Cox Test & Encompassing Tests¶

Quick Reference

Functions: panelbox.diagnostics.specification.encompassing.cox_test() panelbox.diagnostics.specification.encompassing.wald_encompassing_test() panelbox.diagnostics.specification.encompassing.likelihood_ratio_test() Result: EncompassingResult H₀ (Cox): Both models fit equally well H₀ (Wald/LR): Restricted model is adequate (restrictions valid) Stata equivalent: lrtest (for LR test) R equivalent: lmtest::encomptest(), lmtest::lrtest()

What They Test¶

This module provides three tests for comparing model adequacy:

Test	Models	Compares	Based On
Cox test	Non-nested	Whether models fit equally well	Log-likelihood difference
Wald encompassing	Nested	Whether additional parameters are needed	RSS or likelihood
Likelihood ratio	Nested	Whether restrictions are valid	Log-likelihood ratio

The Encompassing Principle¶

A model encompasses another if it can explain everything the other model explains plus more. If Model 1 encompasses Model 2, there is no reason to prefer Model 2.

Cox Test¶

What It Tests¶

The Cox test (1961) compares two non-nested models using their log-likelihoods. Unlike the J-test, which uses fitted values in augmented regressions, the Cox test directly compares how well each model predicts the data in a likelihood framework.

Quick Example¶

from panelbox.models.static.pooled_ols import PooledOLS
from panelbox.diagnostics.specification.encompassing import cox_test
from panelbox.datasets import load_grunfeld

data = load_grunfeld()

# Two competing specifications
model1 = PooledOLS("invest ~ value", data, "firm", "year")
result1 = model1.fit()

model2 = PooledOLS("invest ~ capital", data, "firm", "year")
result2 = model2.fit()

# Cox test
cox_result = cox_test(
    result1, result2,
    model1_name="Value Model",
    model2_name="Capital Model"
)

print(cox_result.interpretation())

Mathematical Details¶

The Cox test statistic is:

\[T_{Cox} = \frac{\ell_1 - \ell_2}{\text{SE}(\ell_1 - \ell_2)}\]

where \(\ell_1\) and \(\ell_2\) are the maximized log-likelihoods of Models 1 and 2.

Under H₀ (both models fit equally well), \(T_{Cox} \sim N(0, 1)\). A two-tailed p-value is computed.

The standard error is estimated from the variance of the log-likelihood difference, using the squared residual ratio:

\[\text{Var} = \text{Var}\left[\log(e_{1t}^2) - \log(e_{2t}^2)\right] / n\]

Interpretation¶

P-value	Decision	Interpretation
p < 0.05	Reject H₀	Models fit significantly differently
p \(\geq\) 0.05	Fail to reject	No significant difference in fit

When rejected, the sign of the statistic indicates which model fits better:

\(T_{Cox} > 0\): Model 1 has higher log-likelihood (better fit)
\(T_{Cox} < 0\): Model 2 has higher log-likelihood (better fit)

Parameters¶

cox_test(result1, result2, model1_name=None, model2_name=None)

Parameter	Type	Default	Description
`result1`	fitted result	required	First model (must have `.llf`, `.resid`)
`result2`	fitted result	required	Second model
`model1_name`	`str` or `None`	`"Model 1"`	Display name for Model 1
`model2_name`	`str` or `None`	`"Model 2"`	Display name for Model 2

Cox Test Metadata¶

Key	Type	Description
`llf1`	`float`	Log-likelihood of Model 1
`llf2`	`float`	Log-likelihood of Model 2
`llf_diff`	`float`	Difference in log-likelihoods
`se_diff`	`float`	Standard error of the difference

Wald Encompassing Test¶

What It Tests¶

The Wald encompassing test compares nested models by testing whether the additional parameters in the unrestricted model are jointly significant. It answers: does the fuller model significantly improve upon the restricted one?

