Binding Functions and Identification in Indirect Inference

Why the Binding Function Matters

Indirect inference connects a structural model to an auxiliary model through the estimates the auxiliary procedure would produce under data generated by the structural model. The key object is the binding function: it maps each candidate structural parameter to the population value of the auxiliary estimate. Identification asks whether different structural parameters produce distinguishable auxiliary values. If nearby parameter values produce almost the same values, the model may be formally identified yet difficult to estimate reliably.

This distinction matters before standard errors or confidence intervals are considered. A precise computational match to auxiliary estimates cannot resolve a parameter direction that the binding function barely changes. The binding function and its Jacobian help separate the question “is the parameter locally determined?” from “how sensitively does the auxiliary information respond to it?”

Definition and Identification

Definition: Let \(\theta\in\Theta\subset\mathbb{R}^p\) denote the structural parameter and let \(\hat{\beta}_n\in\mathbb{R}^k\) be the auxiliary estimator computed from a sample of size \(n\). The binding function is \[ b(\theta)=\operatorname{plim}_{n\to\infty}\hat{\beta}_n \] when the data are generated by the structural model at parameter value \(\theta\), under the specified design and auxiliary estimation procedure. The vector \(b(\theta)\) is the population auxiliary value implied by \(\theta\).

The definition presumes that this probability limit exists and is well-defined for the parameter values under consideration. In applications, the binding function may have no closed form and can be evaluated by simulating data from the structural model and estimating the auxiliary model on those data. Its meaning still concerns the population limit; finite simulation output is an approximation to that object.

Suppose the observed data are generated at \(\theta_0\), and the auxiliary estimator is consistent for its binding value. Then \(\hat{\beta}_n\) converges to \(b(\theta_0)\). The population matching question is whether the equation \(b(\theta)=b(\theta_0)\) singles out \(\theta_0\).

Definition: The structural parameter \(\theta_0\) is locally identified through the binding function if there is a neighborhood \(N\) of \(\theta_0\) such that, for every \(\theta\in N\), \(b(\theta)=b(\theta_0)\) implies \(\theta=\theta_0\). It is globally identified on \(\Theta\) if the same implication holds for every \(\theta\in\Theta\).

Local identification rules out other observationally equivalent parameter values close to \(\theta_0\); it does not rule out distant values with the same binding value. Nor does identification alone guarantee that finite samples provide much information: a binding function can be one-to-one but nearly flat in some direction.

The Jacobian and a Local Identification Result

Assume that \(\theta_0\) is an interior point of the parameter space and that \(b\) is continuously differentiable on an open neighborhood of \(\theta_0\). Its Jacobian at \(\theta_0\) is the \(k\times p\) matrix

J(\theta_0)=\frac{\partial b(\theta_0)}{\partial\theta'}

The \(j\)th column describes how the auxiliary population values change, to first order, when structural parameter \(\theta_j\) changes. A standard regular condition for local identification is that \(J(\theta_0)\) has full column rank \(p\). This requires at least as many auxiliary components as structural parameters, \(k\ge p\).

Theorem: Suppose \(b:\Theta\to\mathbb{R}^k\) is continuously differentiable on an open neighborhood of the interior point \(\theta_0\), and \(J(\theta_0)\) has full column rank \(p\). Then \(\theta_0\) is locally identified through \(b\).

For \(k=p\), full column rank means \(J(\theta_0)\) is nonsingular. The inverse function theorem applies because \(b\) is continuously differentiable on an open neighborhood and its derivative at \(\theta_0\) is nonsingular. It gives a neighborhood on which \(b\) is one-to-one.

For \(k>p\), choose \(p\) rows of \(J(\theta_0)\) that form a nonsingular \(p\times p\) matrix; full column rank ensures such a choice. Let \(S\) select the corresponding \(p\) components of \(b\), and define \(g(\theta)=Sb(\theta)\). This function is continuously differentiable on the same open neighborhood, and its derivative at \(\theta_0\) is nonsingular. The inverse function theorem makes \(g\) one-to-one in a neighborhood of \(\theta_0\). If \(b(\theta)=b(\theta_0)\), then \(g(\theta)=g(\theta_0)\), so \(\theta=\theta_0\) there. Thus \(b\) is locally identifying.

