Indirect Inference with Auxiliary Models

Why Match an Auxiliary Model?

Some structural econometric models are straightforward to simulate but difficult to estimate directly: their likelihood may be unavailable, inconvenient to evaluate, or costly to compute. Indirect inference offers another route. It extracts a compact set of statistics from the observed data by fitting an auxiliary model, then asks which structural parameter values produce simulated data with similar auxiliary estimates.

The auxiliary model is a tool for summarizing data, not necessarily the structural model that generated them. For example, an econometrician might simulate data from a nonlinear dynamic model but summarize both the observed and simulated series using a simple linear regression. The target is not to claim that this regression is structurally correct. It is to choose structural parameters whose simulated data reproduce its estimates.

Definition: Let \(\theta\) denote a vector of structural parameters, and let \(y\) denote the observed sample. An auxiliary model is a specified statistical model fitted to a dataset to produce a finite-dimensional estimate \(\hat{\beta}(y)\). In indirect inference, the observed estimate \(\hat{\beta}(y)\) is compared with estimates from data simulated under candidate values of \(\theta\). The structural estimate minimizes a weighted discrepancy between them.

For this construction, assume the structural model can generate data at any admissible \(\theta\). Use the same sample size and relevant observed design or conditioning information for simulated data as for the observed sample. Apply the same auxiliary-model specification, estimation method, and data transformations to every dataset. These conditions make the comparison meaningful: differences in auxiliary estimates should reflect the structural parameter choice, rather than differences in how the datasets were processed.

The Indirect Inference Criterion

Write the auxiliary estimate from the observed sample as \(\hat{\beta}_{\mathrm{obs}}=\hat{\beta}(y)\). At a candidate \(\theta\), simulate \(R\) datasets and fit the auxiliary model to each one. Denote the resulting estimates by \(\hat{\beta}_r(\theta)\), for \(r=1,\ldots,R\). Their average is

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

Let \(W\) be a chosen positive-definite weighting matrix. The simulated discrepancy and the indirect inference criterion are

d_R(\theta)=\hat{\beta}_{\mathrm{obs}}-\bar{\beta}_{R}(\theta), \qquad Q_R(\theta)=d_R(\theta)'W d_R(\theta).

The estimator is a value of \(\theta\) that minimizes \(Q_R(\theta)\) over the admissible parameter space. In practice, this minimization may be carried out with a grid, a derivative-free optimizer, or a gradient-based method when the simulated criterion is sufficiently smooth. The exact search method does not change the target: find structural parameters whose simulated auxiliary estimates are close to the observed auxiliary estimates, after accounting for the weighting.

Key result: If \(W\) is positive definite, then \(Q_R(\theta)\geq 0\), and \(Q_R(\theta)=0\) exactly when \(\bar{\beta}_{R}(\theta)=\hat{\beta}_{\mathrm{obs}}\). When exact matching is unavailable, minimizing \(Q_R\) finds the candidate with the smallest weighted squared discrepancy among those considered.

The weighting matrix matters when the auxiliary estimate has several components. A discrepancy of 0.1 may be large for an estimate usually varying by 0.02, but small for one usually varying by 2. If \(W\) is diagonal, each component receives its own weight. More generally, off-diagonal entries allow the criterion to reflect covariance among auxiliary estimates. A common principle is to give greater influence to discrepancies in components measured precisely, though the choice of \(W\) depends on the estimation objective and should be stated clearly.

Worked Examples

Worked Example: A Scalar Search With an Auxiliary Regression

Hypothetical setup. Suppose a structural model has one parameter \(\theta\) and generates an outcome \(y\) from two observed regressors \(x\) and \(z\):

y=\theta x+\theta^2 z+\varepsilon.

The econometrician simulates this model but uses the deliberately simpler auxiliary regression of \(y\) on \(x\) alone. Suppose \(x\) and \(z\) are included in every simulated sample with the same design as in the observed sample, the errors have mean zero and are independent of the regressors, and each dataset has \(n=500\) observations. Assume the observed auxiliary slope is \(0.700\). For a transparent first search, use one common set of \(R=100\) simulation draws at each candidate and average the 100 auxiliary slopes.