Quick Example¶

from panelbox.diagnostics.specification.encompassing import wald_encompassing_test

# Restricted model: fewer variables
model_r = PooledOLS("invest ~ value", data, "firm", "year")
result_r = model_r.fit()

# Unrestricted model: more variables
model_u = PooledOLS("invest ~ value + capital", data, "firm", "year")
result_u = model_u.fit()

# Wald encompassing test
wald_result = wald_encompassing_test(
    result_r, result_u,
    model_restricted_name="Value Only",
    model_unrestricted_name="Value + Capital"
)

print(wald_result.interpretation())

Mathematical Details¶

The Wald statistic is based on the improvement in residual sum of squares:

\[W = \frac{(RSS_r - RSS_u) / \hat{\sigma}_u^2}{1}\]

where \(\hat{\sigma}_u^2 = RSS_u / (n - k_u)\) is the unrestricted variance estimate.

Under H₀ (restrictions valid), \(W \sim \chi^2(q)\) where \(q = k_u - k_r\) is the number of restrictions.

When RSS is not available but log-likelihoods are, the test uses the likelihood-ratio approach: \(W = 2(\ell_u - \ell_r)\).

Parameters¶

wald_encompassing_test(
    result_restricted, result_unrestricted,
    model_restricted_name=None, model_unrestricted_name=None
)

Parameter	Type	Default	Description
`result_restricted`	fitted result	required	Restricted (nested) model
`result_unrestricted`	fitted result	required	Unrestricted (full) model
`model_restricted_name`	`str` or `None`	`"Restricted Model"`	Display name
`model_unrestricted_name`	`str` or `None`	`"Unrestricted Model"`	Display name

Model Ordering

The unrestricted model must have more parameters than the restricted model. PanelBox raises a ValueError if this condition is not met.

Likelihood Ratio Test¶

What It Tests¶

The likelihood ratio (LR) test compares nested models estimated by maximum likelihood. It tests whether the restricted model's constraints are supported by the data.

Quick Example¶

from panelbox.diagnostics.specification.encompassing import likelihood_ratio_test

# Restricted and unrestricted models (must have .llf attribute)
lr_result = likelihood_ratio_test(
    result_r, result_u,
    model_restricted_name="Base Model",
    model_unrestricted_name="Full Model"
)

print(lr_result.interpretation())
print(f"LR statistic: {lr_result.statistic:.4f}")
print(f"P-value: {lr_result.pvalue:.4f}")
print(f"Degrees of freedom: {lr_result.df}")

Mathematical Details¶

The LR statistic is:

\[LR = -2(\ell_r - \ell_u) = 2(\ell_u - \ell_r)\]

Under H₀ (restrictions valid), \(LR \sim \chi^2(q)\) where \(q\) is the number of restrictions (difference in parameter counts).

Parameters¶

likelihood_ratio_test(
    result_restricted, result_unrestricted,
    model_restricted_name=None, model_unrestricted_name=None
)

Parameter	Type	Default	Description
`result_restricted`	fitted result	required	Restricted model (must have `.llf`)
`result_unrestricted`	fitted result	required	Unrestricted model (must have `.llf`)
`model_restricted_name`	`str` or `None`	`"Restricted Model"`	Display name
`model_unrestricted_name`	`str` or `None`	`"Unrestricted Model"`	Display name

LR Test Metadata¶

Key	Type	Description
`llf_restricted`	`float`	Log-likelihood of restricted model
`llf_unrestricted`	`float`	Log-likelihood of unrestricted model
`llf_diff`	`float`	Difference (\(\ell_u - \ell_r\))
`k_restricted`	`int`	Parameters in restricted model
`k_unrestricted`	`int`	Parameters in unrestricted model
`num_restrictions`	`int`	Number of restrictions (\(q\))

EncompassingResult Interface¶

All three tests return EncompassingResult objects:

result.test_name          # str   -- Name of the test
result.statistic          # float -- Test statistic
result.pvalue             # float -- P-value
result.df                 # float or None -- Degrees of freedom
result.null_hypothesis    # str   -- H₀ description
result.alternative        # str   -- H₁ description
result.model1_name        # str   -- Name of first model
result.model2_name        # str   -- Name of second model
result.additional_info    # dict  -- Test-specific metadata

# Methods
result.interpretation()   # str   -- Human-readable interpretation
result.summary()          # pd.DataFrame -- Summary table

Choosing the Right Test¶

Situation	Test	Reason
Non-nested models, have residuals	J-Test	Most common for regression models
Non-nested models, have log-likelihoods	Cox test	Likelihood-based comparison
Nested models, OLS	Wald encompassing or F-test	Tests joint significance of extra variables
Nested models, MLE	Likelihood ratio	Standard for likelihood-based models

