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Choosing What to Calibrate and What to Estimate in Structural Models
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A common practice in structural model estimation is to calibrate some structural parameters---that is, to hold them fixed at prespecified values---and estimate the rest.\footnote{Throughout the paper, a parameter is calibrated if it is held fixed during the estimation problem. This usage is narrower than in some applied and econometric work, where calibration refers to choosing parameters to match empirical moments, i.e. to minimum-distance estimation; see, for example, cocci2021standard. In our terminology, the latter is estimation.} This approach is often justified when full estimation is computationally infeasible jorgensen2023sensitivity, when the model is not fully identified iskrev2019expect, or when population values are known. In practice, however, the calibration--estimation partition is rarely chosen by a formal econometric criterion. It is often inherited from previous studies, justified informally, or dictated by computational convenience.
This paper provides an econometric criterion for choosing what to calibrate and what to estimate in a general class of structural models. We treat the calibration--estimation split as a partition-selection problem. For each admissible partition, we construct a scalar sensitivity statistic that measures the local response of the object of interest to perturbations of the calibrated parameters. We then select the admissible partition that minimizes this sensitivity. The resulting rule chooses the partition that delivers the smallest worst-case local bias induced by calibration error. This perspective connects the paper to robust estimation under local misspecification, especially bonhomme2022minimizing.
A natural alternative would be to select the partition by minimizing an MSE criterion that trades off calibration bias and sampling variance. We focus instead on bias, through the sensitivity statistic, because implementing an MSE criterion requires the full joint variance--covariance matrix of the stacked moments or reduced-form estimates. This requirement is often restrictive when different moments are estimated from different datasets and not all cross-covariances are identified.
A key feature of the implementation is that the least-sensitive partition is computed relative to two inputs. The first is a reference point for the structural parameters, typically obtained from a first-stage estimation. This reference point is where local sensitivities are evaluated. It also allows the researcher to compare candidate partitions without re-estimating the model separately for each one. The second input is a normalization of the calibrated parameters. This normalization specifies the scale on which calibration errors are measured and makes perturbations comparable across parameters expressed in different units. Hence, the least-sensitive partition should be interpreted as conditional on both the reference point and the chosen normalization of calibration errors.
The method is developed in a minimum-distance framework newey1994large, broad enough to cover several estimation approaches commonly used in structural work, including the Generalized Method of Moments (GMM), Classical Minimum Distance (CMD), Simulated Method of Moments (SMM), or Indirect Inference (II).
We illustrate the importance of partition choice in three settings. The first two are New Keynesian models, based on an2007bayesian and nakamura2018high. The third is the dynamic discrete-choice model in kalouptsidi2021identification. Across these examples, we show that the calibration--estimation partition has large implications for robustness. In the Nakamura--Steinsson application (Section (ref)), where the object of interest is the monetary policy's information-effect parameter, the sensitivity statistic varies substantially across admissible partitions. For some partitions, miscalibrations of only a few percent are sufficient to overturn the qualitative conclusion about the object of interest. By contrast, miscalibrations of comparable magnitude generate only a small worst-case bias under the least-sensitive partition. These results show that fragility can be avoided purely by choosing more carefully which parameters to fix and which to estimate.
The paper is related to several strands of literature. It builds most directly on work on sensitivity analysis for structural models. iskrev2019expect and jorgensen2023sensitivity study how estimates and model outputs respond to perturbations in calibrated parameters, while lau2024sensitivity develops local and global sensitivity measures for dynamic discrete choice models with fixed parameters. Relative to jorgensen2023sensitivity, our proposed statistic is scalar and independent of calibrated parameter units. Related work studies sensitivity to moments or identifying assumptions, including andrews2017measuring, honore2020informativeness, and escanciano2013set. Our contribution differs in that sensitivity is not the final object of interest. Instead, it is the input into a decision rule that selects which parameters should be fixed and which should be estimated.
The paper is also related to the literature on identification and partial identification of counterfactuals in dynamic discrete choice models. Building on aguirregabiria2010another, aguirregabiria2014identification, norets2014semiparametric, and arcidiacono2020identifying, kalouptsidi2021identification characterize which counterfactuals are point identified despite payoff nonidentification, while kalouptsidi2026counterfactual characterize the corresponding identified set for the counterfactual. We reverse the direction of this exercise. In that literature, the calibration or identifying strategy is taken as given, and the question is which counterfactuals are point or set identified. Here, the counterfactual exercise is taken as given, and the question is which normalization—or calibration–estimation partition—makes that same object of interest least-sensitive to local calibration error.
The minimax interpretation connects the paper to the literature on estimation and inference under local misspecification (e.g., newey1985generalized,conley2012plausibly,kitamura2013robustness). The closest paper is bonhomme2022minimizing, who choose an estimator, through its influence function, to minimize worst-case mean squared error under local misspecification. We differ in two main respects: our criterion targets worst-case bias rather than worst-case MSE, and our decision variable is the calibration--estimation partition rather than the influence function. Also closely related is armstrong2021sensitivity, who show that GMM weights should account for both sampling variance and misspecification bias when moments may be only approximately valid. We apply a similar local-robustness logic to a different margin: the choice of which parameters to calibrate and which to estimate.
The remainder of the paper is organized as follows. Section (ref) presents two motivational examples. Section (ref) defines the framework and discusses the construction of the partition-selection methodology in detail. Section (ref) offers a risk-minimization interpretation and relates the methodology to bonhomme2022minimizing. Section (ref) provides a step-by-step implementation guide. Section (ref) applies the method to nakamura2018high, and Section (ref) presents Monte Carlo evidence.
This section illustrates the importance of choosing the calibration–estimation partition. In both examples, the model requires fixing at least one structural parameter before the remaining parameters can be estimated. The central point is that different admissible choices of the fixed parameter can generate very different sensitivity of the object of interest to calibration error.
Consider the entry--exit model in kalouptsidi2021identification. In each period, a firm chooses whether to be inactive, \(a=0\), or active, \(a=1\). The state is \(x=(k,w)\), where \(k\in\{0,1\}\) is the firm's lagged activity status and \(w\in\{w_L,w_M,w_H\}\) is an exogenous demand state. Up to the idiosyncratic choice shock, flow payoffs are \[ \pi_a(k,w;\eta)=
\qquad \eta=(ec,fc,sv). \] Here \(ec\) is the entry cost, \(fc\) is the fixed cost, and \(sv\) is the scrap value. Given \(\eta\), the choice-specific value and CCP are \[
\] where the second expression uses the type-I extreme-value normalization.\footnote{The numerical primitives are those in kalouptsidi2021identification. Writing \(\xi\) for the inverse-demand slope, residual demand is \(P_t=w_t-\xi Q_t\), marginal cost is \(c=11\), \(\xi=1.5\), \(w=(20,17,12)\), and \(\bar\pi(w)=(w-c)^2/(4\xi)\). The remaining primitives are \(\rho=0.95\), \(ec=9\), \(fc=5.5\), \(sv=10\), and \[ F_w=
. \] The idiosyncratic shocks are type-I extreme value.}
The counterfactual policy is a \(20\%\) proportional tax on the active operating payoff. Let \(\tau=0.20\). The counterfactual flow payoff is \[ \pi^\tau_0(k,w;\eta)=\pi_0(k,w;\eta), \qquad \pi^\tau_1(k,w;\eta) = (1-\tau)\,k\{\bar\pi(w)-fc\}-(1-k)ec . \] The object of interest is the counterfactual CCP vector, \[ p^\tau_1(x;\eta). \] The model is not point identified without one additional normalization: one of \(ec\), \(fc\), or \(sv\) must be fixed aguirregabiria2014identification. This is not merely a computational detail. Observationally equivalent parameterizations of the model can imply different values of \(p^\tau_1(x;\eta)\). In applications, researchers most often fix either the scrap value or the fixed cost. Table (ref) summarizes this choice in related studies.
