm{d}}\varvec{\Theta }_{i}, \end{aligned}$$

(A.19)

$$\begin{aligned}= & {} \int p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{i},\varvec{\Theta }_{i}\right) p\left( \varvec{\Theta }_{i}|M_{i}\right) {\mathrm{d}}\varvec{\Theta }_{i} \end{aligned}$$

(A.20)

where $t=1$ is evaluated by:

$$\begin{aligned} p\left( {\mathbf {Y}}_{1}^{o}|M_{i}\right) =\int p\left( {\mathbf {Y}}_{1}^{o}\right) p\left( \varvec{\Theta }_{i}|M_{i}\right) {\mathrm{d}}\varvec{\Theta }_{i}, \end{aligned}$$

(A.21)

which is entirely driven by the marginal data density: $p\left( {\mathbf {Y}}_{1}^{o}\right)$. Given this value, we then approximate $p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{i}\right)$ for dates $t=2,\dots ,T$ by the Monte Carlo average:

$$\begin{aligned} \widehat{p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{i}\right) }=R^{-1}\sum _{r=1}^{R}p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{i},\varvec{\Theta }_{i,t-1}^{\left( r\right) }\right) , \end{aligned}$$

(A.22)

where $\left\{ \varvec{\Theta }_{i,t-1}^{\left( r\right) }\right\} _{r=1}^{R}$ is a sequence of draws from then Metropolis–Hastings within Gibbs sampler described in “Appendix A.1.”

Finally, to see how the predictive likelihood is related to the Bayes factor, note that the marginal likelihood of the model $M_{i}$ is given by:

$$\begin{aligned} p\left( {\mathbf {Y}}_{T}^{o}|M_{i}\right) =\prod _{t=1}^{T}\widehat{p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{i}\right) }. \end{aligned}$$

(A.23)

Thus, the Bayes factor between models $M_{i}$ and $M_{j}$ is:

$$\begin{aligned} BF_{i,j}=\prod _{t=1}^{T}\frac{\widehat{p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{i}\right) }}{\widehat{p\left( {\mathbf {Y}}_{t}|{\mathbf {Y}}_{t-1}^{o},M_{j}\right) }}. \end{aligned}$$

(A.24)