Efficient Method of Moments (EMM)

Definition

Efficient Method of Moments (EMM, Gallant-Tauchen 1996) is a simulation-based estimator for structural dynamic models. It is a two-step procedure:

1. Auxiliary step: fit a flexible semi-nonparametric (SNP) density to the observed data and compute its score vector at the fitted parameters.
2. Structural step: simulate from the structural model at candidate parameters θ, evaluate the auxiliary score on the simulated path, and choose θ to make the simulated score as close to zero as possible (in a metric that uses the inverse of the auxiliary information matrix as the optimal weighting).

When the auxiliary SNP model nests the true conditional density of the data, EMM achieves the same asymptotic efficiency as full maximum likelihood, while remaining feasible for structural models whose own likelihood is intractable.

Intuition

EMM is “indirect inference with a sufficient auxiliary statistic”. You don’t have a likelihood for your structural model (because regime paths, latent states, or non-linearities make it intractable). But you do have an arbitrary-fidelity density estimator (SNP) for the data. The SNP score vector summarizes everything the data has to say about a flexible class of densities. EMM says: tune your structural model so that simulated data, fed through the same SNP estimator, would have produced the same score. If the SNP class is rich enough to span the structural model’s true density, this matching is asymptotically as informative as the likelihood itself.

Formal notation

• Observed data: $y_1$, …$,y_T$
• SNP auxiliary density: f̃$(y_t$ | $y_{t-1}$, …$,y_{t-L}$; β), with β fitted by maximum likelihood on the data, β̂_T
• Auxiliary score: $s_t(\beta )=\partial logf$̃ $/\partial \beta$
• Structural model: $y_t$ ~ $p(\cdot$ ; θ), simulated path of length N at θ: ỹ${}_1(\theta )$, …, ỹ_N(θ)
• Simulated score: $m_N(\theta ,\beta$̂$)=(1/N){\Sigma }_t{s}_t($ỹ(θ); β̂_T)
• EMM estimator: θ̂ = argmin${}_{\theta }{m}_N(\theta ,\beta$̂)’ Î_T^{-1} $m_N(\theta ,\beta$̂), where Î_T is the estimated auxiliary information matrix

Variants

• GMM with sufficient moments: when the auxiliary model is parametric and known to be correctly specified, EMM reduces to GMM with the score moments (see Gourieroux-Monfort indirect inference).
• SMM (simulated method of moments): same simulation step but with hand-picked moment conditions instead of SNP scores; less efficient when the moments do not span the likelihood.
• Indirect inference (Gourieroux-Monfort-Renault 1993): matches auxiliary parameter estimates rather than scores. EMM is the score-matching specialization.
• Bansal-Zhou (2002) application: SNP auxiliary fitted to a yield panel; structural model is regime-switching CIR; simulated structural yields fed through the same SNP to compute simulated scores.

Comparison

vs. Maximum Likelihood: EMM is asymptotically as efficient as MLE when the SNP class is rich enough. EMM is the only feasible option when the structural likelihood is intractable.

vs. GMM with hand-picked moments: EMM dominates because the SNP score is approximately a sufficient statistic, while hand-picked moments may discard information.

vs. Particle-filter likelihood (e.g. RBPF): particle-filter likelihood gives noisy unbiased Monte Carlo estimates of the true structural likelihood and is the natural alternative for regime-switching state-space models. EMM does not require an explicit filter; particle-filter approaches require one but exploit the no-arbitrage structure more directly.

When to use

• Structural dynamic models with intractable likelihood (latent states, regime switching, non-Gaussian shocks).
• When you have a long enough time series for the SNP auxiliary to be reliably fit (rule of thumb: T > a few hundred).
• When you want efficiency close to MLE without writing a custom filter.
• Bansal-Zhou (2002) is the canonical asset-pricing application.

Known limitations

• EMM efficiency depends on the SNP auxiliary being a good density approximation; misspecification of the auxiliary model contaminates the structural inference.
• The two-step nature inflates standard errors relative to one-step likelihood maximization.
• Simulation noise from the structural step requires careful choice of simulation length N relative to T; common rule $N\ge$ 10T.
• Optimization over the structural parameter θ is non-convex and frequently multi-modal in practice; global search is required.

Open problems

• Bias correction and small-sample inference for EMM in the presence of weak identification.
• Diagnostics for SNP misspecification that propagate cleanly to the structural standard errors.

Key papers

• bansal-zhou-2002-term-structure-regime-shifts — uses EMM with an SNP auxiliary to estimate a regime-switching CIR term-structure model on US Treasury yields.

My understanding

EMM is the precedessor / cousin to the modern simulation-based inference (SBI) and particle-filter likelihood approaches we use in our project. For the CRE asset-pricing model, we deliberately chose a different path — Rao-Blackwellized particle filter (RBPF) for the likelihood, then BOBYQA / global optimization on the resulting noisy MC objective — because the RBPF exploits the conditionally Gaussian structure of the model in a way EMM cannot. The Bansal-Zhou (2002) paper is the anchor reference for the EMM-with-regimes route, and it establishes that the route works on US Treasury yields. Our reasons for not using EMM are: (1) the SNP class would have to be very rich to span our 4-compound-regime, 3-asset, NK-driven joint distribution, and (2) we need a per-evaluation likelihood for global optimization, which RBPF gives us natively.