Leave-One-Out Estimator (HC2)

• Leave-One-Out Estimator (HC2) is a heteroskedasticity-robust covariance estimator that adjusts squared residuals by the factor 1/(1-h_i) to counteract bias in OLS regression.
• It specifically addresses the underestimation of error variance caused by high leverage points, thereby mitigating the shrinkage of raw residuals.
• Its extensions include customized degrees-of-freedom adjustments and adaptations to high-dimensional and clustered settings to enhance inferential accuracy.

The leave-one-out estimator commonly denoted HC2 is a heteroskedasticity-robust covariance estimator for ordinary least squares that replaces raw squared residuals by leverage-adjusted squared residuals, thereby correcting the first-order shrinkage induced by fitting the same observation that is later used in the variance estimate. In current econometric usage, HC2 is sometimes described as a leave-one-out estimator because of its leverage correction and because, in some score-variance formulations, an exact leave-one-out construction coincides with HC2; however, the exact leave-one-out prediction-error identity is algebraically associated with HC3 rather than HC2. Recent work places HC2 at the center of a broader discussion about leverage, effective sample size, and test calibration: its variance formula has strong theoretical motivation, but naive $t$-tests based on HC2 with default degrees of freedom can be substantially liberal in realistic samples, while customized degrees-of-freedom rules and related leave-out generalizations substantially improve inference (Kranz, 2024, Halkiewicz, 6 Jul 2026).

1. Linear-model definition and algebraic form

HC2 is defined in the standard linear regression model

$y = X\beta + \varepsilon,$

with independent errors $\varepsilon_i$ satisfying $\mathbb E[\varepsilon_i]=0$ and $\operatorname{Var}(\varepsilon_i)=\sigma_i^2$. The true covariance matrix of the OLS estimator is

$$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$

The HC estimators considered in the modern literature all take the sandwich form

$$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$

where $\hat\varepsilon_i$ is the OLS residual and $\alpha_i^\tau$ is an estimator-specific adjustment factor. For HC2,

$\alpha_i^{HC2}=\frac{1}{1-h_i},$

where

$y = X\beta + \varepsilon,$0

Hence

$y = X\beta + \varepsilon,$1

This is the MacKinnon–White leverage-adjusted member of the HC family (Kranz, 2024).

A compact comparison with the closely related HC estimators is useful because most inferential debates about HC2 are fundamentally comparative.

Estimator	Adjustment factor $y = X\beta + \varepsilon,$2	Brief characterization
HC0	$y = X\beta + \varepsilon,$3	Raw residual squares
HC1	$y = X\beta + \varepsilon,$4	Global df inflation
HC2	$y = X\beta + \varepsilon,$5	First-order leverage correction
HC3	$y = X\beta + \varepsilon,$6	Stronger leave-out-style correction
HC4	$y = X\beta + \varepsilon,$7	High-leverage penalty

Within this taxonomy, HC2 differs from HC0 and HC1 by using observation-specific leverage correction rather than a global rescaling, and it differs from HC3 and HC4 by applying only a first-order correction rather than a stronger nonlinear penalty for high-leverage points. The same source situates HC2 historically between HC0 and HC3: HC0 is the original Eicker–Huber–White form, HC1–HC3 were introduced by MacKinnon and White to improve small-sample behavior, and HC4 was proposed by Cribari-Neto for aggressive high-leverage adjustment (Kranz, 2024).

2. Leverage, residual shrinkage, and the meaning of “leave-one-out”

The central object in HC2 is the leverage $y = X\beta + \varepsilon,$8, with $y = X\beta + \varepsilon,$9 and $\varepsilon_i$0. Leverage measures how strongly the fitted value at observation $\varepsilon_i$1 reuses that same observation. High-leverage points therefore generate artificially small residuals. Under homoskedasticity,

$\varepsilon_i$2

so raw squared residuals systematically understate the error variance when $\varepsilon_i$3 is large. HC2 rescales by $\varepsilon_i$4, undoing this shrinkage and yielding an unbiased estimator of $\varepsilon_i$5 in the homoskedastic case (Kranz, 2024).