The rank condition is sufficient for this regular local identification result, not necessary for every possible form of local identification. For instance, \(b(\theta)=\theta^3\) is one-to-one around zero even though its derivative at zero is zero. Such cases are not covered by the full-rank result and can have nonstandard behavior.

Worked Example: Recovering Two Parameters From Two Auxiliary Values

Suppose, hypothetically, that a two-parameter structural model has \(\theta=(\theta_1,\theta_2)'\), and its auxiliary estimates have binding function \(b(\theta)=(\theta_1+\theta_2,\theta_1-\theta_2)'\). The observed auxiliary estimates are \(\hat{\beta}=(0.8,0.2)'\). Assume the parameter space contains the solution as an interior point and this linear binding function applies throughout a neighborhood.

The population matching equations are

\theta_1+\theta_2=0.8,\qquad \theta_1-\theta_2=0.2.

Adding the equations gives \(2\theta_1=1.0\), so \(\theta_1=0.5\). Subtracting the second equation from the first gives \(2\theta_2=0.6\), so \(\theta_2=0.3\). The matched parameter is therefore \((0.5,0.3)'\), in the units used for the two structural parameters.

The Jacobian is

J(\theta)= \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}, \qquad \det J(\theta)=-2\ne0.

The binding function is continuously differentiable and its Jacobian is nonsingular, so the inverse function theorem gives local one-to-one behavior. In fact, this linear function is one-to-one globally. The calculation illustrates the key point: two auxiliary components distinguish the two structural parameters because their responses are linearly independent.

Overidentification and Local Curvature

When \(k>p\), the binding function supplies more auxiliary components than there are structural parameters. This can improve the information available about the parameter, but it does not guarantee identification: the components may respond to parameters in redundant ways. Full column rank is the relevant condition. A common population matching criterion is a weighted squared distance, with a symmetric positive-definite weight matrix \(W\):

Q(\theta)=\bigl[b(\theta_0)-b(\theta)\bigr]'W\bigl[b(\theta_0)-b(\theta)\bigr].

Near \(\theta_0\), differentiability gives the first-order expansion \(b(\theta_0+h)=b(\theta_0)+J(\theta_0)h+o(\|h\|)\). Consequently, the leading quadratic term in the criterion is \(h'J(\theta_0)'WJ(\theta_0)h\). If \(J(\theta_0)\) has full column rank, this term is positive for every nonzero \(h\).

h'J'WJh=(Jh)'W(Jh)>0\quad\text{for }h\ne0.

Here \(Jh\ne0\) follows from full column rank, and positive definiteness of \(W\) makes the quadratic form positive. This is a local curvature justification for why the full-rank condition is useful. It is not a claim that the sample criterion has this exact quadratic form or that the condition alone describes finite-sample precision.

Worked Example: Checking Rank With Three Auxiliary Components

Consider the hypothetical binding function \(b(\theta_1,\theta_2)=(\theta_1,\theta_2,\theta_1+2\theta_2)'\). Suppose the observed auxiliary vector is \(\hat{\beta}=(0.4,-0.2,0.5)'\), and use the identity matrix as the weight matrix. The parameter space is open around the match.

The Jacobian and its cross-product are

J= \begin{pmatrix} 1&0\\ 0&1\\ 1&2 \end{pmatrix}, \qquad J'J= \begin{pmatrix} 2&2\\ 2&5 \end{pmatrix}.

The determinant of \(J'J\) is \(2(5)-2(2)=6\), which is nonzero, so \(J\) has full column rank. The binding function is linear and therefore continuously differentiable; the rank condition establishes local identification.

For these data, minimizing the squared distance \(\|\hat{\beta}-b(\theta)\|^2\) gives the normal equations \(J'J\theta=J'\hat{\beta}\). Direct calculation yields

J'\hat{\beta}= \begin{pmatrix}0.9\\0.8\end{pmatrix}, \qquad \hat{\theta} = \frac{1}{6} \begin{pmatrix}5&-2\\-2&2\end{pmatrix} \begin{pmatrix}0.9\\0.8\end{pmatrix} = \begin{pmatrix}0.4833\\-0.03333\end{pmatrix}.

