*“We shall renounce . . . the subterfuges*.”

*Cesare Zavattine, Cinema *(1940), quoted in Torunn Haaland, *Italian Neorealist Cinema*, Edinburgh University Press, 2012, p.33.

Three schools of thought can be identified by the structural profit equation used to prepare transfer pricing reports.

We raise misgivings about the “best practices” employed by these schools because they produce biased and inconsistent measures of operating profits.

We offer a solution for creating defensible transfer pricing reports based on economics and statistics principles.

#### Profit Margin School

Taking the OECD guidelines *ad litteram*, members of the *profit margin school* estimate the *structural* equation:

##### (1) P = μ S

where the variable P denotes “net” profits (*i*.*e*., operating profits after depreciation and amortization), and S denotes net sales.

Hereafter, we disregard the error term of the regression equations (1), (2), (3), (5), (6) and (7) to simplify exposition.

The unknown parameter to be determined is the *profit margin*, μ, which is estimated in “best practices” by using quartiles from comparable data about net operating profits and net sales. In Compustat mnemonics, P = OIADP and S = SALE or REVT.

Unless two-stage least squares regression is used, the estimated slope coefficient (μ = profit margin) of equation (1) is biased and inconsistent. Quartiles do not solve this unreliability problem.

Two-stage least squares regression requires that we find an “instrumental variable” for net sales.

In RoyaltyStat®, we automated two-stage least squares procedures that can find total cost or property, plant & equipment as the best proxy variable for equation (1). See Hill, Griffiths & Lim, 2018, § 10.3.6 (*Estimation using two-stage least squares*), pp. 495–496.

#### Return on Assets School

Members of the *return on assets school* also follow the OECD guidelines *à la lettre*.

The profit equation of the return on assets schools is expressed:

##### (2) P = ρ A

where A denotes operating assets, measured in *ad hoc* ways.

In equation (2), we should use Compustat mnemonics A = PPENT(Net), *i.e*., property, plant & equipment (net of accumulated depreciation), because this is the only composite balance sheet account of *depreciable assets*.

In economics principles, the definition of operating assets (also called “capital stock”) must be consistent with the “perpetual inventory” equation. See Klein (1962), pp. 88, 193: “It is important to make the capital [operating assets] concept consistent with the investment concept…. In order to equate the rate of change in the capital stock [∆ PPENT] to investment [CAPX – DFXA], the latter must be measured *net*, that is, after capital consumption allowances [DFXA] have been subtracted from gross investment expenditures [CAPX].”

Unless two-stage least squares regression is used, the estimated slope coefficient (ρ = return on operating assets) of equation (2) is biased and inconsistent. Again, using quartiles does not resolve the bias and minimum variance problems.

#### Synthetic (Hybrid) School

The hybrid school employs a *combination* of the structural equations of the profit margin and the return on assets schools. The profit equation of the synthetic (hybrid) school can be expressed:

##### (3) P = μ S + ρ A

Equation (3) may be appealing because it combines elements of two rival schools of thought; however, unless two-stage least squares regression is used, the estimated coefficients (μ and ρ) of equation (3) are biased and inconsistent.

Using two-stage least squares regression in equation (3) is complex because the model has two explanatory variables. We must find “instruments” (proxy variables) for both variables.

This task is easier said than done; thus, we prefer to use reduced form equations and apply OLS (ordinary least squares).

#### Misconceptions in Transfer Pricing

*Les* *trois petits cochons* equations (1), (2), and (3) are cases of ill-trained economics practices.

First, total cost may be the most relevant variable explaining profits, yet the three models estimate *structural* equations, which *omit total cost*.

As a result, the estimates of the coefficients μ and ρ are biased and inconsistent. See Hill, Griffiths & Lim, 2018, § 6.3.2 (*Model specification: omitted variables*), pp. 275–277.

Second, the pseudo “best practice” amounts to a computation of the coefficients μ and ρ using quartiles of comparable data. This practice is based on several untested assumptions about the comparable data, including the assumptions that the adopted structural profit equation has no intercept, the relationship is linear (*e*.*g*., instead of the pervasive and testable power function), and the residual errors are well-behaved (no serial correlation, no heteroscedasticity).

In economics, we study the *relationship* *between two or more variables*, and this applied science cannot be reduced to univariate (single variable) calculations of quartiles of the selected profit indicator without exploratory (diagnostic) data analysis.

This “best practices” misuse of quartiles ignores the need for the theoretical specification of the model to be estimated, and it ignores satisfying the objective that to be convincing and satisfy regulatory (most reliable estimate) requirements we must demonstrate the desirable properties of the estimates.

In short, to be defensible, we must demonstrate the desirable statistical properties of the estimated coefficients μ and ρ. See Hill, Griffiths & Lim, 2018, § 2.5 (*Gauss-Markov theorem*), pp. 72–73 and Appendix 2F. See also Wonnacott, 1969, § 7.2 (Desirable properties of estimators (No bias, efficiency, consistency)), pp. 134-139.

#### Perseus Escapes from the Labyrinth

We must escape the unreliable “best practice” of computing quartiles of the structural equations (1), (2), or (3). Computation of the quartiles of single variables (like computing the quartiles of the selected profit indicator) is naïve and does not require economics or statistics expertise.

We offer a better alternative for those wishing to create defensible transfer pricing reports. This option can protect against biased and inconsistent results by estimating reduced-form (and not structural) equations. See Marschak, 1953, p. 26.