Figure (ref) plots the object of interest as a function of the fixed parameter. For each curve, either \(sv\) or \(fc\) is fixed at a value around its population value, the remaining cost parameters are then estimated, and the counterfactual CCP is recomputed. The horizontal axis is centered at the true value of the fixed parameter, so zero corresponds to correct calibration. The horizontal line is the population counterfactual CCP; the vertical distance from this line is the bias induced by miscalibration.
The figure shows that fixing the scrap value produces a relatively flat curve, while fixing the fixed cost produces a much steeper one. Thus, two common ways of identifying the same entry--exit model imply very different robustness properties for the same counterfactual object.
This motivational example is related to kalouptsidi2021identification and kalouptsidi2026counterfactual. Those papers take the calibration strategy as given and ask which counterfactuals are point identified, or what set of counterfactual values is implied when point identification fails. We ask the dual question. We take the counterfactual object, $p^\tau_1(x;\eta)$, as given and compare the admissible calibration strategies themselves. In this example, fixing sv, fc, or ec generates a different local range for the same counterfactual CCP under calibration errors; our method selects the strategy that makes this local range smallest.
The same issue arises in a standard macroeconomic environment. Consider the three-equation forward-looking New Keynesian model used in an2007bayesian:
The shocks \(\varepsilon_{r,t}\), \(\varepsilon_{g,t}\), and \(\varepsilon_{z,t}\) are mutually uncorrelated, with variances \(\sigma_r^2\), \(\sigma_g^2\), and \(\sigma_z^2\)\footnote{We use the calibration in an2007bayesian as the data-generating process: \[ (\rho,\tau,\kappa,\psi_1,\psi_2,\rho_r,\rho_g,\rho_z,\sigma_r^2,\sigma_g^2,\sigma_z^2) = (0.9975,2,0.33,1.5,0.125,0.75,0.95,0.9,0.04,0.36,0.09). \]}.
Let the object of interest be \(\psi_1\), the coefficient on inflation in the Taylor rule. In this model, \(\psi_1\) cannot be jointly identified with all elements of \(\{\psi_2,\rho_r,\sigma_r^2\}\); one of these three parameters must be fixed in order to estimate the others iskrev2010local,komunjer2011dynamic,qu2012identification,qu2017global,kocikecki2023solution. The researcher must therefore decide whether to fix \(\psi_2\), \(\rho_r\), or \(\sigma_r^2\).
Because these parameters have different units, Figure (ref) reports miscalibrations in normalized percent-of-range units. Thus, \(x=0\) means correct calibration, and \(x=0.1\) means that the fixed parameter is set \(10\%\) of its plausible range above its baseline value\footnote{To compute the normalization ranges we use the priors for \(\psi_2\), \(\rho_r\), and \(\sigma_r\) in an2007bayesian. We compute its respective $0.5\%$ and $99.5\%$ quantiles. Obtaining: \[ \psi_2 \in [0,0.98], \qquad \rho_r \in [0,1], \qquad \sigma_r^2 \in [0,0.35]. \]}. For each curve, we fix one parameter at nearby values, re-estimate the remaining parameters by MLE, and record the implied value of \(\psi_1\).
The true value is \(\psi_1=1.5\). Hence, the vertical distance from \(1.5\) is the bias induced by miscalibration. Fixing \(\psi_2\) is relatively robust: the curve is nearly flat around zero. Fixing \(\rho_r\) or \(\sigma_r^2\), in contrast, can generate large changes in \(\psi_1\) even for small normalized perturbations.
These two examples motivate the criterion developed below. Full re-estimation curves are informative, but they are costly to compute, difficult to compare when the target or the calibrated block is multidimensional, and sensitive to the units of the fixed parameters. The statistic \(K_S\) replaces these curves with a local, scalar measure of worst-case sensitivity of the object of interest to calibration error.
Let \(\eta \in \mathcal N \subset \mathbb R^{d_\eta}\) denote the structural parameter vector and let $ \gamma=\Gamma(\eta)\in\mathbb R^{d_\gamma} $ denote the object of interest. A partition is an index set $S\subseteq\{1,\dots,d_\eta\}$ selecting which components of $\eta$ are estimated. Define \[ \alpha := (\eta_j)_{j\in S}, \qquad \beta := (\eta_j)_{j\in S^c}, \] so that $\eta=(\alpha,\beta)$, where $\alpha$ denotes the estimated parameters and $\beta$ the calibrated (fixed) parameters\footnote{Equivalently, $S$ induces $\mathcal N=\mathcal A\times\mathcal B$ with $\mathcal A:=\prod_{j\in S}\mathcal N_j$ and $\mathcal B:=\prod_{j\in S^c}\mathcal N_j$, where $\mathcal N_j$ is the parameter space associated with the $j$th element of $\eta$.}. For a given partition, fixing \(\beta\) and solving the minimum-distance problem yields \(\alpha(\beta;S)\), which in turn induces the partition-specific map \[ \gamma(\beta;S)=\Gamma(\alpha(\beta;S),\beta). \] This induced map is the central object for partition selection: it describes how the object of interest changes when the fixed parameters are perturbed and the remaining parameters are re-estimated.
Our comparison is local. Partitions are evaluated around a reference point \(\eta^*\), typically obtained from a first-stage estimation, which provides the common baseline at which sensitivities and admissibility are assessed.
The goal of this section is to compare these induced maps across partitions and select the one that is least-sensitive to local miscalibration of the fixed parameters. Formally, after defining the admissible set of partitions $\mathcal{S}_{\mathrm{AD}}$, and the sensitivity statistic \(K_S\), we choose \[ S^* \in \arg\min_{S\in \mathcal{S}_{\mathrm{AD}}} K_S. \] The rest of the section builds the ingredients of this criterion step by step. Subsection (ref) introduces the conditional minimum-distance problem, the induced map \(\gamma(\beta;S)\), and the reference point $\eta^*$. Subsection (ref) derives the sensitivity statistic \(K_S\). Subsection (ref) defines the admissible set, \(\mathcal{S}_{\mathrm{AD}}\), and the least-sensitive partition $S^*$.
We study the case in which a structural economic model implies a system of $d_g$ equations \[ g(\eta): \mathcal{N} \longrightarrow \mathcal{G}\subseteq \mathbb{R}^{d_g}. \]
The identified set is defined by all structural parameters that solve the model equations:
Throughout the paper we assume the following conditions about the system of equations and the object of interest are satisfied.
Given $\beta$, the conditional minimum-distance solution for the estimated parameters $\alpha$ is given by
where $W$ is a positive-definite weighting matrix.
To compute the sensitivity statistics and the set of admissible partitions, we work around a reference point $\eta^*$, which may differ from the true parameter $\eta_0$. Typically, it is obtained from a first-stage estimation. For any partition we write $\eta^*=(\alpha^\ast,\beta^\ast)$ for its corresponding relabeled components. See Section (ref) for a detailed discussion on the reference point.
