Cappuccio-Lubian-Raggi (2006) Investigating Asymmetry in US Stock Market Indexes

Summary

Cappuccio, Lubian, and Raggi (2006) extend the standard discrete-time stochastic volatility (SV) model by replacing Gaussian return shocks with a Skew-GED distribution — built via the Azzalini (1985) device applied to the Generalized Error Distribution (GED). Two new parameters: $\kappa$ (skewness; negative = left-skew) and $\nu$ (tail thickness; $\nu=2$ is Gaussian). The model nests Gaussian ($\nu=2,\kappa=0$), Skew-Normal ($\nu=2,\kappa\neq0$), and GED ($\kappa=0$) as special cases. Markov chain Monte Carlo (MCMC) estimation uses delayed-rejection Metropolis-Hastings (MH; Tierney-Mira 1999) for log-volatilities and Adaptive-Rejection Metropolis Sampling (ARMS) for $\kappa$ and $\nu$. Specification tests use Savage-Dickey density ratios, computed from MCMC output via kernel smoothing at the restriction point. Applied to daily and weekly returns on DJ30 (Dow Jones 30), S&P500, and Nasdaq (from Datastream). Key result: heavy tails are pervasive in daily data (GED model preferred for DJ30/S&P500; Skew-GED only needed for Nasdaq); asymmetry is pronounced at weekly frequency with all indexes displaying negative skewness; Gaussianity rejected in all cases.

Key Claims

• Skew-GED construction (Azzalini 1985 device on GED base): For a symmetric GED variable $Z$ with density $f(z) \propto \exp(-|z|^\nu/2)$, the Skew-GED variable $X$ has density $f(x;\kappa,\nu) \propto [1 + \text{sign}(\kappa x)\,\Phi_\nu(|\kappa x|^{1/\nu})] f(x)$ where $\Phi_\nu$ is the GED cumulative distribution function (CDF). Parameter $\kappa>0$ produces right-skew; $\kappa<0$ produces left-skew. Special cases: Normal ($\nu=2,\kappa=0$), Skew-Normal ($\nu=2,\kappa\neq0$), GED ($\kappa=0$).
• SV model: Returns $y_t = \mu\exp(h_t/2)\,\varepsilon_t$ with $h_{t+1} = \alpha + \phi(h_t - \alpha) + \sigma_\eta\eta_t$, $\eta_t \sim \mathcal{N}(0,1)$, and return shock $\varepsilon_t \sim \text{Skew-GED}(\kappa,\nu)$, uncorrelated with $\eta_t$ (no leverage effect). Unconditional skewness and kurtosis decompose into the distribution-of-$\varepsilon_t$ component times a factor $\exp(\sigma_\eta^2/(1-\phi^2))$ from log-volatility variability.
• Persistence: Posterior mean of $\phi \approx 0.991$–$0.993$ (daily), $0.977$–$0.993$ (weekly) — near-integrated-GARCH (IGARCH) across all three indexes.
• Tail thickness: $\hat\nu \approx 1.49$–$1.80$ (daily), $1.63$–$1.94$ (weekly) — heavy tails far from Gaussian ($\nu=2$).
• Asymmetry results: Daily data — $\hat\kappa \approx 0$ for DJ30/S&P500 (Bayes factors favor GED over Skew-GED); $\hat\kappa \approx -0.54$ for Nasdaq (left-skew supported). Weekly data — decisive left-skew across all indexes: $\hat\kappa \approx -0.82$ (DJ30), $-1.00$ (S&P500), $-1.06$ (Nasdaq).
• Gaussianity universally rejected: Joint hypothesis ($\nu=2, \kappa=0$) has negligible Savage-Dickey ratio in all cases.
• Frequency dependence of asymmetry: Asymmetry is a weekly phenomenon — stronger at weekly frequency than daily — mirroring the frequency-dependence of other stylized facts (volatility clustering disappears monthly).
• Savage-Dickey density ratio for nested models: $\text{SD}_{j|1} = p(\theta_j = \theta_j^0 | y) / p(\theta_j = \theta_j^0)$, where the numerator is estimated by kernel smoothing of MCMC output at the restriction point and the denominator is read from the flat prior. Kass-Raftery (1995) scale: substantial evidence SD > 3.2; strong > 10; decisive > 100.

Concepts Introduced or Extended

• Stochastic Volatility — Skew-GED return error extension; ARMS for non-standard shape parameters; no-leverage variant
• Stylized Facts of Asset Returns — confirms fat tails and conditional left-skewness; asymmetry is frequency-dependent

Entities Mentioned

• Jacquier-Polson-Rossi (2004) — cited as prior SV fat-tails work
• Kim-Shephard-Chib (1998) — cited as standard SV baseline
• Eraker (2001) — cited for uncorrelated SV specification convention
• Tierney (1994) — MCMC convergence theory
• Bollerslev-Engle-Nelson (1994) — ARCH/GARCH survey background

Quotes

"The estimation results are consistent with the presence of substantial asymmetry and heavy tails in the distribution of US stock market indexes."

"Daily data provide strong evidence in favour of heavy tails and some mixed evidence in favour of asymmetry. The picture is turned upside down when taking into account weekly data where there is a much more neat evidence of asymmetry."

My Take

A clean extension of the Jacquier-Polson-Rossi / Kim-Shephard-Chib (JPR/KSC) framework with a well-motivated non-Gaussian error distribution. The Skew-GED nesting is convenient for specification testing via Savage-Dickey ratios without requiring non-nested test statistics. The main limitation is the absence of a leverage effect (return and volatility shocks are independent by assumption) — all asymmetry resides in the conditional error distribution rather than in the return-volatility correlation. This is consistent with finding stronger asymmetry at weekly frequency, where the leverage channel would be less prominent than at daily frequency. The delayed-rejection MH sampler and ARMS for non-log-concave conditionals are methodologically useful. Results confirm the standard finding that Gaussianity is inadequate for high-frequency financial returns, and that heavy tails and skewness are complementary features rather than substitutes.