A Multiple Indicators Model for Volatility Using Intra-Daily Data

Summary

Engle and Gallo (2006) introduce the Multiplicative Error Model (MEM) and apply it as a joint system — the Multiple Indicators Model (MIM) — to three daily volatility proxies: absolute returns, the high-low daily range, and realized volatility computed from intra-daily data. The paper shows that modeling the three indicators as a cross-equation system with shared dynamics outperforms three independent single-equation specifications in forecast accuracy and produces superior 22-day-ahead predictions that track the Chicago Board Options Exchange (CBOE) volatility index (VIX).

Key Claims

• The MEM $x_t = \mu_t \varepsilon_t$ with $E[\varepsilon_t | \mathcal{F}_{t-1}] = 1$ and $\varepsilon_t \sim \text{Gamma}(a,a)$ nests GARCH as a special case; quasi-maximum likelihood estimation (QMLE) with sandwich standard errors is consistent under the weaker assumption of correct conditional mean and variance specifications.
• The dynamic equation for the conditional mean is $\mu_t = \omega + \sum_i \alpha_i x_{t-i} + \sum_j \beta_j \mu_{t-j}$, analogous to GARCH(p,q) but for non-negative observables.
• Cross-indicator lags are significant: realized volatility Granger-causes absolute returns and range; range Granger-causes absolute returns.
• Multi-step forecasts converge to the unconditional mean at the rate of the largest characteristic root of the system coefficient matrix $A$. Unlike single-indicator GARCH (monotonic), the MIM system can produce non-monotonic forecasts (overshooting/undershooting) at intermediate horizons due to complex conjugate roots.
• Out-of-sample 22-day-ahead MEM compound forecasts track the CBOE VIX well (S&P 500, January–November 1998); VIX regressions confirm the system specification adds explanatory power over single-indicator models.
• Bayesian information criterion (BIC) general-to-specific selection across all cross-equation lag combinations identifies the parsimonious MIM.

Concepts Introduced or Extended

• Multiplicative Error Model and Multiple Indicators Model
• GARCH and BEKK-GARCH
• Value at Risk

Entities Mentioned

• Robert Engle
• Giampiero Gallo
• Tim Bollerslev

Quotes

"The MEM provides a unified framework for modelling non-negative time series in which volatility indicators from different sources are combined without the need to take logarithms or square roots."

My Take

The MEM framework is elegant and under-used relative to GARCH variants. The key insight is that any non-negative series with a unit-mean multiplicative error can be treated like a GARCH variance: the Gamma log-likelihood objective just replaces the Gaussian one. The MIM cross-equation structure is the multivariate analogue of BEKK (Baba-Engle-Kraft-Kroner) but for observables rather than squared residuals, which makes estimation and interpretation simpler. The VIX comparison is informal — no formal forecast evaluation tests (Diebold-Mariano) are reported — but the visual and regression evidence is suggestive. The non-monotonic multi-step forecasts are genuinely novel and important for risk management at horizons of 1–4 weeks.