Throughout the main text, we focus on cases in which the system of equations $g(\cdot)$ does not depend on the partition S, except through the relabeling of coordinates induced by the split $(\alpha, \beta)$. This condition is satisfied in many applications of GMM, CMD, SMM, and II. There are, however, important cases in which the system of equations does depend on the partition. A leading example is MLE: deciding which parameters are estimated and which are fixed changes the score equations used in the estimation problem. In Appendix (ref) we extend the methodology to the MLE case.
We now formalize the local sensitivity measure used to compare candidate partitions. We proceed in three steps: derive how the MD solution responds to perturbations in the fixed parameters, translate this into sensitivity of the object of interest, and normalize the result so it can be compared across partitions.
The Jacobians of the system of equations, evaluated at the reference point, are denoted by: \[ G_\alpha:=\frac{\partial g(\alpha^\ast,\beta^\ast)}{\partial \alpha}, \qquad G_\beta:=\frac{\partial g(\alpha^\ast,\beta^\ast)}{\partial \beta}. \] Throughout the paper, we suppress the dependence of these Jacobians on the reference point and on the partition $S$.
jorgensen2023sensitivity derives equivalent expressions in the GMM setting. Since the derivative of the object of interest depends on the units of the fixed parameters, we normalize them using the diagonal matrix \[ \Sigma_S:=\operatorname{diag}\bigl((\Delta_j)_{j\in S^c}\bigr), \] where \(\Delta_j\) determines the scale used for parameter \(j\). Section (ref) provides further discussion on how to construct it in practice.
Note that $K_S$ depends on the reference point $\eta^*$, because the derivative $\partial\alpha(\beta^\ast;S)/\partial\beta$ is obtained from the IFT using Jacobians evaluated at $\eta^*$. It also depends on the normalization matrix.
Under range normalization, $\Sigma_S$ plays two roles: $(i)$ it removes dependence on arbitrary units of measurement, and $(ii)$ it incorporates external information about plausible parameter values through bounds, priors, or confidence intervals. As a result, $K_S$ captures both the model-implied sensitivity of $\gamma$ to $\beta$ and the uncertainty surrounding $\beta^\ast$.
This subsection completes the partition-selection rule. We first define the admissible class of partitions and then select, within that class, the partition that minimizes the sensitivity statistic $K_S$.
Admissibility imposes three requirements: the object of interest must depend nontrivially on the estimated parameters, the estimated parameters must be conditionally locally identified, and the estimation error in the object of interest must vanish in mean square as $n$ grows.
Assumption (ref)$.i)$ ensures the object of interest is not only (locally) explained by the fixed parameters. In the case where a particular structural parameter is the object of interest it would rule out partitions where that parameter is fixed.
Assumption (ref)$.ii)$ would be the typical regular local identification rank condition for the estimated parameters.
Third, Section (ref) shows that the least-sensitive partition minimizes worst-case local bias and, under an additional higher-order condition, also minimizes worst-case mean squared error asymptotically. For this MSE interpretation, the sampling variation in the estimated object of interest must be asymptotically negligible, so that the criterion is driven by calibration bias. We therefore restrict attention to partitions satisfying Assumption (ref)$.iii)$.
Finally, let $\mathcal R$ encode any hard researcher restrictions (for example, parameters that must always be estimated or always fixed, or upper and lower bounds on the number of estimated parameters). We define the admissible set as \[ \mathcal S_{\mathrm{AD}}(\eta^*)\; := \;\{S\in \mathcal{R} \; \textit{ s.t. } \; \textit{Assumption }\ref{ass: admissibility} \textit{ holds}\}. \]
Given a reference point $\eta^*$, and a normalization matrix $\Sigma_S$ we evaluate the sensitivity statistic $K_S(\eta^*,\Sigma_S)$ for each partition $S\in\mathcal S_{\mathrm{AD}}(\eta^*)$ and choose a partition with the smallest value.
This section first gives a local minimax-bias justification and then derives an asymptotic worst-case-MSE interpretation. For a given admissible partition $S$, we interpret the choice of partition through a minimax criterion over a local neighborhood of miscalibrations, and relate this perspective to bonhomme2022minimizing's approach to local robust estimation under misspecification.
For a fixed partition $S$, define the local neighborhood around $\beta^*$ by
The factor $\sqrt{|S^c|}$ scales the radius by the number of fixed parameters, so that local perturbations are comparable across partitions with different cardinalities. Thus, $\epsilon$ denotes the magnitude of the normalized miscalibration of a single fixed parameter. For a given $\epsilon$-neighborhood, define the worst-case bias by
The following lemma connects worst-case bias to the sensitivity statistic $K_S$.
The next theorem shows that minimizing $K_S$ is equivalent to minimizing worst-case local bias.
In the boundary case where all parameters are estimated ($|S^c| = 0$), no parameters are subject to calibration error. The worst-case bias is exactly zero, making the full-estimation partition strictly preferred under our criterion (for full estimation $K_S=0$) whenever it is empirically admissible.
Theorem (ref) also yields a worst-case mean squared error interpretation. Let
where \(\mathbb{E}[\cdot]\) is taken under the fixed sampling law governing \(\widehat\gamma(\beta^*;S)\).
When $\widehat{\gamma}(\beta^*;S)$ satisfies the decay condition in (ref), namely \[ \mathbb{E}\!\left[ \left\lVert \widehat{\gamma}(\beta^*;S)-\gamma(\beta^*;S) \right\rVert_2^2 \right]=o(1), \] the sampling component of the target estimator is asymptotically negligible. Hence, for fixed $\epsilon$,
uniformly over $\beta_0\in B_\epsilon(\beta^*;S)$.
This decay requirement is closely related to weak identification. Section (ref) explains how we operationalize this restriction using spectral Jacobian information, following antoine2024gmm and forneron2024detecting.
This interpretation is closely related to bonhomme2022minimizing's approach to robust estimation under misspecification.\footnote{Their notion of misspecification is more general than the classical formulation considered here and includes our setting as a special case.} In both frameworks, the objective can be viewed as minimizing a worst-case mean squared error. The main difference is the decision variable: bonhomme2022minimizing optimize over influence functions, whereas we optimize over parameter partitions. A second difference is the asymptotic regime. Their misspecification neighborhoods shrink with the sample size, so bias and variance are asymptotically comparable, whereas in our setting the neighborhood is not indexed by $n$, so the asymptotic MSE is driven by the bias component.
A natural alternative to our sensitivity criterion is to select the partition by minimizing an MSE criterion. Nevertheless, implementing an MSE ranking would require a credible estimate of the full joint covariance matrix of all moments or reduced-form estimates. This is often unavailable when the moments are constructed from multiple datasets, as in many CMD applications. One could use an MSE-based criterion when this covariance matrix is available, or when a reliable joint bootstrap is feasible. In the nakamura2018high application, however, different datasets are used to compute the reduced-form parameters entering the CMD estimation, which motivates our sensitivity-based approach.
This section describes how to implement the partition-selection procedure in finite samples. We begin by presenting the full algorithm.
The rest of this section discusses the three implementation choices that enter the algorithm: how to obtain the reference point, how to construct the admissible set of partitions, and how to choose the normalization matrix used in the sensitivity computation.
Formal results on the consistency of the estimated least–sensitive partition are given in Appendix (ref).
