Bootstrap Functions

These standalone functions are the lower-level interface; in most cases using SW2023Model.bootstrap() is more convenient.

sw2023.bootstrap_sw(X, Y, B=200, alpha=0.05, direction='mean', method='HMS', log_transform=True, standardize=True, bandwidth_method='silverman', seed=None, verbose=True)[source]

Pairs bootstrap confidence intervals for SW2023Model.

To correctly compute individual observation CIs:

  • Fix the original direction vector d

  • Evaluate moment functions of resampled data at original Z points.

Parameters:

  • X (Original data)

  • Y (Original data)

  • B (Number of bootstrap iterations)

  • alpha (Significance level (1-alpha confidence interval))

  • direction (Direction vector specification ('mean' | 'median' | array))

  • method ('HMS' | 'SVKZ')

  • log_transform (bool)

  • standardize (bool)

  • bandwidth_method ('silverman' | 'loocv' ('silverman' recommended for bootstrap))

  • seed (Random seed)

  • verbose (Print progress)

Returns:

phi_hat_point : (n,) original frontier estimates phi_hat_ci : (n, 2) frontier confidence intervals eff_mean_point : float mean efficiency estimate eff_mean_ci : (2,) mean efficiency confidence interval eff_individual_point: (n,) individual efficiency estimates eff_individual_ci : (n, 2) individual efficiency confidence intervals sigma_eta_ci : (n, 2) sigma_eta confidence intervals B, alpha : parameters used

Return type:

BootstrapResult

sw2023.bootstrap_panel(X, Y, firm_id, time_id, B=200, alpha=0.05, direction='mean', method='HMS', log_transform=True, standardize=True, bandwidth_method='silverman', seed=None, verbose=True)[source]

Cluster bootstrap confidence intervals for PanelSW2023.

Resample by firm unit to preserve within-panel time-series dependence. Individual observation CIs are corrected in the same way as bootstrap_sw: resample, transform with same d, then evaluate at original Z points.

Parameters:

  • X (Original data (n_total × ...))

  • Y (Original data (n_total × ...))

  • firm_id (Firm and time period identifiers (n_total,))

  • time_id (Firm and time period identifiers (n_total,))

  • (rest (same as bootstrap_sw))

Returns:

Cluster bootstrap confidence intervals evaluated at the original observation points.

Return type:

BootstrapResult

sw2023.test_r3_significance(X, Y, direction='mean', method='HMS', log_transform=True, standardize=True, bandwidth_method='silverman', B=999, seed=None, verbose=True)[source]

Test whether inefficiency (eta) genuinely depends on observable covariates Z.

Null hypothesis H0: E(epsilon^3|Z) = const (spatially uniform inefficiency). Alternative H1: E(epsilon^3|Z) differs across Z (heterogeneous inefficiency).

PSVKZ(2024) Section 3.1 wild bootstrap (Rademacher perturbation):

  1. Estimate r_hat_3(Z_i); compute residuals eta_i = eps_hat^3_i - r_hat_3(Z_i).

  2. Center: eta^c_i = eta_i - mean(eta).

  3. Wild perturbation: eta*_i = eta^c_i x V_i, V_i ~ Rademacher(+-1, 50% each).

  4. Bootstrap response: eps_hat^3*_i = r_hat_3(Z_i) + eta*_i.

  5. Re-estimate r_hat*_3(Z) and compute T*_b.

  6. p-value = #{T*_b >= T_obs} / B.

Test statistic (variance-weighted average heterogeneity): T = Var(r_hat_3) / Var(eps_hat^3).

Parameters:

  • X (Original data)

  • Y (Original data)

  • direction (Direction vector)

  • method ('HMS' | 'SVKZ')

  • bandwidth_method (Bandwidth method)

  • B (Number of bootstrap iterations (default 999))

  • seed (Random seed)

  • verbose (Print output)

Returns:

statistic : float observed test statistic T p_value : float bootstrap p-value r3_hat : (n,) estimated r_hat_3(Z) T_boot : (B,) bootstrap distribution of T B : int

Return type:

SignificanceTestResult