Multiplicative Error Model and Multiple Indicators Model

Definition

The Multiplicative Error Model (MEM), introduced by Engle and Gallo (2006), models any non-negative time series $x_t$ as the product of a time-varying conditional mean $\mu_t$ and a unit-mean multiplicative error $\varepsilon_t$ . The Multiple Indicators Model (MIM) extends MEM to a joint system of several volatility proxies linked by cross-equation dynamics.

Key Ideas

The MEM nests generalized autoregressive conditional heteroskedasticity (GARCH) as a special case: GARCH models the conditional variance $\sigma_t^2 = \mu_t$ implicitly through the identity $\varepsilon_{t-1}^2 = \mu_{t-1}\cdot(\eta_{t-1}^2)$ where $\eta_t^2$ is Gamma distributed with unit mean.
By treating observed volatility proxies (absolute returns, range, realized variance) as the dependent variable rather than latent quantities, the MEM eliminates the need for data augmentation or filtering.
Cross-equation lags in the MIM exploit the fact that different proxies measure the same latent volatility at different levels of measurement error: realized volatility (most precise) Granger-causes absolute returns (noisiest) and daily range.
Multi-step forecasts from the MIM system can be non-monotonic at intermediate horizons — overshooting or undershooting the unconditional mean — due to complex conjugate characteristic roots of the system coefficient matrix.

How It Works

Univariate MEM

Let $x_t$ be a non-negative observable (e.g. $|r_t|$ , daily high-low range, or realized volatility). The MEM specifies:

$x_t = \mu_t \cdot \varepsilon_t, \qquad E[\varepsilon_t | \mathcal{F}_{t-1}] = 1$

with GARCH-like dynamics for the conditional mean:

$\mu_t = \omega + \sum_{i=1}^{q} \alpha_i x_{t-i} + \sum_{j=1}^{p} \beta_j \mu_{t-j}, \qquad \omega > 0,\; \alpha_i, \beta_j \geq 0$

The error term follows a Gamma distribution: $\varepsilon_t \sim \text{Gamma}(a, a)$ with density

$f(\varepsilon) = \frac{a^a}{\Gamma(a)} \varepsilon^{a-1} e^{-a\varepsilon}, \qquad E[\varepsilon_t] = 1, \quad \text{Var}(\varepsilon_t) = 1/a$

When $a \to \infty$ the Gamma approaches a point mass at 1 (deterministic); for $a < \infty$ the dispersion parameter $1/a$ quantifies overdispersion.

Connection to GARCH. Setting $x_t = \varepsilon_t^2$ where $\varepsilon_t = r_t$ (the demeaned return), and writing $\eta_t = \varepsilon_t/\sigma_t \sim \mathcal{N}(0,1)$ , the conditional variance recursion $\sigma_t^2 = \mu_t$ matches the GARCH(p,q) variance equation. The Gamma log-likelihood objective replaces the Gaussian one; the two coincide only when returns are exactly Gaussian.

Estimation. Quasi-maximum likelihood estimator (QMLE) maximizing the Gamma log-likelihood $\ell(\theta) = \sum_t [a\log a - \log\Gamma(a) + (a-1)\log x_t - a\log\mu_t - ax_t/\mu_t]$ with sandwich (robust) standard errors. Consistent and asymptotically normal under correct conditional mean and variance specifications, even if the Gamma distributional assumption is misspecified.

Multiple Indicators Model (MIM)

Engle and Gallo (2006) apply MEM jointly to three daily S&P 500 volatility proxies:

Absolute returns $|r_t|$ : cheapest proxy, highest measurement noise.
Daily high-low range $hl_t = \log H_t - \log L_t$ : more efficient than squared returns (Garman-Klass 1980; Alizadeh-Brandt-Diebold 2002; approximately $8\times$ more efficient).
Realized volatility $v_t = \sum_j r_{t,j}^2$ : 5-minute intra-daily squared returns; most accurate proxy.

