Indirect Inference With Simulated Auxiliary Models

Why Match an Auxiliary Model?

A structural model may describe behavior in enough detail to simulate data, while its likelihood is unavailable or too difficult to evaluate. This does not prevent estimation. One option is to fit a simpler, tractable auxiliary model to the observed data, then ask which structural parameter values generate simulated data whose auxiliary estimates resemble the observed estimates.

The auxiliary model is a measurement device for features of the data, not necessarily the true data-generating model. For example, a linear regression can summarize persistence in a series generated by a nonlinear structural process. Indirect inference uses that summary as a bridge between the structural parameters and the observed sample. Its central requirement is not that the auxiliary model be structurally correct, but that the selected auxiliary estimates respond informatively to changes in the structural parameters.

Definition: Let \(\theta\) denote the structural parameter vector, and let \(\hat{\beta}_{\mathrm{obs}}\) be an estimate from an auxiliary model fitted to the observed sample. For each proposed \(\theta\), simulate data from the structural model and compute the same auxiliary estimate, denoted \(\hat{\beta}_{r}(\theta)\) for replication \(r\). The binding function \(b(\theta)\) is the probability limit of that auxiliary estimator under data generated at \(\theta\). Indirect inference estimates \(\theta\) by matching \(\hat{\beta}_{\mathrm{obs}}\) to simulated auxiliary estimates, which approximate \(b(\theta)\).

The Matching Criterion

Suppose the auxiliary estimate has \(q\) components and the structural parameter has \(k\) components. For each candidate \(\theta\), simulate \(R\) samples and average their auxiliary estimates:

\bar{\beta}_{R}(\theta)=\frac{1}{R}\sum_{r=1}^{R}\hat{\beta}_{r}(\theta).

A standard indirect-inference estimator minimizes a weighted squared distance:

\hat{\theta}_{\mathrm{II}} = \arg\min_{\theta\in\Theta} \left[\hat{\beta}_{\mathrm{obs}}-\bar{\beta}_{R}(\theta)\right]' W \left[\hat{\beta}_{\mathrm{obs}}-\bar{\beta}_{R}(\theta)\right],

where \(W\) is a symmetric positive-definite weight matrix. If there is one auxiliary estimate, the distance is a squared difference, possibly rescaled. With several estimates, \(W\) determines how discrepancies are combined. The objective is evaluated using simulated data, so the structural likelihood itself need not be calculated.

Conditions for a meaningful match: The structural model must define a simulator for each \(\theta\) under consideration. The observed and simulated samples must be processed using the same auxiliary model and estimation procedure. The auxiliary estimates must have well-defined probability limits. Finally, the binding function must distinguish the structural parameters: in particular, different relevant values of \(\theta\) should not produce the same binding-function value. If \(q>k\), exact matching may be impossible, so the weighting rule determines the best-fitting compromise.

The observed sample and each simulated sample should have compatible sample sizes and designs. If the observed data include covariates or a particular pattern of timing, the simulation procedure should reproduce the relevant design features so that differences in auxiliary estimates reflect the structural parameter rather than an avoidable mismatch in data construction.

Why Matching Can Identify the Structural Parameter

The logic can be expressed through the binding function. Suppose the true structural parameter is \(\theta_0\), and the auxiliary estimator from observed data converges to \(b(\theta_0)\). For simulated data generated at candidate \(\theta\), the corresponding auxiliary estimates converge to \(b(\theta)\). Thus, as sampling and simulation become sufficiently informative, the indirect-inference distance approaches a criterion comparing \(b(\theta_0)\) with \(b(\theta)\).

Q(\theta) = \left[b(\theta_0)-b(\theta)\right]' W \left[b(\theta_0)-b(\theta)\right].

Because \(W\) is positive definite, \(Q(\theta)\geq 0\), and \(Q(\theta)=0\) exactly when \(b(\theta)=b(\theta_0)\). If the binding function is one-to-one over the parameter space of interest, that equality implies \(\theta=\theta_0\). This is the identification argument: the auxiliary estimates must carry enough information to distinguish the structural parameters.

Define the auxiliary estimator.
Choose a tractable model and specify exactly which estimate or estimates will be matched.

Estimate from observed data.
Fit the auxiliary model once to the observed sample to obtain \(\hat{\beta}_{\mathrm{obs}}\).

Simulate at a candidate.
Generate data under the structural model at \(\theta\), preserving the relevant sample design, and re-estimate the same auxiliary model.

Compare and optimize.
Combine the auxiliary-estimate discrepancies using \(W\), then search over \(\theta\) for the smallest criterion.

Worked Applications

Worked Example: Matching a Single Auxiliary Estimate

Consider a hypothetical structural model for a behavioral outcome. Its likelihood is unavailable, but its simulator can generate an outcome series at any proposed structural parameter \(\theta\). An auxiliary linear regression estimates the series' persistence coefficient. The estimate from the observed data is \(0.48\). For three candidate structural values, the averages of the auxiliary estimates across simulations are \(0.39\) at \(\theta=0.35\), \(0.49\) at \(\theta=0.45\), and \(0.60\) at \(\theta=0.55\). These numbers are dimensionless persistence estimates.

With a scalar auxiliary estimate and \(W=1\), the criterion is the squared discrepancy:

Q(\theta)=\left[0.48-\bar{\beta}_{R}(\theta)\right]^2.

