TST: Keep next states within the interpolation domain in test and benchmark models

[oyamad]

Summary

The 2-D Santos (1999, Sec. 7.3) test model was verified against the paper: the model equations, the reduction to univariate maximization via the intratemporal FOC (Eq. (7.4)), all parameter values, the domain, and the analytical value/policy functions are correct. One genuine flaw was found in the numerical setup, shared by the benchmark models:

During the inner maximization, candidate actions can map the next state outside the interpolation domain (e.g. leisure x -> 0 gives k' ~ z*A*k^alpha, up to ~30 against k_max = 10; in the 1-D growth model s' can even go negative), where the fitted value function is extrapolated. For the linear and spline bases (the ones Santos actually uses) extrapolation is locally linear and benign, but for the Chebyshev cases a degree-40+ polynomial extrapolated that far is astronomically large, and correctness relied on the extrapolated values happening to repel the optimizer. Under VFI, the 1-D stochastic benchmark model in fact diverged silently (see #96 for the convergence-check bug that masked this).

Changes:

• test/test_cdp_multidim.jl: g clamps k' into [k_min, k_max], with a comment explaining why; the true optimal policy maps the domain well inside [0.51, 4.66], so the analytical checks are unaffected.
• test/test_cdp_multidim.jl: the spline policy_tol is written as 1.92e-4 * 100 and documented — Table 20's 1.92e-4 is for Santos's shape-preserving/cubic spline DP, while this collocation config (quadratic over both k and logz; cubic over logz is infeasible with 3 nodes) attains ~1.1e-2. Also corrects the spline-degree comment ("linear over logz" -> quadratic).
• benchmark/benchmarks.jl: the 1-D model's action bounds are tightened using the quadrature shock extrema so that s' stays in the domain for all shocks (keeping the transition smooth), while the 2-D model clamps k' in g (its domain-respecting action bound has no closed form). With this, VFI on the 1-D Chebyshev model genuinely converges (365 iterations, agreeing with PFI to ~3e-7) instead of silently diverging. Note: this changes what the VFI benchmarks measure, so their timings are not comparable with runs before this PR.

Verification

• Full test suite passes.
• All six basis x method combinations of the Santos model converge with errors within the stated tolerances (measured: linear g_err = 0.136 vs. Table 16's 1.12e-1 bound scaled by mesh; spline g_err = 1.09e-2; Chebyshev g_err = 3.3e-4).
• Benchmark suite smoke-runs cleanly (21 benchmarks).

🤖 Generated with Claude Code (Claude Fable 5)

[Copilot]

Pull request overview

This PR updates the test and benchmark model definitions to prevent candidate actions during inner maximization from mapping next-period states outside the interpolation domain, avoiding unstable extrapolation (especially for high-degree Chebyshev bases) and improving VFI robustness.

Changes:

• Clamp next-period capital/state in the Santos 2-D test model transition to stay within the interpolation domain.
• Clarify the spline basis-degree comment and document the rationale behind the spline policy tolerance value.
• Apply the same domain-clamping approach to both benchmark models and update benchmark model function signatures/call sites accordingly.

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 1 comment.

File	Description
test/test_cdp_multidim.jl	Clamp next-state capital in the transition and clarify spline basis/tolerance rationale in the Santos 2-D test.
benchmark/benchmarks.jl	Clamp next-state variables in benchmark model transitions and update model constructors/call sites to pass domain bounds.