The following hypothetical simulation results are the mean auxiliary slopes at four candidate parameter values:

Candidate \(\theta\)	Mean simulated auxiliary slope	Difference from observed slope
0.200	0.520	−0.180
0.400	0.660	−0.040
0.500	0.710	0.010
0.600	0.750	0.050

Suppose the chosen weight is \(W=1/(0.080)^2=156.25\). This scales the slope discrepancy by a reference scale of \(0.080\). For \(\theta=0.500\), the criterion is

$$ Q_R(0.500) =156.25(0.700-0.710)^2 =156.25(0.000100) =0.0156. $$

For comparison, \(Q_R(0.400)=156.25(0.040)^2=0.250\), and \(Q_R(0.600)=156.25(0.050)^2=0.391\). Among these candidates, \(\theta=0.500\) gives the smallest criterion. This is a grid-search conclusion, not a claim that \(0.500\) is the exact minimizer over every possible value. The auxiliary slope is measured in units of \(y\) per unit of \(x\); matching it does not turn the auxiliary regression into the structural equation.

Worked Example: Weighting Two Auxiliary Estimates

Hypothetical setup. An observed time series has been summarized by an auxiliary autoregression. Its estimated intercept and slope are \(\hat{\beta}_{\mathrm{obs}}=(0.30,0.80)'\), where the intercept is in outcome units and the slope is dimensionless. Two structural parameter candidates produce the following average simulated auxiliary estimates:

Candidate	Mean simulated intercept	Mean simulated slope
A	0.29	0.65
B	0.22	0.79

First use the identity matrix \(W=I\). Candidate A has discrepancy \(d_A=(0.01,0.15)'\), so

$$ Q_A=0.01^2+0.15^2=0.0001+0.0225=0.0226. $$

Candidate B has \(d_B=(0.08,0.01)'\), giving

$$ Q_B=0.08^2+0.01^2=0.0064+0.0001=0.0065. $$

With equal numerical weighting, candidate B has the smaller criterion. Now suppose the econometrician chooses \(W=\mathrm{diag}(4,1)\), giving more weight to the intercept discrepancy. Then

$$ Q_A=4(0.01)^2+(0.15)^2=0.0004+0.0225=0.0229, $$

whereas

$$ Q_B=4(0.08)^2+(0.01)^2=0.0256+0.0001=0.0257. $$

Under this weighting, candidate A is preferred. The reversal is not an arithmetic inconsistency: changing \(W\) changes which discrepancies matter most. A researcher should justify the scale and substantive role of each component rather than treat the weighting choice as a cosmetic detail.

Worked Example: Averaging Replications During a Two-Parameter Search

Hypothetical setup. A structural dynamic model has parameters \(\theta=(\rho,\sigma)\), where \(\rho\) is dimensionless and \(\sigma\) is an innovation standard deviation measured in outcome units. The auxiliary model is an autoregression; its estimated slope and residual standard deviation are the two components of \(\hat{\beta}\). Suppose the observed estimates are \((0.70,1.20)'\), and simulations use the same sample length, starting-value rule, and auxiliary estimation procedure as the observed data. At each candidate, average the auxiliary estimates over \(R=50\) simulated datasets.

For illustration, suppose the resulting average estimates are:

Candidate \((\rho,\sigma)\)	Mean simulated slope	Mean simulated residual SD
(0.40, 0.80)	0.40	0.84
(0.70, 1.20)	0.68	1.18
(0.90, 1.40)	0.86	1.39

Use \(W=\mathrm{diag}(100,25)\), corresponding to scaling the slope discrepancy by \(0.10\) and the residual-standard-deviation discrepancy by \(0.20\). At \((0.70,1.20)\), the discrepancies are \((0.02,0.02)'\), so

$$ Q_R(0.70,1.20) =100(0.70-0.68)^2+25(1.20-1.18)^2 =100(0.0004)+25(0.0004) =0.0500. $$

At \((0.40,0.80)\), the discrepancies are \((0.30,0.36)'\), giving \(Q_R=100(0.30)^2+25(0.36)^2=9+3.24=12.24\). At \((0.90,1.40)\), they are \((-0.16,-0.19)'\), giving \(Q_R=100(0.16)^2+25(0.19)^2=2.56+0.9025=3.4625\). The middle candidate has the smallest criterion on this grid. A practical search would usually evaluate additional candidates around it before reporting a numerical estimate.

