Efficient Method of Moments

Summary

The Efficient Method of Moments (EMM) combines indirect inference with a semi-nonparametric (SNP) auxiliary model that is flexible enough to approximate any stationary Markovian density. By increasing the SNP model’s complexity with sample size, EMM achieves asymptotic efficiency equal to MLE — something that standard indirect inference with a fixed auxiliary model cannot guarantee. The SNP density uses a squared Hermite polynomial expansion to capture departures from normality, embedded within a location-scale model that handles dynamic heterogeneity.

Overview

The central limitation of Indirect Inference is that efficiency depends on the auxiliary model’s ability to approximate the structural model. If the auxiliary model is too simple, information is lost. EMM resolves this by using a flexible, data-driven auxiliary model — the SNP density of Gallant and Nychka (1987) — as the score generator.

The SNP Density

Definition: Semi-Nonparametric (SNP) Conditional Density

The SNP model represents any conditional density as:

$$h_q^{\ast }(y_t|z_t;{\lambda }_q)=\frac{[\mathcal{P}(u_t,z_t){]}^2\varphi (u_t)/|\det (S_t)|}{\int [\mathcal{P}(v,z_t){]}^2\varphi (v)dv}$$

where:

• $z_t=[y_{t-1}^{\prime },\dots ,y_{t-l}^{\prime }{]}^{\prime }$ is the conditioning vector
• $u_t=S_t^{-1}(y_t-{\mu }_t)$ is the standardized residual
• ${\mu }_t$ is a location function and $S_t$ is a scale function
• $\varphi (\cdot )$ is the standard multivariate normal density
• $\mathcal{P}(u_t,z_t)$ is a polynomial in $u_t$ with coefficients depending on $z_t$
• ${\lambda }_q$ is the full parameter vector of dimension $q$

The integration constant ensures that $h_q^{\ast }$ integrates to unity.

Components of the SNP Model

Location function (vector autoregression):

$${\mu }_t=b_0+\sum _{i=1}^{l_{\mu }}{B}_i{y}_{t-i}$$

Scale function (ARCH-type):

$$\text{vech}(S_t)=c_0+\sum _{i=1}^{l_s}{C}_i|y_{t-i}-{\mu }_{t-i}|$$

where $\text{vech}(S_t)$ is the vector containing the $n(n+1)/2$ distinct elements of the scale matrix.

Polynomial expansion (Hermite-type):

$$\mathcal{P}(u_t,z_t)=\sum _{|\alpha |=0}^{k_u}{a}_{\alpha }(z_t)u_t^{\alpha }$$

where $u^{\alpha }={\prod }_{i=1}^n{u}_i^{{\alpha }_i}$, $|\alpha |={\sum }_{i=1}^n|{\alpha }_i|$, and $k_u$ controls the degree of the polynomial. The coefficients $a_{\alpha }(z_t)$ are themselves polynomials in $z_t$ of degree $k_z$.

Special Cases of the SNP Model

• $k_u=0$: The SNP model reduces to a Gaussian VAR-ARCH specification (leading term captures first two moments)
• $k_z=0$: Deviations from normality are independent of past values
• $k_u=k_z=0$: Pure Gaussian with dynamic location and scale

Increasing $k_u$ captures non-normality and possible heterogeneity in higher-order moments. Increasing $k_z$ allows the shape of the distribution to depend on past values.

The EMM Estimator

The EMM estimator is a special case of score-based indirect inference where the auxiliary model is the SNP density:

Definition: EMM Estimator

1. Estimate the SNP auxiliary model on the observed data:

   $${\tilde{\lambda }}_T=\arg \underset{\lambda }{\max }{T}^{-1}\sum _{t=1}^T\log h_q^{\ast }(y_t|z_t;\lambda )$$

