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Using a distributed lag model to determine arm’s length profits in transfer pricing

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Transfer pricing reports often lack valid economic principles, relying instead on ad hoc specification. This faulty practice is permitted by misconceived transfer pricing regulations based on the OECD model.

Following the issuance of US transfer pricing regulations introducing the comparable profits method (which we were a major developer), the OECD ordained a specified profit model under the transactional net margin method (TNMM) in which “net [operating] profits” (hereafter profits) are proportional to revenue, costs or assets. Today, quartiles of profit margins (expressed as profits divided by revenue) derived from accepted comparable enterprises is the dominant profit indicator used by tax administrators and controlled taxpayers under the TNMM.

However, the OECD TNMM specified profit model may not be the most reliable method to determine arm’s length profits used to benchmark transactions between related parties. In many circumstances, a distributed lag or another specified regression model (which is allowed under the TNMM) can produce more reliable results.

Regression analysis (including distributed lag models) based on selected comparable enterprise data, and not simple OECD formulae, can provide more defensible (or scientifically acceptable) basis to determine reliable arm’s length profits. RoyaltyStat®’s interactive database of company (enterprise) financials allows the subscriber to select the best fit among various statistical (including regression) models.

Contemporaneous profits, sales

In what follows, we abstract from the random disturbance and consider the causal part of the OECD TNMM profit model.

Let P = Profits and S = Sales (revenue) of a selected enterprise which is comparable to the tested party regarding functions performed, assets employed, and risks assumed with respect to identified controlled transactions, such as distribution or retail of inventory goods.

The OECD prescribed profit model can be posited as a simple linear regression:

     (1)  P(t) = β S(t),

where the audit year profits are proportional to the same year sales, considering the audit year and two or more prior years of data.

The OECD prescription (1) forces a zero intercept (α = 0) à priori, which is misconceived because the actual intercept may be nonzero; thus, arm’s length profits can be distorted using model (1) because of a missing intercept alone.

Prior years’ revenue

In fact, audit year profits may depend on the corresponding fiscal yearand prior year revenues. Thus, a more general formula than (1) can allow for the audit year and past year revenues to affect target profits.

In symbols, a series of previous year revenues, combined with the audit year revenue, may generate uncontrolled profits to provide more reliable taxable income benchmarks for a controlled tested party under transfer pricing compliance obligations:

     (2)  P(t) = α + β S(t) + β1 S(t – 1) + β2 S(t – 2) + β3 S(t – 3) + …

Two key differences separate model (2) from (1). First, we let the selected comparable enterprise data determine the intercept value. Second, we recognize that a series of prior year revenues, instead of a single audit year revenue, can determine audit year profits. Hence, model (2) is more general than (1).

Regression estimation of multiple parameters (α, β, β1, β2, etc.) in (2) is awkward, but from experience we know that these partial coefficients decline exponentially, which means that distant beta coefficients can exert less weight than more recent coefficients on the dependent variable, P(t). Thus, following Koyck (1954, see below), we can assume decreasing weights:

     (3)  βk = β λk,

where 0 < λ < 1, with period index k = 1, 2, 3, …, and obtain β= β λ, β2 = β λ2, β3 = β λ3, etc. Since λ is raised to higher powers, the coefficients of earlier periods become smaller (less relevant) as we reach farther back into the past.

Distributed lag model in transfer pricing

Eq. (3) is called a Koyck transformation, which is a well-known distributed lag model of consumption and investment, including activities generating intangibles such as advertising and research and development expenses. See L. Koyck, Distributed Lags and Investment Analysis, North-Holland, 1954, pp. 11, 16, 18, 20, 22, 39.

This Koyck device (3) is used to transform an infinite geometric lag model (2) into a finite model with a lagged dependent variable (see model (4)).

While this makes for parsimonious parameter estimation, the transformed model is likely to have serial correlation among the regression residuals; thus, OLS (ordinary least squares) estimates may not be adequate. Instead of OLS, we can apply Cochrane-Orcutt or Prais-Winsten estimation.

Many flowers may bloom, and certain economists like to apply more complex and subjective instrumental variables (IV) estimates. For more details, see Jan Kmenta, Elements of Econometrics (2nd edition), Macmillan, 1986, Chapter 11-4 (Distributed lag models). But the OLS estimates obtained from (4) below are consistent and the errors estimates are valid because Henri Theil showed them to be asymptotic standard errors. For proofs, please read Henri Theil, Principles of Econometrics (1971), John Wiley & Sons, pp. 260-261, 417.

Using Koyck’s well-accepted assumption (3), we transform munificent model (2) into parsimonious model (4):

      (4)  P(t) = µ + λ P(t − 1) + β S(t),

where µ = α (1 – λ) is the transformed intercept.

The short-run profit margin is β and the long-rum profit margin is β / (1 – λ).

Reliable data to find comparables

Model (4) provides more information than the OECD TNMM model (1). First, the intercept of (4) is not forced to zero. Second, model (4) is dynamic, measured by the lagged dependent variable, and we can test the stability of P(t) over the selected period. Third, if the intercept is not zero the profit margin is not a simple ratio of profits to revenue because, if we divide (4) by R(t), we obtain a reciprocal function:

     (5)  M(t) = P(t) / S(t ) = β + µ / S(t ) + λ [P(t − 1) / S(t )]

Equation (5) means that when the intercept is statistically different from zero, the profit margin, M(t), is inversely proportional to S(t ), company size measured by sales revenue, and directly proportional to the profit margin of the previous year.

In addition to be the premier database of royalty rates extracted from license agreements, including over 20,500 license agreements with disclosed royalty rates, RoyaltyStat contains another premier database with over 36,000 listed company financials. This company financials database has been programmed with the Cochrane-Orcutt and Prais-Winsten algorithms for interactive or online estimation of the parameters of distributed lag model (4).

Again, see Kmenta, pp. 313-317 (Cochrane-Orcutt estimation) and pp. 318-320 (Prais-Winsten estimation) and Theil (1971), pp. 260-261, 417 about using OLS estimation.

More reliable models

Like Lévi-Strauss in Tristes Tropiques (1955), we consider it harmful for granfino regulators to overreach beyond their competence to force a special profit model à priori, thereby creating a nursery school of rote transfer pricing analysis in which controlled taxable income is assessed as if science (economics and statistics knowledge) doesn’t have a place in tax compliance.

Simple prescription and arbitrary procedures are unlikely to produce reliable arm’s length results, and the economic analyst must have the competence and discretion to make reliable model determination, subject to a Daubert challenge. Under certain circumstances, quartiles of simple profit margins calculated from selected comparables are unlikely to pass any significance test of reliability. Economics and statistics competence require more expertise than blind knowledge of quartiles or routine use of arbitrary asset-based adjustments to the reported profits of the accepted comparables.

Using an acceptable principle of model reliability, which is also embedded in the OECD transfer pricing guidelines, and in its progenitor US transfer pricing regulations (for which we assume major responsibility), is an important escape from unreliable controlled taxable income assessment.

The distributed lag model (4) presented here is an important step to confront routine “best practices” (an oxymoron) with accepted scientific procedures in which model selection is determined by knowledge-based falsifiable propositions instead of being predetermined by regulatory grace.

Forcing model (1), like the OECD and US transfer pricing rules prescribe, without appeal to empirical significance testing, is likely to produce distorted measures of arm’s length profits. Again, we appeal to knowledge-based data analysis against the prevailing mechanical application of misconceived or simplistic transfer pricing rules. Analysis of the selected comparable company data dictates finding the best fit model to be adopted, including model (1), (4), or a more reliable regression model specification.