Ordinary least squares (OLS) produce two things: an estimate of the slope, and an estimate of how precisely that slope is known. The first is sound, based on matrix algebra inversion. The second (standard error) rests on an assumption that is seldom true of benchmark comparables — that every comparable’s deviation from the fitted line is drawn from a distribution of the same width. In practice, larger comparables deviate by larger absolute amounts. When that assumption fails, the reported standard error is unreliable, and with it the interval placed around the estimate.
HC3 repairs the standard error without touching the slope estimate. The point estimate is unchanged; only the stated uncertainty is corrected. The standard error correction takes the form:
(11)\quad \operatorname{Var}(\hat{\beta})
=
(X'X)^{-1}M(X'X)^{-1}\\
M
=
\sum_i
\left[
\frac{e_i^2}{(1-h_i)^2}
\right]
X_iX_i'where eᵢ is the residual for comparable i and hᵢ is its leverage — the extent to which that observation, by virtue of its position on the X-axis, pulls the fitted line toward itself. M is the middle matrix of the sandwich.
Why HC3 rather than HC0, HC1, or HC2
Four variants of the middle matrix are in circulation. HC0 uses eᵢ² unmodified. HC1 applies a degrees-of-freedom scale. HC2 divides by (1 − hᵢ). HC3 divides by (1 − hᵢ)², and is the only one of the four corrections that behaves well at the small sample sizes prevailing in transfer pricing: benchmark sets of five to fifteen comparables, N under thirty. At those sizes, the unmodified residuals are systematically too small at high-leverage points, and HC0 and HC1 understate uncertainty as a result. HC3 corrects for this and errs toward the wider interval.
The practical consequence for a corporate tested party is that the reported interval is not narrowed by an untested assumption about the dispersion of the comparables. The estimator is a documented, standard choice, not an arbitrary one, and it is the conservative choice among the available alternatives — which is the relevant property when the analysis is examined by an adverse party.
Application within EdgarStat
The estimator is applied to the reduced-form operating profit markup equation:
\mathrm{REVT}
=
\lambda_0+\lambda_1\mathrm{XOPR},
\qquad
\mathrm{XOPR}
=
\mathrm{COGS}+\mathrm{XSGA}estimated across the benchmark set, each comparable contributing three years of reported data. λ₁ is the markup on total operating expense; λ₀ is the intercept that ratio-based profit indicators assume to be zero without testing it. The estimated markup is reported as:
\hat{\lambda}_1
\pm
SE(\hat{\lambda}_1)and it is that standard error — not the coefficient — that HC3 corrects.
The correction is not decorative in this regtession specification of the selected PLI. Unequal variance is structural here rather than incidental: a comparable with fifty million dollars of operating expense and one with five billion cannot produce residuals of similar magnitude, and the large comparable carries high leverage, sitting far out on the X-axis where it exerts disproportionate pull on the fitted line. These are the observations whose residuals ordinary least squares understate most severely, and the division by (1 − hᵢ)² is aimed at correcting them. Using conventional (OLS) standard errors on this equation would report a confidence interval narrowed by an assumption the data contradict — a defect an examining agent is entitled to find, and one that is avoidable at no cost to the point estimate.
Essential references
White, H. (1980). “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.” Econometrica 48(4), 817–838. The origin of the method. Establishes that valid standard errors can be obtained without specifying the form of the unequal variances — only that they exist. This is why the estimator survives cross-examination: it does not require the analyst to model something unobservable.
MacKinnon, J. G., and White, H. (1985). “Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties.” Journal of Econometrics 29(3), 305–325. Introduces HC1, HC2, and HC3, and shows by simulation that White’s original estimator is unreliable in small samples. The source of HC3 itself.
Long, J. S., and Ervin, L. H. (2000). “Using Heteroscedasticity Consistent Standard Errors in the Linear Regression Model.” The American Statistician 54(3), 217–224. The applied recommendation: HC3 should be the default whenever N is below roughly 250. Transfer pricing benchmark sets sit an order of magnitude below that threshold, which places this specification squarely inside the recommendation rather than at its margin.
Greene, W. H. Econometric Analysis. Chapter on the classical linear regression model, section on robust estimation of the covariance matrix. Included to establish that HC3 standard error is textbook material rather than a specialist construction. Section and page vary by edition and should be verified against the edition cited.