Multiple Year Data and the Law of Large Numbers

1. What the regulation says

Treas. Reg. § 1.482-1(f)(2)(iii) prescribes no count of years. Paragraph (A) admits data for “one or more years before or after the year under review.” Paragraph (B) ties the extent of multiple year data to the method and the issue: completeness of data for the year under review, the business cycle of the industry, and the life cycle of the product or intangible. Paragraph (B) then makes multiple year data mandatory, not optional, for the comparable profits method of § 1.482-5 and § 1.482-9(f). Paragraph (C) permits data from other years to test whether the economic conditions behind the tested party’s results had a comparable effect over a comparable period on the uncontrolled comparables. Paragraph (D) authorizes a range built from multiple year averages whenever averaging would reduce short-term variation unrelated to transfer pricing.

The operative words are before or after, business cycles, and reduce the effect of short-term variations. Each is a statistical criterion. None is a calendar rule. The regulation’s own examples use three years (Example 4 of (f)(2)(iii)(E), Example 2 of § 1.482-5(e)) and five years (Example 3 of (f)(2)(iii)(E)); these are illustrations chosen for arithmetic brevity, not prescriptions.

2. The two-year convention

Transfer pricing practice has converted a permission into a ceiling. The boilerplate benchmark takes the taxable year under review plus the two preceding years, computes a three-year average (or a weighted average) of a ratio profit indicator for each comparable, and reports the interquartile range of those averages. Three years is treated as the multiple year period, and two years before the audit year is treated as the horizon of admissible evidence.

The three-year count did have a statistical origin, and I can speak to it, since I proposed the multiple year provision in 1992. Three years was calibrated to the inventory cycle: the Kitchin cycle of about 36 to 40 months (Kitchin 1923; Schumpeter 1939). A three-year average of a comparable’s operating results spans one inventory cycle and cancels the swing in gross and operating margins that inventory accumulation and liquidation produce. Three years was the minimum period over which that one source of short-term variation could be expected to average out. It was never a ceiling, and it was never designed to control the fixed-investment cycle, or the product life cycle, each of which runs longer.

The anchoring of the window to the two years before the audit year has a different and purely administrative origin. Contemporaneous documentation under § 1.6662-6 is prepared before the return is filed, when the comparables’ data for the audit year are incomplete and the following years do not exist. The two prior years are what the preparer has on hand. The convention then migrated from documentation into audit, appeals and litigation, where it has no such excuse: an examination opens two to four years after filing, and by then the “after” years that paragraph (A) admits sit in the SEC database beside the “before” years.

Three features mark the convention as contrived. First, the count is fixed in advance, independent of the industry, the product life cycle, and the data available, which are the three criteria paragraph (B) names. Second, the window is anchored to the filing calendar, not to the business cycle the regulation invokes. Third, the convention truncates the same panel in both dimensions at once: a short T (years) multiplied by a small n (comparables, often 5 to 15 after the screening) yields a sample size N = nT of 15 to 45 company-years, from which a 25th and a 75th percentile are then asserted as the arm’s length range.

3. The law of large numbers

Paragraph (D) states the purpose of multiple year averages: to reduce the effect of short-term variation unrelated to transfer pricing. That purpose has a name in statistics. It is the law of large numbers, first proved by James Bernoulli in Ars Conjectandi (1713) and given its modern inequality form by Chebyshev (Hald 1990; Davidson and MacKinnon 2004).

Write the annual profit indicator of comparable i in year t as the sum of a persistent component and a transitory disturbance:

\quad
(1)\qquad
y(i,t)=\mu(i)+\varepsilon(i,t),
\qquad
E[\varepsilon(i,t)]=0,
\qquad
\operatorname{Var}[\varepsilon(i,t)]=\sigma^2 .

The persistent component \mu(i) is what the comparable contributes to the arm’s length range. The disturbance \varepsilon(i,t) is the short-term variation the regulation wants removed. The T-year average A(i) of comparable i is an estimator of \mu(i), and under independent disturbances its variance is \sigma^2/T:

\quad
(2)\qquad
A(i)=\frac{1}{T}\sum_{t=1}^{T}y(i,t),
\qquad
\operatorname{Var}[A(i)]=\frac{\sigma^2}{T} .

Chebyshev’s inequality bounds the probability that the average misses \mu(i) by more than any tolerance \epsilon:

\quad
(3)\qquad
P\bigl(\,|A(i)-\mu(i)|>\epsilon\,\bigr)\le\frac{\sigma^2}{T\epsilon^2} .

