Why Auxiliary-Statistic Choice Matters
Indirect inference matches statistics calculated from observed data to the same statistics calculated from data simulated under a structural model. The auxiliary statistics are therefore the bridge between the structural parameters and the estimation criterion. That bridge can be informative or uninformative: a statistic that barely changes when a parameter changes provides little leverage for estimating it, while several statistics that respond in nearly the same way may fail to distinguish parameters from one another.
Choosing statistics is not just a matter of finding measures that are easy to calculate. Their response to parameters must be considered alongside their sampling variability and their redundancy. The binding function describes the model-implied response; its derivative summarizes local sensitivity. Its covariance describes how precisely the statistics are measured. Both are needed to assess identification and precision.
The relevant sensitivity is local to the parameter value under consideration. A statistic may respond strongly to a parameter in one region and weakly in another. For multiple parameters, the issue is not only whether each column of \(J\) is large: its columns must also describe distinguishable directions of change. If the columns are linearly dependent, some parameter changes produce the same local movement in the selected statistics.
Identification and Sensitivity
Suppose the observed-data statistic vector is \(\hat s\), and the model’s binding function is \(b(\theta)\). A basic indirect-inference estimator minimizes a weighted discrepancy:
where \(W\) is a positive definite weighting matrix. The statistic vector must contain enough independent local information to distinguish the \(p\) structural parameters. Under continuous differentiability near \(\theta_0\), full column rank of \(J(\theta_0)\) is a sufficient first-order condition for local identification, but it is not necessary in general: \(b(\theta)=\theta^3\) is locally one-to-one at \(\theta_0=0\) even though \(b'(0)=0\). This condition is about the selected summaries and their binding function; it does not by itself establish global identification.
Worked Example: A Redundant Statistic Does Not Add a Parameter Direction
Consider two structural parameters, \(\theta_1\) and \(\theta_2\). Suppose the first two candidate auxiliary statistics have binding functions \(b_1(\theta)=\theta_1+\theta_2\) and \(b_2(\theta)=2\theta_1+2\theta_2\). A third candidate has binding function \(b_3(\theta)=\theta_1-\theta_2\). Which statistics provide local identification?
Using the first two statistics. Their Jacobian is
The second row is twice the first, so the matrix has rank 1, not 2. Both statistics respond only to the sum \(\theta_1+\theta_2\); neither distinguishes a change in \(\theta_1\) from an offsetting change in \(\theta_2\). These two statistics do not locally identify both parameters.
Adding the third statistic. Using \(b_1\) and \(b_3\) gives
This Jacobian has rank 2, so the two statistics distinguish local changes in both parameters. For example, if the observed summaries are \(1.4\) and \(0.2\), the matching equations are \(\theta_1+\theta_2=1.4\) and \(\theta_1-\theta_2=0.2\). Adding gives \(2\theta_1=1.6\), so \(\theta_1=0.8\); subtracting the second equation from the first gives \(2\theta_2=1.2\), so \(\theta_2=0.6\).
The example is about the geometry of the binding function, not the numerical size of the summaries. The first pair can have large derivatives and still be redundant. The added statistic is useful because it supplies a direction of response that is not already present.
Full column rank is an identification check, not a complete measure of precision. Even with full rank, the columns of \(J\) may be close to linearly dependent, or the statistics may be noisy. In practice, sensitivity should be judged relative to the scale and covariance of the statistics. A derivative measured in large units is not automatically more informative than a smaller derivative measured with much less noise.
How Sensitivity and Sampling Variability Determine Precision
For a local precision calculation, assume the selected statistics satisfy \(\sqrt{n}[\hat s-b(\theta_0)]\overset{d}{\longrightarrow}N(0,\Sigma)\), where \(\Sigma\) is finite and positive definite, the binding function is differentiable near \(\theta_0\), and \(J(\theta_0)\) has full column rank. Suppose also that the estimator is a consistent interior solution to the weighted minimum-distance problem and that \(W\) converges to a positive definite matrix. Under these regularity conditions, a first-order expansion gives the local covariance.
The result follows by expanding the binding function around \(\theta_0\). Write \(\delta=\hat\theta-\theta_0\). Differentiability gives \(b(\hat\theta)=b(\theta_0)+J\delta+o(\|\delta\|)\). Ignoring the higher-order remainder in the first-order approximation, minimizing the criterion is equivalent to minimizing \([\hat s-b(\theta_0)-J\delta]'W[\hat s-b(\theta_0)-J\delta]\). Differentiating with respect to \(\delta\) and setting the derivative to zero yields
Applying the assumed limiting covariance \(\Sigma\) to this linear expression gives the stated sandwich covariance. Under \(W=\Sigma^{-1}\), multiplication simplifies because \(W\Sigma W=W\), producing \((J'\Sigma^{-1}J)^{-1}\). Thus, the optimal-weighting result from minimum distance makes the roles of sensitivity and noise especially clear: information increases when changes in the parameters produce statistically distinguishable changes in the summaries.
Comparing Sensitivity at Different Parameter Values
A binding function can be nonlinear, so a statistic’s sensitivity can change across the parameter space. The following example uses a familiar auxiliary summary only to illustrate that point. It does not presume that a particular statistic is universally best.
Worked Example: Lag Autocorrelations in an AR(1) Model
Suppose a stationary AR(1) process has structural parameter \(\rho\), with \(|\rho|<1\), and consider the first- and second-lag autocorrelations as candidate auxiliary statistics. Their population binding functions are \(b_1(\rho)=\rho\) and \(b_2(\rho)=\rho^2\). The sensitivities are \(b_1'(\rho)=1\) and \(b_2'(\rho)=2\rho\).
