Spillover Asymmetry Measure (SAM) Overview

• SAM is a quantitative index that measures asymmetries in volatility spillovers between positive (good) and negative (bad) shocks in financial markets.
• It is constructed using realized semivariances, VAR decompositions, and tail-risk copula or nonlinear impulse response methods to separate upward and downward risk transmission.
• The measure informs risk management and portfolio allocation decisions by highlighting differences in shock propagation during periods of market stress and crisis.

The Spillover Asymmetry Measure (SAM) is a quantitative index designed to capture and quantify asymmetries in the transmission of shocks—especially volatility and extreme risk—across assets or sectors in financial markets. By construction, SAM contrasts the intensity and direction of "good" (positive-return, upside) versus "bad" (negative-return, downside) spillovers, with values calibrated to indicate the dominance of either. Its formalization and many variants derive from generalizations of the Diebold–Yilmaz connectedness framework, asymmetric variance decompositions, and tail-risk copula methods, addressing the empirical regularity that downside risks frequently propagate more forcefully than upside risks within and across financial systems (Barunik et al., 2013, Hatemi-J, 2024, Huber et al., 2024, Dai et al., 2023).

1. Formal Definition and Theoretical Basis

At the core of SAM is the comparison between spillover measures constructed from positive and negative return innovations or, more generally, favorable versus adverse shocks. In the canonical form introduced by Baruník, Kočenda, and Vácha, SAM is defined as:

$$\mathrm{SAM} = 100 \times \frac{\mathcal{S}^+ - \mathcal{S}^-}{\frac{1}{2}(\mathcal{S}^+ + \mathcal{S}^-)}$$

where $\mathcal{S}^+$ and $\mathcal{S}^-$ are spillover indices derived from realized semivariances for positive (good) and negative (bad) returns, respectively. The index is zero under symmetric spillovers, positive if good volatility dominates, and negative if bad volatility dominates (Barunik et al., 2013, Hatemi-J, 2024). Extensions to a pairwise or network context are articulated in asymmetric variance decomposition frameworks, yielding analogues such as:

$$SAM^H = \frac{S^{H,+} - S^{H,-}}{S^{H,+} + S^{H,-}}$$

where $S^{H,\pm}$ are total spillovers from positive or negative VAR innovations over horizon $H$ (Hatemi-J, 2024).

In frameworks based on nonlinear impulse response analysis, as in multi-country settings, SAM generalizes to

$$SAM_{i,h}(\sigma) = \frac{\gamma_{i,h}^{(\sigma)} - \gamma_{i,h}^{(-\sigma)}}{|\gamma_{i,h}^{(\sigma)}| + |\gamma_{i,h}^{(-\sigma)}|}$$

with $\gamma_{i,h}^{(\pm \sigma)}$ denoting the $h$-step-ahead generalized impulse response of variable $i$ to a shock of sign $\pm \sigma$ (Huber et al., 2024).

2. Methodological Construction

Realized Semivariance and VAR Decomposition

Following Baruník, Kočenda, and Vácha (Barunik et al., 2013), realized variance is split into positive and negative semivariances using high-frequency returns. Letting $r_i$ denote intraday returns,

$$RS_t^- = \sum_{i=1}^n r_i^2 \mathbb{I}_{\{r_i<0\}}, \quad RS_t^+ = \sum_{i=1}^n r_i^2 \mathbb{I}_{\{r_i>0\}}$$

Sectoral spillover indices $\mathcal{S}^-$ and $\mathcal{S}^+$ are computed via Diebold–Yilmaz-type variance decompositions:

• Stack $\mathbf{RS}_t^-$ and $\mathbf{RS}_t^+$ into multivariate VAR($p$) models.
• Invert to obtain MA($\infty$) representations.
• Compute forecast error variance decompositions (FEVD) for a chosen horizon $H$.
• Normalize and aggregate off-diagonal FEVD terms to construct spillover indices.

Asymmetric Connectedness (Hatemi-J Framework)

Hatemi-J’s methodology (Hatemi-J, 2024) preprocesses series to separate positive and negative innovations, estimates separate VAR models on each, and calculates FEVDs accordingly. The pairwise or aggregate SAM is then constructed by comparing these FEVDs for positive and negative shocks, allowing for full network-level asymmetry quantification.

Copula–CoVaR-Based SAM

For extreme risk spillovers, copula-CoVaR methods compare conditional value-at-risk (CoVaR) for downside (joint tail probability at 0.05) and upside (0.95) events. The SAM at time $t$ is defined as

$SAM_t = \frac{CoVaR_{0.05,0.05,t}/VaR_{0.05,t} - CoVaR_{0.95,0.95,t}/VaR_{0.95,t}}$

with tests of $SAM_t=0$ indicating whether downside risk spillovers statistically exceed upside spillovers (Dai et al., 2023).

Nonlinear Impulse Response (Huber et al.)

SAM is computed from generalized impulse responses to positive and negative shocks in nonlinear multi-country models, directly contrasting the magnitude and duration of spillovers under different shock regimes (Huber et al., 2024).

