Better to give than to receive: Predictive directional measurement of volatility spillovers

Diebold–Yilmaz (2012): Generalized VAR Spillover Framework

A. Tujuan & Intuisi Utama

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Kerangka Diebold–Yilmaz (2012) bertujuan mengukur dan memantau spillover (penularan) volatilitas/ketidakpastian antar pasar/variabel. Keunggulan kunci artikel ini adalah penggunaan Generalized VAR variance decomposition yang tidak bergantung pada urutan variabel (order-invariant), sehingga menghindari bias dari identifikasi Cholesky.

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Diebold–Yilmaz (2012) provides a framework to measure and monitor spillovers (transmission) of volatility/uncertainty across markets/variables. Its key advantage is using generalized VAR variance decomposition, which is order-invariant, avoiding the ordering bias inherent in Cholesky identification.

B. Spesifikasi Model VAR(p)

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Sistem dinamika antar-variabel dimodelkan dengan VAR(p): $\mathbf{x}_t = \sum_{k=1}^{p}\mathbf{\Phi}_k \mathbf{x}_{t-k} + \boldsymbol{\varepsilon}_t, \quad \boldsymbol{\varepsilon}_t \sim (0,\boldsymbol{\Sigma})$

Keterangan:

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

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Dynamic interactions are modeled using a VAR(p): $\mathbf{x}_t = \sum_{k=1}^{p}\mathbf{\Phi}_k \mathbf{x}_{t-k} + \boldsymbol{\varepsilon}_t, \quad \boldsymbol{\varepsilon}_t \sim (0,\boldsymbol{\Sigma})$

Where:

$\mathbf{x}_t$ : $N\times 1$ vector of endogenous variables
$\mathbf{\Phi}_k$ : lag- $k$ coefficient matrices
$p$ : optimal lag length
$\boldsymbol{\varepsilon}_t$ : innovations (shocks)
$\boldsymbol{\Sigma}$ : innovation covariance matrix

C. Representasi Moving Average (MA)

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VAR dapat dinyatakan sebagai MA tak hingga: $\mathbf{x}_t=\sum_{h=0}^{\infty}\mathbf{A}_h\boldsymbol{\varepsilon}_{t-h}, \quad \mathbf{A}_0=\mathbf{I}$

Representasi ini penting karena FEVD dihitung dari respons dinamis $\mathbf{A}_h$ Ah.

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The VAR admits an infinite MA representation: $\mathbf{x}_t=\sum_{h=0}^{\infty}\mathbf{A}_h\boldsymbol{\varepsilon}_{t-h}, \quad \mathbf{A}_0=\mathbf{I}$

This matters because FEVD is computed from dynamic responses $\mathbf{A}_h$ Ah.

D. Generalized FEVD (GFEVD) – Inti Metode

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Untuk menghindari masalah urutan variabel, digunakan Generalized FEVD. Kontribusi shock variabel $j$ j terhadap kesalahan prediksi variabel $i$ i pada horizon $H$ H: $\theta^{g}_{ij}(H)= \frac{ \sigma^{-1}_{jj}\sum_{h=0}^{H-1}( \mathbf{e}_i’\mathbf{A}_h\boldsymbol{\Sigma}\mathbf{e}_j )^2 }{ \sum_{h=0}^{H-1}( \mathbf{e}_i’\mathbf{A}_h\boldsymbol{\Sigma}\mathbf{A}_h’\mathbf{e}_i ) }$

Karena GFEVD tidak harus menjumlah 1 per baris, dilakukan normalisasi: $\tilde{\theta}^{g}_{ij}(H)= \frac{\theta^{g}_{ij}(H)}{\sum_{j=1}^{N}\theta^{g}_{ij}(H)}, \quad \sum_{j=1}^{N}\tilde{\theta}^{g}_{ij}(H)=1$