The bound falls with T and truncating T at three arrests the convergence at its first steps. The standard error of a three-year average is 0.577\sigma; a five-year average, 0.447\sigma; a ten-year average, 0.316\sigma. The two-year convention discards two-thirds of the variance reduction that ten years of readily available SEC data would deliver.

Annual profit indicators are serially correlated, which makes the convention worse, not better. Under first-order autocorrelation \rho, the variance of the T-year average is (Appendix A):

\quad
(4)\qquad
\operatorname{Var}[A(i)]=\frac{\sigma^2}{T}\left[\,1+2\sum_{k=1}^{T-1}\Bigl(1-\frac{k}{T}\Bigr)\rho^{k}\right].

At \rho = 0.5, three years of autocorrelated data carry the information of 1.6 independent observations, and the standard error of the three-year average is 0.782\sigma rather than 0.577\sigma. The convention does not deliver even the three independent draws it appears to promise.

The consequence for the range follows. Each comparable’s average is its persistent indicator plus estimation noise, and the dispersion of the averages across the n comparables is the sum of the two variances:

\quad
(5)\qquad
\operatorname{Var}_i[A(i)]=\tau^2+\frac{\sigma^2}{T},

where \tau^2 is the structural dispersion of \mu(i) across comparables. The interquartile range of three-year averages measures 1.349 times the square root of this sum (under normality). It is inflated by \sigma^2/3 above the dispersion the regulation means to capture. A longer T shrinks the noise term toward zero and leaves the structural dispersion in place. A short T widens the range by construction, which explains its appeal to those who prefer ranges wide enough to make profit shifting difficult to detect.

4. Business cycles and the arithmetic of three years

Paragraph (B) names the business cycle as a reason to consider multiple year data. The inventory cycle is the shortest of the cycles that affect operating margins, and three years covers it. The others it does not cover. The NBER reference cycles since 1945 average about six years from peak to peak, with expansions of five years on average and recessions of under one year; the fixed-investment (Juglar) cycle that drives capital-intensive industries runs seven to eleven years. A window of the audit year and the two preceding years will, in most cases, fall inside a single expansion; it will, in some cases, straddle a recession and nothing else. In neither case does the average cancel the cyclical component of the comparables’ results beyond the inventory swing, which is what the regulation asks the average to do.

The product life cycle, the second criterion in paragraph (B), is longer still for the pharmaceutical, semiconductor and software industries where transfer pricing disputes concentrate. A patented compound earns for a decade or more; a three-year slice taken at launch, at maturity, or at patent expiry gives three different ranges from the same comparable set.

The convention thus retains the one cycle the three-year period was built for and ignores the two criteria paragraph (B) states beyond it. It keeps the form of the multiple year provision and discards its function. Three years is the floor that controls inventory variation; the period needed to control the cycle of the tested party’s industry and the life cycle of its product is a separate finding, and in the industries where disputes concentrate it is longer.

5. The comparable period is an empirical question

The regulation’s criterion in paragraph (C) is that the same economic conditions had a comparable effect over a comparable period. That is a hypothesis about the data, and it is testable. The length of the comparable period T should be determined by three findings, in this order:

None of these findings yields the number three as a rule. Each may yield three in a particular case, and that case should then say why. The burden the regulation places on the analyst is to show that the period chosen is the one over which the economic conditions were comparable; a fixed window shows nothing.

Two further points follow from paragraph (A). The years after the audit year are admissible, and at the time of examination they are available. A tested party’s loss in the audit year is a different fact if the two following years show recovery than if they show persistence, which is the point of Example 3. And paragraph (D) permits the range to be built from multiple year averages only where averaging reduces short-term variation; an analyst who cannot show the reduction has not satisfied the paragraph.

6. Conclusion

The multiple year provision was written as a reliability provision. When I proposed it in 1992 I had the law of large numbers in mind: average over enough years that the transitory variation unrelated to transfer pricing cancels, and compare persistent results with persistent results. Three years was the floor, set to one inventory cycle. The two-year convention inverts the provision. It converts the floor into a ceiling, independent of the industry, the product and the data; it anchors the window to the filing calendar rather than to the inventory cycle; and it inflates the reported range by the estimation noise \sigma^2/3 that a longer period would remove.

The best method rule of § 1.482-1(c) asks which application provides the most reliable measure of an arm’s length result. Between a three-year window chosen by habit and a period chosen by the data, with the stability of the profit relation tested and reported, the second is more reliable by the regulation’s own criterion. A range built on the two-year convention should be expected to fail that test when an opposing economist pools the full panel and reports the slope with its standard error.