At \(\rho=0.2\), the sensitivities are \(1\) and \(0.4\). If each estimated autocorrelation has an approximate standard error of \(0.03\), the local delta-method standard error for the parameter estimate obtained by inverting the first summary is \(0.03/1=0.030\). For the second summary it is approximately \(0.03/0.4=0.075\). Near \(0.2\), the second-lag statistic is less sensitive and, under this equal-standard-error illustration, less precise for estimating \(\rho\).
At \(\rho=0.8\), the sensitivities are \(1\) and \(1.6\). With the same illustrative standard errors, the corresponding local standard errors are \(0.03/1=0.030\) and \(0.03/1.6=0.01875\). Near \(0.8\), the second-lag statistic is more sensitive and yields a smaller local standard error under the stated comparison.
For a scalar statistic with binding function \(b(\rho)\), the local inverse estimate has approximate variance \(\operatorname{Var}(\hat s)/[n^0 b'(\rho)^2]\) when \(\operatorname{Var}(\hat s)\) is already expressed at the sample size in question; equivalently, if \(\operatorname{Var}(\hat s)\approx \sigma_s^2/n\), the variance is approximately \(\sigma_s^2/[n b'(\rho)^2]\). The numerical comparison above uses standard errors for the estimated summaries at the given sample size, so dividing by the absolute derivative is appropriate. A statistic’s derivative alone is not enough if its standard error differs substantially.
Competing Statistics: Sensitivity Is Not the Whole Story
When several summaries are available, their joint covariance matters as well as their separate derivatives. Statistics calculated from the same data are often correlated. A pair of individually sensitive summaries may contribute little new information if their sampling variation and parameter responses largely duplicate one another. Conversely, a statistic with a more modest derivative can improve precision if it brings a distinct response and is measured accurately.
Worked Example: Two Calibrated Contrasts with Different Noise
Consider two observed auxiliary summaries, \(y_1=2.4\) and \(y_2=1.3\), and a scalar structural parameter \(\theta\). Compare two candidate statistics, \(q_A=y_1-y_2\) and \(q_B=-y_1+4y_2\). Suppose each candidate’s simulated binding function is \(3\theta\), and suppose the sampling covariance of \((y_1,y_2)'\) is approximately \(I/n\), where \(I\) is the two-dimensional identity matrix. These are hypothetical values used to compare the statistics.
Candidate A. The observed contrast is
Matching \(q_A\) to its binding function \(3\theta\) gives \(\hat\theta_A=1.1/3=0.3667\). The binding-function sensitivity is \(3\). The contrast has coefficient vector \((1,-1)\), so its variance is \((1^2+(-1)^2)/n=2/n\). Dividing the summary variance by the squared sensitivity gives the approximate estimator variance \(2/(9n)=0.2222/n\).
Candidate B. Its observed contrast is
Matching \(q_B\) to \(3\theta\) gives \(\hat\theta_B=2.8/3=0.9333\). Its sensitivity is also \(3\), but its coefficient vector is \((-1,4)\), giving variance \([(-1)^2+4^2]/n=17/n\). Its approximate estimator variance is therefore \(17/(9n)=1.889/n\).
Both contrasts have the same binding-function slope, but their sampling variances differ. Under these assumptions, Candidate A is substantially more precise. The two point estimates also differ; the example does not claim that either observed contrast is a reliable estimate in every application. It demonstrates that sensitivity alone cannot rank the statistics. A full comparison should use the binding-function response and the sampling covariance, including covariance among summaries when they are used jointly.
A Practical Selection Strategy
A disciplined selection process links substantive relevance, identification, and precision. It should be guided by the structural model and the question being estimated, rather than by searching mechanically for the largest derivative.
Choose summaries that represent features of the data the structural model is intended to explain. State their definitions and units before comparing them.
Use simulations or analytic results to characterize how each candidate summary changes over the relevant parameter region. Compute or approximate the Jacobian there.
Assess the Jacobian’s rank and whether its columns are nearly aligned. Repeated responses may add little identification even when individual derivatives are large.
Estimate the covariance of the summaries and assess how much parameter information remains after accounting for their noise and correlation.
Retain statistics that are meaningful for the structural question and whose binding behavior is adequately informative in the region of interest. Report the selection rationale.
Common Mistake: Choosing the Steepest Statistic
A common mistake is to select the statistic with the largest numerical derivative and conclude that it must provide the most precise estimate. That reasoning ignores units: multiplying a statistic by 100 multiplies its derivative by 100 without adding information. It also ignores measurement noise and dependence on other selected statistics.
The correction is to evaluate sensitivity in relation to the statistics’ covariance. With efficient weighting, the local information matrix is \(J'\Sigma^{-1}J\), not a list of raw derivative magnitudes. An invertible rescaling of all summaries, accompanied by the corresponding transformation of their covariance and weighting, does not create information. A poor selection can nevertheless discard useful directions, leave parameters unidentified, or produce imprecise estimates. The comparison must be made for the summaries as a system.
Check Your Understanding
Use the definitions and results above to answer these questions.
- What does the entry \(J_{jk}(\theta)\) measure, and why can its numerical size depend on units?
- Two statistics have binding functions \(b_1(\theta)=\theta_1+\theta_2\) and \(b_2(\theta)=3\theta_1+3\theta_2\). Can they locally identify both parameters? Explain using the Jacobian.
- A scalar auxiliary statistic has a local binding-function derivative of \(0.5\) and an estimated-summary standard error of \(0.02\). What is the approximate local standard error of the inverse estimate?
- Why can a statistic with a smaller derivative still be more useful than one with a larger derivative?
- In the competing-contrast example, why does the equality of the two binding-function slopes not imply equal estimator precision?