Complete Example: Comprehensive Model Comparison¶

from panelbox.models.static.pooled_ols import PooledOLS
from panelbox.diagnostics.specification.davidson_mackinnon import j_test
from panelbox.diagnostics.specification.encompassing import (
    cox_test, wald_encompassing_test, likelihood_ratio_test
)
from panelbox.datasets import load_grunfeld

data = load_grunfeld()

# Three specifications
model_a = PooledOLS("invest ~ value", data, "firm", "year")
result_a = model_a.fit()

model_b = PooledOLS("invest ~ capital", data, "firm", "year")
result_b = model_b.fit()

model_c = PooledOLS("invest ~ value + capital", data, "firm", "year")
result_c = model_c.fit()

# --- Non-nested comparison: Model A vs Model B ---
print("=== Non-Nested: Value vs Capital ===")

# J-test
jtest = j_test(result_a, result_b,
               model1_name="Value", model2_name="Capital")
print(jtest.interpretation())

# Cox test
cox = cox_test(result_a, result_b,
               model1_name="Value", model2_name="Capital")
print(cox.interpretation())

# --- Nested comparison: Model A vs Model C ---
print("\n=== Nested: Value vs Value+Capital ===")

# Wald encompassing
wald = wald_encompassing_test(
    result_a, result_c,
    model_restricted_name="Value Only",
    model_unrestricted_name="Value + Capital"
)
print(wald.interpretation())

# Likelihood ratio
lr = likelihood_ratio_test(
    result_a, result_c,
    model_restricted_name="Value Only",
    model_unrestricted_name="Value + Capital"
)
print(lr.interpretation())

Common Pitfalls¶

Cox Test Requires Log-Likelihood¶

The Cox test requires .llf (log-likelihood) on both model results. If your model does not compute a log-likelihood (e.g., some robust estimators), use the J-test instead.

LR Test Requires Nested Models¶

The likelihood ratio test is only valid for nested models estimated by maximum likelihood on the same sample. Using it with non-nested models or models estimated on different data produces meaningless results.

Unrestricted Must Have More Parameters¶

For Wald and LR tests, the unrestricted model must have strictly more parameters. If the models have the same number of parameters, they are not nested and you should use the Cox or J-test instead.

Multiple Testing¶

When comparing multiple model pairs, p-values should be adjusted for multiple comparisons (e.g., Bonferroni correction). Running many pairwise tests without correction inflates the overall false positive rate.

Difference from J-Test¶

The Cox test and J-test both handle non-nested models but use different approaches:

Feature	J-Test	Cox Test
Approach	Augmented regressions	Log-likelihood comparison
Statistic	t-statistic	z-statistic (Normal)
Two-directional	Yes (forward + reverse)	One comparison
Requires	Fitted values + regressors	Log-likelihood
Interpretation	Which model encompasses	Which model fits better

References¶

Cox, D. R. (1961). Tests of separate families of hypotheses. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1, 105-123.
Mizon, G. E., & Richard, J. F. (1986). The encompassing principle and its application to testing non-nested hypotheses. Econometrica, 54(3), 657-678.
Vuong, Q. H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57(2), 307-333.
Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2^nd ed.). MIT Press. Chapter 18.
Cameron, A. C., & Trivedi, P. K. (2005). Microeconometrics: Methods and Applications. Cambridge University Press. Chapter 8.

Cox Test & Encompassing Tests¶

What They Test¶

The Encompassing Principle¶

Cox Test¶

What It Tests¶

Quick Example¶

Mathematical Details¶

Interpretation¶

Parameters¶

Cox Test Metadata¶

Wald Encompassing Test¶

What It Tests¶

Quick Example¶

Mathematical Details¶

Parameters¶

Likelihood Ratio Test¶

What It Tests¶

Quick Example¶

Mathematical Details¶

Parameters¶

LR Test Metadata¶

EncompassingResult Interface¶

Choosing the Right Test¶

Complete Example: Comprehensive Model Comparison¶

Common Pitfalls¶

Cox Test Requires Log-Likelihood¶

LR Test Requires Nested Models¶

Unrestricted Must Have More Parameters¶

Multiple Testing¶

Difference from J-Test¶

See Also¶

References¶