Diebold–Yilmaz (2012) Spillover Framework

(Modeling & Equations )

1. VAR(p) Model

🇮🇩 Indonesia

$\mathbf{x}_t = \sum_{k=1}^{p} \mathbf{\Phi}_k \mathbf{x}_{t-k} + \boldsymbol{\varepsilon}_t, \quad \boldsymbol{\varepsilon}_t \sim (0,\boldsymbol{\Sigma})$

Keterangan simbol:

$\mathbf{x}_t$ : vektor variabel endogen $(N \times 1)$
$\mathbf{\Phi}_k$ : matriks koefisien VAR lag ke- $k$
$p$ : panjang lag VAR
$\boldsymbol{\varepsilon}_t$ : vektor inovasi (shock)
$\boldsymbol{\Sigma}$ : matriks kovarians inovasi

🇬🇧 English

$\mathbf{x}_t = \sum_{k=1}^{p} \mathbf{\Phi}_k \mathbf{x}_{t-k} + \boldsymbol{\varepsilon}_t, \quad \boldsymbol{\varepsilon}_t \sim (0,\boldsymbol{\Sigma})$

Notation:

$\mathbf{x}_t$ : $N \times 1$ vector of endogenous variables
$\mathbf{\Phi}_k$ : VAR coefficient matrix at lag $k$
$p$ : VAR lag length
$\boldsymbol{\varepsilon}_t$ : innovation vector
$\boldsymbol{\Sigma}$ : covariance matrix of innovations

2. Moving Average (MA) Representation

🇮🇩 Indonesia

$\mathbf{x}_t = \sum_{h=0}^{\infty} \mathbf{A}_h \boldsymbol{\varepsilon}_{t-h}, \quad \mathbf{A}_0 = \mathbf{I}$ $\mathbf{A}_h = \mathbf{\Phi}_1 \mathbf{A}_{h-1} + \mathbf{\Phi}_2 \mathbf{A}_{h-2} + \cdots + \mathbf{\Phi}_p \mathbf{A}_{h-p}$

🇬🇧 English

3. Generalized Forecast Error Variance Decomposition (GFEVD)

🇮🇩 Indonesia

$\theta^{g}_{ij}(H) = \frac{ \sigma_{jj}^{-1} \sum_{h=0}^{H-1} \left( \mathbf{e}_i^{\prime} \mathbf{A}_h \boldsymbol{\Sigma} \mathbf{e}_j \right)^2 }{ \sum_{h=0}^{H-1} \left( \mathbf{e}_i^{\prime} \mathbf{A}_h \boldsymbol{\Sigma} \mathbf{A}_h^{\prime} \mathbf{e}_i \right) }$

Keterangan simbol:

$\theta^{g}_{ij}(H)$ : kontribusi shock $j$ terhadap varians kesalahan prediksi $i$ i
$H$ : horizon peramalan
$\mathbf{e}_i$ : vektor seleksi (1 pada posisi $i$ i, 0 lainnya)
$\sigma_{jj}$ : elemen diagonal ke- $j$ dari $\boldsymbol{\Sigma}$

🇬🇧 English

4. Normalization of GFEVD

🇮🇩 Indonesia

$\tilde{\theta}^{g}_{ij}(H) = \frac{ \theta^{g}_{ij}(H) }{ \sum_{j=1}^{N} \theta^{g}_{ij}(H) }$ $\sum_{j=1}^{N} \tilde{\theta}^{g}_{ij}(H) = 1$

🇬🇧 English

$\tilde{\theta}^{g}_{ij}(H) = \frac{ \theta^{g}_{ij}(H) }{ \sum_{j=1}^{N} \theta^{g}_{ij}(H) }$ $\sum_{j=1}^{N} \tilde{\theta}^{g}_{ij}(H) = 1$

5. Total Spillover Index

🇮🇩 Indonesia

$S^{g}(H) = \frac{ \sum_{i \neq j} \tilde{\theta}^{g}_{ij}(H) }{ N } \times 100$

🇬🇧 English

$S^{g}(H) = \frac{ \sum_{i \neq j} \tilde{\theta}^{g}_{ij}(H) }{ N } \times 100$

6. Directional Spillovers

6.1 FROM Others

🇮🇩 $S^{g}_{i \leftarrow \cdot}(H) = \frac{ \sum_{j \neq i} \tilde{\theta}^{g}_{ij}(H) }{ N } \times 100$

🇬🇧 $S^{g}_{i \leftarrow \cdot}(H) = \frac{ \sum_{j \neq i} \tilde{\theta}^{g}_{ij}(H) }{ N } \times 100$

6.2 TO Others

🇮🇩 $S^{g}_{i \rightarrow \cdot}(H) = \frac{ \sum_{j \neq i} \tilde{\theta}^{g}_{ji}(H) }{ N } \times 100$

🇬🇧 $S^{g}_{i \rightarrow \cdot}(H) = \frac{ \sum_{j \neq i} \tilde{\theta}^{g}_{ji}(H) }{ N } \times 100$

7. Net Spillover

🇮🇩 Indonesia

$S^{g}_{i}(H) = S^{g}_{i \rightarrow \cdot}(H) – S^{g}_{i \leftarrow \cdot}(H)$

🇬🇧 English

$S^{g}_{i}(H) = S^{g}_{i \rightarrow \cdot}(H) – S^{g}_{i \leftarrow \cdot}(H)$

8. Net Pairwise Spillover

🇮🇩 Indonesia

$S^{g}_{ij}(H) = \frac{ \tilde{\theta}^{g}_{ji}(H) – \tilde{\theta}^{g}_{ij}(H) }{ N } \times 100$

🇬🇧 English

$S^{g}_{ij}(H) = \frac{ \tilde{\theta}^{g}_{ji}(H) – \tilde{\theta}^{g}_{ij}(H) }{ N } \times 100$

9. Rolling Window Estimation

🇮🇩 Indonesia

$\text{VAR}_t = \{ \mathbf{x}_{t-w+1}, \ldots, \mathbf{x}_t \}$

🇬🇧 English

$\text{VAR}_t = \{ \mathbf{x}_{t-w+1}, \ldots, \mathbf{x}_t \}$