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Indirect inference and simulated method of moments

Fit a simulatable model through auxiliary estimates or direct moments, and see why identification, weighting and simulation error determine what the match means.

Indirect inference estimates a structural model by making estimates from a simpler auxiliary model agree between observed data and simulated data. The simulated method of moments (SMM) instead directly matches selected data moments to simulated counterparts. Both help a quant estimate a model that can generate scenarios even when its likelihood or theoretical moments are difficult to evaluate.
The distinction is the target: fitted auxiliary coefficients versus direct moment summaries. They belong to the same simulation-based minimum-distance family, and can overlap when auxiliary estimates are themselves moment summaries. Simulation supplies predictions for estimation; it does not automatically validate the model. Duffie and Singleton develop simulated moment matching for dynamic asset-pricing models.

Three objects that must stay separate

Suppose a structural volatility model has parameters θ\theta governing its latent variance dynamics. An auxiliary GARCH model fitted to returns has parameters β\beta, such as its intercept and lag coefficients. Those GARCH coefficients describe the auxiliary fit; they are not interchangeable with θ\theta.

Fit the auxiliary model to observed returns to obtain β^T\hat\beta_T. For each candidate θ\theta, simulate HH return histories of length TT, apply the same auxiliary estimation procedure to each, and average their estimates:
βˉT,H(θ)=1H∑h=1Hβ^T(h)(θ).\bar\beta_{T,H}(\theta) =\frac{1}{H}\sum_{h=1}^{H}\hat\beta_T^{(h)}(\theta).

One parameter-matching form of indirect inference chooses

θ^II=argmin⁡θ∈Θ[β^T−βˉT,H(θ)]TWT[β^T−βˉT,H(θ)].\hat\theta_{\mathrm{II}} =\operatorname*{argmin}_{\theta\in\Theta} \left[\hat\beta_T-\bar\beta_{T,H}(\theta)\right]^{\mathsf T} W_T \left[\hat\beta_T-\bar\beta_{T,H}(\theta)\right].
The binding function b(θ)b(\theta) is the probability limit of the auxiliary estimate when data come from structural parameter θ\theta. It is a population map, distinct from its finite simulation approximation. An auxiliary model may be misspecified and still provide informative targets; consistent recovery of structural parameters additionally requires the structural model and identification assumptions to hold. Frazier and Renault, Introduction and §2 discuss these objects and a stochastic-volatility/GARCH example, including the problems auxiliary constraints can create.
SMM uses the same distance construction with observed and simulated moments in place of auxiliary parameter estimates. It extends method of moments and GMM by approximating model-implied moments through simulation.

An oracle to check the simulation against

Use a deliberately easy, dimensionless model so every answer has an analytical check:

Y=μ+σε,ε∼N(0,1),μ∈R,σ>0.Y=\mu+\sigma\varepsilon, \qquad \varepsilon\sim\mathcal{N}(0,1), \qquad \mu\in\mathbb{R},\quad\sigma>0.

Stipulate observed targets of mean 22 and variance 33. These are illustrative inputs, not market estimates. With identity weighting, the population objective is

Q(μ,σ)=(μ−2)2+(σ2−3)2.Q(\mu,\sigma)=(\mu-2)^2+(\sigma^2-3)^2.

Both squared terms are nonnegative. They vanish together only at μ=2\mu=2 and σ=3≈1.7321\sigma=\sqrt{3}\approx1.7321, so the minimum is exactly zero. Restricting σ>0\sigma>0 removes the otherwise equivalent negative root.

This calculation uses exact population moments: it is an oracle for SMM, rather than a demonstration that this tractable normal model needs simulation. A practical SMM objective replaces those exact moments by simulated moments.
Two population restriction plots with mu on the horizontal axis and positive sigma on the vertical axis. Matching mean 2 leaves a vertical line at mu 2. Adding variance 3 selects sigma square root 3, approximately 1.7321. A separate simulation using 400 seeded shocks matches both simulated targets at mu approximately 1.9539 and sigma approximately 1.7147.
Two population restriction plots with mu on the horizontal axis and positive sigma on the vertical axis. Matching mean 2 leaves a vertical line at mu 2. Adding variance 3 selects sigma square root 3, approximately 1.7321. A separate simulation using 400 seeded shocks matches both simulated targets at mu approximately 1.9539 and sigma approximately 1.7147.
Population identification in the synthetic normal model, followed by a separate finite-simulation fit. Lines are parameter restrictions, not confidence regions.
Now generate 400400 standard-normal shocks with Python's random.Random(20261030) and gauss(0, 1), holding these shocks fixed as parameters change. Calculate simulated variance with divisor 400400.
CalculationMeanVariance
Stipulated targets2.0000003.000000
Simulation at population oracle2.0465333.060990
Same shocks, parameters adjusted to fit2.0000003.000000

Writing the realised shock mean as εˉ\bar\varepsilon and its divisor-400400 variance as vεv_\varepsilon, the simulation fit is available without an optimiser:

σ^sim=3/vε≈1.714709,μ^sim=2−σ^simεˉ≈1.953933.\hat\sigma_{\mathrm{sim}}=\sqrt{3/v_\varepsilon}\approx1.714709, \qquad \hat\mu_{\mathrm{sim}}=2-\hat\sigma_{\mathrm{sim}}\bar\varepsilon \approx1.953933.

The exact fit to these simulated targets does not equal the population solution. It absorbs the realised simulation error. A different seed changes that finite-simulation answer; more simulation can reduce this error but cannot remove uncertainty in the observed targets.

What makes the match informative?

Identification comes from the chosen targets. In the oracle example, matching the mean alone fixes μ=2\mu=2 and says nothing about σ\sigma. Matching variance adds the missing restriction. For indirect inference, distinct structural parameters must produce distinct relevant auxiliary limits; in regular differentiable settings, full column rank of the binding-function Jacobian is the local rank condition. Counting at least as many auxiliary coefficients as structural parameters is necessary in that regular setting but does not establish identification. Nearly flat directions also make recovery unstable. Shalizi's lecture, “Sensitivity” and “Indirect Inference” develops this connection.
Simulation must represent the observed experiment. Sampling frequency, observation rules, missingness and initial-state treatment matter. Moment existence and suitable laws of large numbers are needed; dynamic models require control of dependence and starting values. Numerical approximations must be controlled as well: matching a discretised model can estimate that approximation rather than the intended continuous-time model. These issues are explicit in Duffie and Singleton, §§2–4.
The distance needs a declared metric. Take WTW_T symmetric positive definite on the matched targets here. Identity weighting is convenient, but changing measurement units changes its priorities. Covariance-based weighting can account for target precision and correlation when its estimate and regularity conditions are justified. Standard errors must account for both data variation and simulation variation, including serial dependence where relevant. More simulation is not a substitute for more observed data.
Finally, a small fitted distance measures agreement through the selected targets. It does not establish normality, reproduce extreme losses, or prove an economic mechanism correct. Check features not used for fitting and evaluate parameter recovery on synthetic data before interpreting a fitted parameter economically. Shalizi's model-checking discussion explains why fitted summaries alone are weak diagnostics.
For the underlying normal mean/variance estimation and its contrast with likelihood fitting, see Estimation and Maximum Likelihood (public preview; full lesson requires membership).

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