Starting from a first-stage calibration $(S_1,\beta_1)$, the reference point is \[ \widehat{\eta} := \bigl(\widehat\alpha(\beta_1;S_1),\beta_1\bigr), \] where $\widehat\alpha(\beta_1;S_1)$ is the minimum-distance estimate obtained under the first-stage partition and calibration. We assume that the sample reference point satisfies \[ \widehat{\eta} \xrightarrow{p} \eta^*. \]
A primitive condition ensuring that $\eta^*\in\mathcal N_0$, as required in Assumption (ref), is that the first-stage calibration is compatible with the population equations. That is, there exists $\alpha\in\mathcal A$ such that
This assumption that the reference point lies in the identified set is crucial for three reasons. First, it guarantees comparability across partitions: for any candidate partition $S$ calibrating the same $\beta^*$, the conditional MD solution $\alpha(\beta^*;S)$ coincides with $\alpha^*$. This ensures that the points at which the local derivatives are evaluated for different MD problems are comparable. Second, it allows us to avoid re-estimating the model for every candidate partition. Third, it allows us to omit the Hessian terms in the IFT formula, which are asymptotically negligible at a reference point in the identified set. The asymptotic justification of the direct derivative computation, which is detailed in the next subsection, relies on these properties.
In overidentified GMM or CMD settings, the condition in ((ref)) is empirically testable. One can use identification-robust calibration-validity tests, as in alegre2023robust, or related weak-identification robust methods such as forneron2024detecting.
The statistic $\widehat{D}_{\alpha \beta,S}$, proposed in the algorithm, should be interpreted as a direct plug-in computation of the local derivative, not as the exact finite-sample derivative of every partition-specific re-estimation map. To see the distinction, fix a candidate partition \(S\) and write the reference point under the \(S\)-specific ordering as \[ \widehat \eta = (\widehat\alpha,\widehat\beta). \] A literal finite-sample computation of the slope of the re-estimation map would first solve \[ \widehat\alpha(\widehat\beta;S) \in \arg\min_{\alpha\in A_S} \widehat g(\alpha,\widehat\beta;S)' \widehat W \widehat g(\alpha,\widehat\beta;S), \] and then differentiate the corresponding first-order conditions with respect to the fixed parameters. Doing this separately for each partition would require re-estimating the model many times. Importantly, $\widehat\alpha$, the $S$-coordinates of the reference point, need not coincide with $\widehat\alpha(\widehat\beta;S)$, the MD solution obtained after fixing $\widehat\beta$.
The exact finite-sample derivative would also include additional Hessian terms weighted by the residuals \(\widehat g(\widehat{\alpha}(\widehat{\beta}),\widehat{\beta})\); under Assumption (ref), which places the reference point in the identified set, these terms vanish asymptotically.
The algorithm avoids this re-estimation step. Instead, for each partition \(S\), it uses the derivative bundle evaluated once at the common reference point \(\widehat\eta\) and computes \[ \widehat D_{\alpha\beta}(S) = - \left( \widehat G_{\alpha}(\widehat\eta;S)' \widehat W \widehat G_{\alpha}(\widehat\eta;S) \right)^{-1} \widehat G_{\alpha}(\widehat\eta;S)' \widehat W \widehat G_{\beta}(\widehat\eta;S). \] The justification is asymptotic. Under the regularity conditions in Appendix (ref), and using the reference point in the identified set, \(\widehat D_{\alpha\beta}(S)\) provides a consistent asymptotic approximation to the population slope, \[ \left. \frac{\partial\widehat\alpha(\beta;S)}{\partial \beta'} \right|_{\beta=\widehat\beta} = \widehat D_{\alpha\beta}(S)+o_p(1), \qquad \widehat D_{\alpha\beta}(S) = \frac{\partial \alpha(\beta^*;S)}{\partial \beta'}+o_p(1). \] Therefore, re-estimating \(\widehat\alpha(\widehat\beta;S)\) for every candidate partition would only change the derivative by an \(o_p(1)\) term. The same argument applies to the derivative of the target object.
The goal of this subsection is to explain how we implement the conditional local identification and declining-variance conditions used in the definition of the admissible set of partitions. We implement both restrictions using singular values of the weighted Jacobian of the estimating equations. The basic idea is that small singular values of \(W^{1/2}G_{\alpha}\) make the inverse problem unstable, which can prevent the variance of the induced estimator of \(\gamma\) from declining with the sample size.
For a candidate partition \(S\), let \[ C_S := (W^{1/2}G_{\alpha})^+, \] where \((\cdot)^+\) denotes the Moore--Penrose generalized inverse. The norm \(\|\cdot\|_2\) denotes the spectral norm, i.e., the largest singular value of a matrix. When \(W^{1/2}G_{\alpha}\) has full column rank,
Thus, small singular values of the weighted Jacobian correspond to a large Moore--Penrose inverse and indicate that the partition is close to a weak-identification region.
To keep notation light, we suppress the local-sequence index in the main text.\footnote{Strictly speaking, the population Jacobian and the matrix \(C_S\) should be indexed by \(n\) along the local sequence. Appendix (ref) gives the formal local-sequence argument.}
Under local curvature restrictions on \(g\) and regularity conditions on \(\Gamma\), Appendix (ref) shows that if \[ \|C_S\|_2=o(\sqrt{n}), \] then
Equations (ref) and (ref) motivate a hard-thresholding rule for the singular values of the estimated weighted Jacobian. For each partition \(S\), let \[ \widehat\sigma_{1,S}\geq \widehat\sigma_{2,S}\geq \cdots \] denote the singular values of \(\widehat W^{1/2}\widehat G_{\alpha,S}\). We estimate the conditional rank of partition \(S\) by
where the threshold follows forneron2024detecting. A partition is kept only if \[ \widehat r_S=|S|. \] This rule removes partitions with exact rank failure and partitions whose weighted Jacobian is numerically close to rank deficient. The estimated admissible set is therefore \[ \widehat{\mathcal S}_{AD}(\widehat{\eta}) = \left\{ S\in \mathcal S\cap \mathcal{R}: \widehat r_S=|S| \right\}. \]
The normalization matrix $\Sigma_S$ defines the metric used to measure local miscalibrations of the fixed parameters. Its purpose is to make perturbations comparable across structural parameters that may be expressed in different units. This subsection explains how we construct this metric, and how we conduct robustness analysis.
Before evaluating candidate partitions, we assign each structural parameter $j$ an interval \[ [\eta_{j,\min},\eta_{j,\max}]. \] We use the range $\eta_{j,\max}-\eta_{j,\min}$. To obtain the ranges, we can use one of the following options.
Because the normalization matrix encodes what the researcher regards as a plausible calibration error, robustness to this choice is an important part of the implementation. The baseline normalization should therefore be interpreted as the main specification, not as the only admissible metric. When several normalizations are equally defensible, the researcher should verify whether the selected partition and the substantive conclusions are stable across them.
A first robustness exercise is to replace the baseline normalization by a family of plausible normalization matrices. Let \(\mathcal{M}_S\) denote the set of normalization matrices considered plausible for partition \(S\). Instead of selecting the partition that minimizes \(K_S(\eta^*,\Sigma_S)\) for a single matrix, one can compute \[ S^{rob} \in \arg\min_{S\in \mathcal{S}_{\mathrm{AD}}(\eta^*)} \max_{\Sigma_S\in \mathcal{M}_S} K_S(\eta^*,\Sigma_S). \]
A second exercise is diagnostic. For each partition, define the normalized Jacobian \[ A_S := D_{\gamma\beta,S}\Sigma_S, \qquad D_{\gamma\beta,S}:=\frac{\partial \gamma(\beta^*;S)}{\partial \beta'}. \] Let \(v_S\) be a right singular vector associated with the largest singular value of \(A_S\), equivalently an eigenvector of \(A_S'A_S\) associated with its largest eigenvalue. The squared entries of $v_S$ quantify the proportion of the worst-case normalized perturbation allocated to each calibrated parameter.