First, and this step cannot be forgotten, we define the accounting equation that holds for every selected (comparable) company and the tested party:

##### (4) S = [C + δ A(−1)] + P

where the variable A is delayed one period. Equation (4) is an accounting identity and thus is not subject to random errors.

The unknown parameter δ denotes the depreciation rate; thus, the measure of operating assets must be restricted to *depreciable assets*.

In Compustat mnemonic, C = XOPR is total cost, which is the sum of COGS and XSGA. Some companies report COGS or XSGA as a combined figure, and do not report these two income statement accounts separately.

Second, we couple the accounting equation (4) with the structural equation (3), but we delay the operating assets to the beginning of the period:

##### (5) P = μ S + ρ A(−1)

Like equations (1), (2), and (3), we cannot estimate the parameters of equation (5) unless two-stage least squares regression is used.

Before the regression estimation, we must consider that equation (5) is *coupled* with an accounting identity equation (4). Thus, we substitute (4) into (5) and obtain:

##### S = [C + δ A(−1)] + [μ S + ρ A(−1)]

##### S − μ S = C + (δ + ρ) A(−1)

##### (6) S = λ_{1} C + λ_{2} A(−1)

where the regression parameters are λ_{1} = 1 / (1 – μ) and λ_{2} = (δ + ρ) / (1 – μ).

The parameter [1 / (1 – μ)] ≥ 1 is the operating profit markup, and the indirect least squares (ILS) parameter μ = (λ_{1} − 1) / λ_{1} is the profit margin.

For an *idealized* single product company, equation (6) is the unit price equation.

Let Q denote quantity sold and p denote unit (average) price, such that S = p Q. We divide (6) by Q and obtain:

##### (6a) p = λ_{1} (C / Q) + λ_{2} (A(−1) / Q)

The reduced form profit margin is easy to calculate from λ_{1}— that is, μ = (λ_{1} − 1) / λ_{1}. However, calculation of the return on assets from λ_{2} = (δ + ρ) / (1 – μ) requires prior knowledge of the depreciation rate.

We can write (6a) using a more recognized price markup equation:

##### (6b) p = λ_{1} c + λ_{2} k(−1)

where the explanatory variables are c = (C / Q) = average cost and k = (A(−1) / Q) = asset intensity, or the “capital/output” ratio.

#### Defensible Transfer Pricing Reports

The application of equations (1), (2), or (3) using quartiles is pedestrian and (as stated above) does not require economics expertise. But the pervasive misuse of quartiles does not make this naïve and unreliable practice credible. To call this misuse of quartiles a “best practice” is a gross misnomer.

To create defensible transfer pricing reports, we must use more reliable regression methods to determine the unbiased estimates of the coefficients (profit indicators), and we must disclose the standard errors of the estimated coefficients. See Eisenhart (1952).

Against Camp (see note below), we should replace structural equations (1), (2), or (3) with the reduced form equation (6). Else, we can estimate equation (1), (2) or (5) using two-stage least squares.

We should test the *regression equation* (6) to determine if the coefficient λ_{2} = (δ + ρ) / (1 – μ) is significant at a selected confidence level. If this partial regression coefficient λ_{2} is not significant, we can determine the comparable profit indicator by using the more parsimonious (reduced form) model:

##### (7) S = λ_{1} C

Using regression equation (6), we can select (with statistical confidence) the most reliable comparable profit indicator to benchmark the controlled tested party—if the comparable operating profit is determined by total cost alone (equation (7)) or by the double effects of total cost and operating assets (equation (6)). In turn, operating assets must consist of depreciable assets, and not be defined willy-nilly.

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**Notes**: Camp is a group engaged in promoting (or defending) a practice, doctrine, or position. Against Camp means that we oppose “anything goes” (arbitrary) practices such as the measurement of comparable profit indicators without economic theory or without verifying statistical bias and reliability (minimum variance). See Sontag, 1966, pp. 275-292 (“Camp is esoteric—something of a private code”).

Marschak, 1953, p. 26. describes *structural* relationships. “In economics, the condition that constitutes a structure are (1) a set of relations describing human behavior and institutions as well as technological laws and involving, in general, nonobservable random disturbances and nonobservable random errors of measurement; (2) the joint probability distribution of these random quantities…. Thus, practice requires theory.” In simultaneous equations models, structure refers to postulated (“autonomous,” not “derived”) relationships like the postulated profit equations (1), (2), (3) and (5) above.

#### References

Churchill Eisenhart, “The Reliability of Measured Values — Fundamental Concepts,” National Bureau of Standards (NBS) Report 1100-53-1199, April 15, 1952.

Carter Hill, William Griffiths & Guay Lim, *Principles of Econometrics* (5^{th} edition), Wiley, 2018. The Kindle version of this book has an excellent transcription of the equations.

Lawrence Klein, *Introduction to Econometrics*, Prentice-Hall, 1962.

Jacob Marschak, “Economic Measurements for Policy and Prediction” in William Hood & Tjalling Koopmans, *Studies in Econometric Method*, John Wiley, 1953.

Ednaldo Silva, “No Tax Parity in Balance Sheet Adjustments to Profits,” *Tax Notes Federal*, Vol. 167, No. 5, May 4, 2020.

Susan Sontag, “Notes on ‘Camp’” in her book, *Against Interpretation and Other Essays*, Farrar, Straus & Giroux, 1966.

Thomas Wonnacott & Ronald Wonnacott, *Introductory Statistics*, Wiley, 1969.