3. Estimation Considerations

Key empirical choices affect SAM’s consistency and interpretability:

• Data Frequency: High-frequency (e.g., 5-minute) returns for variance-based SAM; monthly or daily for macro-financial systems.
• VAR Order: Selected via information criteria (AIC, SIC); separate models for $\pm$ shocks double parameter estimation.
• Forecast Horizon: Typically 10 periods (days or months); sensitivity to $H$ is reported.
• Rolling Window: Windows (e.g., 200 trading days) accommodate time-varying dynamics.
• Covariance Estimation: Based on VAR residuals within each window.
• Bootstrap Inference: Empirical distribution of SAM computed via residual or block-bootstrap resampling for confidence intervals or hypothesis testing (Barunik et al., 2013, Hatemi-J, 2024).
• Copula Inference: Marginal distribution filtering through ARMA–GARCH–skewed-t; copula parameter estimation via IFM or MLE for tail dependence (Dai et al., 2023).
• Impulse Response Monte Carlo: Posterior draws in Bayesian structural models using nonlinear basis expansions (Huber et al., 2024).

4. Empirical Patterns and Sectoral Heterogeneity

SAM uncovers significant variation in spillover asymmetry across time, sectors, and markets:

• US Equity Market (High-frequency): Aggregate portfolios exhibit little asymmetry (SAM ≈ 0), but disaggregated sector analysis reveals persistent and often alternating asymmetries. For example, consumer staples, healthcare, and telecom sectors experienced statistically significant swings between good- and bad-dominant spillover regimes, while financials and energy remained more symmetric outside crisis periods (Barunik et al., 2013).
• Financial Crisis Dynamics: A pronounced shift toward negative (downside) spillover dominance was detected during the 2007–2009 crisis across most sectors.
• Cross-Market Connectedness: In international financial markets, negative-shock spillovers outsize positive ones, with the US driving negative asymmetry and the Euro-Area contributing more to positive spillovers. Empirical point estimates report SAM values as negative (e.g., SAM ≈ –0.30), denoting bad-volatility spillover dominance (Hatemi-J, 2024).
• Extreme Risk Spillovers: Agricultural commodity analysis shows statistically significant downside (left-tail) spillover dominance for soybean and maize futures into spot markets, but symmetry for wheat and rice (Dai et al., 2023).
• Nonlinear International Transmission: Nonlinear multi-country models demonstrate that adverse shocks consistently trigger larger spillovers (SAM near –1), with substantial persistence over horizons, especially following large shocks and in contractionary regimes (Huber et al., 2024).

5. Economic Interpretation and Portfolio Implications

SAM operationalizes the differential network transmission of positive versus negative shocks. Positive SAM implies upside volatility shocks are more contagious, suggesting diminished diversification in rallies and limited risk mitigation through sectoral tilting. Negative SAM (bad-volatility-dominant) indicates increased contagion in downturns, motivating the demand for tail risk hedges and dynamic reallocation. Time-variation in SAM offers real-time insights for risk monitoring, guiding adjustments in hedging, capital buffers, and cross-asset positioning (Barunik et al., 2013, Hatemi-J, 2024). In copula-CoVaR settings, persistent SAM > 0 denotes systemic relevance of left-tail comovement, a critical input for systemic risk surveillance and macroprudential regulation (Dai et al., 2023).

6. Comparison to Symmetric Connectedness and Methodological Extensions

SAM explicitly corrects for a limitation of symmetric connectedness measures (Diebold–Yilmaz framework), which average out sign effects, thus potentially obscuring critical features of risk propagation. Asymmetric decomposition dynamically distinguishes the network response to sign and size of shocks—features empirically validated in volatility clustering, leverage effects, and nonlinear financial contagion phenomena (Hatemi-J, 2024, Huber et al., 2024).

Extensions under consideration include: cross-effects between positive and negative shocks (mixed spillovers), time-varying parameter models for nonstationary asymmetry, and the application of high-frequency methods to macro-financial data. Limitations include sensitivity to model specification, especially in shock splitting and copula choice, and the increase in estimation variance associated with doubling model dimension for separate positive/negative components.

7. Empirical Summary and Significance

The following table summarizes empirical findings from representative studies:

Study/Market	Asymmetry Magnitude & Direction	Typical Interpretation
US Large-Cap Stocks (Barunik et al., 2013)	Aggregate: SAM ≈ 0; Sectors: oscillating sign, often significant	Only sectorally are spillovers asymmetric; bad and good volatility alternate in influence
Global Markets (US/EU/China) (Hatemi-J, 2024)	SAM ≈ –0.30 overall; US dominates negative-side	Downside spillovers stronger globally
Agricultural Commodities (Dai et al., 2023)	Soy/Maize: downside > upside (significant); Wheat/Rice: symmetric	Tail risk propagation is commodity-specific
International (Nonlinear VAR) (Huber et al., 2024)	Adverse shocks: SAM ≈ –1 at peak; persistent over time	Contraction responses much larger than expansion

A robust empirical pattern is the episodic or structural dominance of bad-volatility (negative) spillovers, particularly during market stress periods and in specific sectors or markets. However, positive dominance and reversals are also observed, underlining the necessity of tracking SAM dynamically rather than presuming unidirectional asymmetry.

---

References:

• "Asymmetric connectedness of stocks: How does bad and good volatility spill over the U.S. stock market?" (Barunik et al., 2013)
• "On the Asymmetric Volatility Connectedness" (Hatemi-J, 2024)
• "Asymmetries in Financial Spillovers" (Huber et al., 2024)
• "Tail dependence structure and extreme risk spillover effects between the international agricultural futures and spot markets" (Dai et al., 2023)