This leverage logic explains both the appeal and the ambiguity of the phrase “leave-one-out estimator.” In the most literal residual sense, the exact leave-one-out prediction error satisfies

$\varepsilon_i$6

so its square corresponds to HC3-style scaling, not HC2-style scaling. The modern heteroskedasticity-robust inference literature therefore emphasizes that HC3, not HC2, is exactly equivalent to jackknife or leave-one-out residuals, whereas HC2 is a milder correction motivated by the bias of $\varepsilon_i$7 under leverage (Kranz, 2024).

At the same time, there is a second, score-based sense in which HC2 is literally leave-one-out. In the scalar-regressor fixed-effect setting, the Cattaneo–Jansson–Newey leave-one-out score-variance estimator

$\varepsilon_i$8

reduces, via the Sherman–Morrison identity,

$\varepsilon_i$9

to

$\mathbb E[\varepsilon_i]=0$0

which is exactly HC2. The terminology is therefore not merely rhetorical; it depends on whether the reference point is the leave-one-out residual itself or the leave-one-out score covariance (Halkiewicz, 6 Jul 2026).

A further complication appears at the boundary $\mathbb E[\varepsilon_i]=0$1. Then $\mathbb E[\varepsilon_i]=0$2 and the HC2 term takes the indeterminate form $\mathbb E[\varepsilon_i]=0$3. In applied work this is not exceptional: one study reports that 15.6% of regressions in its sample contain at least one observation with $\mathbb E[\varepsilon_i]=0$4. The conventional remedy follows the Moore–Penrose convention and sets the corresponding HC2 contribution to zero; an alternative is to omit such observations entirely from the variance computation and from $\mathbb E[\varepsilon_i]=0$5, which can materially improve finite-sample performance for some HC2-based procedures (Kranz, 2024).

3. Degrees of freedom, partial leverage, and coefficient-specific calibration

HC2 specifies the covariance estimator, but valid inference also depends on how the resulting $\mathbb E[\varepsilon_i]=0$6-statistic is calibrated. The conventional choice is

$\mathbb E[\varepsilon_i]=0$7

with comparison to a $\mathbb E[\varepsilon_i]=0$8 reference distribution. Modern evidence shows that this default rule is often inadequate, because the sampling distribution of the statistic depends not only on residual scale correction but also on how informative the data are for the specific coefficient being tested (Kranz, 2024).

One route is the Bell–McCaffrey approach. In the specification denoted HC2-BM, the variance estimator remains HC2, but the degrees of freedom are replaced by a coefficient-specific Satterthwaite approximation $\mathbb E[\varepsilon_i]=0$9. The main text of the recent large-scale Monte Carlo study does not reproduce the full Bell–McCaffrey formula, but it presents the generic variance-ratio representation

$\operatorname{Var}(\varepsilon_i)=\sigma_i^2$0

and uses the HC2 version as a benchmark best-practice refinement (Kranz, 2024).

A second route is the paper’s partial-leverage adjustment. For coefficient $\operatorname{Var}(\varepsilon_i)=\sigma_i^2$1, define the Frisch–Waugh–Lovell residual regressor $\operatorname{Var}(\varepsilon_i)=\sigma_i^2$2 and the partial leverage

$\operatorname{Var}(\varepsilon_i)=\sigma_i^2$3

The associated effective sample size is

$\operatorname{Var}(\varepsilon_i)=\sigma_i^2$4

which lies in $\operatorname{Var}(\varepsilon_i)=\sigma_i^2$5. When partial leverage is diffuse, $\operatorname{Var}(\varepsilon_i)=\sigma_i^2$6; when it is concentrated on one or a few observations, $\operatorname{Var}(\varepsilon_i)=\sigma_i^2$7 can be close to 1. The HC2-PL procedure keeps the HC2 variance estimator but replaces the default degrees of freedom by

$\operatorname{Var}(\varepsilon_i)=\sigma_i^2$8

This construction separates two issues that are often conflated. HC2 corrects the residual scale for leverage, whereas the partial-leverage rule corrects the reference distribution when the effective number of informative observations for $\operatorname{Var}(\varepsilon_i)=\sigma_i^2$9 is far smaller than the nominal sample size (Kranz, 2024).

