SFA Diagnostics¶

Quick Reference

Module: panelbox.frontier.tests Import: from panelbox.frontier.tests import inefficiency_presence_test, skewness_test, vuong_test, lr_test, wald_test Stata equivalent: Post-estimation tests R equivalent: frontier::efficiencies(), custom tests

Overview¶

After estimating a stochastic frontier model, rigorous diagnostics are essential to validate the specification and assess the reliability of efficiency estimates. PanelBox provides a comprehensive suite of diagnostic tests organized into five categories:

Inefficiency presence -- Is SFA necessary, or is OLS sufficient?
Distribution selection -- Which distributional assumption fits best?
Variance decomposition -- How much variation is due to inefficiency vs noise?
Functional form -- Are returns to scale constant? Is Translog needed?
Panel model selection -- TFE vs TRE, time-varying vs time-invariant?

This page documents each test with its statistical foundation, PanelBox API, and interpretation guidelines.

Diagnostic Checklist¶

Before reporting SFA results, verify each of these items:

Step	Test	What to Check
1	Skewness test	OLS residuals have correct skewness sign
2	Inefficiency presence	LR test rejects \(H_0: \sigma_u^2 = 0\)
3	Distribution comparison	AIC/BIC or Vuong test supports chosen distribution
4	Variance decomposition	\(\gamma\) is in a reasonable range
5	Mean efficiency	Values are plausible for the industry
6	Convergence	Optimizer converged successfully

Detailed Guide¶

1. Testing for Inefficiency Presence¶

The fundamental question: Is SFA necessary, or is OLS sufficient?

Hypotheses:

\(H_0: \sigma_u^2 = 0\) (no inefficiency; OLS is adequate)
\(H_1: \sigma_u^2 > 0\) (inefficiency is present; SFA is needed)

Since \(\sigma_u^2\) is constrained to be non-negative (boundary parameter), the standard LR test distribution does not apply. Under \(H_0\), the LR statistic follows a mixed chi-square distribution (Kodde & Palm, 1986):

For half-normal or exponential: \(LR \sim \frac{1}{2}\chi^2(0) + \frac{1}{2}\chi^2(1)\)
For truncated-normal: \(LR \sim \frac{1}{2}\chi^2(1) + \frac{1}{2}\chi^2(2)\)

from panelbox.frontier.tests import inefficiency_presence_test

test = inefficiency_presence_test(
    loglik_sfa=result.loglik,
    loglik_ols=ols_loglik,
    residuals_ols=ols_residuals,
    frontier_type="production",
    distribution="half_normal",
)

print(f"LR statistic: {test['lr_statistic']:.4f}")
print(f"P-value (mixed chi-sq): {test['pvalue']:.4f}")
print(f"Conclusion: {test['conclusion']}")
print(f"Skewness: {test['skewness']:.4f}")

if test['skewness_warning']:
    print(f"WARNING: {test['skewness_warning']}")

The function also performs a skewness diagnostic:

Production frontier: OLS residuals should be negatively skewed
Cost frontier: OLS residuals should be positively skewed

Wrong skewness suggests misspecification (wrong frontier type, outliers, or no inefficiency).

Automatic Diagnostics

The inefficiency presence test is automatically included in result.summary(include_diagnostics=True) when OLS log-likelihood information is available.

2. Skewness Test¶

A quick preliminary diagnostic before fitting SFA:

from panelbox.frontier.tests import skewness_test

skew = skewness_test(
    residuals=ols_residuals,
    frontier_type="production",
)

print(f"Skewness: {skew['skewness']:.4f}")
print(f"Expected sign: {skew['expected_sign']}")
print(f"Correct sign: {skew['correct_sign']}")

if skew['warning']:
    print(f"WARNING: {skew['warning']}")

3. Distribution Selection¶

Nested Distributions: LR Test¶

For nested distribution pairs (e.g., half-normal \(\subset\) truncated-normal when \(\mu = 0\)):

from panelbox.frontier.tests import compare_nested_distributions

lr = compare_nested_distributions(
    loglik_restricted=result_halfnormal.loglik,
    loglik_unrestricted=result_truncnormal.loglik,
    dist_restricted="half_normal",
    dist_unrestricted="truncated_normal",
)

print(f"LR statistic: {lr['lr_statistic']:.4f}")
print(f"P-value: {lr['pvalue']:.4f}")
print(f"Preferred: {lr['conclusion']}")

Common nested pairs:

