This study used the TVP-VAR connectedness approach developed by Antonakakis et al. (2020) based on Diebold and Yilmaz’s (2014) connectedness approach. The structure of the estimated TVP-VAR model, allowing for the variance-covariance matrix to change over time, is defined by the following three sets of equations:
\({y}_{t}={A}_{t}{z}_{t-1}+{\epsilon }_{t} {\epsilon }_{t}\left|{{\Omega }}_{t-1}\sim N\right.\) (0,\({ {\Sigma }}_{t})\) (1)
vec(\({A}_{t})=\) vec(\({A}_{t-1})+{{\xi }}_{t} {{\xi }}_{t}\left|{{\Omega }}_{t-1}\sim N \right.\)(0,\({ {\Xi }}_{t})\) (2)
with
$${z}_{t-1}= \left(\begin{array}{c}{y}_{t-1}\\ {y}_{t-2}\\ .\\ .\\ .\\ {y}_{t-p}\end{array}\right) {A}_{t}^{{\prime }}=\left(\begin{array}{c}{A}_{1t}\\ {A}_{2t}\\ .\\ .\\ .\\ {A}_{pt}\end{array}\right)$$
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Here, \({{\Omega }}_{t-1}\) denotes the available information until t-1. \({y}_{t}\), \({z}_{t-1}\) and \({\epsilon }_{t}\)represents m x 1, mp x 1 and m x 1 vectors, including the current and lagged values of CO2 emissions respectively. \({A}_{t}\) and \({A}_{t}^{{\prime }}\) show the m x mp and m x m dimensional matrices, respectively. \({{\xi }}_{t}\) is the m2p x 1 dimensional vector. Time-varying variance-covariance matrices are denoted by \({{\Sigma }}_{t}\) and \({{\Xi }}_{t}\) are with m x m and m2p x m2p dimensions respectively. Finally, vec(\({A}_{t})\) represents the vectorized form of \({A}_{t}\) with a m2p x 1 dimension.
In the TVP-VAR connectedness approach, connectedness measures are computed from generalized impulse response functions (GIRF) and generalized forecast error variance decompositions (GFEVD) (Diebold & Yilmaz, 2014). In order to achieve this, the TVP-VAR model is converted into the vector moving average (VMA) form. The representation, based on the Wold theorem, is given by:
$${y}_{t}={J}^{{\prime }}({M}_{t}\left({z}_{t-2}+{\eta }_{t-1}\right)+{n}_{t})$$
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$$={J}^{{\prime }}({M}_{t}\left({M}_{t}\left({z}_{t-3}+{\eta }_{t-2}\right)+{n}_{t-1}\right)+{n}_{t})$$
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$$={J}^{{\prime }}({M}_{t}^{k-1}{z}_{t-k-1}+\sum _{j=0}^{k}{M}_{t}^{j}{\eta }_{t-j})$$
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with
$${M}_{t}=\left(\begin{array}{cc}{A}_{t}& .\\ {I}_{m(p-1)}& {0}_{m\left(p-1\right) x m}\end{array}\right) {n}_{t}= \left(\begin{array}{c}{ϵ}_{t}\\ 0\\ .\\ .\\ .\\ 0\end{array}\right)= J{ϵ}_{t} J=\left(\begin{array}{c}I\\ 0\\ .\\ .\\ .\\ 0\end{array}\right)$$
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Here, \({M}_{t}\) represents the mp x pm dimensional matrix whereas \(J\) denotes the mp x m dimensional matrix and \({n}_{t}\) stands for the mp x 1 dimensional vector.
Taking the limit as k approaches ∞ gives the following:
$${y}_{t}=\underset{k\to {\infty }}{\text{lim}}{J}^{{\prime }}({M}_{t}^{k-1}{z}_{t-k-1}+\sum _{j=0}^{k}{M}_{t}^{j}{\eta }_{t-j})=\sum _{j=0}^{{\infty }}{{J}^{{\prime }}M}_{t}^{j}{\eta }_{t-j}$$
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$${y}_{t}=\sum _{j=0}^{{\infty }}{{J}^{{\prime }}M}_{t}^{j}J{ϵ}_{t-j} {B}_{jt}= {{J}^{{\prime }}M}_{t}^{j}J, j=\text{0,1},\dots$$
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$${y}_{t}=\sum _{j=0}^{{\infty }}{B}_{jt}{ϵ}_{t-j}$$
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Here \({B}_{jt}\) indicates an m x m dimensional matrix.