This distinction is especially important in designs with sparse indicators or highly localized identifying variation. The same study gives the representation

$$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$0

which makes clear that a coefficient can be driven almost entirely by a single error term when partial leverage is concentrated. In that regime, asymptotic $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$1-approximations based on $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$2 become unreliable even if the variance estimator itself is leverage-aware. This is the principal reason HC2 can be theoretically attractive yet inferentially liberal when paired with default degrees of freedom (Kranz, 2024).

4. Empirical use and finite-sample evidence

An extensive code audit of 4,420 reproduction packages from leading economics journals found 40,571 regressions that explicitly requested heteroskedasticity-robust standard errors; 98.1% used Stata’s default HC1 specification. This establishes the practical backdrop for current discussion of HC2: the method is well known in theory but rarely used in routine empirical work, and inference is still dominated by HC1 defaults (Kranz, 2024).

The same study evaluates HC2 in a Monte Carlo design built from 608 OLS regressions taken from 155 reproduction packages, restricted to regressions with $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$3. Across 3,280 coefficient-specific testing situations and $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$4 Monte Carlo replications per regression, it computes 5% rejection rates and summarizes size distortions by “excess” and “lack.” In this environment, baseline HC2 with $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$5 degrees of freedom is less liberal than HC1, but it still exhibits substantial over-rejection. Among HC1–HC4 with default degrees of freedom, the ordering of average excess is HC1 worst, then HC2, then HC4, then HC3. A notable empirical finding is that homoskedastic IID standard errors are more conservative on average than HC1 and HC2 in this sample of regressions with $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$6 (Kranz, 2024).

Customized degrees of freedom materially alter this picture. HC2-BM improves clearly on baseline HC2 and outperforms HC1–HC4 with default $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$7 calibration on average. The paper’s own HC2-PL procedure performs still better: HC2-PL and HC1-PL “outperform all others on average,” with HC2-PL best overall. Its average excess is close to the simulated ideal test, its average lack remains mildly conservative, and it nearly eliminates extreme rejection frequencies above 50%. The dominant predictor of severe distortion for conventional HC methods is small $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$8, that is, highly concentrated partial leverage (Kranz, 2024).

Maximum leverage matters independently. For HC1–HC4 and HC2-BM, the most extreme over-rejection episodes are concentrated in designs with $$\operatorname{Var}(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \sigma_i^2 x_i x_i^\top\right)(X^\top X)^{-1}.$$9. If leverage-one observations are dropped from the variance calculation rather than assigned zero contribution, the performance of traditional specifications improves sharply and many extreme failures disappear. Under that alternative handling, the ranking between HC2-BM and HC2-PL becomes less stark, and both perform well; HC2-PL is notably less sensitive to how leverage-one observations are treated (Kranz, 2024).

The same comparison includes wild bootstrap variants. In the regressions studied there, wild bootstrap methods improve on naive HC1 and HC2 with $$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$0 degrees of freedom, but they are still outperformed by HC2-BM, HC1-PL, and HC2-PL. A practical implication stated explicitly in that study is that, for many applications, effort devoted to customized degrees-of-freedom correction is more cost-effective than generic bootstrap refinement (Kranz, 2024).

5. Saturated fixed effects and high-dimensional asymptotics

Recent analysis of saturated fixed-effect designs extends the interpretation of HC2 beyond conventional small-sample OLS. In the model

$$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$1

with an expanding fixed-effect space of dimension $$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$2, the within estimator is based on $$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$3 and residualized regressor $$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$4. The corresponding HC estimators become

$$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$5

where $$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$6 is the full regression leverage including both the fixed effects and the treatment regressor (Halkiewicz, 6 Jul 2026).

