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

Diebold–Yilmaz (2012): Generalized VAR Spillover Framework

---

A. Tujuan & Intuisi Utama

🇮🇩
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.

🇬🇧
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)

🇮🇩
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

🇬🇧
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)

🇮🇩
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​.

🇬🇧
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

🇮🇩
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:$$\tilde{\theta}^{g}_{ij}(H)= \frac{\theta^{g}_{ij}(H)}{\sum_{j=1}^{N}\theta^{g}_{ij}(H)}, \quad \sum_{j=1}^{N}\tilde{\theta}^{g}_{ij}(H)=1$$

🇬🇧
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

🇮🇩
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.

🇬🇧
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)

🇮🇩
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

🇬🇧
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)

🇮🇩

• 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.

🇬🇧

• 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.