Indirect Inference and Simulated Method of Moments

Some models — a stochastic volatility model with several interacting latent factors, say, or an agent-based market simulation — are easy to simulate from but have no tractable likelihood function, so maximum likelihood is off the table. Simulated Method of Moments (SMM) and Indirect Inference solve this by turning simulation itself into the estimation tool: pick trial parameter values, simulate a long fake dataset from the model, and adjust the parameters until statistics computed on the simulated data line up with the same statistics computed on the real data.

Simulated Method of Moments matches moments directly — the simulated data's mean, variance, autocorrelation, or any other chosen statistic is compared to the real data's, and parameters are tuned to close the gap, exactly like ordinary Method of Moments except the theoretical moments (which can't be computed in closed form here) are replaced with simulated ones.

Indirect Inference takes an extra step: instead of matching raw statistics, both the real data and the simulated data are fed through a simpler auxiliary model (something easy to fit, like an ARMA model, even if it's not the "true" model), and the parameters of the complex model are tuned until the auxiliary model's fitted coefficients match on both datasets. This is useful when the complex model doesn't have an obvious short list of moments to match, but does produce data that looks reasonably like the auxiliary model's world.

Both approaches trade a tractable likelihood for a tractable simulator, and both require the simulator to be fast enough to run many times during the parameter search — which is often the real practical bottleneck.