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Experimental second-order one-asset HANK

solve_hank_second_order adds quadratic state terms and anticipated-variance corrections to a small one-asset HANK policy. It retains every household marginal control and the complete distribution basis subject to mass, income and bond-market restrictions. Reduced bases, two assets, productive capital, global accuracy and welfare are outside this implementation.

import numpy as np
from puremacro.models import (
    calibrate_one_asset_hank, solve_hank_reduced_policy,
    solve_hank_second_order,
)

ss = calibrate_one_asset_hank(n_a=20, n_e=2, a_max=120.)
linear = solve_hank_reduced_policy(
    ss, distribution_rank=None, marginal_rank=None, shock_std=0.0005,
)
policy = solve_hank_second_order(linear)
x = policy.state(monetary_shock=0.0005)
audit = policy.audit(x, tolerance=1e-5, quadrature_order=3)
assert audit.accepted
assert audit.diagnostics["mass_error"] < 1e-10
assert audit.diagnostics["budget_error"] < 1e-8
assert audit.diagnostics["equation_error"] < 4e-7
assert audit.diagnostics["euler_kkt_is_grid_diagnostic"]

Timing and approximation

The state includes the distribution coordinates, inherited nominal interest rate and current monetary shock. It is observed after the current innovation. For state x, the conditional policy is

\[ y = Gx + \tfrac12 G_{xx}[x,x] + \tfrac12 G_{\sigma\sigma}\sigma^2, \qquad E[x'\mid x] = Hx + \tfrac12 H_{xx}[x,x] + \tfrac12 H_{\sigma\sigma}\sigma^2. \]

The next innovation enters only the monetary-shock coordinate. Household expected marginal values are averaged before the current backward step; the nonlinear Phillips continuation is averaged separately. The state and variance coefficient equations are solved separately, including future policy curvature in the variance equation. Tensors use the one-half Taylor convention and do not already contain the chosen innovation variance.

Centered derivatives at difference_step and half that step must agree. Richardson extrapolation follows this check. The solver rejects inconsistent first derivatives, unresolved curvature, singular variance equations and excessive state/workspace dimensions. It does not smooth policy kinks or add a ridge. Agreement of two steps is evidence about the local calculation, not a proof of global smoothness across borrowing or interpolation boundaries.

The state/variance expansion follows Schmitt-Grohé and Uribe. Simulation keeps first- and second-order state components separate, following the order-consistent treatment discussed by Kim, Kim, Schaumburg and Sims. Quadratic terms use only first-order states; nonlinear feedback of quadratic corrections would introduce unintended higher-order terms.

Anticipated risk versus state dispersion

mean = policy.mean_decomposition()
np.testing.assert_allclose(
    mean["state_mean"], mean["state_risk"] + mean["state_dispersion"],
)
np.testing.assert_allclose(
    mean["control_mean"], mean["control_risk"] + mean["control_dispersion"],
)
zero = policy.state()
np.testing.assert_allclose(
    policy.controls(zero, shock_std=0.0005),
    4 * policy.controls(zero, shock_std=0.00025),
)
path = policy.simulate(np.array([0.0002, -0.0001, 0., 0.]))
assert path.converged
assert path.diagnostics["pruned"]

state_risk is the limit under anticipated risk and zero realized innovations. state_dispersion accounts for the curvature of the state rule evaluated at the first-order stationary covariance. Their sum is the mean of the pruned approximation. Control means are coordinates: the last three are log inflation, log wage and log output deviations. They are not transformed level means or household welfare. Reusing a nonlinear utility formula on these objects does not supply an audited welfare measure.

What the checks establish

controls and next_state evaluate polynomials; audit independently evaluates the nonlinear discrete household, distribution and market equations using Gaussian quadrature. simulate audits every pruned step. Failed equation, probability or positive-domain checks remain failures. Distributions are never clipped or renormalized to rescue a second-order path. Pruning does not ensure distribution feasibility.

Euler/KKT residuals are reported separately from finite-grid equation acceptance. At the demonstrated 20-asset grid they are around 0.004, despite much smaller discrete equation errors. Household-grid convergence is not certified.

The 12-asset signed-shock cases and the 20-asset positive-shock cases show cubic equation-error decay above their numerical floor. At 20 assets, positive shock/innovation scales of 0.001, 0.0005 and 0.00025 give maximum scaled errors of roughly 2.34e-6, 2.91e-7 and 3.62e-8. Exact polynomial fixtures and a separate dense coefficient solve check the state/variance algebra. Independently solved finite nonlinear equilibria provide an additional comparison; they retain their own first-order terminal boundary. Stochastic horizon sensitivity remains material, and agreement with one such tree does not establish infinite-horizon or global accuracy.

The 20-asset negative-shock cases at -0.001 and -0.0005 fail the unchanged 1e-5 audit tolerance: distribution-equation errors are approximately 2.55e-4 and 3.69e-5. The smaller -0.00025 and -0.000125 cases pass. Thus 22 of the 24 amplitude cases pass; both rejected cases remain in the record. Local accuracy must be checked for each shock direction and size, even when the coefficient equations and derivative-refinement checks pass.

The stricter 1e-9 finite-reference check fails on the 12-asset calibration's approximately 1.47e-9 equilibrium floor. Those failures are retained; the 20-asset references meet the same tolerance. No tolerance was relaxed.

Reproduce the amplitude, quadrature, mean and nonlinear-reference records with tools/reference_validation/benchmark_hank_second_order.py. Inspect all failed cases and horizon comparisons before extending the approximation.