These contribution measures identify which calibrated parameters drive the sensitivity statistic. A targeted robustness check is then to widen the normalization bounds only for the parameters with the largest contributions and recompute \(K_S\), the selected partition, and the implied worst-case bounds. This stress test asks whether the empirical conclusion depends on tight bounds for the parameters that matter most for worst-case miscalibration. If the selected partition and the substantive conclusions remain stable after widening those influential bounds, the robustness of the result is more credible.
nakamura2018high provide evidence of monetary non-neutrality using high-frequency responses of real interest rates, expected inflation, and output growth to Federal Reserve announcements. A central implication of their analysis is that monetary policy affects the economy not only through conventional channels, but also by shifting market beliefs about broader economic fundamentals via information effects. They study this mechanism by extending a New Keynesian model to incorporate an information effect and estimating the model by classical minimum distance (CMD), matching model-implied impulse response functions (IRFs) to reduced-form IRFs obtained via local projections.
We use their environment as a laboratory for our partition-selection criterion. Our object of interest is the information-effect parameter $\gamma$. Throughout, we hold fixed the model, the CMD objective, and the reduced-form moments; what varies across implementations is only the partition of structural parameters into estimated and calibrated (fixed) components. In this setting, the mapping $g(\cdot;\, S)$ depends on $S$ only through the relabeling of estimated and fixed parameters, while its functional form remains constant across partitions. The number of reduced-form parameters matched in CMD is $d_{m}=33$, and the number of structural parameters is $d_{\eta}=10$.
This application illustrates why we use the sensitivity-based statistic rather than a plug-in MSE criterion. The CMD reduced-form objects are constructed from different datasets, so the full joint covariance matrix of the stacked reduced-form parameters is not identified. In particular, the cross-covariances across moment blocks estimated from different datasets are not generally identified from the reported reduced-form estimates.
Table (ref) reports the structural parameters and the values reported in nakamura2018high. We evaluate all local objects at the empirical reference point implied by their estimates, and we denote the corresponding estimate of the information effect by $\widehat \gamma$. alegre2023robust test the Nakamura--Steinsson benchmark calibration on a feasible subset of moments and do not reject it, supporting its use as the local reference point.
In this New Keynesian model, credible external information about the calibrated parameters is available, so we can use it to enrich the sensitivity analysis by tightening the bounds used in the normalization.
Because ad hoc modifications of the normalization ranges could potentially change the analysis in arbitrary ways, it is important to be transparent about how these tighter bounds are chosen. It is also important to conduct robustness checks for any parameter whose normalization plays a central role in determining the selected partition.
We proceed as follows. First, we explain, parameter by parameter, how the informative bounds are chosen. We then present the main results. Finally, we conduct a robustness exercise for the discount-factor bounds in Subsection (ref), since this parameter is a key driver of the main results.
We follow schorfheide2000loss and set $\rho\sim \mathrm{Beta}(\mu=0.993,\ \sigma=0.002)$. The normalization bounds are the $1\%$ and $99\%$ quantiles of this prior. Similar tight priors are standard in estimated DSGE work; see, e.g., lubik2005bayesian, smets2007shocks, and an2007bayesian. This reflects the view that the discount factor is tightly pinned down by the steady-state relationship with the steady-state intereste rate.
We follow smets2007shocks and set $\delta\sim \mathrm{Beta}(\mu=0.5,\ \sigma=0.1)$, taking the $0.5\%$ and $99.5\%$ quantiles as bounds. Closely related priors are used by rabanal2005comparing, and lubik2005bayesian.
Following smets2007shocks, we place a Normal prior on $1/\sigma$:
We take the $2.5\%$ and $97.5\%$ quantiles of $1/\sigma$ and invert them to obtain the implied bounds $[\sigma_{\min},\sigma_{\max}]$ (noting that inversion typically yields an asymmetric interval in $\sigma$).
As in smets2007shocks, we set $b\sim \mathrm{Beta}(\mu=0.7,\ \sigma=0.1)$ and use the $1\%$ and $99\%$ quantiles as bounds. These values capture the empirically plausible region in which habits provide internal propagation without implying nearly unit root consumption dynamics.
For the autoregressive roots governing the monetary shock, we use the natural positive bounds implied by stationarity and the model's parametric restrictions.
We follow smets2007shocks by anchoring $\omega$ to more primitive objects: the labor share $a$ and the Frisch elasticity $v$. Using the model-implied mapping $\omega= (v^{-1}+1-a)/a$, we propagate uncertainty from $(a,v)$ into $\omega$. Concretely, we set $a\in\{1/2,\ 3/4\}$ and
and compute the implied bounds on $\omega$ from conservative quantiles of $v$ ($2.5\%$ and $97.5\%$) and the endpoints for $a$. This construction yields economically interpretable variation in $\omega$ while remaining grounded in well-studied labor-supply and factor-share objects.
We discipline the range for the Phillips-curve slope using the full-sample tradeable-demand IV estimate from hazell2022slope. Since the units differ, we map it into nakamura2018high units using the following model-implied relationship: \[ \kappa\zeta = \frac{v}{1+\omega\theta}\kappa^{Hazell}, \] where \(v\) is the Frisch labor-supply elasticity, \(\omega\) is the elasticity of marginal cost with respect to output, and \(\theta\) is the elasticity of substitution across varieties. Using \(v=1\), which corresponds to a labor share of $2/3$ when $\omega=2$, the confidence interval yields \[ \kappa\zeta\in [6.30,52.73]\times 10^{-5}. \]
We use the steady-state relationship between the markup $M$ and $\theta$,
and choose $M\in[1.05,\ 1.30]$, i.e.\ markups between $5\%$ and $30\%$. We then map this interval into bounds for $\theta$. This pins down $\theta$ using a primitive object (markups) with a clear economic interpretation and avoids implausible values that would imply either near-perfect competition or excessively high steady-state profit shares.
Using the hard-thresholding rank estimator proposed in Section (ref), we estimate the set of admissible partitions. In the present application, this procedure yields a total of 60 admissible partitions. For readability, we present a representative subset in the main text; full tables are reported in (ref).
For each admissible partition $S$, we compute the sensitivity statistic $\widehat{K}(\widehat{\beta};\,S)$, which measures the magnitude of worst-case local bias in $\gamma$ induced by small miscalibrations. To aid interpretation, we report $5\%$ worst-case bounds, defined as $\widehat \gamma \pm 0.05\,\widehat{K}(\widehat{\beta};\,S)$, which correspond to a first-order approximation of the worst-case bias of size $\epsilon = 5\%$.
Table (ref) reports these statistics for the representative subset of admissible partitions. For comparison, we also report the implementation in nakamura2018high (labeled Original), even though it is not admissible under our rank hard-thresholding criterion.
Table (ref) reports the sensitivity statistic \(\widehat K(\widehat\beta;S)\) for a representative subset of admissible partitions. Across the full set of admissible partitions, \(\omega\), \(\theta\), and the Phillips-curve slope \(\kappa_\zeta\) are never estimated. This suggests that these parameters are weakly identified in the local rank sense used to construct the admissible set.