Restricted	Unrestricted	Restriction	df
half-normal	truncated-normal	\(\mu = 0\)	1 (mixed \(\chi^2\))
exponential	gamma	\(P = 1\)	1

Non-Nested Distributions: Vuong Test¶

For non-nested distributions (e.g., half-normal vs exponential):

from panelbox.frontier.tests import vuong_test

vuong = vuong_test(
    loglik1=ll_halfnormal_obs,  # Observation-level log-likelihoods
    loglik2=ll_exponential_obs,
    model1_name="half_normal",
    model2_name="exponential",
)

print(f"Vuong statistic: {vuong['statistic']:.4f}")
print(f"P-value (two-sided): {vuong['pvalue_two_sided']:.4f}")
print(f"Preferred: {vuong['conclusion']}")

Vuong Test Requirements

The Vuong test requires observation-level log-likelihoods (arrays), not just the total log-likelihood value. Both models must be estimated on the same sample, and the sample should be large (\(N \gg 30\)).

Automatic Distribution Comparison¶

Compare multiple distributions at once using AIC/BIC:

comparison = result.compare_distributions(
    distributions=["half_normal", "exponential", "truncated_normal"]
)
print(comparison[["Distribution", "Log-Likelihood", "AIC", "BIC", "Mean Efficiency"]])

4. Variance Decomposition¶

The parameter \(\gamma = \sigma_u^2 / (\sigma_v^2 + \sigma_u^2)\) measures the share of total variance attributable to inefficiency:

var_decomp = result.variance_decomposition(ci_level=0.95, method="delta")

print(f"gamma: {var_decomp['gamma']:.4f}")
print(f"95% CI: [{var_decomp['gamma_ci'][0]:.4f}, {var_decomp['gamma_ci'][1]:.4f}]")
print(f"lambda: {var_decomp['lambda_param']:.4f}")
print(f"95% CI: [{var_decomp['lambda_ci'][0]:.4f}, {var_decomp['lambda_ci'][1]:.4f}]")
print(var_decomp['interpretation'])

Interpretation:

\(\gamma\) Value	Interpretation
\(\gamma < 0.1\)	Inefficiency negligible. OLS may be adequate.
\(0.1 \leq \gamma \leq 0.3\)	Inefficiency minor but present.
\(0.3 < \gamma < 0.7\)	Both noise and inefficiency are important.
\(0.7 \leq \gamma \leq 0.9\)	Inefficiency is dominant.
\(\gamma > 0.9\)	Nearly deterministic frontier. Check specification.

Confidence interval methods:

Method	API	Characteristics
Delta method	`method="delta"`	Fast, analytical; default
Bootstrap	`method="bootstrap"`	Robust, slower

Three-Component Decomposition (TRE)¶

For True Random Effects models with heterogeneity:

var_decomp = tre_result.variance_decomposition()

print(f"Noise (gamma_v):         {var_decomp['gamma_v']:.4f}")
print(f"Inefficiency (gamma_u):  {var_decomp['gamma_u']:.4f}")
print(f"Heterogeneity (gamma_w): {var_decomp['gamma_w']:.4f}")
# gamma_v + gamma_u + gamma_w = 1

5. Functional Form Tests¶

Returns to Scale¶

Test \(H_0: RTS = \sum_j \beta_j = 1\) (constant returns to scale):

rts = result.returns_to_scale_test(
    input_vars=["log_labor", "log_capital"],
    alpha=0.05,
)

print(f"RTS = {rts['rts']:.4f} +/- {rts['rts_se']:.4f}")
print(f"Wald statistic: {rts['test_statistic']:.4f}")
print(f"P-value: {rts['pvalue']:.4f}")
print(f"Conclusion: {rts['conclusion']}")  # 'CRS', 'IRS', or 'DRS'
print(rts['interpretation'])

Elasticities¶

# Cobb-Douglas: constant elasticities = coefficients
elas = result.elasticities(input_vars=["log_labor", "log_capital"])
print(elas)

# Translog: varying elasticities
elas_tl = result.elasticities(translog=True, translog_vars=["ln_K", "ln_L"])
print(elas_tl.describe())

Cobb-Douglas vs Translog¶

comparison = cd_result.compare_functional_form(translog_result, alpha=0.05)

print(f"LR statistic: {comparison['lr_statistic']:.4f}")
print(f"df: {comparison['df']}")
print(f"P-value: {comparison['pvalue']:.4f}")
print(f"Preferred: {comparison['conclusion']}")