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To avoid ordering dependence, the framework uses Generalized FEVD. The contribution of shocks from $j$ j to the $H$ H-step-ahead forecast error variance of $i$ i: $\theta^{g}_{ij}(H)= \frac{ \sigma^{-1}_{jj}\sum_{h=0}^{H-1}( \mathbf{e}_i’\mathbf{A}_h\boldsymbol{\Sigma}\mathbf{e}_j )^2 }{ \sum_{h=0}^{H-1}( \mathbf{e}_i’\mathbf{A}_h\boldsymbol{\Sigma}\mathbf{A}_h’\mathbf{e}_i ) }$

Because generalized shares may not sum to 1 row-wise, they are normalized: $\tilde{\theta}^{g}_{ij}(H)= \frac{\theta^{g}_{ij}(H)}{\sum_{j=1}^{N}\theta^{g}_{ij}(H)}, \quad \sum_{j=1}^{N}\tilde{\theta}^{g}_{ij}(H)=1$

E. Total Spillover Index

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Indeks spillover total mengukur proporsi kontribusi lintas-variabel (off-diagonal): $S^{g}(H)=\frac{\sum_{i\neq j}\tilde{\theta}^{g}_{ij}(H)}{N}\times 100$

Interpretasi: semakin tinggi $S^{g}(H)$ Sg(H), semakin kuat keterkaitan sistemik (contagion) antar variabel/pasar.

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The total spillover index measures the share of cross-variable (off-diagonal) contributions: $S^{g}(H)=\frac{\sum_{i\neq j}\tilde{\theta}^{g}_{ij}(H)}{N}\times 100$

Interpretation: higher $S^{g}(H)$ Sg(H) implies stronger systemic interconnectedness/contagion.

F. Directional Spillovers (Arah Spillover)

1) FROM Others (diterima)

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

Mengukur seberapa besar variabel $i$ i menerima shock dari variabel lain.

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

Measures how much variable $i$ i receives shocks from others.

2) TO Others (dikirim)

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

Mengukur seberapa besar variabel $i$ i mengirim shock ke variabel lain.

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

Measures how much variable $i$ i transmits shocks to others.

G. Net Spillover (Posisi Transmitter vs Receiver)

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

$S^{g}_{i}(H)>0$ Sig(H)>0: $i$ i adalah net transmitter
$S^{g}_{i}(H)<0$ Sig(H)<0: $i$ i adalah net receiver

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

$>0$ >0: $i$ i is a net transmitter
$<0$ <0: $i$ i is a net receiver

H. Net Pairwise Spillover (Dua Arah i vs j)

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

Jika positif → dominasi spillover i → j (secara neto). Jika negatif → dominasi j → i.

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

Positive values indicate net dominance i → j; negative values indicate net dominance j → i.

I. Analisis Dinamis (Rolling Window)

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Untuk melihat perubahan spillover dari waktu ke waktu (mis. sebelum–saat–sesudah krisis), VAR dan GFEVD diestimasi pada rolling window (jendela geser) dengan panjang tetap. Output utamanya biasanya:

grafik Total Spillover
grafik Net Spillover per variabel
(opsional) Pairwise Net Spillover antar pasangan variabel

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To capture time variation (e.g., pre-/during-/post-crisis), the VAR and GFEVD are estimated using a fixed-length rolling window. Typical outputs include:

Total spillover plot
variable-specific net spillover plots
(optional) pairwise net spillover plots

J. Catatan Implementasi untuk Paper Anda (Praktis)

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Pilih $p$ (lag VAR) dan $H$ (forecast horizon).
Pastikan variabel stasioner (atau gunakan transformasi yang sesuai).
Laporkan: $S^g(H)$ , TO/FROM, NET, dan (bila perlu) pairwise.
Interpretasi kebijakan: identifikasi variabel yang menjadi sumber guncangan saat episode krisis.

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Choose $p$ p (VAR lags) and $H$ H (forecast horizon).
Ensure stationarity (or apply appropriate transformations).
Report: $S^g(H)$ , directional TO/FROM, NET, and (if needed) pairwise.
Policy interpretation: identify which variables act as shock transmitters during crisis episodes.