Conditional on this admissibility restriction, the choice of partition matters substantially. The least-sensitive partition, \(S^{\ast}\), estimates \((\delta,\sigma,\rho_1,\rho_2,\gamma,b)\) and fixes \((\rho,\omega,\theta,\kappa_\zeta)\). It delivers \(\widehat K(\widehat\beta;S^{\ast})=0.06\), so a \(5\%\) normalized miscalibration generates only the local worst-case interval \([0.67,0.68]\). At the opposite end of the admissible set, the statistic rises to \(11.07\), producing the much wider interval \([0.12,1.23]\). The original Nakamura--Steinsson implementation is not admissible under the rank criterion and has \(\widehat K=1.82\). Hence the original partition is not in the extreme tail, but it is far from the least-sensitive admissible partition.
This heterogeneity is not well summarized by the number of estimated parameters. Among the largest admissible partitions, changing the identity of a single calibrated parameter moves the statistic from \(0.06\) to values above one. Fixing \(\rho\), as in \(S^{\ast}\), is much less costly than fixing \(\sigma\), \(b\), or \(\delta\). The contribution table clarifies why: in the low- and middle-sensitivity partitions, worst-case bias is typically concentrated in one preference or propagation parameter, such as \(\rho\), \(\sigma\), \(\delta\), or \(b\). Robustness therefore depends less on estimating many parameters per se than on estimating the parameters through which calibration errors are most strongly transmitted to \(\gamma\).
The role of the discount factor illustrates this point. In \(S^{\ast}\), almost all remaining sensitivity comes from the calibrated discount factor \(\rho\). This might suggest that the result is mechanically driven by the tight normalization range imposed on \(\rho\). The comparison across partitions argues against this interpretation. Several low-sensitivity alternatives, such as \(S_2\), \(S_3\), and \(S_4\), estimate \(\rho\) and still deliver small values of \(\widehat K\). Moreover, the robustness exercise in Table (ref) shows directly how much the discount-factor range can be widened before \(S^{\ast}\) ceases to be the least-sensitive partition. Thus, the tight bound on \(\rho\) helps explain the ranking, but it does not trivially determine the conclusion.
Finally, the high-sensitivity tail has a clear economic structure. Extreme fragility appears when the persistence of the monetary-policy shock is imposed by calibration rather than disciplined by the data. Once both persistence roots are fixed, \(\widehat K\) jumps into the range between roughly \(9\) and \(11\), and the contribution decomposition shows that \(\rho_1\), with a smaller role for \(\rho_2\), accounts for almost all of the statistic. Thus, for the information-effect parameter, the key positive prescription is to estimate the monetary-shock persistence parameters, especially \(\rho_1\). By contrast, calibrating the weakly identified Phillips-curve and markup-related block \((\omega,\theta,\kappa_\zeta)\) appears comparatively harmless for local robustness.
Table (ref) decomposes \(\widehat K(\widehat\beta;S)\) into the relative contribution of each fixed parameter. The main message is that sensitivity is typically highly concentrated: in most partitions, one fixed parameter accounts for almost all of the statistic. In the least-sensitive partition \(S^{\ast}\), essentially all residual sensitivity comes from the discount factor \(\rho\), which accounts for \(99.6\%\) of the total. In \(S_2\), the contribution is entirely due to \(\sigma\). In \(S_3\), most of the contribution comes from \(\sigma\) (\(70.7\%\)), with a secondary role for nominal rigidity \(\delta\) (\(22.9\%\)). The same concentration appears among admissible partitions that estimate the maximum number of parameters: in \(S_{12}\), the sensitivity is entirely driven by habit formation \(b\), whereas in \(S_{14}\), most of the sensitivity comes from \(\delta\), with smaller contributions from \(\omega\) and \(\theta\).
Another important feature is that the parameters that are always fixed because of weak-identification concerns---\(\omega\), \(\theta\), and \(\kappa_\zeta\)---have close to zero contribution to worst-case bias. In particular, the Phillips curve slope \(\kappa_\zeta\) contributes essentially zero, and \(\omega\) and \(\theta\) are almost never dominant sources of sensitivity. Thus, although these parameters must be calibrated, local miscalibration of them would not have large effects on the object of interest. By contrast, in the low- and middle-sensitivity regions of the admissible set, most of the sensitivity comes from preference and propagation parameters such as \(\rho\), \(\sigma\), \(\delta\), and \(b\).
The high-sensitivity tail has a different structure. Once monetary-shock persistence is fixed rather than estimated, the contribution of \(\rho_1\) becomes dominant. In \(S_{39}\), for example, \(\rho_1\) accounts for \(87.1\%\) of total sensitivity. When both persistence roots are fixed, as in \(S_{46}\), \(\rho_1\) accounts for \(80.0\%\) and \(\rho_2\) for \(18.1\%\). The same pattern appears in the most fragile partitions: in \(S_{58}\), \(\rho_1\) and \(\rho_2\) account for \(85.2\%\) and \(14.2\%\), respectively; in \(S_{59}\), for \(77.4\%\) and \(22.6\%\); and in \(S_{60}\), for \(82.9\%\) and \(15.0\%\). The economic interpretation is therefore clear: the most fragile partition strategies are those that force the persistence of the monetary shock to come from calibration rather than estimation. From a robustness perspective, the most valuable discipline is to estimate the shock-persistence parameters, especially \(\rho_1\), while calibration errors in the slope of the Phillips curve, \(\kappa_\zeta\), and other parameters such as \(\omega\) and \(\theta\) are comparatively harmless.
Because the estimated admissible set depends on the singular-value cutoff used in the rank estimator, Appendix (ref) reports a robustness exercise with respect to this implementation choice. In that exercise, we replace the baseline cutoff $\tau_n=(\log n/n)^{1/2}$ by $\tau_n(a)=(\log n/n)^a$ for $a\in\{0.50,0.60,0.70,0.80,0.90,1.00\}$, recompute the admissible set, and rank partitions again by the sensitivity statistic. The least-sensitive partition remains $S^*$ for all thresholds considered. For $a\leq 0.80$, both the admissible set and the top-five ranking are unchanged; more permissive cutoffs admit additional partitions but do not overturn the main ranking.
The discount-factor normalization is a natural place to check robustness. $\rho$ maps into the steady-state annual nominal interest rate $r$ according to
Thus, small numerical changes in the range for $\rho$ can correspond to large economic changes in steady-state interest rates.
The exercise below asks how far the discount-factor bound must be relaxed before the partition ranking changes. For each value of $\rho_{\min}$ we recompute the sensitivity statistics for the least-sensitive partition, its closest competitors, and the original partition, and report the annual rate implied by $\rho_{\min}$. The baseline selected partition remains least-sensitive until the lower bound is pushed below roughly $\rho_{\min}=0.945$. At that point, the implied steady-state annual nominal rate is above $23\%$, which is difficult to interpret as a plausible benchmark for the economies typically targeted by this class of models.
This section uses Monte Carlo simulations to study the finite-sample implications of the partition-selection criterion in the nakamura2018high application. For details in the implementation of the Monte Carlo simulation see (ref).
