RESET Test

Quick Reference

Class: panelbox.validation.specification.reset.RESETTest Result: ValidationTestResult H₀: Model is correctly specified (linear functional form is appropriate) H₁: Nonlinear terms are needed (functional form misspecification) Stata equivalent: estat ovtest R equivalent: lmtest::resettest()

What It Tests

The Ramsey (1969) Regression Equation Specification Error Test (RESET) checks whether the functional form of a linear model is correct. It detects omitted nonlinearities by testing whether powers of the fitted values have explanatory power beyond the original regressors.

If the true relationship is nonlinear -- for example, quadratic, logarithmic, or involving interactions -- the fitted values from the linear model will partially capture the nonlinearity through their powers. A significant test indicates the linear specification is inadequate.

What RESET Does Not Tell You

RESET detects that a nonlinearity exists but does not identify which specific nonlinearity is missing. Rejection tells you the model is wrong, not how to fix it.

Quick Example

from panelbox.models.static.fixed_effects import FixedEffects
from panelbox.validation.specification.reset import RESETTest
from panelbox.datasets import load_grunfeld

data = load_grunfeld()

# Estimate model
fe = FixedEffects("invest ~ value + capital", data, "firm", "year")
results = fe.fit()

# Run RESET test with default powers [2, 3]
reset = RESETTest(results)
result = reset.run(alpha=0.05)
print(result.summary())

# Access results
print(f"F-statistic: {result.statistic:.4f}")
print(f"P-value: {result.pvalue:.4f}")
print(f"Degrees of freedom: {result.df}")

Interpretation

P-value	Decision	Interpretation	Action
p < 0.01	Strong rejection	Strong evidence of misspecification	Transform variables or add nonlinear terms
0.01 \(\leq\) p < 0.05	Rejection	Moderate evidence of misspecification	Consider alternative functional forms
0.05 \(\leq\) p < 0.10	Borderline	Weak evidence	Investigate further
p \(\geq\) 0.10	Fail to reject	No evidence of misspecification	Linear form appears adequate

What to Do When RESET Rejects

When the test indicates misspecification, consider these remedies:

Add PolynomialsLog TransformAdd Interactions

# Add squared terms
data["value_sq"] = data["value"] ** 2
fe = FixedEffects("invest ~ value + value_sq + capital",
                  data, "firm", "year")

import numpy as np
data["log_value"] = np.log(data["value"])
data["log_capital"] = np.log(data["capital"])
fe = FixedEffects("invest ~ log_value + log_capital",
                  data, "firm", "year")

data["value_capital"] = data["value"] * data["capital"]
fe = FixedEffects("invest ~ value + capital + value_capital",
                  data, "firm", "year")

Mathematical Details

Test Procedure

1. Estimate the original model: \(y = X\beta + \varepsilon\)
2. Compute fitted values: \(\hat{y} = X\hat{\beta}\)
3. Estimate augmented model: \(y = X\beta + \delta_2 \hat{y}^2 + \delta_3 \hat{y}^3 + u\)
4. Test: \(H_0: \delta_2 = \delta_3 = 0\) using an F-test

F-Statistic

The test uses a Wald/F-statistic for joint significance of the power terms:

\[F = \frac{W / q}{1} \sim F(q, n - k - q)\]

where:

• \(W\) = Wald statistic from joint test on \(\delta\) coefficients
• \(q\) = number of added power terms (e.g., 2 for powers [2, 3])
• \(n\) = number of observations
• \(k\) = number of original parameters

Why Powers of Fitted Values?

The fitted values \(\hat{y} = X\hat{\beta}\) are linear combinations of the regressors. Their powers \(\hat{y}^2, \hat{y}^3\) approximate general nonlinear functions of \(X\). If these powers are significant, the linear model misses some nonlinear relationship.

