nnpiv.diagnostics.relative_wellposedness_effective_diagnostic

nnpiv.diagnostics.relative_wellposedness_effective_diagnostic(A, C, C_prime, e_g, *, feature_map='rff', n_features=300, gamma='auto', poly_degree=3, poly_include_bias=False, ridge_alpha=1.0, projection_ridge=1e-08, eta=1e-06, eta_mode='sigma_i', random_state=123, feature_builder=None, feature_matrix=None, mask_s=None, mask_t=None, return_details=False)

Compute the post-estimation effective-direction diagnostic kappa_eff.

Parameters

• A (array-like of shape (n,) or (n, d_a)) – First-stage endogenous block used to construct feature functions.

• C (array-like of shape (n,) or (n, d_c)) – Second-stage instrument block.

• C_prime (array-like of shape (n,) or (n, d_cp)) – First-stage instrument block.

• e_g (array-like of shape (n,) or (n, 1)) – Estimated first-stage error direction, typically \(\widehat g - g_0\).

• feature_map – Same interpretation as in relative_wellposedness_diagnostic().

• n_features – Same interpretation as in relative_wellposedness_diagnostic().

• gamma – Same interpretation as in relative_wellposedness_diagnostic().

• poly_degree – Same interpretation as in relative_wellposedness_diagnostic().

• poly_include_bias – Same interpretation as in relative_wellposedness_diagnostic().

• ridge_alpha – Same interpretation as in relative_wellposedness_diagnostic().

• eta – Same interpretation as in relative_wellposedness_diagnostic().

• eta_mode – Same interpretation as in relative_wellposedness_diagnostic().

• random_state – Same interpretation as in relative_wellposedness_diagnostic().

• feature_builder – Same interpretation as in relative_wellposedness_diagnostic().

• feature_matrix – Same interpretation as in relative_wellposedness_diagnostic().

• projection_ridge (float, default=1e-8) – Ridge used when projecting e_g into the feature span.

• mask_s (array-like, optional) – Optional stage-specific subset selectors, using the same mask/index formats accepted by relative_wellposedness_diagnostic().

• mask_t (array-like, optional) – Optional stage-specific subset selectors, using the same mask/index formats accepted by relative_wellposedness_diagnostic().

• return_details (bool, default=False) – If True, include projected coefficients and intermediate matrices.

Returns

Dictionary containing kappa_eff, regularized counterpart kappa_eff_reg, associated norms, and metadata.

Return type

dict

Notes

The core quantity is

\[\kappa_{\mathrm{eff}} = \frac{\|T_g e_g\|_2}{\|S e_g\|_2},\]

computed after projecting e_g onto the selected finite feature span.