The goal of the simulations is to assess how informative the $K_S$-based sensitivity ranking is for finite-sample bias, variance, and MSE. We generate repeated samples, of sizes $n=150$ and $n=500$, from a data-generating process (DGP) calibrated to the reference parameter vector used in the empirical exercise. The Monte Carlo setting is useful because it allows us to compute finite-sample distributions and relevant statistics---such as the variance and mean squared error (MSE) of $\hat{\gamma}$---that are difficult to obtain in the empirical application, where the reduced-form parameters entering the CMD estimation are constructed from different datasets and their joint variance--covariance matrix is not readily available.
The exercise proceeds as follows. For each admissible partition $S$, we first compute the local worst-case direction at the reference point and then scale the perturbation by the miscalibration size $\varepsilon \in \{5\%,10\%,15\%\}$. We then re-estimate the model under partition $S$ using this approximated worst-case miscalibration. Repeating this procedure across Monte Carlo samples delivers, for each partition, the finite-sample distribution of $\hat{\gamma}$ under its own worst-case miscalibration. We summarize these distributions by reporting the variance and the corresponding worst-case MSE, and by plotting histograms that make the differences across partitions visually transparent.
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Table (ref) delivers three main lessons. First, the $K_S$-based ranking is highly informative about finite-sample worst-case bias. The least-sensitive partition, $S^{\star}$, has the smallest absolute bias and the lowest MSE for every miscalibration size and for both sample sizes. This is not mechanically imposed by the criterion, which is based on a local worst-case bias approximation rather than on simulated finite-sample MSE. The Monte Carlo results therefore support the interpretation of $K_S$ as a useful finite-sample diagnostic of calibration risk.
Second, the results make clear that sampling variance is not the main source of the differences across partitions. Increasing the sample size from $n=150$ to $n=500$ reduces the variance of $\hat{\gamma}$ substantially, but it leaves the miscalibration-induced bias largely unchanged. As a result, larger samples do not eliminate the consequences of fixing sensitive parameters at incorrect values. Instead, they make the bias component more visible. For $S^{\star}$, the MSE remains small even at $\varepsilon=15\%$, whereas the MSE of more fragile partitions grows rapidly as the miscalibration size increases.
Third, the deterioration in performance is strongly partition-specific. Partitions such as $S_2$, $S_3$, and $S_6$ generate moderate biases relative to the most fragile cases, but they still perform worse than $S^{\star}$ across the table. By contrast, partitions such as $S_4$, $S_5$, and the original Nakamura--Steinsson implementation can generate very large worst-case biases, especially at $\varepsilon=10\%$ and $\varepsilon=15\%$. The original partition is therefore not fragile because of sampling variability; it is fragile because the fixed block contains parameters through which calibration errors are strongly transmitted to $\gamma$.
Figure (ref) visualizes the same pattern. For very small miscalibrations, the distributions of \(\hat\gamma\) under the admissible partitions are relatively close to one another, so sampling variation can obscure the gains from reducing worst-case bias. As \(\epsilon\) increases, the distributions separate sharply. The original partition moves far away from the true value, while the distribution under \(S^*\) remains much closer to it. Increasing the sample size tightens the distributions, but it does not remove the displacement generated by fixing miscalibrated parameters. This explains why the advantage of \(S^*\) becomes clearer for larger samples and larger miscalibrations: sampling uncertainty falls, while calibration bias remains.
Figure (ref) reports the worst-case information effect, $\gamma$, as a function of the miscalibration size $\epsilon$ for the main partitions. The figure summarizes how the target object responds under increasing local calibration error, highlighting substantial differences in robustness across partitions: the selected partition remains comparatively stable, while the original implementation exhibits a pronounced deterioration as $\epsilon$ increases, with intermediate partitions lying between these extremes. Overall, the plot provides a direct visual counterpart to the $K_S$-based ranking, illustrating how sensitivity translates into divergence in worst-case outcomes.
The Monte Carlo results above use the baseline admissible set generated by $\tau_n=(\log n/n)^{1/2}$. Appendix (ref) repeats the ranking and the worst-case Monte Carlo comparison under alternative rank thresholds. The results are substantively stable: $S^*$ remains the top-ranked partition according to $K_S$ for every cutoff, and the finite-sample MSE patterns continue to support the main conclusion that choosing the least-sensitive partition sharply reduces calibration risk.
This paper studies a common but under-formalized choice in structural estimation: which parameters should be estimated and which should be calibrated. The main contribution is to treat this choice as a partition-selection problem. For each admissible calibration--estimation partition, the paper constructs a scalar sensitivity statistic that measures the worst-case local effect of calibration errors on the object of interest. The selected partition is the one that minimizes this statistic, and therefore minimizes worst-case local bias induced by plausible miscalibration of the fixed parameters.
The strength of the approach is that it turns sensitivity analysis into a decision rule. Rather than reporting that some calibrated parameters matter, the method provides a criterion for deciding which parameters should be fixed and which should be disciplined by the data. The rule is easy to implement, requires only local derivative information, avoids repeated re-estimation across partitions, and applies to a broad class of structural models. The applications show that this choice can matter substantially: partitions that look similar from the perspective of identification or computational convenience can imply very different robustness properties for the same target object.
The analysis also clarifies the limits of the method. The criterion is local: it evaluates robustness around a reference parameter vector and is informative when calibration errors are plausibly small or moderate. If the reference point is far from the relevant population value, or if large calibration errors are empirically plausible, global sensitivity analysis is more appropriate. A second limitation is that the normalization matrix is part of the economic content of the exercise. It defines what counts as a plausible and comparable miscalibration across parameters. Different normalizations can therefore lead to different selected partitions.
These limitations are not defects of the approach, but conditions for its interpretation. Structural work with calibrated parameters already relies on the assumption that the fixed values are credible approximations. The proposed statistic makes this assumption explicit and forces the researcher to state the scale on which calibration errors are judged. Robustness checks over alternative normalizations, together with the contribution decomposition of the sensitivity statistic, provide a transparent way to assess whether the selected partition depends on fragile choices about plausible parameter ranges.
Overall, the paper's message is that calibration is not merely a computational convenience. It is an econometric choice with direct consequences for the credibility of structural conclusions. By choosing the calibration--estimation partition through the lens of worst-case local bias, researchers can make this choice more systematic, transparent, and aligned with the object of interest.
This appendix uses the notation of the main text. Unless otherwise stated, all local objects are evaluated at the reference point \(\eta^*\). For a partition \(S\), write \(\eta=(\alpha_S,\beta_S)\), where \(\alpha_S\) collects the estimated coordinates and \(\beta_S\) collects the fixed coordinates. When no confusion can arise, the subscript \(S\) is suppressed and we write \((\alpha,\beta)\). The Jacobian blocks are \[ G_{\alpha,S}:=\frac{\partial g(\eta^*;S)}{\partial \alpha'}, \qquad G_{\beta,S}:=\frac{\partial g(\eta^*;S)}{\partial \beta'}, \] and \[ \Gamma_{\alpha,S}:=\frac{\partial \Gamma(\eta^*)}{\partial \alpha'}, \qquad \Gamma_{\beta,S}:=\frac{\partial \Gamma(\eta^*)}{\partial \beta'}. \] For any partition satisfying the rank condition in the main text, define
We omit the $S$ subscript when the context is clear.
The next lemma records the only differentiability fact needed repeatedly in the appendix and is used in the proof of Lemma (ref).
