Robust Standard Errors (HC0-HC3)¶

Quick Reference

Class: panelbox.standard_errors.RobustStandardErrors Convenience: panelbox.standard_errors.robust_covariance() Model integration: model.fit(cov_type="robust") or model.fit(cov_type="hc1") Stata equivalent: vce(robust) (= HC1), vce(hc2), vce(hc3) R equivalent: sandwich::vcovHC(type="HC1")

Overview¶

Classical OLS standard errors assume homoskedasticity: \(\text{Var}(\varepsilon_i) = \sigma^2\) for all observations. When this assumption fails --- and it almost always does in applied work --- the resulting standard errors, t-statistics, and p-values are invalid.

Heteroskedasticity-robust (HC) standard errors, introduced by White (1980), provide valid inference regardless of the error variance structure. PanelBox implements four HC variants (HC0--HC3) with progressively better finite-sample properties.

When to Use¶

Errors have non-constant variance across observations
You suspect heteroskedasticity but don't know its form
As a baseline robustness check for any linear model
When you have a cross-sectional dataset or pooled panel

When NOT to use

Robust HC standard errors do not account for within-cluster correlation (use clustered SE) or serial correlation (use Newey-West or Driscoll-Kraay). In panel data, clustering is almost always preferred over simple HC.

Quick Example¶

from panelbox.standard_errors import RobustStandardErrors, robust_covariance

# Class-based approach
rse = RobustStandardErrors(X, resid)
result = rse.hc1()

print(f"Method: {result.method}")
print(f"Standard errors: {result.std_errors}")
print(f"n_obs: {result.n_obs}, n_params: {result.n_params}")

# Convenience function (equivalent)
result = robust_covariance(X, resid, method="HC1")
print(result.std_errors)

# Via model.fit()
from panelbox.models import PooledOLS
model = PooledOLS("y ~ x1 + x2", data, entity="firm", time="year")
results = model.fit(cov_type="robust")  # Uses HC1 by default
print(results.summary())

HC Variants¶

All four variants use the sandwich formula \(V = (X'X)^{-1} \hat{\Omega} (X'X)^{-1}\), where the meat \(\hat{\Omega}\) differs:

Type	Meat \(\hat{\Omega}\)	Correction	Best For	Reference
HC0	\(\sum \hat{e}_i^2 x_i x_i'\)	None	Large samples	White (1980)
HC1	\(\frac{n}{n-k} \sum \hat{e}_i^2 x_i x_i'\)	Degrees of freedom	General use (default)	---
HC2	\(\sum \frac{\hat{e}_i^2}{1-h_{ii}} x_i x_i'\)	Leverage-adjusted	Moderate samples	---
HC3	\(\sum \frac{\hat{e}_i^2}{(1-h_{ii})^2} x_i x_i'\)	Aggressive leverage	Small samples	MacKinnon & White (1985)

HC0: White's Original Estimator¶

The original White (1980) estimator. Consistent but can be biased downward in finite samples:

\[ \hat{\Omega}_{HC0} = \sum_{i=1}^{n} \hat{e}_i^2 x_i x_i' \]

result = rse.hc0()

HC1: Degrees-of-Freedom Correction¶

Applies a simple \(n/(n-k)\) scaling to HC0. This is the default in PanelBox and Stata's vce(robust):

\[ \hat{\Omega}_{HC1} = \frac{n}{n-k} \sum_{i=1}^{n} \hat{e}_i^2 x_i x_i' \]

result = rse.hc1()
# Or equivalently:
result = rse.compute(method="HC1")

HC2: Leverage-Adjusted¶

Divides each squared residual by \(1 - h_{ii}\), where \(h_{ii}\) is the leverage (hat value) of observation \(i\):

\[ \hat{\Omega}_{HC2} = \sum_{i=1}^{n} \frac{\hat{e}_i^2}{1 - h_{ii}} x_i x_i' \]

result = rse.hc2()
print(f"Leverage values: {result.leverage}")

