1. Reliable PLI
The interquartile range is not an estimator. A benchmark set of n comparables produces a distribution of ratios m = Y/X: operating margin, markup, return on assets. If the underlying relation is Y = a + bX, then
(1)\quad
m(i)=\frac{Y(i)}{X(i)}=b+\frac{a}{X(i)}and the ratio carries a bias that does not vanish as n grows. Its mean bias is a/H, where H is the harmonic mean of X. At the p-th quantile the bias is a/Q(1-p) of X, because the quantiles of the ratio invert the rank order of X, so the interquartile range is distorted by
(2)\quad
a\left[
\frac{1}{X(0.25)}
-
\frac{1}{X(0.75)}
\right]The arm’s length range then measures company size rather than profitability.
No covariance estimator repairs this. MSE = Bias^2 + Variance, and the two terms are separate. If the answer to an audit challenge is a more conservative standard error, the second term has been corrected and the first left untouched.
What HC3 is actually for. Estimate the reduced-form markup equation on the pooled panel of n comparables over 3 years:
(3)\quad REVT=L_0+L_1XOPR+e, \qquad XOPR=COGS+XSGA
and report the slope with its standard uncertainty, L1 +/- SE(L1). The open question is which standard error. Under heteroskedasticity the sandwich
(4)\quad
V(L)
=
(X'X)^{-1}M(X'X)^{-1},
\qquad
M=X'WX,
\qquad
W=\operatorname{diag}(w_1,\ldots,w_N)is common to every candidate; they differ only in the diagonal element w(i), where h(i) = x(i)’ (X’X)^-1 x(i) is the leverage of observation i
(5)\quad
\begin{array}{l}
\mathrm{OLS}:\quad w(i)=s^2 \\[4pt]
\mathrm{HC0}:\quad w(i)=e(i)^2 \\[4pt]
\mathrm{HC1}:\quad w(i)=\frac{N}{N-k}\,e(i)^2 \\[4pt]
\mathrm{HC2}:\quad w(i)=\frac{e(i)^2}{1-h(i)} \\[4pt]
\mathrm{HC3}:\quad w(i)=\frac{e(i)^2}{\left(1-h(i)\right)^2}
\end{array}Under homoskedasticity E[e(i)^2] = (1 - h(i)) * \sigma^2. HC0 therefore understates each observation’s variance contribution by exactly its leverage. In a benchmark set of n = 10 firms (N = 30, k = 2, mean leverage 0.067) one large comparable can carry h(i) near 0.40: HC0 discounts
it by 40 percent, while HC3 inflates it by 1/(1 - h(i))^2, approximately 2.78. That is the entire small-sample argument.
It is also not an exotic choice. MacKinnon and White (1985) showed that HC3 approximates the jackknife estimator; Long and Ervin (2000) recommend it as the default whenever N is below 250. It is in the standard textbooks.
One qualification, stated up front. HC3 treats the N = 3n observations as independent draws. Three annual observations on the same comparable are not. Positive within-firm correlation means SE(L1) as reported is a lower bound, and the interval is narrower than the data warrant. Cluster-robust covariance is the textbook remedy, but at fewer than 30 clusters its own asymptotics are worse than the defect it corrects. The workable answer is arithmetic rather than rhetorical: compute the firm-clustered standard error alongside HC3 and disclose both. Where they are close, the objection is answered. Where they are not, that is information the report should contain.
The regulatory point. OECD Transfer Pricing Guidelines (2022), paragraph 3.57, permits statistical narrowing only where the range contains a sizeable number of observations. Five to fifteen comparables is not a sizeable number. The interquartile range is routinely applied at sample sizes the Guidelines do not contemplate, to a statistic that is biased by construction.
Report the slope and its standard error. That is a claim someone can test.
2. Standard-error correction matrix
LinkedIn does not render tables in body text. This belongs in the accompanying graphic or as the first carousel card.
| Estimator | w(i) in M = X’WX | Small-sample behaviour (N = 3n, n < 30) | Cross-examination exposure |
|---|---|---|---|
| OLS | s^2 | Valid only under constant error variance; corporate financials scale with firm size | The assumption is stated, never tested |
| HC0 (White) | e(i)^2 | Downward biased by the factor (1 – h(i)); asymptotic justification only | Overstates precision, and the bias is computable |
| HC1 | [N/(N-k)] e(i)^2 | Degrees-of-freedom scaling only; does not address leverage | Cosmetic correction |
| HC2 | e(i)^2 / (1-h(i)) | Unbiased under homoskedasticity | Defensible, but less conservative than HC3 |
| HC3 | e(i)^2 / (1-h(i))^2 | Approximates the jackknife; recommended for N below 250 | Selected: conservative direction, standard citation trail |
| HC4 | e(i)^2 / (1-h(i))^d, d=min{, N h(i)/k} | Designed for extreme leverage | Available if one comparable dominates; harder to explain |
3. References
Heteroskedasticity-consistent covariance
White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48(4), 817-838. Origin of the sandwich estimator; HC0. The result is asymptotic, which is precisely the objection at n below 30.
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, HC3; shows HC3 approximates the jackknife and dominates in Monte Carlo at small N. This citation carries the argument.
Chesher, A., and Jewitt, I. (1987). The bias of a heteroskedasticity consistent covariance matrix estimator. Econometrica 55(5), 1217-1222. Quantifies the downward bias of HC0 as a function of leverage. Use when an opposing expert asserts that HC0 is sufficient.
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 practitioner recommendation: HC3 for N below 250. Non-technical enough to hand to counsel.
Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics and Data Analysis 45(2), 215-233. HC4 for high-leverage designs. Cite to show the alternative was considered and set aside, not overlooked.
Textbook
Davidson, R., and MacKinnon, J. G. (1993). Estimation and Inference in Econometrics. Oxford University Press. Treats the HCCME as standard practice, which answers the objection that regression is not used in transfer pricing at its root.
Regulatory
OECD (2022). Transfer Pricing Guidelines for Multinational Enterprises and Tax Administrations. Paris: OECD Publishing. Paragraph 3.57 conditions statistical narrowing on a sizeable number of observations; paragraph 3.62 addresses measures of central tendency.
Treas. Reg. section 1.482-1(e)(2)(iii)(C). The interquartile range provision.
Page ranges in the five econometrics entries should be verified against the journals before publication.
Named the mechanism. Squaring the leverage factor in the denominator describes the formula without saying why. E[e(i)^2] = (1 - h(i)) \sigma^2 is the reason, and it fits in one sentence.
Notation. Lowercase n for the count of comparables, capital N = 3n for the sample size, k = 2 parameters. Compustat mnemonics in majuscules. The middle matrix is M; the colloquial name for it is not used.