Because each candidate is simulated, Monte Carlo variation can make one criterion appear smaller just by chance. Using the same underlying simulation draws across nearby candidates, when feasible, can make comparisons less noisy: differences in the criterion then reflect changes in \((\rho,\sigma)\) more than unrelated changes in random draws. The replication average also reduces simulation variation relative to relying on one simulated dataset.

Why the Construction Works

The procedure has a direct optimization interpretation. Define the discrepancy vector \(d_R(\theta)\) as the observed auxiliary estimate minus its simulated average. If \(W\) is symmetric positive definite, then a factorization \(W=C'C\) exists. Consequently,

Q_R(\theta) =d_R(\theta)'C'Cd_R(\theta) =\lVert C d_R(\theta)\rVert^2.

Thus minimizing \(Q_R\) is equivalent to minimizing the squared length of the transformed discrepancy. If \(W\) is diagonal with entries \(1/a_j^2\), then the criterion is

Q_R(\theta)=\sum_{j=1}^{k} \left(\frac{\hat{\beta}_{\mathrm{obs},j}-\bar{\beta}_{R,j}(\theta)}{a_j}\right)^2,

where \(k\) is the number of auxiliary estimates and \(a_j\) is the scale used for component \(j\). Each term is a squared, scaled mismatch. This explains why both the auxiliary estimates and the weighting must be specified to define the estimator.

At a high level, if simulations faithfully reproduce the structural model, and the simulated auxiliary estimates are accurate enough, parameters that reproduce the observed auxiliary estimates are candidates for explaining the observed data. This is a matching principle, not a guarantee that every structural parameter can be recovered from every auxiliary model. The information in the auxiliary estimates, the model’s behavior across parameter values, and the quality of the search all matter.

Practical Choices and Common Pitfalls

A useful auxiliary model should yield well-defined estimates on both the observed and simulated samples and summarize features that vary meaningfully with the structural parameters. It can include several coefficients or other estimates, but adding components is not automatically beneficial: redundant or unstable estimates can make the criterion difficult to optimize. The analyst should report the auxiliary specification, how its estimates are computed, the simulated sample design, the number of replications, and the weighting matrix.

Increasing \(R\) generally makes the simulated average less sensitive to simulation noise, but it also increases computation. Keeping simulation draws fixed across candidate evaluations can improve the stability of numerical comparisons. If the optimizer uses an estimated gradient, the simulation design should be chosen so that changes in the objective are not dominated by fresh random variation. These steps stabilize the search; they do not by themselves establish statistical uncertainty for the final estimator.

Mistake: matching different statistics. Comparing an observed regression coefficient with a simulated mean of the outcome is not the defined auxiliary-estimate comparison. Fit the same auxiliary model in both settings and compare corresponding estimates.
Mistake: assuming a small criterion proves a good structural model. A small value says that the selected auxiliary estimates match under the chosen weighting. It does not establish that the full simulated data resemble the observed data in every respect.
Mistake: ignoring scale and units. Applying equal numerical weights to estimates on very different scales can let one component dominate. State and motivate the scaling or weighting used.
Mistake: reporting the best grid point as an exact optimum. A grid only compares its listed candidates. Refine the search near the best point or use an appropriate optimizer before treating the result as a numerical minimum.

Key takeaway: Indirect inference fits an auxiliary model to observed data, repeats that same estimation on data simulated at candidate structural parameters, and selects parameters by minimizing a weighted discrepancy between the observed and simulated auxiliary estimates.

Check Your Understanding

Answer each question using the definitions and calculations in this tutorial.

Why must the auxiliary model and its estimation procedure be applied consistently to observed and simulated datasets?
If an observed auxiliary estimate is \(0.40\), a candidate produces a simulated average of \(0.46\), and the scalar weight is \(25\), what is the criterion value?
In a two-component auxiliary estimate, what changes when the weighting matrix is changed from the identity matrix to a diagonal matrix with a larger first entry?
Why is the best candidate in a finite parameter grid not necessarily the exact indirect inference estimate?
What does a small indirect inference criterion establish, and what does it not establish about the structural model?