Let \(S^*:=S^*(\eta^*)\). After selecting \(S^*\), a researcher may use a nearby alternative calibration \(\widetilde\beta\), re-estimate the free parameters under the same partition, and form \[ \widetilde\eta:=\bigl(\alpha(\widetilde\beta;S^*),\widetilde\beta\bigr). \] The relevant question is whether the least-sensitive partition remains the same when the criterion is re-evaluated at \(\widetilde\eta\).
In practice, the maintained requirement is that \(\widetilde\eta=(\alpha(\widetilde\beta;S^*),\widetilde\beta)\) lies in the neighborhood \(U\). This can be checked by recomputing the least-sensitive partition at \(\widetilde\eta\).
This subsection gives the local-sequence argument underlying the variance-decay restriction in the definition of \(\mathcal S_{\mathrm{AD}}(\eta^*)\). Let \[ g_n:\mathcal A_S\times\mathcal B_S\to\mathbb R^{d_g} \] denote the population system along the \(n\)th element of a local population sequence, and let \[ \eta_n^\ast:=(\alpha_n^\ast,\beta_n^\ast) \] satisfy \[ g_n(\alpha_n^\ast,\beta_n^\ast)=0. \] The fixed-population case in the main text is obtained by setting \(g_n\equiv g\), \(\alpha_n^\ast\equiv\alpha^\ast\), and \(\beta_n^\ast\equiv\beta^\ast\).
Throughout this subsection, the fixed component is held at its population reference value \(\beta_n^\ast\). Thus, the stochastic object studied here is the conditional minimum-distance estimator at fixed \(\beta_n^\ast\), which is the object entering the variance-decay admissibility condition. The use of a sample reference point in plug-in derivative computations is handled separately.
Define \[ \widehat\alpha_n \equiv \widehat\alpha_n(\beta_n^\ast) \in \arg\min_{\alpha\in\mathcal A_S} \widehat g_n(\alpha,\beta_n^\ast)' \widehat W \widehat g_n(\alpha,\beta_n^\ast). \] Equivalently, the sample point studied in this subsection is \[ \widehat\eta_n^\ast := (\widehat\alpha_n(\beta_n^\ast),\beta_n^\ast). \]
For \(\alpha\in\mathcal A_S\), define the sample equation error at the fixed population component by \[ e_n(\alpha,\beta_n^\ast) := \widehat g_n(\alpha,\beta_n^\ast) - g_n(\alpha,\beta_n^\ast). \] Let \[ G_n := \frac{\partial g_n(\alpha_n^\ast,\beta_n^\ast)}{\partial\alpha'}, \qquad C_n := \left(W_n^{1/2}G_n\right)^+. \]
Assumption (ref)(iii) is a higher-order uniform convergence condition for the Jacobians used in both the sensitivity statistic and the admissible-set computation. For moment functions \(g(\eta;S)=E[m(X_i,\eta;S)]\), primitive sufficient conditions follow from Lemma 2.4 of newey1994large: compactness of \(N\), stationarity and ergodicity of \(\{X_i\}\), continuity in \(\eta\) of the derivative of \(m\), and an integrable dominating function for that derivative.
This subsection formalizes why Algorithm in Section (ref) can compute the local derivative at the common reference point rather than re-estimating the model separately for every candidate partition. Fix a partition \(S\). For a generic fixed value \(\beta\), define \[ \widehat\alpha(\beta;S) \in \arg\min_{\alpha\in A_S} \widehat g(\alpha,\beta;S)'\widehat W\widehat g(\alpha,\beta;S). \] Let \[ \widehat G_{\alpha}(\eta;S):=\frac{\partial\widehat g(\eta;S)}{\partial\alpha'}, \qquad \widehat G_{\beta}(\eta;S):=\frac{\partial\widehat g(\eta;S)}{\partial\beta'}. \] For the \(S\)-specific ordering of the common reference point, write \(\widehat\eta=(\widehat\alpha_{S},\widehat\beta_{S})\). The direct derivative used by the algorithm is
The examples below clarify when the population system \(g(\eta;S)\) depends on the partition.
In CMD, and in GMM settings where the moment conditions do not depend on the partition, a primitive condition for Assumption (ref) to hold is that the first-stage calibration is compatible with the population equations, i.e., there exists \(\alpha\) such that \(g(\alpha,\beta_1;S_1)=0\).
In MLE, by contrast, \(g_P(\eta;S)\) typically depends on \(S\) because \(S\) selects a subset of score equations. A convenient sufficient condition for Assumption (ref) is that \(\eta^*\) satisfies the mean-zero restriction for the full score vector, not only for the score components associated with the first-stage estimated parameters. Appendix (ref) constructs such a reference point.
Suppose the researcher first estimates the model by MLE under a calibration strategy \((S_1,\beta_1)\) and obtains \[ \widetilde\eta_n:=\bigl(\widehat\alpha_n(\beta_1;S_1),\beta_1\bigr). \] As discussed above, the MLE moment map \(g_P(\eta;S)\) depends on \(S\). Consequently, \(\widetilde\eta_n\) need not satisfy the mean-zero restrictions for score components corresponding to parameters that were fixed in the first stage. This may violate Assumption (ref) when the reference point must be observationally equivalent across candidate partitions.
Let \[ S_{\mathrm{full}}:=\{1,\ldots,d_\eta\}, \qquad s(X_i,\eta):=\frac{\partial}{\partial\eta}\log f(X_i\mid \eta), \] and define the full-score sample and population moments by \[ \widehat g_n(\eta;S_{\mathrm{full}}):=\frac{1}{n}\sum_{i=1}^n s(X_i,\eta), \qquad g(\eta;S_{\mathrm{full}}):=E_P[s(X_i,\eta)]. \] Let \[ Q_n(\eta):=\widehat g_n(\eta;S_{\mathrm{full}})'W_n\widehat g_n(\eta;S_{\mathrm{full}}), \qquad Q(\eta):=g(\eta;S_{\mathrm{full}})'Wg(\eta;S_{\mathrm{full}}), \] where \(W_n\overset{p}{\to}W\) and \(W\) is symmetric positive definite. To avoid confusion with the weighting matrix, let \(H\) denote a fixed symmetric positive-definite penalty metric and write \(\|x\|_H^2:=x'Hx\). Define
The special case \(H=I_{d_\eta}\) gives the Euclidean penalty.
Define the full-score identified set \[ N_0(S_{\mathrm{full}}):=\{\eta\in N:g(\eta;S_{\mathrm{full}})=0\}. \] Since every partition-specific score system is a subvector of the full score, any \(\eta\in N_0(S_{\mathrm{full}})\) also belongs to \(N_0(S)\) for every partition \(S\). If \(\widetilde\eta_n\overset{p}{\to}\widetilde\eta_0\), define
Thus \(\eta^*\) is the element of the full-score identified set closest to the probability limit of the first-stage estimate.
Assumption (ref) collects the conditions needed for the penalized full-score estimator. Parts (i)--(iii) are standard extremum-estimation conditions. Parts (iv)--(vi) handle the possibility that the full-score problem is set-identified. Part (vii) ensures that the penalty vanishes asymptotically but dominates the \(O_p(n^{-1})\) stochastic variation of \(Q_n\) along flat directions. A convenient deterministic choice is \(\lambda_n=n^{-\kappa}\) with \(0<\kappa<1\). If one also wants the penalty to be negligible at the \(\sqrt n\) scale in well-identified directions, one may choose \(\kappa\in(1/2,1)\), for instance \(\lambda_n=n^{-2/3}\).