Total, Asymmetric and Frequency Connectedness Between Oil and Forex Markets

2.1. Data & Variables

The dataset spans 2007–2017 and includes:

• Oil Market: West Texas Intermediate (WTI) crude oil futures prices (5-minute intervals).
• Forex Market: Exchange rates for major currencies (EUR, GBP, JPY, etc.) against the USD, also at high frequency.
• Core Variable: Realized volatility (RV) calculated from intra-day returns, serving as the measure of market uncertainty.
• Decomposition: Realized semivariances ($RS^+$ and $RS^-$) are computed to capture volatility due to positive and negative returns separately, enabling asymmetry analysis.

2.2. Total Connectedness Framework

The study adopts the Diebold and Yilmaz (2012, 2015) spillover index framework based on Vector Autoregressive (VAR) models and forecast error variance decompositions (FEVD). The Total Connectedness Index quantifies the proportion of forecast error variance in all variables coming from spillovers, as opposed to idiosyncratic shocks.

2.3. Asymmetric & Frequency Decomposition

This is the paper's key methodological contribution:

• Asymmetric Connectedness: By feeding realized semivariances ($RS^+$, $RS^-$) into the connectedness framework, the authors separate spillovers from "good volatility" (positive returns) and "bad volatility" (negative returns).
• Frequency Connectedness: Using the spectral representation of variance decompositions by Baruník and Křehlík (2018), the total connectedness is decomposed into components associated with different frequency bands (e.g., short-term: 1-5 days, long-term: >20 days). This reveals whether spillovers are transient or persistent.

3.1. Total Connectedness Dynamics

The total volatility connectedness between oil and forex markets is significant and time-varying. Key findings:

• Spillovers intensify dramatically during periods of financial distress (e.g., the 2008 Global Financial Crisis, the 2014-2016 oil price crash).
• Divergence in global monetary policy regimes (e.g., Fed tapering) is a key driver of increased forex volatility spillovers.
• Portfolio Insight: Adding oil to a pure forex portfolio decreases the overall connectedness of the portfolio. This suggests oil can act as a diversifier that reduces the portfolio's internal vulnerability to cross-market spillovers.

3.2. Asymmetric Spillover Effects

The magnitude of asymmetric effects is found to be relatively small on average, but the direction is revealing:

• Within the forex market alone, spillovers from negative shocks (bad volatility) dominate those from positive shocks.
• When oil and forex markets are analyzed jointly, positive shocks (good volatility) generate stronger spillovers. This indicates that positive oil market developments might propagate optimism or risk-on sentiment across to currencies.

3.3. Frequency-Dependent Connectedness

This analysis yields perhaps the most nuanced insights:

• Long-term connectedness (associated with lower frequencies) is the most dominant component and exhibits the most dramatic surges during crises.
• Primary Driver: Long-term connectedness is largely driven by uncertainty shocks (e.g., geopolitical events, structural demand changes).
• Secondary Driver: Liquidity shocks also impact long-term connectedness, but to a lesser extent.
• Short-term connectedness is more stable and linked to high-frequency trading and transient news.

5.1. Mathematical Foundation

The core of the frequency connectedness rests on the spectral decomposition of the variance-covariance matrix. For a $K$-variable VAR($p$) system: $\mathbf{Y}_t = \sum_{i=1}^p \Phi_i \mathbf{Y}_{t-i} + \epsilon_t$, with $\epsilon_t \sim (0, \Sigma)$. The spectral density of $\mathbf{Y}_t$ at frequency $\omega$ is: $S_{\mathbf{Y}}(\omega) = \Psi(e^{-i\omega}) \Sigma \Psi'(e^{+i\omega})$, where $\Psi(e^{-i\omega})$ is the Fourier transform of the MA($\infty$) coefficients. The share of the forecast error variance of variable $j$ attributable to shocks in variable $k$ at frequency $\omega$ is given by a spectral version of the FEVD:

$$\theta_{j,k}(\omega) = \frac{\sigma_{kk}^{-1} \sum_{h=0}^{\infty} |\Psi_h(\omega)_{j,k}|^2}{\sum_{k=1}^K \sigma_{kk}^{-1} \sum_{h=0}^{\infty} |\Psi_h(\omega)_{j,k}|^2}$$

where $\Psi_h(\omega)$ are the frequency-response functions. The connectedness measure within a specific frequency band $d = (a, b)$ is then obtained by integrating $\theta_{j,k}(\omega)$ over that band.

5.2. Analytical Framework Example

Case Study: Analyzing the 2014 Oil Price Crash

Objective: Determine how volatility spilled from oil to the Canadian dollar (CAD/USD) and Norwegian krone (NOK/USD) during the 2014-2016 period, distinguishing between short-term trading effects and long-term structural impacts.

1. Data Preparation: Calculate 5-minute realized volatility and semivariances for WTI, CAD/USD, and NOK/USD.
2. Model Estimation: Estimate a daily VAR model for the vector $[RV_{Oil}, RV_{CAD}, RV_{NOK}]$ and separately for $[RS^+_{Oil}, RS^+_{CAD}, ...]$ and $[RS^-_{Oil}, RS^-_{CAD}, ...]$.
3. Frequency Decomposition: Apply the Baruník-Křehlík spectral decomposition to the variance-covariance matrix from the total RV VAR. Define bands: Short-term (1-5 business days), Medium-term (5-20 days), Long-term (20+ days).
4. Interpretation:
   • If spillovers from Oil to CAD are strongest in the Long-term band, it suggests the crash impacted Canada's terms of trade and long-run economic outlook, driving sustained CAD volatility.
   • If the Asymmetric analysis shows $RS^-$ spillovers dominate, it confirms the crisis was driven by negative shocks propagating fear.
   • Comparing the Total Connectedness of a portfolio [CAD, NOK] vs. [CAD, NOK, Oil] would likely show a decrease, illustrating the diversification benefit.