Each indicator gets its own MEM equation; Bayesian Information Criterion (BIC) general-to-specific selection determines which cross-indicator lags to include:

$\mu_t^{(r)} = \omega_r + \sum_i \alpha_{ri} |r_{t-i}| + \sum_j \beta_{rj} \mu_{t-j}^{(r)} + [\text{lags of } hl_{t-i}, v_{t-i}]$ $\mu_t^{(hl)} = \omega_{hl} + \sum_i \alpha_{hi} hl_{t-i} + \sum_j \beta_{hj} \mu_{t-j}^{(hl)} + [\text{lags of } |r_{t-i}|, v_{t-i}]$ $\mu_t^{(v)} = \omega_v + \sum_i \alpha_{vi} v_{t-i} + \sum_j \beta_{vj} \mu_{t-j}^{(v)} + [\text{lags of } |r_{t-i}|, hl_{t-i}]$

Application. S&P 500 daily data, January 4 1988 – December 30 1997. Summary statistics (all in annual percentage terms): $\bar{|r_t|} \approx 4.28$ , $\overline{hl_t} \approx 4.83$ , $\bar{v_t} \approx 7.60$ .

Multi-step Forecasting

For the univariate MEM, $k$ -step-ahead forecasts satisfy $E[x_{t+k}|\mathcal{F}_t] = A^k e_t$ where $A$ is the companion matrix of the system and $e_t$ captures current levels. In the scalar MEM(1,1) with persistence parameter $\alpha + \beta$ , forecasts decay to the unconditional mean $\bar\mu = \omega/(1-\alpha-\beta)$ geometrically, exactly as in GARCH.

For the multivariate MIM, if the system matrix $A$ has complex conjugate eigenvalues, the $k$ -step-ahead path can oscillate — overshooting or undershooting the unconditional mean at intermediate horizons — before converging. The daily range indicator alone exhibits the familiar monotonic GARCH profile; the absolute returns and realized volatility equations, connected by cross-lags, can produce non-monotonic fan-chart forecasts.

VIX Comparison

The 22-day-ahead compound forecast aggregates the MIM one-step predictions:

$F_{t, t+22} = \sum_{k=1}^{22} E[v_{t+k} | \mathcal{F}_t]$

This quantity is comparable to the Chicago Board Options Exchange (CBOE) VIX (S&P 500 implied volatility, annualized), which measures market expectations of 30-day volatility. Out-of-sample regressions of VIX on MIM compound forecasts (January–November 1998) show all cross-indicator system terms jointly significant; the system specification BIC dominates single-indicator base specifications. Augmented Dickey-Fuller (ADF) rejects a unit root for VIX, consistent with mean-reverting volatility expectations.

Why It Matters

Realized volatility is a near-direct observation of the latent variance process, yet MEM treats it identically to more traditional proxies; the framework unifies multiple data sources without imputation.
MEM estimation requires only a standard optimization of the Gamma log-likelihood — no latent-variable filtering, no Markov Chain Monte Carlo (MCMC). Existing GARCH software can be adapted by changing the objective function.
Cross-indicator dynamics provide efficiency gains over single-indicator models and have economic content: the lead-lag structure reveals which proxies contain the freshest volatility signal.
Non-monotonic multi-step forecasts are an empirically important departure from the GARCH benchmark: risk managers using MIM at 5–15-day horizons may face non-standard term structure shapes.

Open Questions

Formal model comparison (Diebold-Mariano tests) between MIM and competing models (heterogeneous autoregressive realized volatility (HAR-RV), GARCH-X — GARCH with exogenous regressors) is not provided in Engle-Gallo (2006); subsequent literature (Corsi 2009 HAR-RV) has partly addressed this.
Extension to asynchronous indicators (e.g., when realized volatility is only available with a lag) requires a modified filtering step.
Optimal aggregation weights across indicators (rather than BIC-selected cross-lags) remain an open specification question.