The asymptotic regime tracks two nondegenerate quantities: the proportional fixed-effect dimension

$$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$7

and the residual treatment variance

$$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$8

Within this framework, three sharp results emerge. First, HC0 is biased downward by a fixed factor $$\hat V^\tau(\hat\beta)=(X^\top X)^{-1}\left(\sum_{i=1}^n \alpha_i^\tau \hat\varepsilon_i^2 x_i x_i^\top\right)(X^\top X)^{-1},$$9, producing over-rejection that worsens with saturation. Second, HC3 over-corrects in the opposite direction by factor $\hat\varepsilon_i$0. Third, HC2 removes the first-order leverage distortion and is asymptotically exact under conditional homoskedasticity or under design-balanced heteroskedasticity (Halkiewicz, 6 Jul 2026).

The paper formalizes the remaining heteroskedastic complication through two variance aggregates, $\hat\varepsilon_i$1 and $\hat\varepsilon_i$2, and an “effective” variance

$\hat\varepsilon_i$3

Under homoskedasticity, $\hat\varepsilon_i$4, so HC2 is asymptotically exact. Under general heteroskedasticity with non-uniform leverage, HC2 retains an additional bias of order

$\hat\varepsilon_i$5

This clarifies the status of HC2 in high-dimensional fixed-effect designs: it corrects the dominant leverage dilution, but if saturation is substantial and heteroskedasticity interacts with leverage imbalance, a residual bias remains (Halkiewicz, 6 Jul 2026).

The same analysis also refines the comparison between HC1 and HC2. Under asymptotically uniform leverage, HC1 and HC2 become equivalent, since the per-observation factor $\hat\varepsilon_i$6 is approximately constant. Under non-uniform leverage, however, HC2 is preferable because its correction remains observation-specific. The empirical application to Piotroski F-Score returns in Central and Eastern European markets exhibits the predicted variance hierarchy in real data: HC0 yields the smallest standard errors, HC3 the largest, and HC2 occupies the theoretically expected middle position, with HC1 often close to HC2 in balanced designs (Halkiewicz, 6 Jul 2026).

6. Related leave-out estimators and broader extensions

HC2 belongs to a larger family of leave-out estimators that target the same leverage problem at different levels of complexity. A key comparison is the leave-one-out individual variance estimator used in work on testing many restrictions under heteroskedasticity: $\hat\varepsilon_i$7 This estimator is explicitly constructed to satisfy

$\hat\varepsilon_i$8

whereas HC2 uses $\hat\varepsilon_i$9 and is not an exactly unbiased estimator of the individual variance $\alpha_i^\tau$0. In that literature, leave-one-out centering is paired with leave-three-out scaling to obtain valid inference for quadratic-form test statistics when the number of tested restrictions is large (Anatolyev et al., 2020).

Cluster dependence leads to an analogous generalization. The leave-cluster-out crossfit estimator

$\alpha_i^\tau$1

is unbiased, consistent, and robust to cluster dependence under high-dimensional asymptotics. When clusters are singletons, this construction nests the leave-one-out crossfit estimator of Kline, Saggio, and Sølvsten, thereby extending leave-one-out logic from heteroskedasticity-only settings to clustered data (Fai, 2022).

A further extension concerns clustered quadratic forms. There, the leave-one-cluster-out estimator debiases plug-in quadratic-form estimators, and consistent variance estimation requires either leave-three-cluster-out correction or a conservative leave-two-cluster-out alternative. The algebra is the cluster analogue of the HC2 idea: the correction depends on blockwise inverses $\alpha_i^\tau$2, which play the role of cluster-level leverage factors just as $\alpha_i^\tau$3 does in ordinary HC2 (Kolesár et al., 14 Feb 2026).

Taken together, these developments clarify the place of HC2. It is the canonical first-order leverage-corrected member of the heteroskedasticity-robust sandwich family. Its theoretical appeal comes from unbiasedness under homoskedasticity and from a clear correction of residual shrinkage. Its limitations arise when finite-sample inference is governed by concentrated leverage or partial leverage rather than by variance bias alone. And its modern descendants—leave-one-out, leave-cluster-out, and higher-order leave-out estimators—show that the underlying principle is broader than the original covariance formula: leverage correction is best understood as a family of debiasing devices whose simplest classical form is HC2 (Kranz, 2024, Halkiewicz, 6 Jul 2026).