The GIRFs (\({\psi }_{ij,t}\left(H\right))\) show the responses of all variables j following a shock in variable i. Here, the differences between an H-step-ahead forecast are calculated with and without variable i being shocked.
This can be computed in the following manner:
$${GIRF}_{t}\left(H,{\delta }_{j,t},{{\Omega }}_{t-1}\right)=E\left({y}_{t+H}|{e}_{j}={\delta }_{j,t},{{\Omega }}_{t-1}\right)-E({y}_{t+J}│{{\Omega }}_{t-1})$$
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$${\psi }_{j,t}\left(H\right)=\frac{{B}_{H,t}{\sum }_{t}{e}_{j}}{\sqrt{{\sum }_{jj,t}}}\frac{{\delta }_{j,t}}{\sqrt{{\sum }_{jj,t}}} {\delta }_{j,t}= \sqrt{{\sum }_{jj,t}}$$
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$${\psi }_{j,t}\left(H\right)={\sum }_{jj,t}^{- \frac{1}{2}}{B}_{H,t}{\sum }_{t}{e}_{j}$$
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Here, \({e}_{j}\) is an m x 1 selection vector with unity in the jth rank, and zero otherwise. GFEVD(\({\stackrel{\sim}{\varphi }}_{ij,t}\left(H\right))\) is pairwise directional connectedness from j to i. It represents the effect variable j has on variable i. It is measured as:
$${\stackrel{\sim}{\varphi }}_{ij,t}\left(H\right)=\frac{\sum _{t=1}^{H-1}{\psi }_{ij,t}^{2}}{\sum _{j=1}^{m}\sum _{t=1}^{H-1}{\psi }_{ij,t}^{2}}$$
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where \(\sum _{j=1}^{m}{\stackrel{\sim}{\varphi }}_{ij,t}\left(H\right)\) = 1 and \(\sum _{i,j=1}^{m}{\stackrel{\sim}{\varphi }}_{ij,t}\left(H\right)\) = m. In Eq. 15, the denominator indicates the cumulative impact of all the shocks, whereas the numerator shows the cumulative impact of a shock of in variable i.
Using Eq. 15, the total connectedness index is constructed as follows:
$${C}_{t}\left(H\right)=\frac{\sum _{i, j=1, i\ne j }^{m}{\stackrel{\sim}{\varnothing }}_{ij,t}\left(H\right)}{\sum _{i, j=1}^{m}{\stackrel{\sim}{\varnothing }}_{ij,t}\left(H\right)}*100=\frac{\sum _{i, j=1, i\ne j }^{m}{\stackrel{\sim}{\varnothing }}_{ij,t}\left(H\right)}{m}*100.$$
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This connectedness index shows how a shock that occurred in one variable spills over to other variables. If variable i transmits its shock to all other variables j, it is labelled “total directional connectedness to others” and calculated as follows:
$${C}_{i\to \text{j},\text{t}}\left(H\right)=\frac{\sum _{j=1, i\ne j }^{m}{\stackrel{\sim}{\varnothing }}_{ij,t}\left(H\right)}{\sum _{j=1}^{m}{\stackrel{\sim}{\varnothing }}_{ij,t}\left(H\right)}*100.$$
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When variable i receives shocks from all variables j, it is labelled “total directional connectedness from others” and calculated as follows:
$${C}_{i\leftarrow \text{j},\text{t}}\left(H\right)=\frac{\sum _{j=1, i\ne j }^{m}{\stackrel{\sim}{\varnothing }}_{ij,t}\left(H\right)}{\sum _{i=1}^{m}{\stackrel{\sim}{\varnothing }}_{ij,t}\left(H\right)}*100.$$
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“Net total directional connectedness,” which is the difference between total directional connectedness to others and total directional connectedness from others, illustrates the impact of variable i on the network. It is written as follows:
$${C}_{i,t}={C}_{i\to \text{j},\text{t}}\left(H\right)-{C}_{i\leftarrow \text{j},\text{t}}\left(H\right)$$
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If \({C}_{i,t}\) is positive, variable i affects the network more than being affected itself. Conversely, if it is negative, variable i is influenced by the network. Lastly, net total directional connectedness can be broken down to measure bidirectional relationships by calculating “net pairwise directional connectedness.”
$${NPDC}_{ij }\left(H\right)={\stackrel{\sim}{\varnothing }}_{jit}\left(H\right)-{\stackrel{\sim}{\varnothing }}_{jit}\left(H\right))*100.$$
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If \({NPDC}_{ij }\left(H\right)\) is higher than zero, it indicates that variable i dominates variable j. When it is lower than zero, it indicates that variable j dominates variable i.