Monetary Policy Experiments and Structural Model Families

Conceptual map

IV-14.01A common experiment protocol
IV-14.02Four structural families
IV-14.03Response objects
IV-14.04Calibration and profiling
IV-14.05Counterfactual comparison

1. Model comparison begins with a common response contract

Structural families become comparable after they receive the same experiment, normalization, observable map, horizon, and loss function.

Index model families by $m\in\{1,\ldots,M\}$ . Family $$m$$ has parameter space $\Theta _{m}$ , equilibrium map $G_{m}$ , experiment $$e$$ , and response vector $r_{m}(\theta ;e)\in \mathbb{R}^{K}$ . The response contract specifies shock units, information set, intervention timing, equilibrium selection, observable transformation, horizon, and whether responses are levels, elasticities, or percentage deviations.

Definition 1 · Family distance

For benchmark response $r^{0}$ and positive-semidefinite weight matrix $$W$$ , define $d_{m}^{2}=\operatorname{inf}_{\theta \in \Theta _{m}}(r_{m}(\theta )-r^{0})^{\top}W(r_{m}(\theta)-r^{0})$ . This is a distance in response space only when $$W$$ is positive definite; a singular $$W$$ defines a seminorm and ignores its null directions.

Business-cycle, sticky-price, search, and heterogeneous-agent models can all enter this contract. Their internal state vectors can differ. Comparison uses declared response objects and preserves family-specific equilibrium restrictions.

2. Calibration, profiling, and estimation answer different questions

Calibration fixes parameters from external evidence or chosen targets. Profiling minimizes a stated response loss over nuisance parameters for each focal parameter. Statistical estimation adds a sampling model and interprets the objective through likelihood or moment restrictions. A numerical minimizer alone does not determine which of these operations occurred.

Proposition 1 · Interior projection condition

Suppose $\Theta _{m}$ is open near an interior minimizer $\hat{\theta}$ , the response map is differentiable there, and $$W$$ is symmetric. If $\hat{\theta}$ minimizes the squared response loss, then $J_{m}(\hat{\theta})^{\top}W[r_{m}(\hat{\theta})-r^{0}]=0$ , where $J_{m}$ is the response Jacobian.

Proof. Differentiate the quadratic objective. Its gradient is twice the displayed Jacobian-weighted residual. An interior differentiable minimum has zero gradient. Boundary minima satisfy a normal-cone condition, so the zero-gradient conclusion need not hold there. ∎

The condition says that the residual is orthogonal, in the $$W$$ geometry, to locally attainable response movements. It is a fit condition rather than evidence that the fitted parameter is economically identified. Rank of the Jacobian, parameter support, and the sampling design still matter.

3. Two families can fit nearby responses and disagree on a target

Take benchmark $r^{0}$ =(1,0.5), Euclidean weighting, and two one-parameter families. Family A has $r_{A}(\theta)=(\theta,0.5\theta)$ , $\theta \in \mathbb{R}$ . It fits exactly at $\theta=1$ . Family B has $r_{B}(\phi)=(\phi,1-\phi)$ , $\phi \in[0,1]$ . Minimizing $(\phi-1)^{2}+(0.5-\phi)^{2}$ gives $\phi$ ̂=0.75, response (0.75,0.25), squared distance 0.125, and distance $\sqrt{0.125}\approx0.353553$ .

Let the target be $\tau _{A}(\theta)=0.5\theta$ and $\tau _{B}(\phi)=1-2\phi$ . At their best fits, the targets are 0.5 and $$-0.5$$ . The disagreement follows from family-specific target maps left unrestricted by the two matched observables. A comparison report should show the fit distance, fitted response, target, and parameter boundary for every family.

4. Normalization and support determine the frontier

Failure case · Shock-size differences presented as mechanism differences

If one model receives a one-standard-deviation shock and another a one-percentage-point shock, response magnitudes are incomparable until the shocks are mapped into common units.

A singular $$W$$ can give zero distance to distinct response vectors. A covariance inverse is unstable when estimated eigenvalues approach zero; regularization changes the metric and must be reported. Compact parameter sets guarantee a minimum for continuous losses, while open or unbounded sets may yield only an infimum. Multiple equilibria require a selection rule. Parameters at inequality boundaries use constrained first-order conditions. Responses at date zero depend on timing conventions, and terminal truncation can distort long-horizon comparisons. Model families selected after viewing the target data require a separate validation sample or an explicit selection adjustment for statistical claims.

5. Implementation, exercises, and sources

Write one machine-readable experiment contract, expose a family adapter returning responses and targets, normalize units before optimization, and retain solver status and equilibrium residuals. Search globally when the objective is nonconvex, verify local derivatives, and trace the profile over economically admissible parameter sets. Distinguish sampling uncertainty, calibration ranges, numerical error, and cross-family variation in the output.

Download the volume verification script →

Exercises

Derive $\phi$ ̂ and the reported distance for Family B.
Replace $$W$$ by $\operatorname{diag}(1,0)$ and characterize the minimizers and fitted distance.
Derive the constrained first-order condition when $\phi$ is restricted to [0,0.6].

Partial solutions

1. The derivative is $4\phi-3$ , so $\phi$ ̂=3/4 and the two residuals are $$-1/4$$ . 2. $\phi=1$ fits the weighted coordinate exactly; the ignored second coordinate receives no penalty. 3. The unconstrained value 0.75 is infeasible, so the minimum occurs at the upper endpoint 0.6 and the outward derivative is consistent with the normal-cone condition.

Robert E. Lucas Jr. (1976), “Econometric Policy Evaluation: A Critique,” Carnegie-Rochester Conference Series on Public Policy 1, 19–46.Policy experiments and structural invariance.
Lars Peter Hansen (1982), “Large Sample Properties of Generalized Method of Moments Estimators,” Econometrica 50, 1029–1054.Weighted moment criteria and statistical estimation.
Chae-Yeon Xon (2026), “Between Models: Counterfactual Frontiers across Causal-Response Experiments.”Related cross-family comparison application.

6. Audit checkpoint

Experiment units, information set, timing, equilibrium selection, observable map, horizon, normalization, parameter domains, family adapters, metric definiteness, covariance conditioning, rank, boundary solutions, global search, numerical residuals, family selection, target map, and separation of sampling, calibration, numerical, and family uncertainty.

7. Scope boundary

The chapter covers finite-dimensional response comparison across declared structural families. Bayesian model averaging, encompassing tests, and full solution methods for particular macroeconomic models require additional development.