At the three candidates, the calculations are:

Q(0.35)=(0.48-0.39)^2=0.09^2=0.0081, \qquad Q(0.45)=(0.48-0.49)^2=(-0.01)^2=0.0001,

$$ Q(0.55)=(0.48-0.60)^2=(-0.12)^2=0.0144. $$

Among these three candidates, \(\theta=0.45\) gives the smallest discrepancy. A full numerical search might examine additional values, so this comparison alone does not establish that \(0.45\) is the global minimum. It does show how simulated auxiliary estimates can guide estimation without evaluating the structural likelihood.

Worked Example: Recovering Two Parameters From Two Auxiliary Estimates

Suppose a hypothetical structural model has two parameters, \(\theta_1\) and \(\theta_2\), and the chosen auxiliary procedure reports two dimensionless estimates. Simulations indicate the binding function \(b(\theta)=(\theta_1+\theta_2,\ \theta_1-\theta_2)'\). The observed auxiliary estimates are \(\hat{\beta}_{\mathrm{obs}}=(1.20,\ 0.30)'\). Assume the simulated estimates are sufficiently precise that this binding function represents their average.

For an exactly identified match, set each simulated auxiliary estimate equal to its observed counterpart:

\theta_1+\theta_2=1.20, \qquad \theta_1-\theta_2=0.30.

Adding the equations gives \(2\theta_1=1.50\), so \(\theta_1=0.75\). Subtracting the second equation from the first gives \(2\theta_2=0.90\), so \(\theta_2=0.45\). Substitution verifies the match:

b(0.75,0.45) = (0.75+0.45,\ 0.75-0.45)' = (1.20,\ 0.30)'.

Here the two auxiliary estimates jointly distinguish the two structural parameters: the sum identifies their combined level, while the difference separates them. If the binding function instead depended only on \(\theta_1+\theta_2\), those two parameters could not be individually recovered from that auxiliary information alone.

Worked Example: An Overidentified Weighted Match

Now suppose a hypothetical structural model has one parameter, \(\theta\), but two auxiliary estimates. Their simulated averages follow \(b(\theta)=(\theta,\ 2\theta)'\), and the observed estimates are \((0.80,\ 1.50)'\). Treat both statistics as dimensionless and use the identity weight matrix \(W=I\). Since there are two targets but one parameter, the two observed estimates need not be matched exactly.

The criterion is

Q(\theta)=(0.80-\theta)^2+(1.50-2\theta)^2.

Expanding and differentiating gives

Q(\theta)=0.80^2+1.50^2-2(0.80)\theta-2(1.50)(2\theta)+5\theta^2,

\frac{dQ(\theta)}{d\theta}=-1.60-6.00+10\theta.

Setting the derivative to zero yields \(10\hat{\theta}_{\mathrm{II}}=7.60\), so \(\hat{\theta}_{\mathrm{II}}=0.760\). The second derivative is \(10>0\), confirming this is the unique minimum. At this value, the simulated auxiliary estimates are \((0.760,\ 1.520)'\), and the residual vector is \((0.040,\ -0.020)'\). Therefore,

$$ Q(0.760)=(0.040)^2+(-0.020)^2=0.0016+0.0004=0.0020. $$

The estimates are close, but not identical: one parameter cannot generally fit two distinct targets exactly. The estimate is a compromise under the stated identity weighting, not evidence that the auxiliary model is the structural model.

Practical Interpretation and a Common Pitfall

Indirect inference matters when structural simulation is feasible but likelihood-based estimation is not. It lets the researcher use informative implications of a structural model while relying on a comparatively simple auxiliary estimator. The resulting estimate is only as informative as the relationship between structural parameters and auxiliary estimates. A small matching distance, by itself, does not prove that the structural model is correct or that its parameters are identified.

Common mistake: “The auxiliary regression is misspecified, so its estimates cannot be used for structural estimation.” This reasoning confuses the auxiliary model's purpose with the structural model's purpose. The auxiliary model need not be true; it must be estimable in both observed and simulated data and provide estimates that respond usefully to \(\theta\). Conversely, calling the auxiliary model a mere summary does not make every summary adequate: if the binding function fails to distinguish parameter values, matching cannot recover them separately.

A second practical caution concerns the objective itself. The weighting matrix affects the compromise when there are more auxiliary estimates than structural parameters. Thus, an overidentified estimate should always be understood relative to the chosen targets and weighting rule. The criterion's numerical value is a discrepancy measure, not a probability that the structural model is true.

Key takeaway: Indirect inference maps structural parameters into auxiliary estimates by simulation, then chooses parameters whose simulated estimates match those from observed data. Identification comes from a sufficiently informative binding function; estimation requires applying the same auxiliary procedure to observed and simulated samples.

Check Your Understanding

Answer each question by referring to the definitions and calculations above.

Why can an auxiliary model be useful even when it is not the structural data-generating model?
What does the binding function \(b(\theta)\) represent, and what feature of it is needed to distinguish structural parameters?
If the observed auxiliary estimate is \(0.50\) and the simulated estimate at a candidate is \(0.44\), what is the scalar criterion when \(W=1\)?
In the two-parameter example, which combinations of the parameters are summarized by the sum and difference statistics?
Why might the minimum criterion be positive when there are more auxiliary estimates than structural parameters?