The implied auxiliary vector is approximately \((0.4833,-0.03333,0.4167)'\). Subtracting it from the observed vector gives residuals \((-0.08333,-0.1667,0.08333)'\). The mismatch is expected: three observed auxiliary components need not lie exactly on the two-dimensional set traced out by \(b(\theta)\). The full-rank result concerns whether parameters are locally distinguished, not whether every sample auxiliary vector can be matched exactly.

Weak Identification and the Jacobian

Even when \(J\) has full column rank, one or more of its directions can be very small. To diagnose this, consider the singular values of the Jacobian, or of a suitably weighted and scaled version of it. A small singular value means there is a direction in structural-parameter space along which a relatively large parameter change produces only a small first-order change in the auxiliary values. The corresponding right singular vector identifies that direction.

A small singular value is a warning about weak local information, not by itself a universal test or a numerical threshold. Its interpretation depends on units: changing a parameter from dollars to cents changes the numerical entries in a Jacobian column. Auxiliary components also have different scales. Rescaling parameters and auxiliary values, or examining \(W^{1/2}J\) for a relevant positive-definite weight \(W\), helps make comparisons meaningful. Scaling should be reported and chosen substantively, not used to disguise a poorly responsive direction.

Worked Example: A Full-Rank but Weak Direction

Suppose the hypothetical binding function is \(b(\theta_1,\theta_2)=(\theta_1,0.02\theta_2)'\), with observed auxiliary estimates \(\hat{\beta}=(0.6,0.012)'\). Assume the parameter space is open around the solution and use the same units for the parameters and auxiliary components as in the stated function.

Matching the first component gives \(\theta_1=0.6\), and matching the second gives \(0.02\theta_2=0.012\), hence \(\theta_2=0.6\). The Jacobian is

J= \begin{pmatrix} 1&0\\ 0&0.02 \end{pmatrix}.

Its singular values are \(1\) and \(0.02\). Both are positive, so the Jacobian has full column rank and the parameter is locally identified. Yet the second direction is much less responsive: a change of \(0.1\) in \(\theta_2\) changes the second binding component by only \(0.02(0.1)=0.002\). Conversely, an auxiliary-value perturbation of \(0.001\) in that component corresponds, under exact matching, to a parameter change of \(0.001/0.02=0.05\) in \(\theta_2\). This sensitivity illustrates why exact local identification need not imply strong information in finite samples.

The numerical comparison is conditional on the stated scales. If either parameter or auxiliary component is expressed in different units, the raw singular values change. A useful diagnostic therefore records the scaling convention and investigates whether the weak direction persists under meaningful standardization.

Practical Diagnostics and a Common Pitfall

In applied work, the binding function and its Jacobian may be computed by simulation or numerical differentiation. Diagnostics should focus on whether the estimated Jacobian is stable enough to support the rank and sensitivity conclusions. For example, vary simulation draws or numerical step sizes, check parameter scaling, and inspect directions associated with the smallest singular values. A near-zero singular value may reflect genuine weak identification, an inadequate simulation design, or numerical error; these possibilities should not be conflated.

Common mistake: “The Jacobian determinant is nonzero at the parameter value, so the binding function must be one-to-one nearby,” without stating the required regularity. A derivative existing throughout a neighborhood is not, by itself, enough for this inverse-function conclusion. The correction is to require continuous differentiability on an open neighborhood and a nonsingular square Jacobian at the point. In the overidentified case, use a nonsingular \(p\times p\) row submatrix and apply the same argument to the selected components. Also, do not interpret a small raw Jacobian entry as weak identification without considering the units and scales of both parameters and auxiliary values.

Key takeaway: The binding function maps structural parameters to population auxiliary estimates. A continuously differentiable binding function with a full-column-rank Jacobian at an interior parameter value is locally identifying. Small singular values, interpreted on meaningful scales, flag directions in which auxiliary information responds weakly.

Check Your Understanding

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

In words, what does the binding function \(b(\theta)\) represent, and what assumption about the auxiliary estimator is needed for its definition?
A binding function maps two structural parameters to three auxiliary components. What Jacobian rank condition gives the regular local identification result?
Why does full column rank imply that some selection of \(p\) auxiliary components can be used with the inverse function theorem?
A Jacobian has full column rank but a very small singular value. What does that suggest, and why should the parameter and auxiliary scales be checked?
Why is a nonsingular Jacobian at a point not enough to invoke the inverse function theorem without a regularity condition on the binding function?