HC3: MacKinnon-White¶

The most conservative variant. Squares the leverage correction, providing the best finite-sample performance:

\[ \hat{\Omega}_{HC3} = \sum_{i=1}^{n} \frac{\hat{e}_i^2}{(1 - h_{ii})^2} x_i x_i' \]

result = rse.hc3()
print(f"Leverage values: {result.leverage}")

Mathematical Details¶

Leverage (Hat Values)¶

The leverage \(h_{ii}\) is the \(i\)-th diagonal element of the hat matrix:

\[ H = X(X'X)^{-1}X' \]

Properties of leverage values:

\(0 \leq h_{ii} \leq 1\)
\(\sum_{i=1}^{n} h_{ii} = k\) (number of parameters)
Average leverage: \(\bar{h} = k/n\)
High-leverage threshold: \(h_{ii} > 2k/n\) or \(3k/n\)

Observations with high leverage have disproportionate influence on the fitted regression. HC2 and HC3 upweight these observations, compensating for the fact that their residuals are artificially small.

Why HC2/HC3 Matter¶

Under homoskedasticity, \(E(\hat{e}_i^2) = \sigma^2(1 - h_{ii})\). This means:

OLS residuals underestimate the true error variance for high-leverage points
HC0 and HC1 inherit this bias
HC2 corrects exactly: \(\hat{e}_i^2 / (1 - h_{ii})\) is unbiased for \(\sigma^2\)
HC3 over-corrects, which gives better finite-sample size in hypothesis testing

Configuration Options¶

RobustStandardErrors Class¶

Parameter	Type	Description
`X`	`np.ndarray`	Design matrix \((n \times k)\)
`resid`	`np.ndarray`	OLS residuals \((n,)\)

RobustCovarianceResult¶

Attribute	Type	Description
`cov_matrix`	`np.ndarray`	Robust covariance matrix \((k \times k)\)
`std_errors`	`np.ndarray`	Robust standard errors \((k,)\)
`method`	`str`	HC variant used (`'HC0'`, `'HC1'`, `'HC2'`, `'HC3'`)
`n_obs`	`int`	Number of observations
`n_params`	`int`	Number of parameters
`leverage`	`np.ndarray` or `None`	Leverage values (HC2, HC3 only)

Comparing All HC Variants¶

from panelbox.standard_errors import RobustStandardErrors

rse = RobustStandardErrors(X, resid)

# Compute all four variants
results = {
    "HC0": rse.hc0(),
    "HC1": rse.hc1(),
    "HC2": rse.hc2(),
    "HC3": rse.hc3(),
}

# Compare standard errors
for method, result in results.items():
    print(f"{method}: {result.std_errors}")

Which HC to choose?

HC1 is the safe default for most applications (matches Stata's robust)
HC3 is recommended for small samples (\(n < 250\)) or when leverage varies substantially
HC2 is a middle ground with exact unbiasedness under homoskedasticity
HC0 is mainly of historical interest; prefer HC1 or higher

Diagnostics¶

Detecting Heteroskedasticity¶

Before applying robust SEs, you can test for heteroskedasticity:

from panelbox.validation.heteroskedasticity.breusch_pagan import BreuschPaganTest
from panelbox.validation.heteroskedasticity.white import WhiteTest

bp_result = BreuschPaganTest(results).run(alpha=0.05)
white_result = WhiteTest(results).run(alpha=0.05)

print(f"Breusch-Pagan: p={bp_result.pvalue:.4f} -> {bp_result.conclusion}")
print(f"White test:    p={white_result.pvalue:.4f} -> {white_result.conclusion}")

Note

Even if tests do not reject homoskedasticity, robust SEs are still valid (just slightly less efficient). Many applied researchers use robust SEs by default.

References¶

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817-838.
MacKinnon, J. G., & White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29(3), 305-325.
Long, J. S., & Ervin, L. H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54(3), 217-224.