6. Panel Model Selection¶

Hausman Test: TFE vs TRE¶

from panelbox.frontier.tests import hausman_test_tfe_tre

hausman = hausman_test_tfe_tre(
    params_tfe=result_tfe.params.values,
    params_tre=result_tre.params.values,
    vcov_tfe=result_tfe.vcov,
    vcov_tre=result_tre.vcov,
    param_names=result_tfe.params.index.tolist(),
)

print(f"Hausman statistic: {hausman['statistic']:.4f}")
print(f"P-value: {hausman['pvalue']:.4f}")
print(f"Recommendation: {hausman['conclusion']}")

LR Test: Nested Models¶

from panelbox.frontier.tests import lr_test

# Test if time-varying is better than time-invariant
lr = lr_test(
    loglik_restricted=result_pitt_lee.loglik,
    loglik_unrestricted=result_bc92.loglik,
    df_diff=1,  # One additional parameter (eta)
)
print(f"LR = {lr['statistic']:.4f}, p = {lr['pvalue']:.4f}")

Wald Test¶

from panelbox.frontier.tests import wald_test
import numpy as np

# Test if beta_1 = beta_2 = 0
R = np.array([[0, 1, 0, 0], [0, 0, 1, 0]])
w = wald_test(params=result.params.values, vcov=result.vcov, R=R)
print(f"Wald = {w['statistic']:.4f}, p = {w['pvalue']:.4f}")

Heterogeneity Significance (TRE)¶

from panelbox.frontier.tests import heterogeneity_significance_test

het = heterogeneity_significance_test(
    sigma_w_sq=result_tre.sigma_w_sq,
    se_sigma_w_sq=result_tre.se["sigma_w_sq"],
)
print(f"z-statistic: {het['statistic']:.4f}")
print(f"P-value: {het['pvalue']:.4f}")
print(f"Conclusion: {het['conclusion']}")

Complete Diagnostic Workflow¶

from panelbox.frontier import StochasticFrontier

# 1. Estimate base model
sf = StochasticFrontier(
    data=df,
    depvar="log_output",
    exog=["log_capital", "log_labor"],
    frontier="production",
    dist="half_normal",
)
result = sf.fit()

# 2. Full summary with diagnostics
print(result.summary(include_diagnostics=True))

# 3. Compare distributions
dist_comp = result.compare_distributions(
    distributions=["half_normal", "exponential", "truncated_normal"]
)
print("\nDistribution Comparison:")
print(dist_comp)

# 4. Variance decomposition
var_decomp = result.variance_decomposition(method="delta")
print(f"\nGamma: {var_decomp['gamma']:.4f}")
print(var_decomp['interpretation'])

# 5. Returns to scale
rts = result.returns_to_scale_test(input_vars=["log_capital", "log_labor"])
print(f"\nRTS = {rts['rts']:.4f} ({rts['conclusion']})")

# 6. Efficiency estimates
eff = result.efficiency(estimator="bc")
print(f"\nMean efficiency: {eff['efficiency'].mean():.4f}")

Available Test Functions¶

Function	Test	H0
`inefficiency_presence_test`	Mixed chi-square LR	\(\sigma_u^2 = 0\)
`skewness_test`	Residual skewness diagnostic	Correct sign for frontier type
`lr_test`	Likelihood ratio	Restricted model adequate
`wald_test`	Wald test for linear restrictions	\(R\theta = r\)
`vuong_test`	Non-nested model comparison	Models equally close to truth
`compare_nested_distributions`	Nested distribution LR	Simpler distribution adequate
`hausman_test_tfe_tre`	Hausman test for TFE vs TRE	TRE is consistent
`heterogeneity_significance_test`	\(\sigma_w^2 > 0\) test	No heterogeneity in TRE

Tutorials¶

Tutorial	Description	Link
SFA Diagnostics	Complete diagnostic workflow

References¶

Kodde, D. A., & Palm, F. C. (1986). Wald criteria for jointly testing equality and inequality restrictions. Econometrica, 1243-1248.
Coelli, T. J. (1995). Estimators and hypothesis tests for a stochastic frontier function: A Monte Carlo analysis. Journal of Productivity Analysis, 6(4), 247-268.
Vuong, Q. H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 307-333.
Hausman, J. A. (1978). Specification tests in econometrics. Econometrica, 1251-1271.
Greene, W. H. (2005). Fixed and random effects in stochastic frontier models. Journal of Productivity Analysis, 23(1), 7-32.
Kumbhakar, S. C., & Lovell, C. A. K. (2000). Stochastic Frontier Analysis. Cambridge University Press.