Appendix A. Variance of a T-year average under serial correlation

Let the disturbances follow a stationary first-order autoregression with coefficient \rho, so that \operatorname{Cov}[\varepsilon(i,t), \varepsilon(i,t-k)] = \sigma^2\rho^{k}. The variance of the T-year average is the sum of all T^2 covariances divided by T^2:

\begin{aligned}
&\text{(A.1)}\quad
\operatorname{Var}[A(i)]=\frac{1}{T^2}\sum_{t=1}^{T}\sum_{s=1}^{T}\sigma^2\rho^{|t-s|}
\\[8pt]
&\phantom{\text{(A.1)}\quad \operatorname{Var}[A(i)]}
=\frac{\sigma^2}{T}\left[\,1+2\sum_{k=1}^{T-1}\Bigl(1-\frac{k}{T}\Bigr)\rho^{k}\right].
\end{aligned}

The bracket is the ratio of the actual variance to the independent-draws variance \sigma^2/T; its reciprocal times T is the effective number of independent observations. Values of the standard error of the average, in units of \sigma:

The reduction from three to ten years is a factor of 0.55 under independence and 0.65 under \rho = 0.5. In both cases the three-year average sits near the top of the curve, where each added year still buys a large reduction in variance.

Appendix B. Pooled regression on n \times T company-year observations

EdgarStat does not average and then rank. It pools the N = nT company-year observations and estimates the reduced-form profit relation by least squares with an intercept:

\quad
\text{(B.1)}\qquad
\mathrm{REVT}(i,t)=\lambda_0+\lambda_1\,\mathrm{XOPR}(i,t)+u(i,t),
\qquad
i=1,\dots,n;\; t=1,\dots,T,

where \mathrm{XOPR} = \mathrm{COGS} + \mathrm{XSGA} is total operating cost and \lambda_1 - 1 is the operating markup. The standard error of the slope falls with the square root of N and with the dispersion of XOPR:

\quad
\text{(B.2)}\qquad
\operatorname{SE}(\hat{\lambda}_1)=\frac{\sigma_u}{\sqrt{N}\,s_X},
\qquad
N=nT .

With n = 10 comparables, moving from T = 3 to T = 10 raises N from 30 to 100 and reduces the standard error by a factor of 0.55; the degrees of freedom rise from 28 to 98. EdgarStat reports the slope with HC3 heteroskedasticity-consistent standard errors and the 68 percent interval \hat{\lambda}_1 \pm \operatorname{SE}(\hat{\lambda}_1) (Weisberg 2014; Davidson and MacKinnon 2004).

Pooling has a second advantage over averaging ratios. A ratio profit indicator carries the intercept of the profit relation: if Y = \lambda_0 + \lambda_1 X then

\quad
\text{(B.3)}\qquad
\frac{Y}{X}=\lambda_1+\frac{\lambda_0}{X},

and the mean ratio differs from \lambda_1 by \lambda_0/H, where H is the harmonic mean of X. Averaging the ratio over years removes none of this bias, which is structural and independent of T. The regression estimates \lambda_1 without the bias and tests \lambda_0 = 0 as a hypothesis rather than assuming it.

The Chow test of Section 5 is the F-statistic comparing the residual sum of squares of the pooled regression with the sum of the residual sums of squares of the regressions on each subperiod, with 2 and N - 4 degrees of freedom for a single break.

References

Davidson, R. and J. G. MacKinnon (2004), Econometric Theory and Methods, Oxford University Press. Laws of large numbers, HAC covariance, and the Chow test in one accessible graduate text.

Hald, A. (1990), A History of Probability and Statistics and Their Applications before 1750, Wiley. Bernoulli’s theorem in Ars Conjectandi (1713) and its reception.

Hald, A. (1998), A History of Mathematical Statistics from 1750 to 1930, Wiley. Chebyshev’s inequality and the Russian school.

Kitchin, J. (1923), “Cycles and Trends in Economic Factors,” Review of Economic Statistics 5(1), 10–16. The 40-month inventory cycle.

National Bureau of Economic Research, US Business Cycle Expansions and Contractions, https://www.nber.org/research/data/us-business-cycle-expansions-and-contractions. Reference cycle dates since 1854.

Schumpeter, J. A. (1939), Business Cycles, McGraw-Hill. The three-cycle scheme: Kitchin (inventory), Juglar (fixed investment), Kondratieff (prices).

Weisberg, S. (2014), Applied Linear Regression, 4th ed., Wiley. Standing textbook reference for least squares with an intercept and heteroskedasticity-consistent standard errors.

26 C.F.R. § 1.482-1(f)(2)(iii) (multiple year data), § 1.482-1(e) (arm’s length range), § 1.482-1(c) (best method rule), § 1.482-5(e) Examples 2 and 3, § 1.6662-6 (documentation).