*In ergodic theory, time and space averages become equal*. Ives Coudène, *Ergodic Theory and Dynamical Systems*, Springer, 2016.

Consider two aspects of statistical reliability principles. First, reliability can be measured by the ratio of the selected parameter *estimate* divided by its *standard error*. We want this reliability ratio to be as high as possible. Second, reliability depends on sample size such that larger samples produce more reliable estimates.

Transfer pricing rules recognize that statistical estimates of the selected profit indicator must consider multiple-year analysis to achieve a reliable measure of arm’s length taxable income. See OECD (2017), ¶¶ 3.75-3.79 and US Treas. Reg. § 1.482-1(f)(2)(iii).

The OECD Transfer Pricing Guidelines (2017), ¶ 3.76 states:

“In order to obtain a complete understanding of the facts and circumstances surrounding the controlled transaction, it generally might be useful to examine data from both the year under examination and prior years.”

The OECD Guidelines (2017), ¶ 3.38 also states:

“The identification of potential comparables has to be made with the objective of finding the most reliable data, recognising that they will not always be perfect.”

In practice, though, multiple-year analysis is reduced to the audit year plus two years before or after the audit year.

Here, we show by one actual (as opposed to imagined) example that multiple-year analysis must consider all the available years of data of each comparable. Otherwise, estimates of the selected comparable profit indicator are not as reliable as the available data permit.

This conclusion is supported by the OECD Guidelines (2017), ¶ 3.57, which states:

“[I]f the range includes a sizeable number of observations, statistical tools that take account of central tendency *to narrow the range* (e.g. the interquartile range or other percentiles) might help to enhance the reliability of the analysis.” Emphases added.

We aim to satisfy two concurrent objectives. First, we need to *stabilize the center estimate* of the selected profit indicator, avoiding wobbly numbers. Second, we aim to *achieve the maximum reliability* *of the center estimate*.

We are confident that regression analysis is the most appropriate statistical algorithm (contrary to rote computing of quartiles) to satisfy these two objectives. We wish to show also that more years of comparable company (enterprise) data are better than using truncated three-year data samples.

We can estimate the operating profit markup model and examine the behavior of the estimated markup and its standard error, measured by the *t*-statistics (which is the ratio of the OLS (ordinary least squares) coefficient estimate divided by its standard error):

##### (1) S(*t*) = α + β C(*t*) + U(*t*)

##### (2) U(*t*) is corrected for serial correlation by the Newey-West algorithm,

and *t* = 1 to *T* years of data.

The variable S(*t*) denotes Net Sales of the selected comparable enterprise in period *t*, and C(*t*) denotes Total Costs (*Lato*) = (COGS + XSGA + (DP − AM)). Kalecki (1954), Chapter 1 (Cost and prices), is the most original reference to the profit markup model.

The residual errors U(*t*) have zero mean and Newey-West corrected variance. See Green (2018), §20.5, pp. 996-999. Wikipedia has a good entry on the Newey-West estimator of the standard errors of the OLS regression coefficients at: https://en.wikipedia.org/wiki/Newey%E2%80%93West_estimator

Since OLS requires a certain minimum degrees of freedom (sample size minus two observations in the bivariate case of model (1)), we start with *T *= 3 years of data considering the most recent to previous values. As an exception, the *t*-statistics = 168.3 for *T *= 3 years of data are from OLS because the number from Newey-West is a large outlier resulting from its standard error being near zero. See the OECD Guidelines (2017), ¶¶ 3.57-3.62 for a discussion of statistical tools.

#### Best Buy (GVKEY 2184)

The (S(*t*) and C(*t*)) data pairs are obtained from RoyaltyStat’s integration of Standard & Poor’s Global (Compustat) database of listed global company financials. We use Best Buy (GVKEY 2184) data from 1983 to 2018 and apply OLS to our bivariate regression model (1):

* *

*t* = 3, β = 1.0813, *t*-statistics = 168.3 *t* = 21, β = 1.0431, *t*-statistics = 171.2

* t* = 4, β = 1.1214, *t*-statistics = 51.8 *t* = 22, β = 1.0450, *t*-statistics = 181.7

* t* = 5, β = 1.1312, *t*-statistics = 34.0 *t* = 23, β = 1.0468, *t*-statistics = 186.4

* t* = 6, β = 0.9857, *t*-statistics = 10.2 *t* = 24, β = 1.0477, *t*-statistics = 195.0

* t* = 7, β = 0.9373, *t*-statistics = 26.4 *t* = 25, β = 1.0479, *t*-statistics = 207.3

* t* = 8, β = 1.0499, *t*-statistics = 33.9 *t* = 26, β = 1.0479, *t*-statistics = 219.4

* t* = 9, β = 1.0703, *t*-statistics = 45.3 *t* = 27, β = 1.0478, *t*-statistics = 230.1

*t* = 10, β = 1.0786, *t*-statistics = 55.3 *t* = 28, β = 1.0477, *t*-statistics = 239.2

*t* = 11, β = 1.0793, *t*-statistics = 60.0 *t* = 29, β = 1.0475, *t*-statistics = 246.8

*t* = 12, β = 1.0623, *t*-statistics = 49.8 *t* = 30, β = 1.0474, *t*-statistics = 253.3

*t* = 13, β = 1.0416, *t*-statistics = 40.7 *t* = 31, β = 1.0473, *t*-statistics = 258.7

*t* = 14, β = 1.0326, *t*-statistics = 55.8 *t* = 32, β = 1.0472, *t*-statistics = 263.5

*t* = 15, β = 1.0313, *t*-statistics = 78.0 *t* = 33, β = 1.0471, *t*-statistics = 267.6

*t* = 16, β = 1.0316, *t*-statistics = 99.1 *t* = 34, β = 1.0470, *t*-statistics = 271.4

*t* = 17, β = 1.0347, *t*-statistics = 117.8 *t* = 35, β = 1.0470, *t*-statistics = 274.7

*t* = 18, β = 1.0364, *t*-statistics = 132.6 *t* = 36, β = 1.0469, *t*-statistics = 277.6

*t* = 19, β = 1.0396, *t*-statistics = 142.5

*t* = 20, β = 1.0413, *t*-statistics = 157.1

To induce rule change, we make a compelling case for the use of all available comparable data in transfer pricing analysis.

The *convergence* of the estimated profit markup coefficients to a stable value and their accompanying *t*-statistics as the comparable company sample size gets larger, and the scatterplots with the superimposed Loess curves, provide strong support to our statistical principle and fact-based position that reliable estimates of arm’s length taxable income require that we use all available data.

Restricting the data to three-year company samples in practice today produces unreliable (unstable) estimates of the selected profit indicator together with inefficient standard errors (and thus very wide ranges).

#### REFERENCES

William Green, *Econometric Analysis* (8th edition), Pearson, 2018.

Michael Kalecki, *Theory of Economic Dynamics*, George Allen & Unwin, 1954.

OECD, *Transfer Pricing Guidelines* (2017). DOI: https://dx.doi.org/10.1787/tpg-2017-en

Published on Oct 17, 2019 10:38:26 AM

**Ednaldo Silva**(Ph.D.) is founder and managing director of RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com

and

**RoyaltyStat**provides premier online databases of**royalty rates**extracted from unredacted license agreementsand

**normalized company financials**(income statement, balance sheet, cash flow). We provide high-quality data, built-in analytical tools, customer training and attentive technical support.