Yu (2005) On Leverage in a Stochastic Volatility Model

Summary

Yu compares two competing discrete-time asymmetric stochastic volatility (ASV) specifications that both claim to model the leverage effect. He shows that ASV1 (Harvey-Shephard 1996, Euler approximation to continuous-time SV) is theoretically superior — it preserves the martingale difference property and admits a clean leverage interpretation — while ASV2 (Jacquier et al. 2004) violates the efficient market hypothesis (EMH) and understates the magnitude of leverage by ~20%. Bayesian Markov Chain Monte Carlo (MCMC) via Bayesian inference Using Gibbs Sampling (BUGS) and marginal likelihood via the Chib (1995) identity and Kitagawa particle filter provide decisive empirical evidence (Bayes factor $\approx$ 4,837) in favour of ASV1 on S&P500 and Center for Research in Security Prices (CRSP) data.

Key Claims

• Two specs, one difference: Both ASV1 and ASV2 have $X_t = \sigma_t u_t$ and $\ln \sigma^2_{t+1} = \alpha + \phi \ln \sigma^2_t + \sigma_\eta v_{t+1}$, but ASV1 sets $\text{corr}(u_t, v_{t+1}) = \rho$ (contemporaneous return shock correlates with next period's volatility shock) while ASV2 sets $\text{corr}(u_t, v_t) = \rho$ (same-period).
• ASV2 violates EMH: $E(X_{t+1}|X_t) \neq 0$ under ASV2 — returns are predictable. Using S&P500 posterior means this implies an annual drift of $-8.75$%; for CRSP, $-15.75$%. ASV1 is a martingale difference sequence.
• Nonlinear state-space leverage interpretation: Orthogonalising ASV1 (eq. 2.3) yields a log-variance equation with an explicit leverage term: $\ln \sigma^2_{t+1} = \alpha + \phi \ln \sigma^2_t + \rho\sigma_\eta \sigma_t^{-1} X_t + \sigma_\eta\sqrt{1-\rho^2}\, w_{t+1}$. Leverage = coefficient on $X_t$: if $\rho < 0$, negative returns raise future log-variance. Analogous orthogonalisation of ASV2 places $\sigma_{t+1}$ on both sides — intractable, no clean leverage reading.
• BUGS estimation: Multi-move algorithms (Kim-Shephard-Chib 1998) require a log-squared transformation that destroys correlation information and cannot be used for asymmetric SV. BUGS (single-move Metropolis-Hastings (MH)) is applied directly on the nonlinear state-space form. Priors: $\phi^* \sim \text{Beta}(20,1.5)$; $\sigma^2_\eta \sim \text{InvGamma}(2.5,0.025)$; $\rho \sim \text{Uniform}(-1,1)$.
• Particle filter for marginal likelihood: Kitagawa (1996) filter evaluates the log-likelihood at posterior means (needed for Chib 1995 marginal likelihood identity). Applicable after re-parameterising both models to have uncorrelated measurement and transition errors.
• S&P500 results (2,022 daily returns, 1980–1987): ASV1 log marginal likelihood = $-2794.6$ vs. ASV2 = $-2803.1$ $\to$ Bayes factor $\approx$ 4,837 (decisive). $\hat\rho = -0.318$ (ASV1); $\hat\rho = -0.256$ (ASV2) (~20% underestimate). $\hat\phi \approx 0.972$.
• CRSP results (2,529 daily returns, 1986–1995): $\hat\rho = -0.394$ (ASV1); decisive evidence against ASV2. Encompassing model (allows both lags): ASV1 decisively beats encompassing; encompassing barely beats ASV2.
• Simulation: MCMC outperforms Quasi-Maximum Likelihood (QML) on all parameters. QML relative efficiency (Root Mean Square Error (RMSE)) = 56–71% of MCMC for $\rho$, $\phi$, $\ln\sigma^2$.
• Single-move sufficiency: Simulation inefficiency factors for asymmetric SV models are 21–27% lower than for the basic SV model — leverage parameterisation improves mixing even with single-move MH.

Concepts Introduced or Extended

• Stochastic Volatility

Entities Mentioned

• Jun Yu

Quotes

"The ASV2 model is not consistent with the efficient market hypothesis because the model is not a martingale difference sequence."

"While the interpretation of the leverage effect using a parameter in the ASV1 model is clear, the strict interpretation of leverage is not obvious in the ASV2 model."

"Since MCMC is a fully likelihood-based method, it always performs better than QML."

My Take

The paper makes a narrow but important methodological point: the timing convention for the leverage correlation matters both theoretically (EMH) and empirically (20% bias in $\rho$). The nonlinear state-space re-parameterisation (eq. 2.3) is the clearest way to understand what leverage means in discrete-time SV, and should be standard exposition. The limitation is that BUGS with single-move MH is computationally slow for large datasets — the paper notes simulation inefficiency factors of 131–218, meaning the effective sample is $\approx$1/150th of the raw chain. For the particle filter likelihood evaluation, Kitagawa's (1996) basic bootstrap filter is used, which is less efficient than the auxiliary particle filter (Pitt-Shephard 1999); this choice was made for implementation simplicity.