Choosing Between SML, MSM, and Indirect Inference

Why the Choice of Estimator Matters

A structural model may be easy to simulate but difficult to evaluate as a likelihood. Or its likelihood may be available while repeatedly optimizing it is costly. In other cases, the model may imply useful moments but not a convenient probability density. These differences matter because simulated maximum likelihood (SML), simulated method of moments (MSM), and indirect inference (II) require different outputs from the model and retain different information about its parameters.

The choice is not a contest with one universally best estimator. It depends on what the model can calculate reliably, which aspects of the observed data are informative about the parameters, and how much simulation error the computation can tolerate. A method that is feasible but weakly identifying may be less useful than a more demanding method with a clearer connection between model parameters and observed features.

Common Setup and the Three Model Outputs

Let \(y_1,\ldots,y_n\) denote observed data generated by a structural model indexed by a parameter vector \(\theta\in\Theta\subseteq\mathbb{R}^q\), where \(q\) is the number of unknown parameters. The target value is \(\theta_0\). Each method also needs appropriate regularity conditions: the chosen model output must be well defined, the parameter must be identified by that output, and the data and simulation approximations must support the estimator’s limiting behavior. The details differ by method.

The outputs required are distinct. SML needs observation-level likelihood contributions, or a way to approximate them. MSM needs empirical statistics and corresponding model-implied moments. II needs an auxiliary estimator that can be applied both to the observed data and to simulated data. In all three methods, simulation approximates a model output; it does not by itself guarantee that the output contains enough information to distinguish parameter values.

Information and Identification

The methods differ in the way they summarize the model’s implications. SML can use the full specified distribution of the observations. Under correct specification and regularity conditions, this can use more information than a small set of moments or auxiliary estimates. But the full distribution must be evaluable with adequate accuracy, and distinct parameter values must not imply the same distribution.

MSM uses a chosen vector of \(k\) moments. Let \(m_n\) be the vector of sample moments and \(m(\theta)\) the corresponding population moments implied by the model. A basic MSM estimator minimizes a weighted discrepancy between them:

where \(\hat m_R(\theta)\) is a simulation-based estimate of \(m(\theta)\) using \(R\) draws, and \(W\) is a positive semidefinite weighting matrix. A sufficient regularity condition for local identification by this criterion is that \(W^{1/2}G(\theta_0)\) have column rank \(q\). Identification requires the selected moments to distinguish the parameters. A sufficient regularity condition for local identification of \(q\) parameters is that the derivative matrix \(G(\theta_0)=\partial m(\theta)/\partial\theta'\) have column rank \(q\). Failure of this rank condition alone does not rule out local identification; the moment map must be checked directly. In particular, at least \(q\) informative moment restrictions are needed; merely choosing \(k\ge q\) moments does not ensure identification.

II replaces the selected raw moments with estimates from an auxiliary model. Write \(\hat\beta_{\text{obs}}\) for the auxiliary estimate in the observed data and \(\hat\beta_R(\theta)\) for the estimate obtained from data simulated at \(\theta\). Its objective has the form

The auxiliary model need not be the structural model. It is a way to extract features of the data that respond to changes in structural parameters. Identification depends on the binding relationship between \(\theta\) and the population auxiliary estimates generated by the model. A sufficient regularity condition for local identification is that the derivative of that relationship have column rank \(q\). Failure of this rank condition alone does not rule out local identification; the binding relationship must be checked directly. Thus, an auxiliary regression that is easy to estimate is not automatically informative enough for II.

Computational Cost and Approximation Error

There is no fixed computational ranking of the methods. SML can be expensive when each likelihood evaluation requires many simulated probabilities or high-dimensional integration, and optimization may require many such evaluations. The earlier material on importance sampling, quasi-Monte Carlo integration, and GHK simulation concerns tools for particular likelihood calculations; these can help, but their success depends on the model and implementation.

MSM may be cheaper when a modest number of moments can be simulated directly. Its cost can rise when many moments are used, their covariance must be estimated, or simulation and optimization must be repeated. II often avoids evaluating a structural likelihood, but may require generating a full simulated data set and fitting the auxiliary model many times for each parameter trial. It can be costly if that auxiliary estimation is itself demanding. The appropriate comparison is therefore the total cost of obtaining a stable, informative objective—not just the cost of one simulation.

Finite simulation affects each method through its own model output. In plug-in SML, an estimated probability may be inserted inside a logarithm. If \(\hat p\) is an unbiased simulation estimate of \(p>0\), \(\hat p>0\) almost surely, and \(E[\log(\hat p)]\) exists, Jensen’s inequality gives

This shows why unbiased simulation of a probability does not generally make its simulated log contribution unbiased. The effect on the final estimator depends on the full procedure; adequate draw counts, smooth simulation, and stable optimization matter.

For MSM, a simulated moment average may be unbiased for a fixed \(\theta\), but optimizing a noisy, finite-draw objective can still affect the estimated minimizer. In II, simulation error perturbs the simulated auxiliary estimates and therefore the binding relationship used for matching. Increasing \(R\), using well-designed draws, and employing common random numbers across nearby parameter values can reduce simulation noise or make objectives smoother. These practices do not repair weak identification, model misspecification, or an unsuitable auxiliary statistic.

Robustness: Numerical Approximation Is Not Model Robustness

Two meanings of robustness should be kept separate. Robustness to simulation approximation concerns how much a finite number of draws changes the computed estimator. Robustness to distributional misspecification concerns whether the estimator remains meaningful when the assumed model is not the true data-generating process. A method can perform well on one dimension and poorly on the other.

SML is particularly dependent on a correctly specified likelihood for its usual likelihood-based interpretation. If its simulated likelihood is inaccurately computed, increasing simulation effort can address that numerical error; it cannot make an incorrect likelihood correct. MSM can be less reliant on the full distribution if its moment restrictions remain valid under departures from that distribution. But this is not automatic: invalid moments or weakly informative moments undermine the estimator, and choosing only a few moments can discard useful information.

II can be useful when a full structural likelihood is unavailable but the model can generate data and an auxiliary estimator captures important features. Its conclusions still depend on the structural simulation, the quality and informativeness of the auxiliary estimates, and a sufficiently accurate binding relationship. None of the three methods is universally robust to finite simulation error. The relevant comparison is which numerical approximation can be controlled most credibly for the chosen model output.

A Practical Decision Framework

A defensible choice starts with the most reliable output available and then checks whether that output identifies the parameters of interest. The following sequence makes the trade-offs explicit.

Common Pitfall

A common mistake is to choose MSM or II simply because the likelihood is difficult, then assume that matching any available summaries will recover the structural parameters. The correction is to check whether those summaries are informative about every parameter being estimated, and to account for finite-simulation error in the summaries or binding function. Conversely, a computable likelihood is not automatically preferable if its numerical approximation is unstable or its identifying information is weak. Method choice is a joint assessment of feasibility, information, computation, and assumptions.