Outer-search numerical error budget

Companion to the continuous-outer ADR. A central finite difference of the moment map decomposes as

Ĝ(h) − G  =  O(h²)                       parameter truncation
           + O(ε_outer  / h)             outer search (mesh + bracket)
           + O(ε_inner  / h)             inner NB-EGM solve
           + O(ε_interp / h)             outer interpolation
           + O(ε_sim    / h)             finite simulation

Every term must have an observable estimate before a Jacobian is accepted; “the steps agree with each other” alone is consistency, not accuracy — three steps can share a bias regime and agree while uniformly wrong.

Error source	Measured by
parameter truncation	comparison of h, h/2, h/4 central differences
outer search	tighter mesh and bracket tolerances
inner NB-EGM	tighter savings-grid/envelope settings and Euler errors
interpolation	exact midpoint validations (interpolant vs exact solve)
simulation	independent scrambles and larger N
branch ties	population mass near keeper/adjuster and local-max ties
moment nonsmoothness	median atoms, empty groups, exact-equality mass

All seven diagnostics are stored in the Jacobian manifest; inference refuses a manifest whose diagnostics are missing or whose configuration does not match the run being reported.

Mesh freezing for derivatives¶

Adaptive node insertion is discrete, so the mesh must not differ arbitrarily between θ+h and θ−h. Derivative protocol: adapt at baseline θ and at pilot perturbations θ ± h_max·e_j; take the per-period union of all resulting nodes; sort, deduplicate, freeze; rerun baseline and every perturbation on the frozen mesh; verify the frozen mesh meets the interpolation-error criterion for every run; on any failure, augment the mesh and restart the entire derivative batch. The frozen nodes are stored in the Jacobian manifest.

Simulation variance¶

The simulation allowance added to the empirical moment variance is the variance of one production-sized simulated moment vector, estimated from independent scrambles/replications; it is not divided by the number of auxiliary replications (the estimator’s objective uses one simulation, not their average).