2. The EMM estimator minimizes:

   $${\hat{\theta }}_{GT}^R=\arg \underset{\theta }{\min }{\hat{g}}_R(\theta ,{\tilde{\lambda }}_T{)}^{\prime }A{\hat{g}}_R(\theta ,{\tilde{\lambda }}_T)$$

   where the simulated scores are:

   $${\hat{g}}_R(\theta ,{\tilde{\lambda }}_T)=\frac{1}{R}\sum _{r=1}^R\frac{1}{T}\sum _{t=1}^T\frac{\partial \log h_q^{\ast }[y_t^{(r)}(\theta )|z_t^{(r)}(\theta );{\tilde{\lambda }}_T]}{\partial \lambda }$$

The EMM estimator searches for $\theta$ such that the SNP scores, evaluated at ${\tilde{\lambda }}_T$, are close to zero when applied to simulated data from the structural model.

Asymptotic Efficiency

Theorem: Asymptotic Efficiency of EMM

If the dimension $q$ of the SNP model increases with the sample size $T$ such that $q\to \infty$ as $T\to \infty$, then under weak conditions the quasi-ML estimate ${\tilde{\lambda }}_T$ is an efficient nonparametric estimate of the true density $h_0(y_t|z_t)$. Consequently:

1. The SNP model is “smoothly embedded” within the structural model in the sense required for asymptotic efficiency of indirect inference
2. The EMM estimator attains the asymptotic efficiency of MLE

Specifically, Gallant and Long (1997) show that the indirect inference estimator with the SNP model as score generator attains the asymptotic efficiency of the ML estimator by increasing the dimension $q$.

Model Selection for the SNP Auxiliary

Determining the adequate specification of the SNP model — choosing $l_{\mu }$, $l_s$, $k_u$, and $k_z$ — is a key practical challenge:

1. Successive expansion: The dimension $q$ is successively increased, and model selection criteria (AIC, BIC/Schwarz) determine the preferred specification
2. Diagnostic tests: After selecting a preferred specification, diagnostic tests on the standardized residuals verify adequacy
3. ARCH vs. polynomial scale: Substituting an ARCH-type scale function for the polynomial scale in the SNP model can reduce the number of parameters while maintaining the autocorrelation structure

Over-Parameterization Risk

Score generators based on an over-parameterized SNP model can lead to a substantial loss of efficiency, especially in smaller samples (Andersen, Chung, and Sørensen, 1998). The dimension of the polynomial ($k_u$) and the number of lags ($l_{\mu }$, $l_s$) should be chosen carefully.

Finite Sample Properties

Andersen, Chung, and Sørensen (1998) conduct a comprehensive Monte Carlo study comparing EMM with GMM and likelihood-based estimators for the SV model:

• EMM is generally more efficient than standard GMM
• Likelihood-based estimators are generally more efficient than EMM, but EMM approaches their efficiency as sample size increases
• ARCH-type scale function in the SNP model improves efficiency relative to the polynomial scale specification, because it more directly captures the autocorrelation in variance that the SV model implies

Connections

• Specializes Indirect Inference by choosing a flexible, data-driven auxiliary model
• Achieves the efficiency that Method of Simulated Moments cannot guarantee without optimal moment selection
• Practical Issues in Simulation Estimation covers variance reduction relevant to EMM implementation

See Also

• Simulation-Based Estimation - Overview — comparison of all methods
• Indirect Inference — general auxiliary model framework
• Method of Simulated Moments — simpler but less efficient alternative
• Practical Issues in Simulation Estimation — implementation guidance

Sources

• tdb136.pdf — Liesenfeld & Breitung (1998), Section 5
• Gallant, A.R. and D.W. Nychka (1987), “Semi-Nonparametric Maximum Likelihood Estimation,” Econometrica 55, 363-390
• Gallant, A.R. and G.E. Tauchen (1996a), “Which Moments to Match?,” Econometric Theory 12, 657-681
• Gallant, A.R. and J.R. Long (1997), “Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Squared,” Biometrica 84, 125-141
• Andersen, T.G., H.J. Chung, and B.E. Sørensen (1998), “Efficient Method of Moments Estimation of a Stochastic Volatility Model: A Monte Carlo Study,” Working Paper, Brown University