Configuration Options

from panelbox.validation.specification.reset import RESETTest

reset = RESETTest(results)

# Default: powers [2, 3] (quadratic and cubic)
result = reset.run(alpha=0.05)

# Only quadratic
result = reset.run(alpha=0.05, powers=[2])

# Quadratic, cubic, and quartic
result = reset.run(alpha=0.05, powers=[2, 3, 4])

Parameters

The .run() method accepts:

Parameter	Type	Default	Description
alpha	float	0.05	Significance level
powers	list[int]	[2, 3]	Powers of fitted values to include (must be \(\geq 2\))

Result Metadata

Key	Type	Description
powers	list[int]	Powers used in the test
gamma_coefficients	dict	Coefficients on power terms
standard_errors	dict	Standard errors for power terms
wald_statistic	float	Wald chi-squared statistic
F_statistic	float	F-statistic (primary test statistic)
df_numerator	int	Numerator degrees of freedom
df_denominator	int	Denominator degrees of freedom
pvalue_chi2	float	P-value from chi-squared approximation
augmented_formula	str	Augmented regression formula

Choosing Powers

The choice of powers affects test sensitivity:

Powers	Detects	Trade-off
[2]	Quadratic misspecification	Low power against cubic
[2, 3]	Quadratic and cubic (default)	Good balance
[2, 3, 4]	Higher-order nonlinearity	May lose power (too many terms)

Recommendation

Use the default powers=[2, 3] unless you have strong reasons to expect a specific form of nonlinearity. Adding too many powers reduces test power.

Complete Example

from panelbox.models.static.fixed_effects import FixedEffects
from panelbox.validation.specification.reset import RESETTest
from panelbox.datasets import load_grunfeld

data = load_grunfeld()

# Linear specification
fe = FixedEffects("invest ~ value + capital", data, "firm", "year")
results = fe.fit()

# Test with different power specifications
for powers in [[2], [2, 3], [2, 3, 4]]:
    reset = RESETTest(results)
    result = reset.run(powers=powers)
    print(f"Powers {powers}: F={result.statistic:.4f}, p={result.pvalue:.4f}")

# If RESET rejects, try log specification
import numpy as np
data["log_invest"] = np.log(data["invest"])
data["log_value"] = np.log(data["value"])
data["log_capital"] = np.log(data["capital"])

fe_log = FixedEffects("log_invest ~ log_value + log_capital",
                      data, "firm", "year")
results_log = fe_log.fit()

reset_log = RESETTest(results_log)
result_log = reset_log.run()
print(f"\nLog specification: F={result_log.statistic:.4f}, "
      f"p={result_log.pvalue:.4f}")

Common Pitfalls

Rejection Is Common

Many economic relationships are inherently nonlinear. RESET rejection does not mean the model is useless -- it means a nonlinear specification might fit better. Always consider whether the improvement in fit is economically meaningful.

Not a Test for Omitted Variables

RESET detects nonlinear misspecification, not omitted linear variables. A model that passes RESET may still suffer from omitted variable bias if a relevant linear regressor is missing.

Panel Data Considerations

PanelBox implements RESET using Pooled OLS with cluster-robust standard errors on the augmented model. This accounts for within-entity correlation when testing the significance of power terms. The test is applicable to any panel model type (FE, RE, Pooled).

Multicollinearity in Augmented Model

Powers of fitted values can be highly correlated with the original regressors. If multicollinearity is severe, the test may be unreliable. Consider using a single power (e.g., powers=[2]) instead.

See Also

• Hausman Test -- FE vs RE specification
• Chow Test -- Structural break detection
• Specification Tests Overview -- All specification tests
• Diagnostics Overview -- Complete diagnostic workflow

References

• Ramsey, J. B. (1969). Tests for specification errors in classical linear least squares regression analysis. Journal of the Royal Statistical Society, Series B, 31(2), 350-371.
• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2^nd ed.). MIT Press.
• Cameron, A. C., & Trivedi, P. K. (2005). Microeconometrics: Methods and Applications. Cambridge University Press. Chapter 8.